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b/figs/uniaxial_support_compliance_test_system.svg @@ -1,226 +1,230 @@ - + - - - + - + - + - + - + - + - - - + - + - + - + - + - - - + - - - - - - - - + - + - + - + - + - + + + + - + - + - + - + - + + + + + + + + + + - + - + - + - - - - - - - + + + + + + + - + - + - + - + - + - + - - - + + + - + - + - - + + - + - - - - - - - - + + + + + + + + - + - + - + - + - + - + - - - + + + - + - + - - - - - + + + + + - + - - + + - + - - - - - - + + + + - + - + + + + + + + + - - + + - + - + - + - + - - - + + + - + diff --git a/phd-thesis.bib b/phd-thesis.bib index bd46ecd..d87aaba 100644 --- a/phd-thesis.bib +++ b/phd-thesis.bib @@ -134,8 +134,8 @@ Barrett, R. and Zhang, L. and Homs-Regojo, R. A. and Favre-Nicolin, V. and Boesecke, P. and Sch{\"u}lli, T. U.}, - title = {The Nanodiffraction Beamline Id01/esrf: a Microscope for - Imaging Strain and Structure}, + title = {{The Nanodiffraction Beamline ID01/ESRF: a Microscope for + Imaging Strain and Structure}}, journal = {Journal of Synchrotron Radiation}, volume = 26, number = 2, diff --git a/phd-thesis.org b/phd-thesis.org index 08944fa..5e7e92b 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -92,7 +92,7 @@ |-------+-------------------------+----------------------------------------------------------| | ms | \ensuremath{m_s} | Mass of the sample | | mn | \ensuremath{m_n} | Mass of the nano-hexapod | -| mh | \ensuremath{m_h} | Mass of the micro-hexapod | +| mh | \ensuremath{m_h} | Mass of the positioning hexapod | | mt | \ensuremath{m_t} | Mass of the micro-station stages | | mg | \ensuremath{m_g} | Mass of the granite | | xf | \ensuremath{x_f} | Floor motion | @@ -384,7 +384,7 @@ It experienced an approximate 100-fold increase with the implementation of acrsh While this enhanced beam quality presents unprecedented scientific opportunities, it concurrently introduces considerable engineering challenges, particularly regarding experimental instrumentation and sample positioning systems. #+name: fig:introduction_moore_law_brillance -#+caption: Evolution of the peak brilliance (expressed in $\text{photons}/s/mm^2/mrad^2/0.1\%BW$) of synchrotron radiation facilities. Note the vertical logarithmic scale. +#+caption: Evolution of the peak brilliance (expressed in $\text{photons}/s/\text{mm}^2/\text{mrad}^2/0.1\%BW$) of synchrotron radiation facilities. Note the vertical logarithmic scale. #+attr_latex: :scale 0.9 #+attr_latex: :options [h!tbp] [[file:figs/introduction_moore_law_brillance.png]] @@ -427,9 +427,9 @@ Detectors are used to capture the X-rays transmitted through or scattered by the Throughout this thesis, the standard acrshort:esrf coordinate system is adopted, wherein the X-axis aligns with the beam direction, Y is transverse horizontal, and Z is vertical upwards against gravity. The specific end-station employed on the ID31 beamline is designated the "micro-station". -As depicted in Figure\nbsp{}ref:fig:introduction_micro_station_dof, it comprises a stack of positioning stages: a translation stage (in blue), a tilt stage (in red), a spindle for continuous rotation (in yellow), and a micro-hexapod (in purple). +As depicted in Figure\nbsp{}ref:fig:introduction_micro_station_dof, it comprises a stack of positioning stages: a translation stage (in blue), a tilt stage (in red), a spindle for continuous rotation (in yellow), and a positioning hexapod (in purple). The sample itself (cyan), potentially housed within complex sample environments (e.g., for high pressure or extreme temperatures), is mounted on top of this assembly. -Each stage serves distinct positioning functions; for example, the micro-hexapod enables fine static adjustments, while the $T_y$ translation and $R_z$ rotation stages are used for specific scanning applications. +Each stage serves distinct positioning functions; for example, the positioning hexapod enables fine static adjustments, while the $T_y$ translation and $R_z$ rotation stages are used for specific scanning applications. #+name: fig:introduction_micro_station #+caption: 3D view of the ID31 Experimal Hutch (\subref{fig:introduction_id31_cad}). There are typically four main elements: the focusing optics in yellow, the sample stage in green, the sample itself in purple and the detector in blue. All these elements are fixed to the same granite. 3D view of the micro-station with associated degrees of freedom (\subref{fig:introduction_micro_station_dof}). @@ -458,7 +458,7 @@ This reconstruction depends critically on maintaining the sample's acrfull:poi w Mapping or scanning experiments, depicted in Figure\nbsp{}ref:fig:introduction_scanning_schematic, typically use focusing optics to have a small beam size at the sample's location. The sample is then translated perpendicular to the beam (along Y and Z axes), while data is collected at each position. -An example\nbsp{}[[cite:&sanchez-cano17_synch_x_ray_fluor_nanop]] of a resulting two-dimensional map, acquired with 20nm step increments, is shown in Figure\nbsp{}ref:fig:introduction_scanning_results. +An example\nbsp{}[[cite:&sanchez-cano17_synch_x_ray_fluor_nanop]] of a resulting two-dimensional map, acquired with $20\,\text{nm}$ step increments, is shown in Figure\nbsp{}ref:fig:introduction_scanning_results. The fidelity and resolution of such images are intrinsically linked to the focused beam size and the positioning precision of the sample relative to the focused beam. Positional instabilities, such as vibrations and thermal drifts, inevitably lead to blurring and distortion in the obtained image. Other advanced imaging modalities practiced on ID31 include reflectivity, diffraction tomography, and small/wide-angle X-ray scattering (SAXS/WAXS). @@ -482,7 +482,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra #+end_figure #+name: fig:introduction_scanning -#+caption: Exemple of a scanning experiment. The sample is scanned in the Y-Z plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of 20nm (\subref{fig:introduction_scanning_results}). +#+caption: Exemple of a scanning experiment. The sample is scanned in the Y-Z plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of $20\,\text{nm}$ (\subref{fig:introduction_scanning_results}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_scanning_schematic} Experimental setup} @@ -525,7 +525,7 @@ The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source s #+end_subfigure #+end_figure -Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of acrshort:esrf, where typical spot sizes were on the order of $10\,\mu m$ [[cite:&riekel89_microf_works_at_esrf]]. +Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of acrshort:esrf, where typical spot sizes were on the order of $10\,\mu\text{m}$ [[cite:&riekel89_microf_works_at_esrf]]. Various technologies, including zone plates, Kirkpatrick-Baez mirrors, and compound refractive lenses, have been developed and refined, each presenting unique advantages and limitations\nbsp{}[[cite:&barrett16_reflec_optic_hard_x_ray]]. The historical reduction in achievable spot sizes is represented in Figure\nbsp{}ref:fig:introduction_moore_law_focus. Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines. @@ -568,7 +568,7 @@ Recent developments in detector technology have yielded sensors with improved sp Historically, detector integration times for scanning and tomography experiments were in the range of 0.1 to 1 second. This extended integration effectively filtered high-frequency vibrations in beam or sample position, resulting in apparently stable but larger beam. -With higher X-ray flux and reduced detector noise, integration times can now be shortened to approximately 1 millisecond, with frame rates exceeding 100 Hz. +With higher X-ray flux and reduced detector noise, integration times can now be shortened to approximately 1 millisecond, with frame rates exceeding $100\,\text{Hz}$. This reduction in integration time has two major implications for positioning requirements. Firstly, for a given spatial sampling ("pixel size"), faster integration necessitates proportionally higher scanning velocities. Secondly, the shorter integration times make the measurements more susceptible to high-frequency vibrations. @@ -663,11 +663,11 @@ The PtiNAMi microscope at DESY P06 (Figure\nbsp{}ref:fig:introduction_stages_sch For applications requiring active compensation of measured errors, particularly with nano-beams, feedback control loops are implemented. Actuation is typically achieved using piezoelectric actuators\nbsp{}[[cite:&nazaretski15_pushin_limit;&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage;&villar18_nanop_esrf_id16a_nano_imagin_beaml;&nazaretski22_new_kirkp_baez_based_scann]], 3-phase linear motors\nbsp{}[[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&engblom18_nanop_resul]], or acrfull:vc actuators\nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]]. -While often omitted, feedback bandwidth for such stages are relatively low (around 1 Hz), primarily targeting the compensation of slow thermal drifts. -More recently, higher bandwidths (up to 100 Hz) have been demonstrated, particularly with the use of voice coil actuators\nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]]. +While often omitted, feedback bandwidth for such stages are relatively low (around $1\,\text{Hz}$), primarily targeting the compensation of slow thermal drifts. +More recently, higher bandwidths (up to $100\,\text{Hz}$) have been demonstrated, particularly with the use of voice coil actuators\nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]]. Figure\nbsp{}ref:fig:introduction_active_stations showcases two end-stations incorporating online metrology and active feedback control. -The ID16A system at acrshort:esrf (Figure\nbsp{}ref:fig:introduction_stages_villar) uses capacitive sensors and a piezoelectric hexapod to compensate for rotation stage errors and to perform accurate scans\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]]. +The ID16A system at acrshort:esrf (Figure\nbsp{}ref:fig:introduction_stages_villar) uses capacitive sensors and a piezoelectric Stewart platform to compensate for rotation stage errors and to perform accurate scans\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]]. Another example, shown in Figure\nbsp{}ref:fig:introduction_stages_nazaretski, employs interferometers and piezoelectric stages to compensate for thermal drifts\nbsp{}[[cite:&nazaretski15_pushin_limit;&nazaretski17_desig_perfor_x_ray_scann]]. A more comprehensive review of actively controlled end-stations is provided in Section\nbsp{}ref:sec:nhexa_platform_review. @@ -693,9 +693,9 @@ For tomography experiments, correcting spindle guiding errors is critical. Correction stages are typically placed either below the spindle\nbsp{}[[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage;&villar18_nanop_esrf_id16a_nano_imagin_beaml;&engblom18_nanop_resul;&nazaretski22_new_kirkp_baez_based_scann;&xu23_high_nsls_ii]] or above it\nbsp{}[[cite:&wang12_autom_marker_full_field_hard;&schroer17_ptynam;&schropp20_ptynam;&geraldes23_sapot_carnaub_sirius_lnls]]. In most reported cases, only translation errors are actively corrected. Payload capacities for these high-precision systems are usually limited, typically handling calibrated samples on the micron scale, although capacities up to 500g have been reported\nbsp{}[[cite:&nazaretski22_new_kirkp_baez_based_scann;&kelly22_delta_robot_long_travel_nano]]. -The system developed in this thesis aims for payload capabilities approximately 100 times heavier (up to 50 kg) than previous stations with similar positioning requirements. +The system developed in this thesis aims for payload capabilities approximately 100 times heavier (up to $50\,\text{kg}$) than previous stations with similar positioning requirements. -End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few acrshortpl:dof with strokes around $100\,\mu m$. +End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few acrshortpl:dof with strokes around $100\,\mu\text{m}$. Recently, acrfull:vc actuators were used to increase the stroke up to $3\,\text{mm}$ [[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]] An alternative strategy involves a "long stroke-short stroke" architecture, illustrated conceptually in Figure\nbsp{}ref:fig:introduction_two_stage_schematic. In this configuration, a high-accuracy, high-bandwidth short-stroke stage is mounted on top of a less precise long-stroke stage. @@ -730,9 +730,9 @@ This necessitates peak-to-peak positioning errors below $200\,\text{nm}$ in $D_y Additionally, the $R_y$ tilt angle error must remain below $0.1\,\text{mdeg}$ ($250\,\text{nrad RMS}$). Given the high frame rates of modern detectors, these specified positioning errors must be maintained even when considering high-frequency vibrations. -These demanding stability requirements must be achieved within the specific context of the ID31 beamline, which necessitates the integration with the existing micro-station, accommodating a wide range of experimental configurations requiring high mobility, and handling substantial payloads up to 50 kg. +These demanding stability requirements must be achieved within the specific context of the ID31 beamline, which necessitates the integration with the existing micro-station, accommodating a wide range of experimental configurations requiring high mobility, and handling substantial payloads up to $50\,\text{kg}$. -The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately $10\,\mu m$ and $10\,\mu\text{rad}$ due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. +The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately $\SI{10}{\mu\m}$ and $\SI{10}{\mu\rad}$ due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. The primary objective of this project is therefore defined as enhancing the positioning accuracy and stability of the ID31 micro-station by roughly two orders of magnitude, to fully leverage the capabilities offered by the ESRF-EBS source and modern detectors, without compromising its existing mobility and payload capacity. @@ -770,7 +770,7 @@ The active stabilization platform, positioned between the micro-station top plat It needs to provide active motion compensation in 5 acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$ and $R_y$). It must possess excellent dynamic properties to enable high-bandwidth control capable of suppressing vibrations and tracking desired trajectories with nanometer-level precision. Consequently, it must be free from backlash and play, and its active components (e.g., actuators) should introduce minimal vibrations. -Critically, it must accommodate payloads up to 50 kg. +Critically, it must accommodate payloads up to $50\,\text{kg}$. A suitable candidate architecture for this platform is the Stewart platform (also known as "hexapod"), a parallel kinematic mechanism capable of 6-DoF motion. Stewart platforms are widely employed in positioning and vibration isolation applications due to their inherent stiffness and potential for high precision. @@ -806,7 +806,7 @@ Several factors complicate the design of robust feedback control for the NASS. First, the system must operate under across diverse experimental conditions, including different scan types (tomography, linear scans) and payloads' inertia. The continuous rotation of the spindle introduces gyroscopic effects that can affect the system dynamics. As actuators of the active platforms rotate relative to stationary sensors, the control kinematics to map the errors in the frame of the active platform is complex. -But perhaps the most significant challenge is the wide variation in payload mass (1 kg up to 50 kg) that the system must accommodate. +But perhaps the most significant challenge is the wide variation in payload mass ($1\,\text{kg}$ up to $50\,\text{kg}$) that the system must accommodate. Designing for robustness against large payload variations typically necessitates larger stability margins, which can compromise achievable performance. Consequently, high-performance positioning stages often work with well-characterized payload, as seen in systems like wafer-scanners or atomic force microscopes. @@ -850,7 +850,7 @@ This methodology, detailed in Section\nbsp{}ref:sec:detail_fem, is presented as ***** Control Robustness by design -The requirement for robust operation across diverse conditions—including payloads up to 50kg, complex underlying dynamics from the micro-station, and varied operational modes like different rotation speeds—presented a critical design challenge. +The requirement for robust operation across diverse conditions—including payloads up to $50\,\text{kg}$, complex underlying dynamics from the micro-station, and varied operational modes like different rotation speeds—presented a critical design challenge. This challenge was met by embedding robustness directly into the active platform's design, rather than depending solely on complex post-design control synthesis techniques such as $\mathcal{H}_\infty\text{-synthesis}$ and $\mu\text{-synthesis}$. Key elements of this strategy included the model-based evaluation of active stage designs to identify architectures inherently easier to control, the incorporation of collocated actuator/sensor pairs to leverage passivity-based guaranteed stability, and the comparison of architecture to combine several sensors such as sensor fusion and High Authority Control / Low Authority Control (HAC-LAC). Furthermore, decoupling strategies for parallel manipulators were compared (Section\nbsp{}ref:sec:detail_control_decoupling), addressing a topic identified as having limited treatment in the literature. @@ -875,7 +875,7 @@ The integration of such filters into feedback control architectures can also lea ***** Experimental validation of the Nano Active Stabilization System The conclusion of this work involved the experimental implementation and validation of the complete NASS on the ID31 beamline. -Experimental results, presented in Section\nbsp{}ref:sec:test_id31, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native $\approx 10\,\mu m$ level down to the target $\approx 100\,nm$ range during representative scientific experiments. +Experimental results, presented in Section\nbsp{}ref:sec:test_id31, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native $\approx 10\,\mu\text{m}$ level down to the target $\approx 100\,\text{nm}$ range during representative scientific experiments. Crucially, robustness to variations in sample mass and diverse experimental conditions was verified. The NASS thus provides a versatile end-station solution, uniquely combining high payload capacity with nanometer-level accuracy, enabling optimal use of the advanced capabilities of the ESRF-EBS beam and associated detectors. To the author's knowledge, this represents the first demonstration of such a 5-DoF active stabilization platform being used to enhance the accuracy of a complex positioning system to this level. @@ -952,9 +952,9 @@ This refined model was then validated through simulations of scientific experime For the active stabilization stage, the Stewart platform architecture was selected after careful evaluation of various options. Section\nbsp{}ref:sec:nhexa examines the kinematic and dynamic properties of this parallel manipulator, exploring its control challenges and developing appropriate control strategies for implementation within the NASS. -The multi-body modeling approach facilitated the seamless integration of the nano-hexapod with the micro-station model. +The multi-body modeling approach facilitated the seamless integration of the active platform with the micro-station model. -Finally, Section\nbsp{}ref:sec:nass validates the NASS concept through closed-loop simulations of tomography experiments. +Finally, Section\nbsp{}ref:sec:nass validates the NASS concept through acrfull:cl simulations of tomography experiments. These simulations incorporate realistic disturbance sources, confirming the viability of the proposed design approach and control strategies. This progressive approach, beginning with easily comprehensible simplified models, proved instrumental in developing a thorough understanding of the physical phenomena at play. @@ -969,11 +969,11 @@ In this report, a uniaxial model of the acrfull:nass is developed and used to ob Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, yielding to similar conclusions. To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section\nbsp{}ref:sec:uniaxial_micro_station_model). -Then, a model of the nano-hexapod is added on top of the micro-station. +Then, a model of the active platform is added on top of the micro-station. With the added sample and sensors, this gives a uniaxial dynamical model of the acrshort:nass that will be used for further analysis (Section\nbsp{}ref:sec:uniaxial_nano_station_model). The disturbances affecting position stability are identified experimentally (Section\nbsp{}ref:sec:uniaxial_disturbances) and included in the model for dynamical noise budgeting (Section\nbsp{}ref:sec:uniaxial_noise_budgeting). -In all the following analysis, three nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate nano-hexapod design. +In all the following analysis, three active platform stiffnesses are considered to better understand the trade-offs and to find the most adequate active platform design. Three sample masses are also considered to verify the robustness of the applied control strategies with respect to a change of sample. To improve the position stability of the sample, an acrfull:haclac strategy is applied. @@ -982,7 +982,7 @@ It consists of first actively damping the plant (the acrshort:lac part), and the Three active damping techniques are studied (Section\nbsp{}ref:sec:uniaxial_active_damping) which are used to both reduce the effect of disturbances and make the system easier to control afterwards. Once the system is well damped, a feedback position controller is applied and the obtained performance is analyzed (Section\nbsp{}ref:sec:uniaxial_position_control). -Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section\nbsp{}ref:sec:uniaxial_support_compliance) and the presence of dynamics between the nano-hexapod and the sample's acrshort:poi (Section\nbsp{}ref:sec:uniaxial_payload_dynamics). +Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section\nbsp{}ref:sec:uniaxial_support_compliance) and the presence of dynamics between the active platform and the sample's acrshort:poi (Section\nbsp{}ref:sec:uniaxial_payload_dynamics). *** Micro Station Model <> @@ -992,11 +992,11 @@ In this section, a uniaxial model of the micro-station is tuned to match measure **** Measured dynamics -The measurement setup is schematically shown in Figure\nbsp{}ref:fig:uniaxial_ustation_meas_dynamics_schematic where two vertical hammer hits are performed, one on the Granite (force $F_{g}$) and the other on the micro-hexapod's top platform (force $F_{h}$). -The vertical inertial motion of the granite $x_{g}$ and the top platform of the micro-hexapod $x_{h}$ are measured using geophones[fn:uniaxial_1]. +The measurement setup is schematically shown in Figure\nbsp{}ref:fig:uniaxial_ustation_meas_dynamics_schematic where two vertical hammer hits are performed, one on the Granite (force $F_{g}$) and the other on the positioning hexapod's top platform (force $F_{h}$). +The vertical inertial motion of the granite $x_{g}$ and the top platform of the positioning hexapod $x_{h}$ are measured using geophones[fn:uniaxial_1]. Three acrfullpl:frf were computed: one from $F_{h}$ to $x_{h}$ (i.e., the compliance of the micro-station), one from $F_{g}$ to $x_{h}$ (or from $F_{h}$ to $x_{g}$) and one from $F_{g}$ to $x_{g}$. -Due to the poor coherence at low frequencies, these acrlongpl:frf will only be shown between 20 and 200Hz (solid lines in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model). +Due to the poor coherence at low frequencies, these acrlongpl:frf will only be shown between 20 and $200\,\text{Hz}$ (solid lines in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model). #+name: fig:micro_station_uniaxial_model #+caption: Schematic of the Micro-Station measurement setup and uniaxial model. @@ -1021,7 +1021,7 @@ The uniaxial model of the micro-station is shown in Figure\nbsp{}ref:fig:uniaxia It consists of a mass spring damper system with three acrshortpl:dof. A mass-spring-damper system represents the granite (with mass $m_g$, stiffness $k_g$ and damping $c_g$). Another mass-spring-damper system represents the different micro-station stages (the $T_y$ stage, the $R_y$ stage and the $R_z$ stage) with mass $m_t$, damping $c_t$ and stiffness $k_t$. -Finally, a third mass-spring-damper system represents the micro-hexapod with mass $m_h$, damping $c_h$ and stiffness $k_h$. +Finally, a third mass-spring-damper system represents the positioning hexapod with mass $m_h$, damping $c_h$ and stiffness $k_h$. The masses of the different stages are estimated from the 3D model, while the stiffnesses are from the data-sheet of the manufacturers. The damping coefficients were tuned to match the damping identified from the measurements. @@ -1031,17 +1031,17 @@ The parameters obtained are summarized in Table\nbsp{}ref:tab:uniaxial_ustation_ #+caption: Physical parameters used for the micro-station uniaxial model #+attr_latex: :environment tabularx :width 0.6\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| *Stage* | *Mass* | *Stiffness* | *Damping* | -|---------------------+-------------------------+----------------------+-----------------------------| -| Micro-Hexapod | $m_h = 15\,\text{kg}$ | $k_h = 61\,N/\mu m$ | $c_h = 3\,\frac{kN}{m/s}$ | -| $T_y$, $R_y$, $R_z$ | $m_t = 1200\,\text{kg}$ | $k_t = 520\,N/\mu m$ | $c_t = 80\,\frac{kN}{m/s}$ | -| Granite | $m_g = 2500\,\text{kg}$ | $k_g = 950\,N/\mu m$ | $c_g = 250\,\frac{kN}{m/s}$ | +| *Stage* | *Mass* | *Stiffness* | *Damping* | +|---------------------+-------------------------+----------------------------+-------------------------------------------| +| Hexapod | $m_h = 15\,\text{kg}$ | $k_h = 61\,\text{N}/\mu\text{m}$ | $c_h = 3\,\frac{\text{kN}}{\text{m/s}}$ | +| $T_y$, $R_y$, $R_z$ | $m_t = 1200\,\text{kg}$ | $k_t = 520\,\text{N}/\mu\text{m}$ | $c_t = 80\,\frac{\text{kN}}{\text{m/s}}$ | +| Granite | $m_g = 2500\,\text{kg}$ | $k_g = 950\,\text{N}/\mu\text{m}$ | $c_g = 250\,\frac{\text{kN}}{\text{m/s}}$ | Two disturbances are considered which are shown in red: the floor motion $x_f$ and the stage vibrations represented by $f_t$. The hammer impacts $F_{h}, F_{g}$ are shown in blue, whereas the measured inertial motions $x_{h}, x_{g}$ are shown in black. **** Comparison of model and measurements -The transfer functions from the forces injected by the hammers to the measured inertial motion of the micro-hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model. +The transfer functions from the forces injected by the hammers to the measured inertial motion of the positioning hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model. Because the uniaxial model has three acrshortpl:dof, only three modes with frequencies at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$ are modeled. Many more modes can be observed in the measurements (see Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model). @@ -1053,18 +1053,18 @@ More accurate models will be used later on. #+attr_latex: :scale 0.8 [[file:figs/uniaxial_comp_frf_meas_model.png]] -*** Nano-Hexapod Model +*** Active Platform Model <> **** Introduction :ignore: -A model of the nano-hexapod and sample is now added on top of the uniaxial model of the micro-station (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). +A model of the active platform and sample is now added on top of the uniaxial model of the micro-station (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). Disturbances (shown in red) are gls:fs the direct forces applied to the sample (for example cable forces), gls:ft representing the vibrations induced when scanning the different stages and gls:xf the floor motion. -The control signal is the force applied by the nano-hexapod $f$ and the measurement is the relative motion between the sample and the granite $d$. -The sample is here considered as a rigid body and rigidly fixed to the nano-hexapod. -The effect of resonances between the sample's acrshort:poi and the nano-hexapod actuator will be considered in Section\nbsp{}ref:sec:uniaxial_payload_dynamics. +The control signal is the force applied by the active platform $f$ and the measurement is the relative motion between the sample and the granite $d$. +The sample is here considered as a rigid body and rigidly fixed to the active platform. +The effect of resonances between the sample's acrshort:poi and the active platform actuator will be considered in Section\nbsp{}ref:sec:uniaxial_payload_dynamics. #+name: fig:uniaxial_model_micro_station_nass_with_tf -#+caption: Uniaxial model of the NASS (\subref{fig:uniaxial_model_micro_station_nass}) with the micro-station shown in black, the nano-hexapod represented in blue and the sample represented in green. Disturbances are shown in red. Extracted transfer function from $f$ to $d$ (\subref{fig:uniaxial_plant_first_params}). +#+caption: Uniaxial model of the NASS (\subref{fig:uniaxial_model_micro_station_nass}) with the micro-station shown in black, the active platform represented in blue and the sample represented in green. Disturbances are shown in red. Extracted transfer function from $f$ to $d$ (\subref{fig:uniaxial_plant_first_params}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_model_micro_station_nass}Uniaxial mass-spring-damper model of the NASS} @@ -1081,18 +1081,18 @@ The effect of resonances between the sample's acrshort:poi and the nano-hexapod #+end_subfigure #+end_figure -**** Nano-Hexapod Parameters -The nano-hexapod is represented by a mass spring damper system (shown in blue in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). +**** Active Platform Parameters +The active platform is represented by a mass spring damper system (shown in blue in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). Its mass gls:mn is set to $15\,\text{kg}$ while its stiffness $k_n$ can vary depending on the chosen architecture/technology. The sample is represented by a mass gls:ms that can vary from $1\,\text{kg}$ up to $50\,\text{kg}$. -As a first example, the nano-hexapod stiffness of is set at $k_n = 10\,N/\mu m$ and the sample mass is chosen at $m_s = 10\,\text{kg}$. +As a first example, the active platform stiffness of is set at $k_n = 10\,\text{N}/\mu\text{m}$ and the sample mass is chosen at $m_s = 10\,\text{kg}$. **** Obtained Dynamic Response The sensitivity to disturbances (i.e., the transfer functions from $x_f,f_t,f_s$ to $d$) can be extracted from the uniaxial model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass and are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_first_params. The /plant/ (i.e., the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure\nbsp{}ref:fig:uniaxial_plant_first_params. -For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass will be considered: three nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) combined with three sample's masses ($m_s = 1\,kg$, $m_s = 25\,kg$ and $m_s = 50\,kg$). +For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass will be considered: three active platform stiffnesses ($k_n = 0.01\,\text{N}/\mu\text{m}$, $k_n = 1\,\text{N}/\mu\text{m}$ and $k_n = 100\,\text{N}/\mu\text{m}$) combined with three sample's masses ($m_s = 1\,\text{kg}$, $m_s = 25\,\text{kg}$ and $m_s = 50\,\text{kg}$). #+name: fig:uniaxial_sensitivity_dist_first_params #+caption: Sensitivity of the relative motion $d$ to disturbances: $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_first_params_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}) @@ -1122,7 +1122,7 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nb <> **** Introduction :ignore: To quantify disturbances (red signals in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), three geophones[fn:uniaxial_2] are used. -One is located on the floor, another one on the granite, and the last one on the micro-hexapod's top platform (see Figure\nbsp{}ref:fig:uniaxial_ustation_meas_disturbances). +One is located on the floor, another one on the granite, and the last one on the positioning hexapod's top platform (see Figure\nbsp{}ref:fig:uniaxial_ustation_meas_disturbances). The geophone located on the floor was used to measure the floor motion $x_f$ while the other two geophones were used to measure vibrations introduced by scanning of the $T_y$ stage and $R_z$ stage (see Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup). #+name: fig:uniaxial_ustation_meas_disturbances_setup @@ -1145,20 +1145,20 @@ The geophone located on the floor was used to measure the floor motion $x_f$ whi **** Ground Motion To acquire the geophone signals, the measurement setup shown in Figure\nbsp{}ref:fig:uniaxial_geophone_meas_chain is used. -The voltage generated by the geophone is amplified using a low noise voltage amplifier[fn:uniaxial_3] with a gain of 60dB before going to the acrfull:adc. +The voltage generated by the geophone is amplified using a low noise voltage amplifier[fn:uniaxial_3] with a gain of $60\,\text{dB}$ before going to the acrfull:adc. This is done to improve the signal-to-noise ratio. To reconstruct the displacement $x_f$ from the measured voltage $\hat{V}_{x_f}$, the transfer function of the measurement chain from $x_f$ to $\hat{V}_{x_f}$ needs to be estimated. -First, the transfer function $G_{geo}$ from the floor motion $x_{f}$ to the generated geophone voltage $V_{x_f}$ is shown in\nbsp{}eqref:eq:uniaxial_geophone_tf, with $T_g = 88\,\frac{V}{m/s}$ the sensitivity of the geophone, $f_0 = \frac{\omega_0}{2\pi} = 2\,\text{Hz}$ its resonance frequency and $\xi = 0.7$ its damping ratio. +First, the transfer function $G_{geo}$ from the floor motion $x_{f}$ to the generated geophone voltage $V_{x_f}$ is shown in\nbsp{}eqref:eq:uniaxial_geophone_tf, with $T_g = 88\,\frac{V}{\text{m/s}}$ the sensitivity of the geophone, $f_0 = \frac{\omega_0}{2\pi} = 2\,\text{Hz}$ its resonance frequency and $\xi = 0.7$ its damping ratio. This model of the geophone was taken from\nbsp{}[[cite:&collette12_review]]. The gain of the voltage amplifier is $V^{\prime}_{x_f}/V_{x_f} = g_0 = 1000$. \begin{equation}\label{eq:uniaxial_geophone_tf} -G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi \omega_0 s + \omega_0^2} \quad \left[ V/m \right] +G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi \omega_0 s + \omega_0^2} \quad \left[ \text{V/m} \right] \end{equation} #+name: fig:uniaxial_geophone_meas_chain -#+caption: Measurement setup for one geophone. The inertial displacement $x$ is converted to a voltage $V$ by the geophone. This voltage is amplified by a factor $g_0 = 60\,dB$ using a low-noise voltage amplifier. It is then converted to a digital value $\hat{V}_x$ using a 16bit ADC. +#+caption: Measurement setup for one geophone. The inertial displacement $x$ is converted to a voltage $V$ by the geophone. This voltage is amplified by a factor $g_0 = 60\,\text{dB}$ using a low-noise voltage amplifier. It is then converted to a digital value $\hat{V}_x$ using a 16bit ADC. [[file:figs/uniaxial_geophone_meas_chain.png]] The acrfull:asd of the floor motion $\Gamma_{x_f}$ can be computed from the acrlong:asd of measured voltage $\Gamma_{\hat{V}_{x_f}}$ using\nbsp{}eqref:eq:uniaxial_asd_floor_motion. @@ -1188,7 +1188,7 @@ The estimated acrshort:asd $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in **** Stage Vibration To estimate the vibrations induced by scanning the micro-station stages, two geophones are used, as shown in Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup. -The vertical relative velocity between the top platform of the micro hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. +The vertical relative velocity between the top platform of the positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. The vibrations induced by the $T_y$ stage are not considered here because they have less amplitude than the vibrations induced by the $R_z$ stage and because the $T_y$ stage can be scanned at lower velocities if the induced vibrations are found to be an issue. The amplitude spectral density of the relative motion with and without the Spindle rotation are compared in Figure\nbsp{}ref:fig:uniaxial_asd_vibration_spindle_rotation. @@ -1196,11 +1196,11 @@ It is shown that the spindle rotation increases the vibrations above $20\,\text{ The sharp peak observed at $24\,\text{Hz}$ is believed to be induced by electromagnetic interference between the currents in the spindle motor phases and the geophone cable because this peak is not observed when rotating the spindle "by hand". #+name: fig:uniaxial_asd_vibration_spindle_rotation -#+caption: Amplitude Spectral Density $\Gamma_{R_z}$ of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating +#+caption: Amplitude Spectral Density $\Gamma_{R_z}$ of the relative motion measured between the granite and the positioning hexapod's top platform during Spindle rotating #+attr_latex: :scale 0.8 [[file:figs/uniaxial_asd_vibration_spindle_rotation.png]] -To compute the equivalent disturbance force $f_t$ (Figure\nbsp{}ref:fig:uniaxial_model_micro_station) that induces such motion, the transfer function $G_{f_t}(s)$ from $f_t$ to the relative motion between the micro-hexapod's top platform and the granite $(x_{h} - x_{g})$ is extracted from the model. +To compute the equivalent disturbance force $f_t$ (Figure\nbsp{}ref:fig:uniaxial_model_micro_station) that induces such motion, the transfer function $G_{f_t}(s)$ from $f_t$ to the relative motion between the positioning hexapod's top platform and the granite $(x_{h} - x_{g})$ is extracted from the model. The amplitude spectral density $\Gamma_{f_{t}}$ of the disturbance force is them computed from\nbsp{}eqref:eq:uniaxial_ft_asd and is shown in Figure\nbsp{}ref:fig:uniaxial_asd_disturbance_force. \begin{equation}\label{eq:uniaxial_ft_asd} @@ -1221,15 +1221,15 @@ This is very useful to identify what is limiting the performance of the system, <> From the uniaxial model of the acrshort:nass (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), the transfer function from the disturbances ($f_s$, $x_f$ and $f_t$) to the displacement $d$ are computed. -This is done for two extreme sample masses $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$ and three nano-hexapod stiffnesses: -- $k_n = 0.01\,N/\mu m$ that represents a voice coil actuator with soft flexible guiding -- $k_n = 1\,N/\mu m$ that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator -- $k_n = 100\,N/\mu m$ that represents a stiff piezoelectric stack actuator +This is done for two extreme sample masses $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$ and three active platform stiffnesses: +- $k_n = 0.01\,\text{N}/\mu\text{m}$ that represents a voice coil actuator with soft flexible guiding +- $k_n = 1\,\text{N}/\mu\text{m}$ that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator +- $k_n = 100\,\text{N}/\mu\text{m}$ that represents a stiff piezoelectric stack actuator -The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the sample mass $m_s = 1\,\text{kg}$ (the same conclusions can be drawn with $m_s = 50\,\text{kg}$): -- The soft nano-hexapod is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs) -- Between the suspension mode of the nano-hexapod (here at 5Hz for the soft nano-hexapod) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft) -- Above the suspension mode of the nano-hexapod, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to $1$ (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf) +The obtained sensitivity to disturbances for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the sample mass $m_s = 1\,\text{kg}$ (the same conclusions can be drawn with $m_s = 50\,\text{kg}$): +- The soft active platform is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs) +- Between the suspension mode of the active platform (here at $5\,\text{Hz}$) and the first mode of the micro-station (here at $70\,\text{Hz}$), the disturbances induced by the stage vibrations are filtered out (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft) +- Above the suspension mode of the active platform, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to $1$ (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf) #+name: fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses #+caption: Sensitivity of $d$ to disturbances for three different nano-hexpod stiffnesses. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}) @@ -1257,15 +1257,15 @@ The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses **** Open-Loop Dynamic Noise Budgeting <> -Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion $d$ for each nano-hexapod and sample configuration. -The acrfull:cas of the relative motion $d$ due to both floor motion $x_f$ and stage vibrations $f_t$ are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three nano-hexapod stiffnesses. -It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft nano-hexapod below $5\,\text{Hz}$. +Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion $d$ for each active platform and sample configuration. +The acrfull:cas of the relative motion $d$ due to both floor motion $x_f$ and stage vibrations $f_t$ are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three active platform stiffnesses. +It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft active platform below $5\,\text{Hz}$. -The total cumulative amplitude spectrum of $d$ for the three nano-hexapod stiffnesses and for the two samples masses are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. +The total cumulative amplitude spectrum of $d$ for the three active platform stiffnesses and for the two samples masses are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. The conclusion is that the sample mass has little effect on the cumulative amplitude spectrum of the relative motion $d$. #+name: fig:uniaxial_cas_d_disturbances -#+caption: Cumulative Amplitude Spectrum of the relative motion $d$. The effect of $x_f$ and $f_t$ are shown in (\subref{fig:uniaxial_cas_d_disturbances_stiffnesses}). The effect of sample mass for the three hexapod stiffnesses is shown in (\subref{fig:uniaxial_cas_d_disturbances_payload_masses}). The control objective of having a residual error of 20 nm RMS is shown by the horizontal black dashed line. +#+caption: Cumulative Amplitude Spectrum of the relative motion $d$. The effect of $x_f$ and $f_t$ are shown in (\subref{fig:uniaxial_cas_d_disturbances_stiffnesses}). The effect of sample mass for the three active platform stiffnesses is shown in (\subref{fig:uniaxial_cas_d_disturbances_payload_masses}). The control objective of having a residual error of $20\,\text{nm RMS}$ is shown by the horizontal black dashed line. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_d_disturbances_stiffnesses}Effect of floor motion $x_f$ and stage disturbances $f_t$} @@ -1274,7 +1274,7 @@ The conclusion is that the sample mass has little effect on the cumulative ampli #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_d_disturbances_stiffnesses.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_d_disturbances_payload_masses}Effect of nano-hexapod stiffness $k_n$ and payload mass $m_s$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_d_disturbances_payload_masses}Effect of active platform stiffness $k_n$ and payload mass $m_s$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1285,24 +1285,24 @@ The conclusion is that the sample mass has little effect on the cumulative ampli **** Conclusion The open-loop residual vibrations of $d$ can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. -This residual vibration of $d$ is found to be in the order of $100\,nm\,\text{RMS}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$), $200\,nm\,\text{RMS}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$) and $1\,\mu m\,\text{RMS}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$). -From this analysis, it may be concluded that the stiffer the nano-hexapod the better. +This residual vibration of $d$ is found to be in the order of $100\,\text{nm RMS}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$), $200\,\text{nm RMS}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$) and $1\,\mu\text{m}\,\text{RMS}$ for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$). +From this analysis, it may be concluded that the stiffer the active platform the better. However, what is more important is the /closed-loop/ residual vibration of $d$ (i.e., while the feedback controller is used). -The goal is to obtain a closed-loop residual vibration $\epsilon_d \approx 20\,nm\,\text{RMS}$ (represented by an horizontal dashed black line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses). -The bandwidth of the feedback controller leading to a closed-loop residual vibration of $20\,nm\,\text{RMS}$ can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. +The goal is to obtain a closed-loop residual vibration $\epsilon_d \approx 20\,\text{nm RMS}$ (represented by an horizontal dashed black line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses). +The bandwidth of the feedback controller leading to a closed-loop residual vibration of $20\,\text{nm RMS}$ can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. -A closed loop bandwidth of $\approx 10\,\text{Hz}$ is found for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$), $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$), and $\approx 100\,\text{Hz}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$). -Therefore, while the /open-loop/ vibration is the lowest for the stiff nano-hexapod, it requires the largest feedback bandwidth to meet the specifications. +A closed loop bandwidth of $\approx 10\,\text{Hz}$ is found for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$), $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$), and $\approx 100\,\text{Hz}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$). +Therefore, while the /open-loop/ vibration is the lowest for the stiff active platform, it requires the largest feedback bandwidth to meet the specifications. -The advantage of the soft nano-hexapod can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft. +The advantage of the soft active platform can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft. *** Active Damping <> **** Introduction :ignore: -In this section, three active damping techniques are applied to the nano-hexapod (see Figure\nbsp{}ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]], Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] and Direct Velocity Feedback (DVF)\nbsp{}[[cite:&karnopp74_vibrat_contr_using_semi_activ_force_gener;&serrand00_multic_feedb_contr_isolat_base_excit_vibrat;&preumont02_force_feedb_versus_accel_feedb]]. +In this section, three active damping techniques are applied to the active platform (see Figure\nbsp{}ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]], Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] and Direct Velocity Feedback (DVF)\nbsp{}[[cite:&karnopp74_vibrat_contr_using_semi_activ_force_gener;&serrand00_multic_feedb_contr_isolat_base_excit_vibrat;&preumont02_force_feedb_versus_accel_feedb]]. -These damping strategies are first described (Section\nbsp{}ref:ssec:uniaxial_active_damping_strategies) and are then compared in terms of achievable damping of the nano-hexapod mode (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping), reduction of the effect of disturbances (i.e., $x_f$, $f_t$ and $f_s$) on the displacement $d$ (Sections\nbsp{}ref:ssec:uniaxial_active_damping_sensitivity_disturbances). +These damping strategies are first described (Section\nbsp{}ref:ssec:uniaxial_active_damping_strategies) and are then compared in terms of achievable damping of the active platform mode (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping), reduction of the effect of disturbances (i.e., $x_f$, $f_t$ and $f_s$) on the displacement $d$ (Sections\nbsp{}ref:ssec:uniaxial_active_damping_sensitivity_disturbances). #+name: fig:uniaxial_active_damping_strategies #+caption: Three active damping strategies. Integral Force Feedback (\subref{fig:uniaxial_active_damping_strategies_iff}) using a force sensor, Relative Damping Control (\subref{fig:uniaxial_active_damping_strategies_rdc}) using a relative displacement sensor, and Direct Velocity Feedback (\subref{fig:uniaxial_active_damping_strategies_dvf}) using a geophone @@ -1419,14 +1419,14 @@ The plant dynamics for all three active damping techniques are shown in Figure\n All have /alternating poles and zeros/ meaning that the phase does not vary by more than $180\,\text{deg}$ which makes the design of a /robust/ damping controller very easy. This alternating poles and zeros property is guaranteed for the IFF and acrshort:rdc cases because the sensors are collocated with the actuator\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7]]. -For the acrshort:dvf controller, this property is not guaranteed, and may be lost if some flexibility between the nano-hexapod and the sample is considered\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 8.4]]. +For the acrshort:dvf controller, this property is not guaranteed, and may be lost if some flexibility between the active platform and the sample is considered\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 8.4]]. -When the nano-hexapod's suspension modes are at frequencies lower than the resonances of the micro-station (blue and red curves in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the IFF and acrshort:dvf transfer functions. -For the stiff nano-hexapod (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques. -Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used. +When the active platform's suspension modes are at frequencies lower than the resonances of the micro-station (blue and red curves in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the IFF and acrshort:dvf transfer functions. +For the stiff active platform (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques. +Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff active platform is used. #+name: fig:uniaxial_plant_active_damping_techniques -#+caption: Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), for three nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$ in blue, $k_n = 1\,N/\mu m$ in red and $k_n = 100\,N/\mu m$ in yellow) and three sample's masses ($m_s = 1\,kg$: solid curves, $m_s = 25\,kg$: dot-dashed curves, and $m_s = 50\,kg$: dashed curves). +#+caption: Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), for three active platform stiffnesses ($k_n = 0.01\,\text{N}/\mu\text{m}$ in blue, $k_n = 1\,\text{N}/\mu\text{m}$ in red and $k_n = 100\,\text{N}/\mu\text{m}$ in yellow) and three sample's masses ($m_s = 1\,\text{kg}$: solid curves, $m_s = 25\,\text{kg}$: dot-dashed curves, and $m_s = 50\,\text{kg}$: dashed curves). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_active_damping_techniques_iff}IFF} @@ -1461,35 +1461,35 @@ This is illustrated in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniqu \xi = \sin(\phi) \end{equation} -The Root Locus for the three nano-hexapod stiffnesses and the three active damping techniques are shown in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques. -All three active damping approaches can lead to /critical damping/ of the nano-hexapod suspension mode (angle $\phi$ can be increased up to 90 degrees). +The Root Locus for the three active platform stiffnesses and the three active damping techniques are shown in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques. +All three active damping approaches can lead to /critical damping/ of the active platform suspension mode (angle $\phi$ can be increased up to 90 degrees). There is even some damping authority on micro-station modes in the following cases: -- IFF with a stiff nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: +- IFF with a stiff active platform (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: This can be understood from the mechanical equivalent of IFF shown in Figure\nbsp{}ref:fig:uniaxial_active_damping_iff_equiv considering an high stiffness $k$. - The micro-station top platform is connected to an inertial mass (the nano-hexapod) through a damper, which dampens the micro-station suspension suspension mode. -- DVF with a stiff nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: - In that case, the "sky hook damper" (see mechanical equivalent of acrshort:dvf in Figure\nbsp{}ref:fig:uniaxial_active_damping_dvf_equiv) is connected to the micro-station top platform through the stiff nano-hexapod. -- RDC with a soft nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode) :: - At the frequency of the micro-station mode, the nano-hexapod top mass behaves as an inertial reference because the suspension mode of the soft nano-hexapod is at much lower frequency. - The micro-station and the nano-hexapod masses are connected through a large damper induced by acrshort:rdc (see mechanical equivalent in Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv) which allows some damping of the micro-station. + The micro-station top platform is connected to an inertial mass (the active platform) through a damper, which dampens the micro-station suspension suspension mode. +- DVF with a stiff active platform (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: + In that case, the "sky hook damper" (see mechanical equivalent of acrshort:dvf in Figure\nbsp{}ref:fig:uniaxial_active_damping_dvf_equiv) is connected to the micro-station top platform through the stiff active platform. +- RDC with a soft active platform (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode) :: + At the frequency of the micro-station mode, the active platform top mass behaves as an inertial reference because the suspension mode of the soft active platform is at much lower frequency. + The micro-station and the active platform masses are connected through a large damper induced by acrshort:rdc (see mechanical equivalent in Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv) which allows some damping of the micro-station. #+name: fig:uniaxial_root_locus_damping_techniques -#+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three nano-hexapod stiffnesses. The Root Loci are zoomed in the suspension mode of the nano-hexapod. +#+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The Root Loci are zoomed in the suspension mode of the active platform. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_root_locus_damping_techniques_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_root_locus_damping_techniques_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1498,7 +1498,7 @@ There is even some damping authority on micro-station modes in the following cas #+end_figure #+name: fig:uniaxial_root_locus_damping_techniques_micro_station_mode -#+caption: Root Locus for the three damping techniques applied with the soft nano-hexapod. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod. +#+caption: Root Locus for the three damping techniques applied with the soft active platform. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the positioning hexapod. #+attr_latex: :scale 0.8 [[file:figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png]] @@ -1509,19 +1509,19 @@ All three active damping techniques yielded similar damped plants. #+caption: Obtained damped transfer function from $f$ to $d$ for the three damping techniques. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_damped_plant_three_active_damping_techniques_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_damped_plant_three_active_damping_techniques_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1532,16 +1532,16 @@ All three active damping techniques yielded similar damped plants. **** Sensitivity to disturbances and Noise Budgeting <> -Reasonable gains are chosen for the three active damping strategies such that the nano-hexapod suspension mode is well damped. +Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped. The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping. -The comparison is done with the nano-hexapod having a stiffness $k_n = 1\,N/\mu m$. +The comparison is done with the active platform having a stiffness $k_n = 1\,\text{N}/\mu\text{m}$. Several conclusions can be drawn by comparing the obtained sensitivity transfer functions: -- IFF degrades the sensitivity to direct forces on the sample (i.e., the compliance) below the resonance of the nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_fs). +- IFF degrades the sensitivity to direct forces on the sample (i.e., the compliance) below the resonance of the active platform (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_fs). This is a well-known effect of using IFF for vibration isolation\nbsp{}[[cite:&collette15_sensor_fusion_method_high_perfor]]. -- RDC degrades the sensitivity to stage vibrations around the nano-hexapod's resonance as compared to the other two methods (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_ft). +- RDC degrades the sensitivity to stage vibrations around the active platform's resonance as compared to the other two methods (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_ft). This is because the equivalent damper in parallel with the actuator (see Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv) increases the transmission of the micro-station vibration to the sample which is not the same for the other two active damping strategies. -- both IFF and acrshort:dvf degrade the sensitivity to floor motion below the resonance of the nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_xf). +- both IFF and acrshort:dvf degrade the sensitivity to floor motion below the resonance of the active platform (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_xf). #+name: fig:uniaxial_sensitivity_dist_active_damping #+caption: Change of sensitivity to disturbance with all three active damping strategies. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_active_damping_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_active_damping_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_active_damping_fs}) @@ -1575,19 +1575,19 @@ All three active damping methods give similar results. #+caption: Comparison of the cumulative amplitude spectrum (CAS) of the distance $d$ for all three active damping techniques (acrshort:ol in black, IFF in blue, RDC in red and DVF in yellow). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.37\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_active_damping_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_active_damping_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1601,7 +1601,7 @@ Three active damping strategies have been studied for the acrfull:nass. Equivalent mechanical representations were derived in Section\nbsp{}ref:ssec:uniaxial_active_damping_strategies which are helpful for understanding the specific effects of each strategy. The plant dynamics were then compared in Section\nbsp{}ref:ssec:uniaxial_active_damping_plants and were found to all have alternating poles and zeros, which helps in the design of the active damping controller. However, this property is not guaranteed for acrshort:dvf. -The achievable damping of the nano-hexapod suspension mode can be made as large as possible for all three active damping techniques (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping). +The achievable damping of the active platform suspension mode can be made as large as possible for all three active damping techniques (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping). Even some damping can be applied to some micro-station modes in specific cases. The obtained damped plants were found to be similar. The damping strategies were then compared in terms of disturbance reduction in Section\nbsp{}ref:ssec:uniaxial_active_damping_sensitivity_disturbances. @@ -1660,29 +1660,29 @@ This control architecture applied to the uniaxial model is shown in Figure\nbsp{ **** Damped Plant Dynamics <> -The damped plants obtained for the three nano-hexapod stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses. -For $k_n = 0.01\,N/\mu m$ and $k_n = 1\,N/\mu m$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). -However, this is not the case for the stiff nano-hexapod ($k_n = 100\,N/\mu m$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). -This is due to the interaction between the micro-station (modeled modes at 70Hz, 140Hz and 320Hz) and the nano-hexapod. +The damped plants obtained for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses. +For $k_n = 0.01\,\text{N}/\mu\text{m}$ and $k_n = 1\,\text{N}/\mu\text{m}$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). +However, this is not the case for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). +This is due to the interaction between the micro-station (modeled modes at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$) and the active platform. This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_compliance. #+name: fig:uniaxial_hac_iff_damped_plants_masses #+caption: Obtained damped plant using Integral Force Feedback for three sample masses #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.37\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_hac_iff_damped_plants_masses_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_hac_iff_damped_plants_masses_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1693,27 +1693,27 @@ This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_ **** Position Feedback Controller <> -The objective is to design high-authority feedback controllers for the three nano-hexapods. +The objective is to design high-authority feedback controllers for the three active platforms. This controller must be robust to the change of sample's mass (from $1\,\text{kg}$ up to $50\,\text{kg}$). The required feedback bandwidths were estimated in Section\nbsp{}ref:sec:uniaxial_noise_budgeting: -- $f_b \approx 10\,\text{Hz}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$). - Near this frequency, the plants (shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft) are equivalent to a mass line (i.e., slope of $-40\,dB/\text{dec}$ and a phase of -180 degrees). - The gain of this mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,kg$ up to $65\,kg$). +- $f_b \approx 10\,\text{Hz}$ for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$). + Near this frequency, the plants (shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft) are equivalent to a mass line (i.e., slope of $-40\,\text{dB/dec}$ and a phase of -180 degrees). + The gain of this mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,\text{kg}$ up to $65\,\text{kg}$). This means that the designed controller will need to have /large gain margins/ to be robust to the change of sample's mass. -- $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$). - Similar to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). - It will probably be easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode. -- $\approx 100\,\text{Hz}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$). - Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). +- $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$). + Similar to the soft active platform, the plants near the crossover frequency are equivalent to a mass line (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). + It will probably be easier to have a little bit more bandwidth in this configuration to be further away from the active platform suspension mode. +- $\approx 100\,\text{Hz}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$). + Contrary to the two first active platform stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). The micro-station is not stiff enough to have a clear stiffness line at this frequency. Therefore, there is both a change of phase and gain depending on the sample mass. This makes the robust design of the controller more complicated. -Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure\nbsp{}ref:fig:uniaxial_nyquist_hac). +Position feedback controllers are designed for each active platform such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure\nbsp{}ref:fig:uniaxial_nyquist_hac). An arbitrary minimum modulus margin of $0.25$ was chosen when designing the controllers. These acrfullpl:hac are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency, and a acrfull:lpf to increase the robustness to high frequency dynamics. -The controllers used for the three nano-hexapod are shown in Equation\nbsp{}eqref:eq:uniaxial_hac_formulas, and the parameters used are summarized in Table\nbsp{}ref:tab:uniaxial_feedback_controller_parameters. +The controllers used for the three active platform are shown in Equation\nbsp{}eqref:eq:uniaxial_hac_formulas, and the parameters used are summarized in Table\nbsp{}ref:tab:uniaxial_feedback_controller_parameters. \begin{subequations} \label{eq:uniaxial_hac_formulas} \begin{align} @@ -1736,17 +1736,17 @@ K_{\text{stiff}}(s) &= g \cdot #+caption: Parameters used for the position feedback controllers #+attr_latex: :environment tabularx :width 0.75\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| | *Soft* | *Moderately stiff* | *Stiff* | -|--------+-------------------------------------------+--------------------------------------------+------------------------------------------| -| *Gain* | $g = 4 \cdot 10^5$ | $g = 3 \cdot 10^6$ | $g = 6 \cdot 10^{12}$ | -| *Lead* | $a = 5$, $\omega_c = 20\,Hz$ | $a = 4$, $\omega_c = 70\,Hz$ | $a = 5$, $\omega_c = 100\,Hz$ | -| *Lag* | $\omega_0 = 5\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_0 = 20\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_i = 0.01\,Hz$ | -| *LPF* | $\omega_l = 200\,Hz$ | $\omega_l = 300\,Hz$ | $\omega_l = 500\,Hz$ | +| | *Soft* | *Moderately stiff* | *Stiff* | +|--------+---------------------------------------------------------+----------------------------------------------------------+--------------------------------------| +| *Gain* | $g = 4 \cdot 10^5$ | $g = 3 \cdot 10^6$ | $g = 6 \cdot 10^{12}$ | +| *Lead* | $a = 5$, $\omega_c = 20\,\text{Hz}$ | $a = 4$, $\omega_c = 70\,\text{Hz}$ | $a = 5$, $\omega_c = 100\,\text{Hz}$ | +| *Lag* | $\omega_0 = 5\,\text{Hz}$, $\omega_i = 0.01\,\text{Hz}$ | $\omega_0 = 20\,\text{Hz}$, $\omega_i = 0.01\,\text{Hz}$ | $\omega_i = 0.01\,\text{Hz}$ | +| *LPF* | $\omega_l = 200\,\text{Hz}$ | $\omega_l = 300\,\text{Hz}$ | $\omega_l = 500\,\text{Hz}$ | -The loop gains corresponding to the designed acrlongpl:hac for the three nano-hexapod are shown in Figure\nbsp{}ref:fig:uniaxial_loop_gain_hac. -We can see that for the soft and moderately stiff nano-hexapod (Figures\nbsp{}ref:fig:uniaxial_nyquist_hac_vc and ref:fig:uniaxial_nyquist_hac_md), the crossover frequency varies significantly with the sample mass. +The loop gains corresponding to the designed acrlongpl:hac for the three active platform are shown in Figure\nbsp{}ref:fig:uniaxial_loop_gain_hac. +We can see that for the soft and moderately stiff active platform (Figures\nbsp{}ref:fig:uniaxial_nyquist_hac_vc and ref:fig:uniaxial_nyquist_hac_md), the crossover frequency varies significantly with the sample mass. This is because the crossover frequency corresponds to the mass line of the plant (whose gain is inversely proportional to the mass). -For the stiff nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_nyquist_hac_pz), it was difficult to achieve the desired closed-loop bandwidth of $\approx 100\,\text{Hz}$. +For the stiff active platform (Figure\nbsp{}ref:fig:uniaxial_nyquist_hac_pz), it was difficult to achieve the desired closed-loop bandwidth of $\approx 100\,\text{Hz}$. A crossover frequency of $\approx 65\,\text{Hz}$ was achieved instead. Note that these controllers were not designed using any optimization methods. @@ -1756,19 +1756,19 @@ The goal is to have a first estimation of the attainable performance. #+caption: Nyquist Plot for the high authority controller. The minimum modulus margin is illustrated by a black circle. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_nyquist_hac_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_nyquist_hac_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1780,19 +1780,19 @@ The goal is to have a first estimation of the attainable performance. #+caption: Loop gains for the High Authority Controllers #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_loop_gain_hac_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_loop_gain_hac_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1804,11 +1804,11 @@ The goal is to have a first estimation of the attainable performance. <> The acrlong:hac are then implemented and the closed-loop sensitivities to disturbances are computed. -These are compared with the open-loop and damped plants cases in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass). +These are compared with the open-loop and damped plants cases in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff active platform with $25\,\text{kg}$ sample's mass). As expected, the sensitivity to disturbances decreased in the controller bandwidth and slightly increased outside this bandwidth. #+name: fig:uniaxial_sensitivity_dist_hac_lac -#+caption: Change of sensitivity to disturbances with acrshort:lac and with acrshort:haclac. A nano-Hexapod with $k_n = 1\,N/\mu m$ and a sample mass of $25\,kg$ is used. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}) +#+caption: Change of sensitivity to disturbances with acrshort:lac and with acrshort:haclac. An active platform with $k_n = 1\,\text{N}/\mu\text{m}$ and a sample mass of $25\,\text{kg}$ is used. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_fs}Direct forces} @@ -1831,27 +1831,27 @@ As expected, the sensitivity to disturbances decreased in the controller bandwid #+end_subfigure #+end_figure -The cumulative amplitude spectrum of the motion $d$ is computed for all nano-hexapod configurations, all sample masses and in the acrfull:ol, damped (IFF) and position controlled (HAC-IFF) cases. +The cumulative amplitude spectrum of the motion $d$ is computed for all active platform configurations, all sample masses and in the acrfull:ol, damped (IFF) and position controlled (HAC-IFF) cases. The results are shown in Figure\nbsp{}ref:fig:uniaxial_cas_hac_lac. -Obtained root mean square values of the distance $d$ are better for the soft nano-hexapod ($\approx 25\,nm$ to $\approx 35\,nm$ depending on the sample's mass) than for the stiffer nano-hexapod (from $\approx 30\,nm$ to $\approx 70\,nm$). +Obtained root mean square values of the distance $d$ are better for the soft active platform ($\approx 25\,\text{nm}$ to $\approx 35\,\text{nm}$ depending on the sample's mass) than for the stiffer active platform (from $\approx 30\,\text{nm}$ to $\approx 70\,\text{nm}$). #+name: fig:uniaxial_cas_hac_lac -#+caption: Cumulative Amplitude Spectrum for all three nano-hexapod stiffnesses - Comparison of OL, IFF and acrshort:haclac cases +#+caption: Cumulative Amplitude Spectrum for all three active platform stiffnesses - Comparison of OL, IFF and acrshort:haclac cases #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.37\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_hac_lac_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_hac_lac_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1861,13 +1861,13 @@ Obtained root mean square values of the distance $d$ are better for the soft nan **** Conclusion -On the basis of the open-loop noise budgeting made in Section\nbsp{}ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,nm\,\text{RMS}$ was estimated. +On the basis of the open-loop noise budgeting made in Section\nbsp{}ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,\text{nm RMS}$ was estimated. To achieve such bandwidth, the acrshort:haclac strategy was followed, which consists of first using an active damping controller (studied in Section\nbsp{}ref:sec:uniaxial_active_damping) and then adding a high authority position feedback controller. In this section, feedback controllers were designed in such a way that the required closed-loop bandwidth was reached while being robust to changes in the payload mass. -The attainable vibration control performances were estimated for the three nano-hexapod stiffnesses and were found to be close to the required values. -However, the stiff nano-hexapod ($k_n = 100\,N/\mu m$) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. -A slight advantage can be given to the soft nano-hexapod as it requires less feedback bandwidth while providing better stability results. +The attainable vibration control performances were estimated for the three active platform stiffnesses and were found to be close to the required values. +However, the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. +A slight advantage can be given to the soft active platform as it requires less feedback bandwidth while providing better stability results. *** Effect of limited micro-station compliance <> @@ -1877,22 +1877,22 @@ In this section, the impact of the compliance of the support (i.e., the micro-st This is a critical point because the dynamics of the micro-station is complex, depends on the considered direction (see measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model) and may vary with position and time. It would be much better to have a plant dynamics that is not impacted by the micro-station. -Therefore, the objective of this section is to obtain some guidance for the design of a nano-hexapod that will not be impacted by the complex micro-station dynamics. +Therefore, the objective of this section is to obtain some guidance for the design of a active platform that will not be impacted by the complex micro-station dynamics. To study this, two models are used (Figure\nbsp{}ref:fig:uniaxial_support_compliance_models). -The first one consists of the nano-hexapod directly fixed on top of the granite, thus neglecting any support compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_nano_hexapod_only). -The second one consists of the nano-hexapod fixed on top of the micro-station having some limited compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) +The first one consists of the active platform directly fixed on top of the granite, thus neglecting any support compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_nano_hexapod_only). +The second one consists of the active platform fixed on top of the micro-station having some limited compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) #+name: fig:uniaxial_support_compliance_models #+caption: Models used to study the effect of limited support compliance #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_support_compliance_nano_hexapod_only}Nano-Hexapod fixed directly on the Granite} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_support_compliance_nano_hexapod_only}Active platform fixed directly on the Granite} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 1 [[file:figs/uniaxial_support_compliance_nano_hexapod_only.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_support_compliance_test_system}Nano-Hexapod fixed on top of the Micro-Station} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_support_compliance_test_system}Active platform fixed on top of the Micro-Station} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 1 @@ -1903,9 +1903,9 @@ The second one consists of the nano-hexapod fixed on top of the micro-station ha **** Neglected support compliance The limited compliance of the micro-station is first neglected and the uniaxial model shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_nano_hexapod_only is used. -The nano-hexapod mass (including the payload) is set at $20\,\text{kg}$ and three hexapod stiffnesses are considered, such that their resonance frequencies are at $\omega_{n} = 10\,\text{Hz}$, $\omega_{n} = 70\,\text{Hz}$ and $\omega_{n} = 400\,\text{Hz}$. +The active platform mass (including the payload) is set at $20\,\text{kg}$ and three active platform stiffnesses are considered, such that their resonance frequencies are at $\omega_{n} = 10\,\text{Hz}$, $\omega_{n} = 70\,\text{Hz}$ and $\omega_{n} = 400\,\text{Hz}$. Obtained transfer functions from $F$ to $L^\prime$ (shown in Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_neglected) are simple second-order low-pass filters. -When neglecting the support compliance, a large feedback bandwidth can be achieved for all three nano-hexapods. +When neglecting the support compliance, a large feedback bandwidth can be achieved for all three active platforms. #+name: fig:uniaxial_effect_support_compliance_neglected #+caption: Obtained transfer functions from $F$ to $L^{\prime}$ when neglecting support compliance @@ -1935,14 +1935,14 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev Some support compliance is now added and the model shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system is used. The parameters of the support (i.e., $m_{\mu}$, $c_{\mu}$ and $k_{\mu}$) are chosen to match the vertical mode at $70\,\text{Hz}$ seen on the micro-station (Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model). -The transfer functions from $F$ to $L$ (i.e., control of the relative motion of the nano-hexapod) and from $L$ to $d$ (i.e., control of the position between the nano-hexapod and the fixed granite) can then be computed. +The transfer functions from $F$ to $L$ (i.e., control of the relative motion of the active platform) and from $L$ to $d$ (i.e., control of the position between the active platform and the fixed granite) can then be computed. -When the relative displacement of the nano-hexapod $L$ is controlled (dynamics shown in Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics), having a stiff nano-hexapod (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_stiff). +When the relative displacement of the active platform $L$ is controlled (dynamics shown in Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics), having a stiff active platform (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_stiff). This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages. In such a case, the control of the piezoelectric stage using its integrated metrology (typically capacitive sensors) is quite simple as the plant is not much affected by the dynamics of the support on which it is fixed. # TODO - Add references of such stations with piezo stages on top, for instance [[cite:&schropp20_ptynam]] -If a soft nano-hexapod is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_soft) which will impact the control robustness and performance. +If a soft active platform is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_soft) which will impact the control robustness and performance. #+name: fig:uniaxial_effect_support_compliance_dynamics #+caption: Effect of the support compliance on the transfer functions from $F$ to $L$ @@ -1970,9 +1970,9 @@ If a soft nano-hexapod is used, the support dynamics appears in the dynamics bet **** Effect of support compliance on $d/F$ -When the motion to be controlled is the relative displacement $d$ between the granite and the nano-hexapod's top platform (which is the case for the acrshort:nass), the effect of the support compliance on the plant dynamics is opposite to that previously observed. -Indeed, using a "soft" nano-hexapod (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_soft). -Conversely, if a "stiff" nano-hexapod is used, the support dynamics appears in the plant dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_stiff). +When the motion to be controlled is the relative displacement $d$ between the granite and the active platform's top platform (which is the case for the acrshort:nass), the effect of the support compliance on the plant dynamics is opposite to that previously observed. +Indeed, using a "soft" active platform (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_soft). +Conversely, if a "stiff" active platform is used, the support dynamics appears in the plant dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_stiff). #+name: fig:uniaxial_effect_support_compliance_dynamics_d #+caption: Effect of the support compliance on the transfer functions from $F$ to $d$ @@ -2001,27 +2001,27 @@ Conversely, if a "stiff" nano-hexapod is used, the support dynamics appears in t **** Conclusion To study the impact of support compliance on plant dynamics, simple models shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_models were used. -Depending on the quantity to be controlled ($L$ or $d$ in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) and on the relative location of $\omega_\nu$ (suspension mode of the nano-hexapod) with respect to $\omega_\mu$ (modes of the support), the interaction between the support and the nano-hexapod dynamics can drastically change (observations made are summarized in Table\nbsp{}ref:tab:uniaxial_effect_compliance). +Depending on the quantity to be controlled ($L$ or $d$ in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) and on the relative location of $\omega_\nu$ (suspension mode of the active platform) with respect to $\omega_\mu$ (modes of the support), the interaction between the support and the active platform dynamics can drastically change (observations made are summarized in Table\nbsp{}ref:tab:uniaxial_effect_compliance). -For the acrfull:nass, having the suspension mode of the nano-hexapod at lower frequencies than the suspension modes of the micro-station would make the plant less dependent on the micro-station dynamics, and therefore easier to control. -Note that the observations made in this section are also affected by the ratio between the support mass $m_{\mu}$ and the nano-hexapod mass $m_n$ (the effect is more pronounced when the ratio $m_n/m_{\mu}$ increases). +For the acrfull:nass, having the suspension mode of the active platform at lower frequencies than the suspension modes of the micro-station would make the plant less dependent on the micro-station dynamics, and therefore easier to control. +Note that the observations made in this section are also affected by the ratio between the support mass $m_{\mu}$ and the active platform mass $m_n$ (the effect is more pronounced when the ratio $m_n/m_{\mu}$ increases). #+name: tab:uniaxial_effect_compliance #+caption: Impact of the support dynamics on the plant dynamics #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| | $\omega_{\nu} \ll \omega_{\mu}$ | $\omega_{\nu} \approx \omega_{\mu}$ | $\omega_{\nu} \gg \omega_{\mu}$ | -|--------+---------------------------------+-------------------------------------+---------------------------------| -| $d/F$ | small | large | large | -| $L/F$ | large | large | small | +| | $\omega_{\nu} \ll \omega_{\mu}$ | $\omega_{\nu} \approx \omega_{\mu}$ | $\omega_{\nu} \gg \omega_{\mu}$ | +|-------+---------------------------------+-------------------------------------+---------------------------------| +| $d/F$ | small | large | large | +| $L/F$ | large | large | small | *** Effect of Payload Dynamics <> **** Introduction :ignore: -Up to this section, the sample was modeled as a mass rigidly fixed to the nano-hexapod (as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic). -However, such a sample may present internal dynamics, and its fixation to the nano-hexapod may have limited stiffness. +Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic). +However, such a sample may present internal dynamics, and its fixation to the active platform may have limited stiffness. To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are used. #+name: fig:uniaxial_payload_dynamics_models @@ -2045,26 +2045,26 @@ To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref **** Impact on plant dynamics <> -To study the impact of the flexibility between the nano-hexapod and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used. +To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used. Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are considered. The resonances of the payload are set at $\omega_s = 20\,\text{Hz}$ and at $\omega_s = 200\,\text{Hz}$ while its mass is either $m_s = 1\,\text{kg}$ or $m_s = 50\,\text{kg}$. -The transfer functions from the nano-hexapod force $f$ to the motion of the nano-hexapod top platform are computed for all the above configurations and are compared for a soft Nano-Hexapod ($k_n = 0.01\,N/\mu m$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. +The transfer functions from the active platform force $f$ to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern). The frequency of the anti-resonance corresponds to the "free" resonance of the sample $\omega_s = \sqrt{k_s/m_s}$. The flexibility of the sample also changes the high frequency gain (the mass line is shifted from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$). #+name: fig:uniaxial_payload_dynamics_soft_nano_hexapod -#+caption: Effect of the payload dynamics on the soft Nano-Hexapod. Light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}) +#+caption: Effect of the payload dynamics on the soft active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,N/\mu m$, $m_s = 1\,kg$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_payload_dynamics_soft_nano_hexapod_light.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,N/\mu m$, $m_s = 50\,kg$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2072,22 +2072,22 @@ The flexibility of the sample also changes the high frequency gain (the mass lin #+end_subfigure #+end_figure -The same transfer functions are now compared when using a stiff nano-hexapod ($k_n = 100\,N/\mu m$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod. -In this case, the sample's resonance $\omega_s$ is smaller than the nano-hexapod resonance $\omega_n$. +The same transfer functions are now compared when using a stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod. +In this case, the sample's resonance $\omega_s$ is smaller than the active platform resonance $\omega_n$. This changes the zero/pole pattern to a pole/zero pattern (the frequency of the zero still being equal to $\omega_s$). -Even though the added sample's flexibility still shifts the high frequency mass line as for the soft nano-hexapod, the dynamics below the nano-hexapod resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy). +Even though the added sample's flexibility still shifts the high frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy). #+name: fig:uniaxial_payload_dynamics_stiff_nano_hexapod -#+caption: Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}) +#+caption: Effect of the payload dynamics on the stiff active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,N/\mu m$, $m_s = 1\,kg$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_payload_dynamics_stiff_nano_hexapod_light.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,N/\mu m$, $m_s = 50\,kg$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2098,18 +2098,18 @@ Even though the added sample's flexibility still shifts the high frequency mass **** Impact on close loop performances <> -Having a flexibility between the measured position (i.e., the top platform of the nano-hexapod) and the acrshort:poi to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). +Having a flexibility between the measured position (i.e., the top platform of the active platform) and the acrshort:poi to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). To estimate whether the sample flexibility is critical for the closed-loop position stability of the sample, the model shown in Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_control is used. -This is the same model that was used in Section\nbsp{}ref:sec:uniaxial_position_control but with an added flexibility between the nano-hexapod and the sample (considered sample modes are at $\omega_s = 20\,\text{Hz}$ and $\omega_n = 200\,\text{Hz}$). +This is the same model that was used in Section\nbsp{}ref:sec:uniaxial_position_control but with an added flexibility between the active platform and the sample (considered sample modes are at $\omega_s = 20\,\text{Hz}$ and $\omega_n = 200\,\text{Hz}$). In this case, the measured (i.e., controlled) distance $d$ is no longer equal to the real performance index (the distance $y$). #+name: fig:uniaxial_sample_flexibility_control -#+caption: Uniaxial model considering some flexibility between the nano-hexapod top platform and the sample. In this case, the measured and controlled distance $d$ is different from the distance $y$ which is the real performance index +#+caption: Uniaxial model considering some flexibility between the active platform top platform and the sample. In this case, the measured and controlled distance $d$ is different from the distance $y$ which is the real performance index [[file:figs/uniaxial_sample_flexibility_control.png]] The system dynamics is computed and IFF is applied using the same gains as those used in Section\nbsp{}ref:sec:uniaxial_active_damping. -Due to the collocation between the nano-hexapod and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample. +Due to the collocation between the active platform and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample. The acrlong:hac used in Section\nbsp{}ref:sec:uniaxial_position_control are then implemented on the damped plants. The obtained closed-loop systems are stable, indicating good robustness. @@ -2117,10 +2117,10 @@ Finally, closed-loop noise budgeting is computed for the obtained closed-loop sy The cumulative amplitude spectrum of the measured distance $d$ (Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_d) shows that the added flexibility at the sample location has very little effect on the control performance. However, the cumulative amplitude spectrum of the distance $y$ (Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_y) shows that the stability of $y$ is degraded when the sample flexibility is considered and is degraded as $\omega_s$ is lowered. -What happens is that above $\omega_s$, even though the motion $d$ can be controlled perfectly, the sample's mass is "isolated" from the motion of the nano-hexapod and the control on $y$ is not effective. +What happens is that above $\omega_s$, even though the motion $d$ can be controlled perfectly, the sample's mass is "isolated" from the motion of the active platform and the control on $y$ is not effective. #+name: fig:uniaxial_sample_flexibility_noise_budget -#+caption: Cumulative Amplitude Spectrum of the distances $d$ and $y$. The effect of the sample's flexibility does not affect much $d$ but is detrimental to the stability of $y$. A sample mass $m_s = 1\,\text{kg}$ and a nano-hexapod stiffness of $100\,N/\mu m$ are used for the simulations. +#+caption: Cumulative Amplitude Spectrum of the distances $d$ and $y$. The effect of the sample's flexibility does not affect much $d$ but is detrimental to the stability of $y$. A sample mass $m_s = 1\,\text{kg}$ and a active platform stiffness of $100\,\text{N}/\mu\text{m}$ are used for the simulations. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_d}Cumulative Amplitude Spectrum of $d$} @@ -2140,15 +2140,15 @@ What happens is that above $\omega_s$, even though the motion $d$ can be control **** Conclusion Payload dynamics is usually a major concern when designing a positioning system. -In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the nano-hexapod $\omega_n$. +In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the active platform $\omega_n$. The larger the sample mass, the larger the effect (i.e., change of high frequency gain, appearance of additional resonances and anti-resonances). A zero/pole pattern is observed if $\omega_s > \omega_n$ and a pole/zero pattern if $\omega_s > \omega_n$. Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in\nbsp{}[[cite:&rankers98_machin Section 4.2]]. -The general conclusion is that the stiffer the nano-hexapod, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload. +The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload. This is why high-bandwidth soft positioning stages are usually restricted to constant and calibrated payloads (CD-player, lithography machines, isolation system for gravitational wave detectors, ...), whereas stiff positioning systems are usually used when the control must be robust to a change of payload's mass (stiff piezo nano-positioning stages for instance). Having some flexibility between the measurement point and the acrshort:poi (i.e., the sample point to be position on the x-ray) also degrades the position stability as shown in Section\nbsp{}ref:ssec:uniaxial_payload_dynamics_effect_stability. -Therefore, it is important to take special care when designing sampling environments, especially if a soft nano-hexapod is used. +Therefore, it is important to take special care when designing sampling environments, especially if a soft active platform is used. *** Conclusion :PROPERTIES: @@ -2160,20 +2160,20 @@ Therefore, it is important to take special care when designing sampling environm In this study, a uniaxial model of the nano-active-stabilization-system was tuned from both dynamical measurements (Section\nbsp{}ref:sec:uniaxial_micro_station_model) and from disturbances measurements (Section\nbsp{}ref:sec:uniaxial_disturbances). -Three active damping techniques can be used to critically damp the nano-hexapod resonances (Section\nbsp{}ref:sec:uniaxial_active_damping). +Three active damping techniques can be used to critically damp the active platform resonances (Section\nbsp{}ref:sec:uniaxial_active_damping). However, this model does not allow the determination of which one is most suited to this application (a comparison of the three active damping techniques is done in Table\nbsp{}ref:tab:comp_active_damping). -Position feedback controllers have been developed for three considered nano-hexapod stiffnesses (Section\nbsp{}ref:sec:uniaxial_position_control). +Position feedback controllers have been developed for three considered active platform stiffnesses (Section\nbsp{}ref:sec:uniaxial_position_control). These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances. -Having a soft nano-hexapod makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section\nbsp{}ref:sec:uniaxial_support_compliance) and requires less position feedback bandwidth to fulfill the requirements. -The moderately stiff nano-hexapod ($k_n = 1\,N/\mu m$) is requiring a higher feedback bandwidth, but still gives acceptable results. -However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performance. +Having a soft active platform makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section\nbsp{}ref:sec:uniaxial_support_compliance) and requires less position feedback bandwidth to fulfill the requirements. +The moderately stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$) is requiring a higher feedback bandwidth, but still gives acceptable results. +However, the stiff active platform is the most complex to control and gives the worst positioning performance. ** Effect of Rotation <> *** Introduction :ignore: -An important aspect of the acrfull:nass is that the nano-hexapod continuously rotates around a vertical axis, whereas the external metrology is not. +An important aspect of the acrfull:nass is that the active platform continuously rotates around a vertical axis, whereas the external metrology is not. Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance. To study these effects, a model of a rotating suspended platform is first presented (Section\nbsp{}ref:sec:rotating_system_description) This model is simple enough to be able to derive its dynamics analytically and to understand its behavior, while still allowing the capture of important physical effects in play. @@ -2191,9 +2191,9 @@ This study of adapting acrshort:iff for the damping of rotating platforms has be It is then shown that acrfull:rdc is less affected by gyroscopic effects (Section\nbsp{}ref:sec:rotating_relative_damp_control). Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, damped plant and closed-loop compliance and transmissibility (Section\nbsp{}ref:sec:rotating_comp_act_damp). -The previous analysis was applied to three considered nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) and the optimal active damping controller was obtained in each case (Section\nbsp{}ref:sec:rotating_nano_hexapod). +The previous analysis was applied to three considered active platform stiffnesses ($k_n = 0.01\,\text{N}/\mu\text{m}$, $k_n = 1\,\text{N}/\mu\text{m}$ and $k_n = 100\,\text{N}/\mu\text{m}$) and the optimal active damping controller was obtained in each case (Section\nbsp{}ref:sec:rotating_nano_hexapod). Up until this section, the study was performed on a very simplistic model that only captures the rotation aspect, and the model parameters were not tuned to correspond to the NASS. -In the last section (Section\nbsp{}ref:sec:rotating_nass), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. +In the last section (Section\nbsp{}ref:sec:rotating_nass), a model of the micro-station is added below the active platform with a rotating spindle and parameters tuned to match the NASS dynamics. The goal is to determine whether the rotation imposes performance limitation on the NASS. *** System Description and Analysis @@ -2853,39 +2853,39 @@ This is very well known characteristics of these common active damping technique #+end_subfigure #+end_figure -*** Rotating Nano-Hexapod +*** Rotating Active Platform <> **** Introduction :ignore: -The previous analysis is now applied to a model representing a rotating nano-hexapod. -Three nano-hexapod stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. +The previous analysis is now applied to a model representing a rotating active platform. +Three active platform stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) because this is the worst identified case scenario in terms of gyroscopic effects. **** Nano-Active-Stabilization-System - Plant Dynamics -For the NASS, the maximum rotating velocity is $\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}$ for a suspended mass on top of the nano-hexapod's actuators equal to $m_n + m_s = \SI{16}{\kilo\gram}$. +For the NASS, the maximum rotating velocity is $\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}$ for a suspended mass on top of the active platform's actuators equal to $m_n + m_s = \SI{16}{\kilo\gram}$. The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \approx \SI{0.6}{\newton\per\mm}$. -The transfer functions from the nano-hexapod actuator force $F_u$ to the displacement of the nano-hexapod in the same direction $d_u$ as well as in the orthogonal direction $d_v$ (coupling) are shown in Figure\nbsp{}ref:fig:rotating_nano_hexapod_dynamics for all three considered nano-hexapod stiffnesses. -The soft nano-hexapod is the most affected by rotation. -This can be seen by the large shift of the resonance frequencies, and by the induced coupling, which is larger than that for the stiffer nano-hexapods. +The transfer functions from the active platform actuator force $F_u$ to the displacement of the active platform in the same direction $d_u$ as well as in the orthogonal direction $d_v$ (coupling) are shown in Figure\nbsp{}ref:fig:rotating_nano_hexapod_dynamics for all three considered active platform stiffnesses. +The soft active platform is the most affected by rotation. +This can be seen by the large shift of the resonance frequencies, and by the induced coupling, which is larger than that for the stiffer active platforms. The coupling (or interaction) in a acrshort:mimo $2 \times 2$ system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). #+name: fig:rotating_nano_hexapod_dynamics -#+caption: Effect of rotation on the nano-hexapod dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity +#+caption: Effect of rotation on the active platform dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nano_hexapod_dynamics_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nano_hexapod_dynamics_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2894,7 +2894,7 @@ The coupling (or interaction) in a acrshort:mimo $2 \times 2$ system can be visu #+end_figure **** Optimal IFF with a High-Pass Filter -Integral Force Feedback with an added acrlong:hpf is applied to the three nano-hexapods. +Integral Force Feedback with an added acrlong:hpf is applied to the three active platforms. First, the parameters ($\omega_i$ and $g$) of the IFF controller that yield the best simultaneous damping are determined from Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain. The IFF parameters are chosen as follows: - for $k_n = \SI{0.01}{\N\per\mu\m}$ (Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain): $\omega_i$ is chosen such that maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. @@ -2907,19 +2907,19 @@ The obtained IFF parameters and the achievable damping are visually shown by lar #+caption: For each value of $\omega_i$, the maximum damping ratio $\xi$ is computed (blue), and the corresponding controller gain is shown (in red). The chosen controller parameters used for further analysis are indicated by the large dots. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_hpf_nass_optimal_gain_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_hpf_nass_optimal_gain_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2931,24 +2931,24 @@ The obtained IFF parameters and the achievable damping are visually shown by lar #+caption: Obtained optimal parameters ($\omega_i$ and $g$) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping of the two modes $\xi$ is also shown. #+attr_latex: :environment tabularx :width 0.3\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| $k_n$ | $\omega_i$ | $g$ | $\xi_\text{opt}$ | -|-----------------+------------+------+------------------| -| $0.01\,N/\mu m$ | 7.3 | 51 | 0.45 | -| $1\,N/\mu m$ | 39 | 427 | 0.93 | -| $100\,N/\mu m$ | 500 | 3775 | 0.94 | +| $k_n$ | $\omega_i$ | $g$ | $\xi_\text{opt}$ | +|-----------------------+------------+------+------------------| +| $0.01\,\text{N}/\mu\text{m}$ | 7.3 | 51 | 0.45 | +| $1\,\text{N}/\mu\text{m}$ | 39 | 427 | 0.93 | +| $100\,\text{N}/\mu\text{m}$ | 500 | 3775 | 0.94 | **** Optimal IFF with Parallel Stiffness -For each considered nano-hexapod stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total nano-hexapod stiffness). -To keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased ($k_a = k_n - k_p$, with $k_n$ the total nano-hexapod stiffness). +For each considered active platform stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total active platform stiffness). +To keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased ($k_a = k_n - k_p$, with $k_n$ the total active platform stiffness). A high-pass filter is also added to limit the low-frequency gain with a cut-off frequency $\omega_i$ equal to one tenth of the system resonance ($\omega_i = \omega_0/10$). The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure\nbsp{}ref:fig:rotating_iff_kp_nass_optimal_gain). -It is shown that the soft nano-hexapod cannot yield good damping because the parallel stiffness cannot be sufficiently large compared to the negative stiffness induced by the rotation. +It is shown that the soft active platform cannot yield good damping because the parallel stiffness cannot be sufficiently large compared to the negative stiffness induced by the rotation. For the two stiff options, the achievable damping decreases when the parallel stiffness is too high, as explained in Section\nbsp{}ref:sec:rotating_iff_parallel_stiffness. Such behavior can be explained by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chapt 7.2]]. -This distance is larger for stiff nano-hexapod because the open-loop pole will be at higher frequencies while the open-loop zero, whereas depends on the value of the parallel stiffness, can only be made large for stiff nano-hexapods. +This distance is larger for stiff active platform because the open-loop pole will be at higher frequencies while the open-loop zero, whereas depends on the value of the parallel stiffness, can only be made large for stiff active platforms. -Let's choose $k_p = 1\,N/mm$, $k_p = 0.01\,N/\mu m$ and $k_p = 1\,N/\mu m$ for the three considered nano-hexapods. +Let's choose $k_p = 1\,\text{N/mm}$, $k_p = 0.01\,\text{N}/\mu\text{m}$ and $k_p = 1\,\text{N}/\mu\text{m}$ for the three considered active platforms. The corresponding optimal controller gains and achievable damping are summarized in Table\nbsp{}ref:tab:rotating_iff_kp_opt_iff_kp_params_nass. #+attr_latex: :options [b]{0.49\linewidth} @@ -2964,16 +2964,16 @@ The corresponding optimal controller gains and achievable damping are summarized #+latex: \centering #+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align cccc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| $k_n$ | $k_p$ | $g$ | $\xi_{\text{opt}}$ | -|-----------------+-----------------+---------+--------------------| -| $0.01\,N/\mu m$ | $1\,N/mm$ | 47.9 | 0.44 | -| $1\,N/\mu m$ | $0.01\,N/\mu m$ | 465.57 | 0.97 | -| $100\,N/\mu m$ | $1\,N/\mu m$ | 4624.25 | 0.99 | +| $k_n$ | $k_p$ | $g$ | $\xi_{\text{opt}}$ | +|-----------------------+-----------------------+---------+--------------------| +| $0.01\,\text{N}/\mu\text{m}$ | $1\,\text{N/mm}$ | 47.9 | 0.44 | +| $1\,\text{N}/\mu\text{m}$ | $0.01\,\text{N}/\mu\text{m}$ | 465.57 | 0.97 | +| $100\,\text{N}/\mu\text{m}$ | $1\,\text{N}/\mu\text{m}$ | 4624.25 | 0.99 | #+latex: \captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses} #+end_minipage **** Optimal Relative Motion Control -For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure\nbsp{}ref:fig:rotating_rdc_optimal_gain). +For each considered active platform stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure\nbsp{}ref:fig:rotating_rdc_optimal_gain). The gain is chosen such that 99% of modal damping is obtained (obtained gains are summarized in Table\nbsp{}ref:tab:rotating_rdc_opt_params_nass). #+attr_latex: :options [b]{0.49\linewidth} @@ -2989,11 +2989,11 @@ The gain is chosen such that 99% of modal damping is obtained (obtained gains ar #+latex: \centering #+attr_latex: :environment tabularx :width 0.6\linewidth :placement [b] :align ccc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| $k_n$ | $g$ | $\xi_{\text{opt}}$ | -|-----------------+-------+--------------------| -| $0.01\,N/\mu m$ | 1600 | 0.99 | -| $1\,N/\mu m$ | 8200 | 0.99 | -| $100\,N/\mu m$ | 80000 | 0.99 | +| $k_n$ | $g$ | $\xi_{\text{opt}}$ | +|-----------------------+-------+--------------------| +| $0.01\,\text{N}/\mu\text{m}$ | 1600 | 0.99 | +| $1\,\text{N}/\mu\text{m}$ | 8200 | 0.99 | +| $100\,\text{N}/\mu\text{m}$ | 80000 | 0.99 | #+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC} #+end_minipage @@ -3002,26 +3002,26 @@ Now that the optimal parameters for the three considered active damping techniqu Similar to what was concluded in the previous analysis: - acrshort:iff adds more coupling below the resonance frequency as compared to the open-loop and acrshort:rdc cases -- All three methods yield good damping, except for acrshort:iff applied on the soft nano-hexapod -- Coupling is smaller for stiff nano-hexapods +- All three methods yield good damping, except for acrshort:iff applied on the soft active platform +- Coupling is smaller for stiff active platforms #+name: fig:rotating_nass_damped_plant_comp -#+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$. +#+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three active platform stiffnesses are considered. For this analysis the rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_damped_plant_comp_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_damped_plant_comp_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3037,14 +3037,14 @@ While quite simplistic, this allowed us to study the effects of rotation and the In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. **** Nano Active Stabilization System model -To have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): +To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): - the floor whose motion is imposed - a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$) - a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_t = \SI{600}{\kg}$) - a spindle (vertical rotation) stage whose rotation is imposed ($m_s = \SI{600}{\kg}$) -- a 2-DoF micro-hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}$, $m_h = \SI{15}{\kg}$) +- a 2-DoF positioning hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}$, $m_h = \SI{15}{\kg}$) -A payload is rigidly fixed to the nano-hexapod and the $x,y$ motion of the payload is measured with respect to the granite. +A payload is rigidly fixed to the active platform and the $x,y$ motion of the payload is measured with respect to the granite. #+name: fig:rotating_nass_model #+caption: 3D view of the Nano-Active-Stabilization-System model. @@ -3056,28 +3056,28 @@ A payload is rigidly fixed to the nano-hexapod and the $x,y$ motion of the paylo The dynamics of the undamped and damped plants are identified using the optimal parameters found in Section\nbsp{}ref:sec:rotating_nano_hexapod. The obtained dynamics are compared in Figure\nbsp{}ref:fig:rotating_nass_plant_comp_stiffness in which the direct terms are shown by the solid curves and the coupling terms are shown by the shaded ones. It can be observed that: -- The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod +- The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft active platform - Damping added using the three proposed techniques is quite high, and the obtained plant is rather easy to control -- There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz) +- There is some coupling between active platform and micro-station dynamics for the stiff active platform (mode at $200\,\text{Hz}$) - The two proposed IFF modifications yield similar results #+name: fig:rotating_nass_plant_comp_stiffness -#+caption: Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology +#+caption: Bode plot of the transfer function from active platform actuator to measured motion by the external metrology #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_plant_comp_stiffness_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_plant_comp_stiffness_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3092,28 +3092,28 @@ Note that only the transfer functions from the disturbances in the $x$ direction Conclusions are similar than those of the uniaxial (non-rotating) model: - Regarding the effect of floor motion and forces applied on the payload: - - The stiffer, the better. This can be seen in Figures\nbsp{}ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces where the magnitudes for the stiff hexapod are lower than those for the soft one + - The stiffer, the better. This can be seen in Figures\nbsp{}ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces where the magnitudes for the stiff active platform are lower than those for the soft one - acrshort:iff degrades the performance at low-frequency compared to acrshort:rdc - Regarding the effect of micro-station vibrations: - - Having a soft nano-hexapod allows filtering of these vibrations between the suspension modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure\nbsp{}ref:fig:rotating_nass_effect_stage_vibration_vc). + - Having a soft active platform allows filtering of these vibrations between the suspension modes of the active platform and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure\nbsp{}ref:fig:rotating_nass_effect_stage_vibration_vc). #+name: fig:rotating_nass_effect_floor_motion -#+caption: Effect of floor motion $x_{f,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. IFF is shown to increase the sensitivity to floor motion at low-frequency. +#+caption: Effect of floor motion $x_{f,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. IFF is shown to increase the sensitivity to floor motion at low-frequency. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_floor_motion_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_floor_motion_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3122,22 +3122,22 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: #+end_figure #+name: fig:rotating_nass_effect_stage_vibration -#+caption: Effect of micro-station vibrations $f_{t,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft nano-hexapod suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc}) +#+caption: Effect of micro-station vibrations $f_{t,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft active platform suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_stage_vibration_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_stage_vibration_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3146,22 +3146,22 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: #+end_figure #+name: fig:rotating_nass_effect_direct_forces -#+caption: Effect of sample forces $f_{s,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low-frequency. +#+caption: Effect of sample forces $f_{s,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. Integral Force Feedback degrades this compliance at low-frequency. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_direct_forces_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_direct_forces_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,\text{N}/\mu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3189,14 +3189,14 @@ It was shown that if the stiffness $k_p$ of the additional springs is larger tha These two modifications were compared with acrlong:rdc. While having very different implementations, both proposed modifications were found to be very similar with respect to the attainable damping and the obtained closed-loop system behavior. -This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters. -As for the uniaxial model, three nano-hexapod stiffnesses values were considered. -The dynamics of the soft nano-hexapod ($k_n = 0.01\,N/\mu m$) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). -In addition, the attainable damping ratio of the soft nano-hexapod when using acrshort:iff is limited by gyroscopic effects. +This study has been applied to a rotating platform that corresponds to the active platform parameters. +As for the uniaxial model, three active platform stiffnesses values were considered. +The dynamics of the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). +In addition, the attainable damping ratio of the soft active platform when using acrshort:iff is limited by gyroscopic effects. To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been considered. -Results are similar to those of the uniaxial model except that come complexity is added for the soft nano-hexapod due to the spindle's rotation. -For the moderately stiff nano-hexapod ($k_n = 1\,N/\mu m$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that showed better results with the uniaxial model. +Results are similar to those of the uniaxial model except that come complexity is added for the soft active platform due to the spindle's rotation. +For the moderately stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model. ** Micro Station - Modal Analysis <> @@ -3298,9 +3298,9 @@ In this modal analysis, it is chosen to measure the response of the structure at <> The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured. -A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod. +A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the positioning hexapod. The positions of the accelerometers are visually shown on a 3D model in Figure\nbsp{}ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the acrshort:poi are summarized in Table\nbsp{}ref:tab:modal_position_accelerometers. -Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure\nbsp{}ref:fig:modal_accelerometer_pictures. +Pictures of the accelerometers fixed to the translation stage and to the positioning hexapod are shown in Figure\nbsp{}ref:fig:modal_accelerometer_pictures. As all key stages of the micro-station are expected to behave as solid bodies, only 6 acrshort:dof can be considered for each solid body. However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section\nbsp{}ref:ssec:modal_solid_body_assumption). @@ -3356,7 +3356,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrsh #+attr_latex: :height 6cm [[file:figs/modal_accelerometers_ty.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod} +#+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_hexapod} Positioning Hexapod} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :height 6cm @@ -3405,7 +3405,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co The "normalized" acrfull:asd of the two signals were computed and shown in Figure\nbsp{}ref:fig:modal_asd_acc_force. Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer). -These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the micro-hexapod). +These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the positioning hexapod). Similar results were obtained for all measured acrshortpl:frf. #+name: fig:modal_raw_meas_asd @@ -3470,7 +3470,7 @@ For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that link \end{bmatrix} \end{equation} -However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod. +However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the positioning hexapod. Therefore, only $6 \times 6 = 36$ acrshortpl:dof are of interest. Therefore, the objective of this section is to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36. @@ -3536,14 +3536,14 @@ The position of each accelerometer with respect to the acrlong:com of the corres #+caption: Center of mass of considered solid bodies with respect to the "point of interest" #+attr_latex: :environment tabularx :width 0.45\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| | $X$ | $Y$ | $Z$ | -|-------------------+-----------------+------------------+--------------------| -| Bottom Granite | $45\,\text{mm}$ | $144\,\text{mm}$ | $-1251\,\text{mm}$ | -| Top granite | $52\,\text{mm}$ | $258\,\text{mm}$ | $-778\,\text{mm}$ | -| Translation stage | $0$ | $14\,\text{mm}$ | $-600\,\text{mm}$ | -| Tilt Stage | $0$ | $-5\,\text{mm}$ | $-628\,\text{mm}$ | -| Spindle | $0$ | $0$ | $-580\,\text{mm}$ | -| Hexapod | $-4\,\text{mm}$ | $6\,\text{mm}$ | $-319\,\text{mm}$ | +| | $X$ | $Y$ | $Z$ | +|---------------------+-----------------+------------------+--------------------| +| Bottom Granite | $45\,\text{mm}$ | $144\,\text{mm}$ | $-1251\,\text{mm}$ | +| Top granite | $52\,\text{mm}$ | $258\,\text{mm}$ | $-778\,\text{mm}$ | +| Translation stage | $0$ | $14\,\text{mm}$ | $-600\,\text{mm}$ | +| Tilt Stage | $0$ | $-5\,\text{mm}$ | $-628\,\text{mm}$ | +| Spindle | $0$ | $0$ | $-580\,\text{mm}$ | +| Positioning Hexapod | $-4\,\text{mm}$ | $6\,\text{mm}$ | $-319\,\text{mm}$ | Using\nbsp{}eqref:eq:modal_cart_to_acc, the frequency response matrix $\bm{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the acrlong:com of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\bm{H}$. @@ -3568,13 +3568,13 @@ From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements $\bm{H}$. This is what is done here to check whether the solid body assumption is correct in the frequency band of interest. -The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure\nbsp{}ref:fig:modal_comp_acc_solid_body_frf). +The comparison is made for the 4 accelerometers fixed on the positioning hexapod (Figure\nbsp{}ref:fig:modal_comp_acc_solid_body_frf). The original acrshortpl:frf and those computed from the CoM responses match well in the frequency range of interest. Similar results were obtained for the other solid bodies, indicating that the solid body assumption is valid and that a multi-body model can be used to represent the dynamics of the micro-station. This also validates the reduction in the number of acrshortpl:dof from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof). #+name: fig:modal_comp_acc_solid_body_frf -#+caption: Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the micro-hexapod are shown. Input is a hammer force applied on the micro-hexapod in the $x$ direction +#+caption: Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the positioning hexapod are shown. Input is a hammer force applied on the positioning hexapod in the $x$ direction #+attr_latex: :scale 0.8 [[file:figs/modal_comp_acc_solid_body_frf.png]] @@ -3662,19 +3662,19 @@ From the obtained modal parameters, the mode shapes are computed and can be disp #+caption: Three obtained mode shape animations #+attr_latex: :options [hbtp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:modal_mode1_animation}$1^{st}$ mode at 11.9 Hz: tilt suspension mode of the granite} +#+attr_latex: :caption \subcaption{\label{fig:modal_mode1_animation}$1^{st}$ mode at $11.9\,\text{Hz}$: tilt suspension mode of the granite} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth [[file:figs/modal_mode1_animation.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:modal_mode6_animation}$6^{th}$ mode at 69.8 Hz: vertical resonance of the spindle} +#+attr_latex: :caption \subcaption{\label{fig:modal_mode6_animation}$6^{th}$ mode at $69.8\,\text{Hz}$: vertical resonance of the spindle} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth [[file:figs/modal_mode6_animation.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:modal_mode13_animation}$13^{th}$ mode at 124.2 Hz: lateral micro-hexapod resonance} +#+attr_latex: :caption \subcaption{\label{fig:modal_mode13_animation}$13^{th}$ mode at $124.2\,\text{Hz}$: lateral hexapod resonance} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth @@ -3735,7 +3735,7 @@ With $\bm{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response o A comparison between original measured acrshortpl:frf and synthesized ones from the modal model is presented in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal. Whether the obtained match is good or bad is quite arbitrary. However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective. -This can be seen in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal_3 that shows the acrshort:frf from the force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction. +This can be seen in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal_3 that shows the acrshort:frf from the force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the positioning hexapod) in the $x$ direction. #+name: fig:modal_comp_acc_frf_modal #+caption: Comparison of the measured FRF with the FRF synthesized from the modal model. @@ -3784,8 +3784,8 @@ However, the measurements are useful for tuning the parameters of the micro-stat From the start of this work, it became increasingly clear that an accurate micro-station model was necessary. -First, during the uniaxial study, it became clear that the micro-station dynamics affects the nano-hexapod dynamics. -Then, using the 3-DoF rotating model, it was discovered that the rotation of the nano-hexapod induces gyroscopic effects that affect the system dynamics and should therefore be modeled. +First, during the uniaxial study, it became clear that the micro-station dynamics affects the active platform dynamics. +Then, using the 3-DoF rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modeled. Finally, a modal analysis of the micro-station showed how complex the dynamics of the station is. The modal analysis also confirm that each stage behaves as a rigid body in the frequency range of interest. Therefore, a multi-body model is a good candidate to accurately represent the micro-station dynamics. @@ -3868,7 +3868,7 @@ It is composed of an air bearing spindle[fn:ustation_2], whose angular position Additional rotary unions and slip-rings are used to be able to pass electrical signals, fluids and gazes through the rotation stage. -***** Micro-Hexapod +***** Positioning Hexapod Finally, a Stewart platform[fn:ustation_3] is used to position the sample. It includes a DC motor and an optical linear encoders in each of the six struts. @@ -3887,7 +3887,7 @@ It can also be used to precisely position the acrfull:poi vertically with respec #+attr_latex: :options [t]{0.49\linewidth} #+begin_minipage #+name: fig:ustation_hexapod_stage -#+caption: Micro Hexapod +#+caption: Positioning Hexapod #+attr_latex: :scale 1 :float nil [[file:figs/ustation_hexapod_stage.png]] #+end_minipage @@ -4090,14 +4090,14 @@ As any motion stage induces parasitic motion in all 6 DoF, the transformation ma The homogeneous transformation matrix corresponding to the micro-station $\bm{T}_{\mu\text{-station}}$ is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation\nbsp{}eqref:eq:ustation_transformation_station. \begin{equation}\label{eq:ustation_transformation_station} -\bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}} +\bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\text{hexapod}} \end{equation} $\bm{T}_{\mu\text{-station}}$ represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite. If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), $\bm{T}_{\mu\text{-station}}$ then represents the pose setpoint of the sample with respect to the granite. The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation\nbsp{}eqref:eq:ustation_transformation_matrices_stages. -The setpoints are $D_y$ for the translation stage, $\theta_y$ for the tilt-stage, $\theta_z$ for the spindle, $[D_{\mu x},\ D_{\mu y}, D_{\mu z}]$ for the micro-hexapod translations and $[\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]$ for the micro-hexapod rotations. +The setpoints are $D_y$ for the translation stage, $\theta_y$ for the tilt-stage, $\theta_z$ for the spindle, $[D_{\mu x},\ D_{\mu y}, D_{\mu z}]$ for the positioning hexapod translations and $[\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]$ for the positioning hexapod rotations. \begin{equation}\label{eq:ustation_transformation_matrices_stages} \begin{align} @@ -4107,7 +4107,7 @@ The setpoints are $D_y$ for the translation stage, $\theta_y$ for the tilt-stage 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad -\bm{T}_{\mu\text{-hexapod}} = +\bm{T}_{\text{hexapod}} = \left[ \begin{array}{ccc|c} & & & D_{\mu x} \\ & \bm{R}_x(\theta_{\mu x}) \bm{R}_y(\theta_{\mu y}) \bm{R}_{z}(\theta_{\mu z}) & & D_{\mu y} \\ @@ -4141,7 +4141,7 @@ The inertia of the solid bodies and the stiffness properties of the guiding mech The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section\nbsp{}ref:ssec:ustation_model_comp_dynamics). # TODO - Add reference to uniaxial model -As the dynamics of the nano-hexapod is impacted by the micro-station compliance, the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. +As the dynamics of the active platform is impacted by the micro-station compliance, the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_model_compliance). **** Multi-Body Model @@ -4185,13 +4185,13 @@ The spring values are summarized in Table\nbsp{}ref:tab:ustation_6dof_stiffness_ #+caption: Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoF joints are defined are indicated in figures found in Section\nbsp{}ref:ssec:ustation_stages #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccccc #+attr_latex: :center t :booktabs t -| *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | -|-------------+-----------------+-----------------+-----------------+-------------------------+------------------------+-------------------------| -| Granite | $5\,kN/\mu m$ | $5\,kN/\mu m$ | $5\,kN/\mu m$ | $25\,Nm/\mu\text{rad}$ | $25\,Nm/\mu\text{rad}$ | $10\,Nm/\mu\text{rad}$ | -| Translation | $200\,N/\mu m$ | - | $200\,N/\mu m$ | $60\,Nm/\mu\text{rad}$ | $90\,Nm/\mu\text{rad}$ | $60\,Nm/\mu\text{rad}$ | -| Tilt | $380\,N/\mu m$ | $400\,N/\mu m$ | $380\,N/\mu m$ | $120\,Nm/\mu\text{rad}$ | - | $120\,Nm/\mu\text{rad}$ | -| Spindle | $700\,N/\mu m$ | $700\,N/\mu m$ | $2\,kN/\mu m$ | $10\,Nm/\mu\text{rad}$ | $10\,Nm/\mu\text{rad}$ | - | -| Hexapod | $10\,N/\mu m$ | $10\,N/\mu m$ | $100\,N/\mu m$ | $1.5\,Nm/rad$ | $1.5\,Nm/rad$ | $0.27\,Nm/rad$ | +| *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | +|-------------+-----------------------------+-----------------------------+-----------------------------+--------------------------------+-------------------------------+--------------------------------| +| Granite | $5\,\text{kN}/\mu\text{m}$ | $5\,\text{kN}/\mu\text{m}$ | $5\,\text{kN}/\mu\text{m}$ | $25\,\text{Nm}/\mu\text{rad}$ | $25\,\text{Nm}/\mu\text{rad}$ | $10\,\text{Nm}/\mu\text{rad}$ | +| Translation | $200\,\text{N}/\mu\text{m}$ | - | $200\,\text{N}/\mu\text{m}$ | $60\,\text{Nm}/\mu\text{rad}$ | $90\,\text{Nm}/\mu\text{rad}$ | $60\,\text{Nm}/\mu\text{rad}$ | +| Tilt | $380\,\text{N}/\mu\text{m}$ | $400\,\text{N}/\mu\text{m}$ | $380\,\text{N}/\mu\text{m}$ | $120\,\text{Nm}/\mu\text{rad}$ | - | $120\,\text{Nm}/\mu\text{rad}$ | +| Spindle | $700\,\text{N}/\mu\text{m}$ | $700\,\text{N}/\mu\text{m}$ | $2\,\text{kN}/\mu\text{m}$ | $10\,\text{Nm}/\mu\text{rad}$ | $10\,\text{Nm}/\mu\text{rad}$ | - | +| Hexapod | $10\,\text{N}/\mu\text{m}$ | $10\,\text{N}/\mu\text{m}$ | $100\,\text{N}/\mu\text{m}$ | $1.5\,\text{Nm/rad}$ | $1.5\,\text{Nm/rad}$ | $0.27\,\text{Nm/rad}$ | **** Comparison with the measured dynamics <> @@ -4237,8 +4237,8 @@ When considering the NASS, the most important dynamical characteristics of the m Therefore, the adopted strategy is to accurately model the micro-station compliance. The micro-station compliance was experimentally measured using the setup illustrated in Figure\nbsp{}ref:fig:ustation_compliance_meas. -Four 3-axis accelerometers were fixed to the micro-hexapod top platform. -The micro-hexapod top platform was impacted at 10 different points. +Four 3-axis accelerometers were fixed to the positioning hexapod top platform. +The positioning hexapod top platform was impacted at 10 different points. For each impact position, 10 impacts were performed to average and improve the data quality. #+name: fig:ustation_compliance_meas @@ -4270,7 +4270,7 @@ Then, the acceleration in the cartesian frame can be computed using\nbsp{}eqref: a_{\mathcal{X}} = \bm{J}_a^{-1} \cdot a_{\mathcal{L}} \end{equation} -Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is computed\nbsp{}eqref:eq:ustation_compliance_force_jacobian and used to convert the individual hammer forces $F_{\mathcal{L}}$ to force and torques $F_{\mathcal{X}}$ applied at the center of the micro-hexapod top plate (defined by frame $\{\mathcal{X}\}$ in Figure\nbsp{}ref:fig:ustation_compliance_meas). +Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is computed\nbsp{}eqref:eq:ustation_compliance_force_jacobian and used to convert the individual hammer forces $F_{\mathcal{L}}$ to force and torques $F_{\mathcal{X}}$ applied at the center of the positioning hexapod top plate (defined by frame $\{\mathcal{X}\}$ in Figure\nbsp{}ref:fig:ustation_compliance_meas). \begin{equation}\label{eq:ustation_compliance_force_jacobian} \bm{J}_F = \left[\begin{smallmatrix} @@ -4340,7 +4340,7 @@ Finally, the obtained disturbance sources are compared in Section\nbsp{}ref:ssec In this section, ground motion is directly measured using geophones. Vibrations induced by scanning the translation stage and the spindle are also measured using dedicated setups. -The tilt stage and the micro-hexapod also have positioning errors; however, they are not modeled here because these two stages are only used for pre-positioning and not for scanning. +The tilt stage and the positioning hexapod also have positioning errors; however, they are not modeled here because these two stages are only used for pre-positioning and not for scanning. Therefore, from a control perspective, they are not important. ***** Ground Motion @@ -4375,15 +4375,15 @@ A similar setup was used to measure the horizontal deviation (i.e. in the $x$ di #+caption: Experimental setup to measure the straightness (vertical deviation) of the translation stage [[file:figs/ustation_errors_ty_setup.png]] -Six scans were performed between $-4.5\,mm$ and $4.5\,mm$. +Six scans were performed between $-4.5\,\text{mm}$ and $4.5\,\text{mm}$. The results for each individual scan are shown in Figure\nbsp{}ref:fig:ustation_errors_dy_vertical. The measurement axis may not be perfectly aligned with the translation stage axis; this, a linear fit is removed from the measurement. The remaining vertical displacement is shown in Figure\nbsp{}ref:fig:ustation_errors_dy_vertical_remove_mean. -A vertical error of $\pm300\,nm$ induced by the translation stage is expected. +A vertical error of $\pm300\,\text{nm}$ induced by the translation stage is expected. Similar result is obtained for the $x$ lateral direction. #+name: fig:ustation_errors_dy -#+caption: Measurement of the linear (vertical) deviation of the Translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}). +#+caption: Measurement of the vertical error of the translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical}Measured vertical error} @@ -4431,7 +4431,7 @@ A large fraction of the radial (Figure\nbsp{}ref:fig:ustation_errors_spindle_rad This is displayed by the dashed circle. After removing the best circular fit from the data, the vibrations induced by the Spindle may be viewed as stochastic disturbances. However, some misalignment between the acrshort:poi of the sample and the rotation axis will be considered because the alignment is not perfect in practice. -The vertical motion induced by scanning the spindle is in the order of $\pm 30\,nm$ (Figure\nbsp{}ref:fig:ustation_errors_spindle_axial). +The vertical motion induced by scanning the spindle is in the order of $\pm 30\,\text{nm}$ (Figure\nbsp{}ref:fig:ustation_errors_spindle_axial). #+name: fig:ustation_errors_spindle #+caption: Measurement of the radial (\subref{fig:ustation_errors_spindle_radial}), axial (\subref{fig:ustation_errors_spindle_axial}) and tilt (\subref{fig:ustation_errors_spindle_tilt}) Spindle errors during a 60rpm spindle rotation. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis. @@ -4495,7 +4495,7 @@ From the measured effect of disturbances in Section\nbsp{}ref:ssec:ustation_dist The obtained power spectral density of the disturbances are shown in Figure\nbsp{}ref:fig:ustation_dist_sources. #+name: fig:ustation_dist_sources -#+caption: Measured spectral density of the micro-station disturbance sources. Ground motion (\subref{fig:ustation_dist_source_ground_motion}), translation stage (\subref{fig:ustation_dist_source_translation_stage}) and spindle (\subref{fig:ustation_dist_source_spindle}). +#+caption: Measured acrshort:asd of the micro-station disturbance sources. Ground motion (\subref{fig:ustation_dist_source_ground_motion}), translation stage (\subref{fig:ustation_dist_source_translation_stage}) and spindle (\subref{fig:ustation_dist_source_spindle}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_ground_motion}Ground Motion} @@ -4561,7 +4561,7 @@ Second, a constant velocity scans with the translation stage was performed and a To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of 60rpm. Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section\nbsp{}ref:sec:ustation_disturbances. -A radial offset of $\approx 1\,\mu m$ between the acrfull:poi and the spindle's rotation axis is introduced to represent what is experimentally observed. +A radial offset of $\approx 1\,\mu\text{m}$ between the acrfull:poi and the spindle's rotation axis is introduced to represent what is experimentally observed. During the 10 second simulation (i.e. 10 spindle turns), the position of the acrshort:poi with respect to the granite was recorded. Results are shown in Figure\nbsp{}ref:fig:ustation_errors_model_spindle. A good correlation with the measurements is observed both for radial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_radial) and axial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_axial). @@ -4588,13 +4588,13 @@ A good correlation with the measurements is observed both for radial errors (Fig <> A second experiment was performed in which the translation stage was scanned at constant velocity. -The translation stage setpoint is configured to have a "triangular" shape with stroke of $\pm 4.5\, mm$. +The translation stage setpoint is configured to have a "triangular" shape with stroke of $\pm 4.5\,\text{mm}$. Both ground motion and translation stage vibrations were included in the simulation. Similar to what was performed for the tomography simulation, the acrfull:poi position with respect to the granite was recorded and compared with the experimental measurements in Figure\nbsp{}ref:fig:ustation_errors_model_dy_vertical. A similar error amplitude was observed, thus indicating that the multi-body model with the included disturbances accurately represented the micro-station behavior in typical scientific experiments. #+name: fig:ustation_errors_model_dy_vertical -#+caption: Vertical errors during a constant-velocity scan of the translation stage. Comparison of the measurements and simulated errors. +#+caption: Vertical errors during a constant-velocity scan of the translation stage. #+attr_latex: :scale 0.8 [[file:figs/ustation_errors_model_dy_vertical.png]] @@ -4612,7 +4612,7 @@ After tuning the model parameters, a good match with the measured compliance was The disturbances affecting the sample position should also be well modeled. After experimentally estimating the disturbances (Section\nbsp{}ref:sec:ustation_disturbances), the multi-body model was finally validated by performing a tomography simulation (Figure\nbsp{}ref:fig:ustation_errors_model_spindle) as well as a simulation in which the translation stage was scanned (Figure\nbsp{}ref:fig:ustation_errors_model_dy_vertical). -** Nano Hexapod - Multi Body Model +** Active Platform - Multi Body Model <> *** Introduction :ignore: @@ -4653,8 +4653,8 @@ To overcome this limitation, external metrology systems have been implemented to A review of existing sample stages with active vibration control reveals various approaches to implementing such feedback systems. In many cases, sample position control is limited to translational acrshortpl:dof. -At NSLS-II, for instance, a system capable of $100\,\mu m$ stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure\nbsp{}ref:fig:nhexa_stages_nazaretski). -Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately 100 Hz (Figure\nbsp{}ref:fig:nhexa_stages_sapoti). +At NSLS-II, for instance, a system capable of $100\,\mu\text{m}$ stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure\nbsp{}ref:fig:nhexa_stages_nazaretski). +Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately $100\,\text{Hz}$ (Figure\nbsp{}ref:fig:nhexa_stages_sapoti). #+name: fig:nhexa_stages_translations #+caption: Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope\nbsp{}[[cite:&nazaretski15_pushin_limit]] at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] at Sirius facility (\subref{fig:nhexa_stages_sapoti}) @@ -4676,8 +4676,8 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac The integration of $R_z$ rotational capability, which is necessary for tomography experiments, introduces additional complexity. At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section\nbsp{}ref:sec:nhexa_stewart_platform) using piezoelectric actuators has been positioned below the spindle (Figure\nbsp{}ref:fig:nhexa_stages_villar). -While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to $50\,\mu m$ due to the inherent constraints of piezoelectric actuators. -In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\mu m$ stroke (Figure\nbsp{}ref:fig:nhexa_stages_schroer). +While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to $50\,\mu\text{m}$ due to the inherent constraints of piezoelectric actuators. +In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\mu\text{m}$ stroke (Figure\nbsp{}ref:fig:nhexa_stages_schroer). However, attempts to implement real-time feedback using YZ external metrology proved challenging, possibly due to the poor dynamical response of the serial stage configuration. #+name: fig:nhexa_stages_spindle @@ -4705,46 +4705,46 @@ Although direct performance comparisons between these systems are challenging du #+caption: End-Stations with integrated feedback loops based on online metrology. The stages used for feedback are indicated in bold font. Stages not used for scanning purposes are ommited or indicated between parentheses. The specifications for the NASS are indicated in the last row. #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccccc #+attr_latex: :placement [!ht] :center t :booktabs t -| *Stacked Stages* | *Specifications* | *Measured DoFs* | *Bandwidth* | *Reference* | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | 3 PID, n/a | APS | -| *XYZ stage (piezo)* | $D_{xyz}: 0.05\,mm$ | $D_{xyz}$ | |\nbsp{}[[cite:&nazaretski15_pushin_limit]] | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | light | Capacitive sensors | $\approx 10\,\text{Hz}$ | ESRF | -| Spindle | $R_z: \pm 90\,\text{deg}$ | $D_{xyz},\ R_{xy}$ | | ID16a | -| *Hexapod (piezo)* | $D_{xyz}: 0.05\,mm$ | | |\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] | -| | $R_{xy}: 500\,\mu\text{rad}$ | | | | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | n/a | PETRA III | -| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,mm$ | $D_{yz}$ | | P06 | -| Spindle | $R_z: 180\,\text{deg}$ | | |\nbsp{}[[cite:&schroer17_ptynam;&schropp20_ptynam]] | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | PID, n/a | PSI | -| Spindle | $R_z: \pm 182\,\text{deg}$ | $D_{yz},\ R_x$ | | OMNY | -| *Tripod (piezo)* | $D_{xyz}: 0.4\,mm$ | | |\nbsp{}[[cite:&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage]] | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | n/a | Soleil | -| (XY stage) | | $D_{xyz},\ R_{xy}$ | | Nanoprobe | -| Spindle | $R_z: 360\,\text{deg}$ | | |\nbsp{}[[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&engblom18_nanop_resul]] | -| *XYZ linear motors* | $D_{xyz}: 0.4\,mm$ | | | | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | up to 0.5kg | Interferometers | n/a | NSLS | -| Spindle | $R_z: 360\,\text{deg}$ | $D_{xyz}$ | | SRX | -| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,mm$ | | |\nbsp{}[[cite:&nazaretski22_new_kirkp_baez_based_scann]] | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | up to 0.35kg | Interferometers | $\approx 100\,\text{Hz}$ | Diamond, I14 | -| *Parallel XYZ VC* | $D_{xyz}: 3\,mm$ | $D_{xyz}$ | |\nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano]] | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | light | Capacitive sensors | $\approx 100\,\text{Hz}$ | LNLS | -| *Parallel XYZ VC* | $D_{xyz}: 3\,mm$ | and interferometers | | CARNAUBA | -| (Spindle) | $R_z: \pm 110 \,\text{deg}$ | $D_{xyz}$ | |\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] | -|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| -| Sample | up to 50kg | $D_{xyz},\ R_{xy}$ | | ESRF | -| *Active Platform* | | | | ID31 | -| (Micro-Hexapod) | | | |\nbsp{}[[cite:&dehaeze18_sampl_stabil_for_tomog_exper;&dehaeze21_mechat_approac_devel_nano_activ_stabil_system]] | -| Spindle | $R_z: 360\,\text{deg}$ | | | | -| Tilt-Stage | $R_y: \pm 3\,\text{deg}$ | | | | -| Translation Stage | $D_y: \pm 10\,mm$ | | | | +| *Stacked Stages* | *Specifications* | *Measured DoFs* | *Bandwidth* | *Reference* | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | 3 PID, n/a | APS | +| *XYZ stage (piezo)* | $D_{xyz}: 0.05\,\text{mm}$ | $D_{xyz}$ | | \nbsp{}[[cite:&nazaretski15_pushin_limit]] | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | light | Capacitive sensors | $\approx 10\,\text{Hz}$ | ESRF | +| Spindle | $R_z: \pm 90\,\text{deg}$ | $D_{xyz},\ R_{xy}$ | | ID16a | +| *Hexapod (piezo)* | $D_{xyz}: 0.05\,\text{mm}$ | | | \nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] | +| | $R_{xy}: 500\,\mu\text{rad}$ | | | | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | n/a | PETRA III | +| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,\text{mm}$ | $D_{yz}$ | | P06 | +| Spindle | $R_z: 180\,\text{deg}$ | | | \nbsp{}[[cite:&schroer17_ptynam;&schropp20_ptynam]] | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | PID, n/a | PSI | +| Spindle | $R_z: \pm 182\,\text{deg}$ | $D_{yz},\ R_x$ | | OMNY | +| *Tripod (piezo)* | $D_{xyz}: 0.4\,\text{mm}$ | | | \nbsp{}[[cite:&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage]] | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | n/a | Soleil | +| (XY stage) | | $D_{xyz},\ R_{xy}$ | | Nanoprobe | +| Spindle | $R_z: 360\,\text{deg}$ | | | \nbsp{}[[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&engblom18_nanop_resul]] | +| *XYZ linear motors* | $D_{xyz}: 0.4\,\text{mm}$ | | | | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | up to $0.5\,\text{kg}$ | Interferometers | n/a | NSLS | +| Spindle | $R_z: 360\,\text{deg}$ | $D_{xyz}$ | | SRX | +| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,\text{mm}$ | | | \nbsp{}[[cite:&nazaretski22_new_kirkp_baez_based_scann]] | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | up to $0.35\,\text{kg}$ | Interferometers | $\approx 100\,\text{Hz}$ | Diamond, I14 | +| *Parallel XYZ VC* | $D_{xyz}: 3\,\text{mm}$ | $D_{xyz}$ | | \nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano]] | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | light | Capacitive sensors | $\approx 100\,\text{Hz}$ | LNLS | +| *Parallel XYZ VC* | $D_{xyz}: 3\,\text{mm}$ | and interferometers | | CARNAUBA | +| (Spindle) | $R_z: \pm 110 \,\text{deg}$ | $D_{xyz}$ | | \nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] | +|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| +| Sample | up to $50\,\text{kg}$ | $D_{xyz},\ R_{xy}$ | | ESRF | +| *Active Platform* | | | | ID31 | +| (Hexapod) | | | | \nbsp{}[[cite:&dehaeze18_sampl_stabil_for_tomog_exper;&dehaeze21_mechat_approac_devel_nano_activ_stabil_system]] | +| Spindle | $R_z: 360\,\text{deg}$ | | | | +| Tilt-Stage | $R_y: \pm 3\,\text{deg}$ | | | | +| Translation Stage | $D_y: \pm 10\,\text{mm}$ | | | | The first key distinction of the NASS is in the continuous rotation of the active vibration platform. This feature introduces significant complexity through gyroscopic effects and real-time changes in the platform orientation, which substantially impact both the system's kinematics and dynamics. @@ -4753,9 +4753,9 @@ In conventional systems, active platforms typically correct spindle positioning The NASS, however, faces a more complex task: it must compensate for positioning errors of the translation and tilt stages in real-time during their operation, including corrections along their primary axes of motion. For instance, when the translation stage moves along Y, the active platform must not only correct for unwanted motions in other directions but also correct the position along Y, which necessitate some synchronization between the control of the long stroke stages and the control of the active platform. -The second major distinguishing feature of the NASS is its capability to handle payload masses up to 50 kg, exceeding typical capacities in the literature by two orders of magnitude. +The second major distinguishing feature of the NASS is its capability to handle payload masses up to $50\,\text{kg}$, exceeding typical capacities in the literature by two orders of magnitude. This substantial increase in payload mass fundamentally alters the system's dynamic behavior, as the sample mass significantly influences the overall system dynamics, in contrast to conventional systems where sample masses are negligible relative to the stage mass. -This characteristic introduces significant control challenges, as the feedback system must remain stable and maintain performance across a wide range of payload masses (from a few kilograms to 50 kg), requiring robust control strategies to handle such large plant variations. +This characteristic introduces significant control challenges, as the feedback system must remain stable and maintain performance across a wide range of payload masses (from a few kilograms to $50\,\text{kg}$), requiring robust control strategies to handle such large plant variations. The NASS also distinguishes itself through its high mobility and versatility, which are achieved through the use of multiple stacked stages (translation stage, tilt stage, spindle, positioning hexapod) that enable a wide range of experimental configurations. The resulting mechanical structure exhibits complex dynamics with multiple resonance modes in the low frequency range. @@ -4767,10 +4767,10 @@ The primary control requirements focus on $[D_y,\ D_z,\ R_y]$ motions; however, <> The choice of the active platform architecture for the NASS requires careful consideration of several critical specifications. -The platform must provide control over five acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\mu m$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of 300 mm diameter and 95 mm height. -It must accommodate payloads up to 50 kg while maintaining high dynamical performance. -For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to 100 Hz. -However, achieving such resonance frequencies with a 50 kg payload would require unrealistic stiffness values of approximately $2000\,N/\mu m$. +The platform must provide control over five acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\mu\text{m}$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of 300 mm diameter and 95 mm height. +It must accommodate payloads up to $50\,\text{kg}$ while maintaining high dynamical performance. +For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to $100\,\text{Hz}$. +However, achieving such resonance frequencies with a $50\,\text{kg}$ payload would require unrealistic stiffness values of approximately $2000\,\text{N}/\mu\text{m}$. This limitation necessitates alternative control approaches, and the High acrfull:haclac strategy is proposed to address this challenge. To this purpose, the design includes force sensors for active damping. Compliant mechanisms must also be used to eliminate friction and backlash, which would otherwise compromise the nano-positioning capabilities. @@ -4843,11 +4843,11 @@ These characteristics make the Stewart platforms particularly valuable in applic For the NASS application, the Stewart platform architecture offers three key advantages. First, as a fully parallel manipulator, all the motion errors of the micro-station can be compensated through the coordinated action of the six actuators. -Second, its compact design compared to serial manipulators makes it ideal for integration on top micro-station where only $95\,mm$ of height is available. +Second, its compact design compared to serial manipulators makes it ideal for integration on top micro-station where only $95\,\text{mm}$ of height is available. Third, the good dynamical properties should enable high-bandwidth positioning control. While Stewart platforms excel in precision and stiffness, they typically exhibit a relatively limited workspace compared to serial manipulators. -However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of $10\,\mu m$. +However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of $10\,\mu\text{m}$. This section provides a comprehensive analysis of the Stewart platform's properties, focusing on aspects crucial for precision positioning applications. The analysis encompasses the platform's kinematic relationships (Section\nbsp{}ref:ssec:nhexa_stewart_platform_kinematics), the use of the Jacobian matrix (Section\nbsp{}ref:ssec:nhexa_stewart_platform_jacobian), static behavior (Section\nbsp{}ref:ssec:nhexa_stewart_platform_static), and dynamic characteristics (Section\nbsp{}ref:ssec:nhexa_stewart_platform_dynamics). @@ -4873,7 +4873,7 @@ To facilitate the rigorous analysis of the Stewart platform, four reference fram Frames $\{F\}$ and $\{M\}$ serve primarily to define the joint locations. In contrast, frames $\{A\}$ and $\{B\}$ are used to describe the relative motion of the two platforms through the position vector ${}^A\bm{P}_B$ of frame $\{B\}$ expressed in frame $\{A\}$ and the rotation matrix ${}^A\bm{R}_B$ expressing the orientation of $\{B\}$ with respect to $\{A\}$. -For the nano-hexapod, frames $\{A\}$ and $\{B\}$ are chosen to be located at the theoretical focus point of the X-ray light which is $150\,mm$ above the top platform, i.e. above $\{M\}$. +For the active platform, frames $\{A\}$ and $\{B\}$ are chosen to be located at the theoretical focus point of the X-ray light which is $150\,\text{mm}$ above the top platform, i.e. above $\{M\}$. The location of the joints and the orientation and length of the struts are crucial for subsequent kinematic, static, and dynamic analyses of the Stewart platform. The center of rotation for the joint fixed to the base is noted $\bm{a}_i$, while $\bm{b}_i$ is used for the top platform joints. @@ -4914,7 +4914,7 @@ The obtained strut lengths are given by\nbsp{}eqref:eq:nhexa_inverse_kinematics. \end{equation} If the position and orientation of the platform lie in the feasible workspace, the solution is unique. -While configurations outside this workspace yield complex numbers, this only becomes relevant for large displacements that far exceed the nano-hexapod's operating range. +While configurations outside this workspace yield complex numbers, this only becomes relevant for large displacements that far exceed the active platform's operating range. ***** Forward Kinematics @@ -4922,7 +4922,7 @@ The forward kinematic problem seeks to determine the platform pose $\bm{\mathcal Unlike inverse kinematics, this presents a significant challenge because it requires solving a system of nonlinear equations. Although various numerical methods exist for solving this problem, they can be computationally intensive and may not guarantee convergence to the correct solution. -For the nano-hexapod application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative. +For the active platform application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative. This approximation, which is developed in subsequent sections through the Jacobian matrix analysis, is particularly useful for real-time control applications. # TODO - Add references @@ -4999,14 +4999,14 @@ The accuracy of the Jacobian-based forward kinematics solution was estimated by For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation\nbsp{}eqref:eq:nhexa_inverse_kinematics. These strut lengths are then used with the Jacobian to estimate the platform pose\nbsp{}eqref:eq:nhexa_forward_kinematics_approximate, from which the error between the estimated and true poses can be calculated, both in terms of position $\epsilon_D$ and orientation $\epsilon_R$. -For motion strokes from $1\,\mu m$ to $10\,mm$, the errors are estimated for all direction of motion, and the worst case errors are shown in Figure\nbsp{}ref:fig:nhexa_forward_kinematics_approximate_errors. -The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\mu m$ as the size of the Stewart platform is here $\approx 100\,mm$), the Jacobian approximation provides excellent accuracy. +For motion strokes from $1\,\mu\text{m}$ to $10\,\text{mm}$, the errors are estimated for all direction of motion, and the worst case errors are shown in Figure\nbsp{}ref:fig:nhexa_forward_kinematics_approximate_errors. +The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\mu\text{m}$ as the size of the Stewart platform is here $\approx 100\,\text{mm}$), the Jacobian approximation provides excellent accuracy. -Since the maximum required stroke of the nano-hexapod ($\approx 100\,\mu m$) is three orders of magnitude smaller than its overall size ($\approx 100\,mm$), the Jacobian matrix can be considered constant throughout the workspace. +Since the maximum required stroke of the active platform ($\approx 100\,\mu\text{m}$) is three orders of magnitude smaller than its overall size ($\approx 100\,\text{mm}$), the Jacobian matrix can be considered constant throughout the workspace. It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy. #+name: fig:nhexa_forward_kinematics_approximate_errors -#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of $100\,mm$ was used to perform this analysis. $\epsilon_D$ corresponds to the distance between the true positioin and the estimated position. $\epsilon_R$ corresponds to the angular motion between the true orientation and the estimated orientation. +#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of $100\,\text{mm}$ was used to perform this analysis. $\epsilon_D$ corresponds to the distance between the true positioin and the estimated position. $\epsilon_R$ corresponds to the angular motion between the true orientation and the estimated orientation. #+attr_latex: :scale 0.8 [[file:figs/nhexa_forward_kinematics_approximate_errors.png]] @@ -5147,7 +5147,7 @@ Through this multi-body modeling approach, each component model (including joint The analysis is structured as follows. First, the multi-body model is developed, and the geometric parameters, inertial properties, and actuator characteristics are established (Section\nbsp{}ref:ssec:nhexa_model_def). The model is then validated through comparison with the analytical equations in a simplified configuration (Section\nbsp{}ref:ssec:nhexa_model_validation). -Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section\nbsp{}ref:ssec:nhexa_model_dynamics). +Finally, the validated model is employed to analyze the active platform dynamics, from which insights for the control system design are derived (Section\nbsp{}ref:ssec:nhexa_model_dynamics). **** Model Definition <> @@ -5155,9 +5155,9 @@ Finally, the validated model is employed to analyze the nano-hexapod dynamics, f The Stewart platform's geometry is defined by two principal coordinate frames (Figure\nbsp{}ref:fig:nhexa_stewart_model_def): a fixed base frame $\{F\}$ and a moving platform frame $\{M\}$. The joints connecting the actuators to these frames are located at positions ${}^F\bm{a}_i$ and ${}^M\bm{b}_i$ respectively. -The acrshort:poi, denoted by frame $\{A\}$, is situated $150\,mm$ above the moving platform frame $\{M\}$. +The acrshort:poi, denoted by frame $\{A\}$, is situated $150\,\text{mm}$ above the moving platform frame $\{M\}$. -The geometric parameters of the nano-hexapod are summarized in Table\nbsp{}ref:tab:nhexa_stewart_model_geometry. +The geometric parameters of the active platform are summarized in Table\nbsp{}ref:tab:nhexa_stewart_model_geometry. These parameters define the positions of all connection points in their respective coordinate frames. From these parameters, key kinematic properties can be derived: the strut orientations $\hat{\bm{s}}_i$, strut lengths $l_i$, and the system's Jacobian matrix $\bm{J}$. @@ -5174,8 +5174,8 @@ From these parameters, key kinematic properties can be derived: the strut orient #+latex: \centering #+attr_latex: :environment tabularx :width 0.75\linewidth :placement [b] :align Xrrr #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| | $\bm{x}$ | $\bm{y}$ | $\bm{z}$ | -|-----------+----------+----------+----------| +| | $\bm{x}$ | $\bm{y}$ | $\bm{z}$ | +|----------------+----------+----------+----------| | ${}^M\bm{O}_B$ | $0$ | $0$ | $150$ | | ${}^F\bm{O}_M$ | $0$ | $0$ | $95$ | | ${}^F\bm{a}_1$ | $-92$ | $-77$ | $20$ | @@ -5196,9 +5196,9 @@ From these parameters, key kinematic properties can be derived: the strut orient ***** Inertia of Plates The fixed base and moving platform were modeled as solid cylindrical bodies. -The base platform was characterized by a radius of $120\,mm$ and thickness of $15\,mm$, matching the dimensions of the micro-hexapod's top platform. -The moving platform was similarly modeled with a radius of $110\,mm$ and thickness of $15\,mm$. -Both platforms were assigned a mass of $5\,kg$. +The base platform was characterized by a radius of $120\,\text{mm}$ and thickness of $15\,\text{mm}$, matching the dimensions of the positioning hexapod's top platform. +The moving platform was similarly modeled with a radius of $110\,\text{mm}$ and thickness of $15\,\text{mm}$. +Both platforms were assigned a mass of $5\,\text{kg}$. ***** Joints @@ -5223,7 +5223,7 @@ This modular approach to actuator modeling allows for future refinements as the #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage #+name: fig:nhexa_actuator_model -#+caption: Model of the nano-hexapod actuators +#+caption: Model of the active platform actuators #+attr_latex: :float nil :scale 0.8 [[file:figs/nhexa_actuator_model.png]] #+end_minipage @@ -5233,11 +5233,11 @@ This modular approach to actuator modeling allows for future refinements as the #+latex: \centering #+attr_latex: :environment tabularx :width 0.5\linewidth :placement [b] :align Xl #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| | Value | -|-------+-----------------| -| $k_a$ | $1\,N/\mu m$ | -| $c_a$ | $50\,N/(m/s)$ | -| $k_p$ | $0.05\,N/\mu m$ | +| | Value | +|-------+------------------------------| +| $k_a$ | $1\,\text{N}/\mu\text{m}$ | +| $c_a$ | $50\,\text{Ns}/\text{m}$ | +| $k_p$ | $0.05\,\text{N}/\mu\text{m}$ | #+latex: \captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters} #+end_minipage @@ -5251,7 +5251,7 @@ A three-dimensional visualization of the model is presented in Figure\nbsp{}ref: #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage #+name: fig:nhexa_stewart_model_input_outputs -#+caption: Nano-Hexapod plant with inputs and outputs. Frames $\{F\}$ and $\{M\}$ can be connected to other elements in the model. +#+caption: Active platform plant with inputs and outputs. Frames $\{F\}$ and $\{M\}$ can be connected to other elements in the model. #+attr_latex: :scale 1 :float nil [[file:figs/nhexa_stewart_model_input_outputs.png]] #+end_minipage @@ -5268,7 +5268,7 @@ The validation of the multi-body model was performed using the simplest Stewart This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness $k_a = 1\,\text{N}/\mu\text{m}$ and damping $c_a = 10\,\text{N}/({\text{m}/\text{s}})$. The geometric parameters remain as specified in Table\nbsp{}ref:tab:nhexa_actuator_parameters. -While the moving platform itself is considered massless, a $10\,\text{kg}$ cylindrical payload is mounted on top with a radius of $r = 110\,mm$ and a height $h = 300\,mm$. +While the moving platform itself is considered massless, a $10\,\text{kg}$ cylindrical payload is mounted on top with a radius of $r = 110\,\text{mm}$ and a height $h = 300\,\text{mm}$. For the analytical model, the stiffness, damping, and mass matrices are defined in\nbsp{}eqref:eq:nhexa_analytical_matrices. @@ -5292,11 +5292,11 @@ The close agreement between both approaches across the frequency spectrum valida #+attr_latex: :scale 0.8 [[file:figs/nhexa_comp_multi_body_analytical.png]] -**** Nano Hexapod Dynamics +**** Active Platform Dynamics <> -Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics was performed. -The model parameters were set according to the specifications outlined in Section\nbsp{}ref:ssec:nhexa_model_def, with a payload mass of $10\,kg$. +Following the validation of the multi-body model, a detailed analysis of the active platform dynamics was performed. +The model parameters were set according to the specifications outlined in Section\nbsp{}ref:ssec:nhexa_model_def, with a payload mass of $10\,\text{kg}$. The transfer functions from actuator forces $\bm{f}$ to both strut displacements $\bm{\mathcal{L}}$ and force measurements $\bm{f}_n$ were derived from the multi-body model. The transfer functions relating actuator forces to strut displacements are presented in Figure\nbsp{}ref:fig:nhexa_multi_body_plant_dL. @@ -5314,7 +5314,7 @@ Each actuator's transfer function to its associated force sensor exhibits altern The inclusion of parallel stiffness introduces an additional complex conjugate zero at low frequency, which was previously observed in the three-degree-of-freedom rotating model. #+name: fig:nhexa_multi_body_plant -#+caption: Bode plot of the transfer functions computed from the nano-hexapod multi-body model +#+caption: Bode plot of the transfer functions computed using the active platform multi-body model #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{\mathcal{L}}$} @@ -5333,7 +5333,7 @@ The inclusion of parallel stiffness introduces an additional complex conjugate z **** Conclusion -The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system. +The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the active platform system. Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform. A key advantage of this modeling approach lies in its flexibility for future refinements. @@ -5369,7 +5369,7 @@ This strategy potentially enables better performance by explicitly accounting fo The choice between these approaches depends significantly on the degree of interaction between the different control channels, and also on the available sensors and actuators. For instance, when using external metrology systems that measure the platform's global position, centralized control becomes necessary because each sensor measurement depends on all actuator inputs. -In the context of the nano-hexapod, two distinct control strategies were examined during the conceptual phase: +In the context of the active platform, two distinct control strategies were examined during the conceptual phase: - Decentralized Integral Force Feedback (IFF), which uses collocated force sensors to implement independent control loops for each strut (Section\nbsp{}ref:ssec:nhexa_control_iff) - acrfull:hac, which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section\nbsp{}ref:ssec:nhexa_control_hac_lac) @@ -5427,11 +5427,11 @@ This is particularly valuable when performance requirements differ between direc However, significant coupling exists between certain acrshortpl:dof, particularly between rotations and translations (e.g., $\epsilon_{R_x}/\mathcal{F}_y$ or $\epsilon_{D_y}/\bm\mathcal{M}_x$). -For the conceptual validation of the nano-hexapod, control in the strut space was selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies. +For the conceptual validation of the acrshort:nass, control in the strut space was selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies. More sophisticated control strategies will be explored during the detailed design phase. #+name: fig:nhexa_plant_frame -#+caption: Bode plot of the transfer functions computed from the nano-hexapod multi-body model +#+caption: Bode plot of the transfer functions computed using the active platform multi-body model #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts} @@ -5612,10 +5612,10 @@ This study establishes the theoretical framework necessary for the subsequent de *** Introduction :ignore: The previous chapters have established crucial foundational elements for the development of the Nano Active Stabilization System (NASS). -The uniaxial model study demonstrated that very stiff nano-hexapod configurations should be avoided due to their high coupling with the micro-station dynamics. -A rotating three-degree-of-freedom model revealed that soft nano-hexapod designs prove unsuitable due to gyroscopic effect induced by the spindle rotation. +The uniaxial model study demonstrated that very stiff active platform configurations should be avoided due to their high coupling with the micro-station dynamics. +A rotating three-degree-of-freedom model revealed that soft active platform designs prove unsuitable due to gyroscopic effect induced by the spindle rotation. To further improve the model accuracy, a multi-body model of the micro-station was developed, which was carefully tuned using experimental modal analysis. -Furthermore, a multi-body model of the nano-hexapod was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure\nbsp{}ref:fig:nass_simscape_model. +Furthermore, a multi-body model of the active platform was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure\nbsp{}ref:fig:nass_simscape_model. #+name: fig:nass_simscape_model #+caption: 3D view of the NASS multi-body model @@ -5624,8 +5624,8 @@ Furthermore, a multi-body model of the nano-hexapod was created, that can then b [[file:figs/nass_simscape_model.jpg]] Building upon these foundations, this chapter presents the validation of the NASS concept. -The investigation begins with the previously established nano-hexapod model with actuator stiffness $k_a = 1\,N/\mu m$. -A thorough examination of the control kinematics is presented in Section\nbsp{}ref:sec:nass_kinematics, detailing how both external metrology and nano-hexapod internal sensors are used in the control architecture. +The investigation begins with the previously established active platform model with actuator stiffness $k_a = 1\,\text{N}/\mu\text{m}$. +A thorough examination of the control kinematics is presented in Section\nbsp{}ref:sec:nass_kinematics, detailing how both external metrology and active platform internal sensors are used in the control architecture. The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section\nbsp{}ref:sec:nass_active_damping), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section\nbsp{}ref:sec:nass_hac). The robustness of the proposed control scheme was evaluated under various operational conditions. @@ -5651,7 +5651,7 @@ For the Nano Active Stabilization System, computing the positioning errors in th First, desired sample pose with respect to a fixed reference frame is computed using the micro-station kinematics as detailed in Section\nbsp{}ref:ssec:nass_ustation_kinematics. This fixed frame is located at the X-ray beam focal point, as it is where the acrshort:poi needs to be positioned. Second, it measures the actual sample pose relative to the same fix frame, described in Section\nbsp{}ref:ssec:nass_sample_pose_error. -Finally, it determines the sample pose error and maps these errors to the nano-hexapod struts, as explained in Section\nbsp{}ref:ssec:nass_error_struts. +Finally, it determines the sample pose error and maps these errors to the active platform struts, as explained in Section\nbsp{}ref:ssec:nass_error_struts. The complete control architecture is described in Section\nbsp{}ref:ssec:nass_control_architecture. @@ -5660,12 +5660,12 @@ The complete control architecture is described in Section\nbsp{}ref:ssec:nass_co The micro-station kinematics enables the computation of the desired sample pose from the reference signals of each micro-station stage. These reference signals consist of the desired lateral position $r_{D_y}$, tilt angle $r_{R_y}$, and spindle angle $r_{R_z}$. -The micro-hexapod pose is defined by six parameters: three translations ($r_{D_{\mu x}}$, $r_{D_{\mu y}}$, $r_{D_{\mu z}}$) and three rotations ($r_{\theta_{\mu x}}$, $r_{\theta_{\mu y}}$, $r_{\theta_{\mu z}}$). +The hexapod pose is defined by six parameters: three translations ($r_{D_{\mu x}}$, $r_{D_{\mu y}}$, $r_{D_{\mu z}}$) and three rotations ($r_{\theta_{\mu x}}$, $r_{\theta_{\mu y}}$, $r_{\theta_{\mu z}}$). Using these reference signals, the desired sample position relative to the fixed frame is expressed through the homogeneous transformation matrix $\bm{T}_{\mu\text{-station}}$, as defined in equation\nbsp{}eqref:eq:nass_sample_ref. \begin{equation}\label{eq:nass_sample_ref} - \bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}} + \bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\text{hexapod}} \end{equation} \begin{equation}\label{eq:nass_ustation_matrices} @@ -5676,7 +5676,7 @@ Using these reference signals, the desired sample position relative to the fixed 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad - \bm{T}_{\mu\text{-hexapod}} = + \bm{T}_{\text{hexapod}} = \left[ \begin{array}{ccc|c} & & & r_{D_{\mu x}} \\ & \bm{R}_x(r_{\theta_{\mu x}}) \bm{R}_y(r_{\theta_{\mu y}}) \bm{R}_{z}(r_{\theta_{\mu z}}) & & r_{D_{\mu y}} \\ @@ -5705,8 +5705,8 @@ Using these reference signals, the desired sample position relative to the fixed The external metrology system measures the sample position relative to the fixed granite. Due to the system's symmetry, this metrology provides measurements for five acrshortpl:dof: three translations ($D_x$, $D_y$, $D_z$) and two rotations ($R_x$, $R_y$). -The sixth acrshort:dof ($R_z$) is still required to compute the errors in the frame of the nano-hexapod struts (i.e. to compute the nano-hexapod inverse kinematics). -This $R_z$ rotation is estimated by combining measurements from the spindle encoder and the nano-hexapod's internal metrology, which consists of relative motion sensors in each strut (note that the micro-hexapod is not used for $R_z$ rotation, and is therefore ignored for $R_z$ estimation). +The sixth acrshort:dof ($R_z$) is still required to compute the errors in the frame of the active platform struts (i.e. to compute the active platform inverse kinematics). +This $R_z$ rotation is estimated by combining measurements from the spindle encoder and the active platform's internal metrology, which consists of relative motion sensors in each strut (note that the positioning hexapod is not used for $R_z$ rotation, and is therefore ignored for $R_z$ estimation). The measured sample pose is represented by the homogeneous transformation matrix $\bm{T}_{\text{sample}}$, as shown in equation\nbsp{}eqref:eq:nass_sample_pose. @@ -5726,7 +5726,7 @@ The measured sample pose is represented by the homogeneous transformation matrix The homogeneous transformation formalism enables straightforward computation of the sample position error. This computation involves the previously computed homogeneous $4 \times 4$ matrices: $\bm{T}_{\mu\text{-station}}$ representing the desired pose, and $\bm{T}_{\text{sample}}$ representing the measured pose. -Their combination yields $\bm{T}_{\text{error}}$, which expresses the position error of the sample in the frame of the rotating nano-hexapod, as shown in equation\nbsp{}eqref:eq:nass_transformation_error. +Their combination yields $\bm{T}_{\text{error}}$, which expresses the position error of the sample in the frame of the rotating active platform, as shown in equation\nbsp{}eqref:eq:nass_transformation_error. \begin{equation}\label{eq:nass_transformation_error} \bm{T}_{\text{error}} = \bm{T}_{\mu\text{-station}}^{-1} \cdot \bm{T}_{\text{sample}} @@ -5746,7 +5746,7 @@ From $\bm{T}_{\text{error}}$, the position and orientation errors $\bm{\epsilon} \end{align} \end{equation} -Finally, these errors are mapped to the strut space using the nano-hexapod Jacobian matrix\nbsp{}eqref:eq:nass_inverse_kinematics. +Finally, these errors are mapped to the strut space using the active platform Jacobian matrix\nbsp{}eqref:eq:nass_inverse_kinematics. \begin{equation}\label{eq:nass_inverse_kinematics} \bm{\epsilon}_{\mathcal{L}} = \bm{J} \cdot \bm{\epsilon}_{\mathcal{X}} @@ -5756,10 +5756,10 @@ Finally, these errors are mapped to the strut space using the nano-hexapod Jacob <> The complete control architecture is summarized in Figure\nbsp{}ref:fig:nass_control_architecture. -The sample pose is measured using external metrology for five acrshortpl:dof, while the sixth acrshort:dof ($R_z$) is estimated by combining measurements from the nano-hexapod encoders and spindle encoder. +The sample pose is measured using external metrology for five acrshortpl:dof, while the sixth acrshort:dof ($R_z$) is estimated by combining measurements from the active platform encoders and spindle encoder. -The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and micro-hexapod. -The position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the nano-hexapod frame. +The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and positioning hexapod. +The position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the active platform frame. Then, the Jacobian matrix $\bm{J}$ maps these errors to individual strut coordinates. For control purposes, force sensors mounted on each strut are used in a decentralized manner for active damping, as detailed in Section\nbsp{}ref:sec:nass_active_damping. @@ -5777,14 +5777,14 @@ Then, the high authority controller uses the computed errors in the frame of the Building on the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the acrshort:haclac strategy. The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3DoF rotating model. -The objective here is to design a decentralized IFF controller that provides good damping of the nano-hexapod modes across payload masses ranging from $1$ to $50\,\text{kg}$ and rotational velocity up to $360\,\text{deg/s}$. -The payloads used for validation have a cylindrical shape with 250 mm height and with masses of 1 kg, 25 kg, and 50 kg. +The objective here is to design a decentralized IFF controller that provides good damping of the active platform modes across payload masses ranging from $1$ to $50\,\text{kg}$ and rotational velocity up to $360\,\text{deg/s}$. +The payloads used for validation have a cylindrical shape with 250 mm height and with masses of $1\,\text{kg}$, $25\,\text{kg}$, and $50\,\text{kg}$. **** IFF Plant <> Transfer functions from actuator forces $f_i$ to force sensor measurements $f_{mi}$ are computed using the multi-body model. -Figure\nbsp{}ref:fig:nass_iff_plant_effect_kp examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity $\Omega_z = 360\,\text{deg/s}$ and with a payload mass of 25 kg. +Figure\nbsp{}ref:fig:nass_iff_plant_effect_kp examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity $\Omega_z = 360\,\text{deg/s}$ and with a payload mass of $25\,\text{kg}$. Without parallel stiffness (Figure\nbsp{}ref:fig:nass_iff_plant_no_kp), the plant dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model. Adding parallel stiffness (Figure\nbsp{}ref:fig:nass_iff_plant_kp) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation. @@ -5792,7 +5792,7 @@ Adding parallel stiffness (Figure\nbsp{}ref:fig:nass_iff_plant_kp) transforms th Although both cases show significant coupling around the resonances, stability is guaranteed by the collocated arrangement of the actuators and sensors\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with]]. #+name: fig:nass_iff_plant_effect_kp -#+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of 25kg. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}) +#+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of $25\,\text{kg}$. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}) #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_no_kp}without parallel stiffness} @@ -5809,7 +5809,7 @@ Although both cases show significant coupling around the resonances, stability i #+end_subfigure #+end_figure -The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,N/\mu m$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). +The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,\text{N}/\mu\text{m}$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). Figure\nbsp{}ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics. The poles and zeros shift in frequency as the payload mass varies. @@ -5892,12 +5892,12 @@ The results demonstrate that the closed-loop poles remain within the left-half p <> **** Introduction :ignore: -The implementation of high-bandwidth position control for the nano-hexapod presents several technical challenges. -The plant dynamics exhibits complex behavior influenced by multiple factors, including payload mass, rotational velocity, and the mechanical coupling between the nano-hexapod and the micro-station. +The implementation of high-bandwidth position control for the active platform presents several technical challenges. +The plant dynamics exhibits complex behavior influenced by multiple factors, including payload mass, rotational velocity, and the mechanical coupling between the active platform and the micro-station. This section presents the development and validation of a centralized control strategy designed to achieve precise sample positioning during high-speed tomography experiments. First, a comprehensive analysis of the plant dynamics is presented in Section\nbsp{}ref:ssec:nass_hac_plant, examining the effects of spindle rotation, payload mass variation, and the implementation of Integral Force Feedback (IFF). -Section\nbsp{}ref:ssec:nass_hac_stiffness validates previous modeling predictions that both overly stiff and compliant nano-hexapod configurations lead to degraded performance. +Section\nbsp{}ref:ssec:nass_hac_stiffness validates previous modeling predictions that both overly stiff and compliant active platform configurations lead to degraded performance. Building upon these findings, Section\nbsp{}ref:ssec:nass_hac_controller presents the design of a robust high-authority controller that maintains stability across varying payload masses while achieving the desired control bandwidth. The performance of the developed control strategy was validated through simulations of tomography experiments in Section\nbsp{}ref:ssec:nass_hac_tomography. @@ -5909,13 +5909,13 @@ Particular attention was paid to the system's behavior under maximum rotational The plant dynamics from force inputs $\bm{f}$ to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were first extracted from the multi-body model without the implementation of the decentralized IFF. The influence of spindle rotation on plant dynamics was investigated, and the results are presented in Figure\nbsp{}ref:fig:nass_undamped_plant_effect_Wz. -While rotational motion introduces coupling effects at low frequencies, these effects remain minimal at operational velocities, owing to the high stiffness characteristics of the nano-hexapod assembly. +While rotational motion introduces coupling effects at low frequencies, these effects remain minimal at operational velocities, owing to the high stiffness characteristics of the active platform assembly. Payload mass emerged as a significant parameter affecting system behavior, as illustrated in Figure\nbsp{}ref:fig:nass_undamped_plant_effect_mass. As expected, increasing the payload mass decreased the resonance frequencies while amplifying coupling at low frequency. These mass-dependent dynamic changes present considerable challenges for control system design, particularly for configurations with high payload masses. -Additional operational parameters were systematically evaluated, including the $R_y$ tilt angle, $R_z$ spindle position, and micro-hexapod position. +Additional operational parameters were systematically evaluated, including the $R_y$ tilt angle, $R_z$ spindle position, and positioning hexapod position. These factors were found to exert negligible influence on the plant dynamics, which can be attributed to the effective mechanical decoupling achieved between the plant and micro-station dynamics. This decoupling characteristic ensures consistent performance across various operational configurations. This also validates the developed control strategy. @@ -5941,13 +5941,13 @@ This also validates the developed control strategy. The Decentralized Integral Force Feedback was implemented in the multi-body model, and transfer functions from force inputs $\bm{f}^\prime$ of the damped plant to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were extracted from this model. The effectiveness of the IFF implementation was first evaluated with a $1\,\text{kg}$ payload, as demonstrated in Figure\nbsp{}ref:fig:nass_comp_undamped_damped_plant_m1. -The results indicate successful damping of the nano-hexapod resonance modes, although a minor increase in low-frequency coupling was observed. +The results indicate successful damping of the active platform resonance modes, although a minor increase in low-frequency coupling was observed. This trade-off was considered acceptable, given the overall improvement in system behavior. The benefits of IFF implementation were further assessed across the full range of payload configurations, and the results are presented in Figure\nbsp{}ref:fig:nass_hac_plants. -For all tested payloads ($1\,\text{kg}$, $25\,\text{kg}$ and $50\,\text{kg}$), the decentralized IFF significantly damped the nano-hexapod modes and therefore simplified the system dynamics. +For all tested payloads ($1\,\text{kg}$, $25\,\text{kg}$ and $50\,\text{kg}$), the decentralized IFF significantly damped the active platform modes and therefore simplified the system dynamics. More importantly, in the vicinity of the desired high authority control bandwidth (i.e. between $10\,\text{Hz}$ and $50\,\text{Hz}$), the damped dynamics (shown in red) exhibited minimal gain and phase variations with frequency. -For the undamped plants (shown in blue), achieving robust control with bandwidth above 10Hz while maintaining stability across different payload masses would be practically impossible. +For the undamped plants (shown in blue), achieving robust control with bandwidth above $10\,\text{Hz}$ while maintaining stability across different payload masses would be practically impossible. #+name: fig:nass_hac_plant #+caption: Effect of Decentralized Integral Force Feedback on the positioning plant for a $1\,\text{kg}$ sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}). @@ -5967,12 +5967,12 @@ For the undamped plants (shown in blue), achieving robust control with bandwidth #+end_subfigure #+end_figure -The coupling between the nano-hexapod and the micro-station was evaluated through a comparative analysis of plant dynamics under two mounting conditions. -In the first configuration, the nano-hexapod was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance. +The coupling between the active platform and the micro-station was evaluated through a comparative analysis of plant dynamics under two mounting conditions. +In the first configuration, the active platform was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance. As illustrated in Figure\nbsp{}ref:fig:nass_effect_ustation_compliance, the complex dynamics of the micro-station were found to have little impact on the plant dynamics. -The only observable difference manifests as additional alternating poles and zeros above 100Hz, a frequency range sufficiently beyond the control bandwidth to avoid interference with the system performance. -This result confirms effective dynamic decoupling between the nano-hexapod and the supporting micro-station structure. +The only observable difference manifests as additional alternating poles and zeros above $100\,\text{Hz}$, a frequency range sufficiently beyond the control bandwidth to avoid interference with the system performance. +This result confirms effective dynamic decoupling between the active platform and the supporting micro-station structure. #+name: fig:nass_effect_ustation_compliance #+caption: Effect of the micro-station limited compliance on the plant dynamics @@ -5980,33 +5980,33 @@ This result confirms effective dynamic decoupling between the nano-hexapod and t #+attr_latex: :scale 0.8 [[file:figs/nass_effect_ustation_compliance.png]] -**** Effect of Nano-Hexapod Stiffness on System Dynamics +**** Effect of Active Platform Stiffness on System Dynamics <> -The influence of nano-hexapod stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models. -These models suggest that a moderate stiffness of approximately $1\,N/\mu m$ would provide better performance than either very stiff or very soft configurations. +The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models. +These models suggest that a moderate stiffness of approximately $1\,\text{N}/\mu\text{m}$ would provide better performance than either very stiff or very soft configurations. -For the stiff nano-hexapod analysis, a system with an actuator stiffness of $100\,N/\mu m$ was simulated with a $25\,\text{kg}$ payload. +For the stiff active platform analysis, a system with an actuator stiffness of $100\,\text{N}/\mu\text{m}$ was simulated with a $25\,\text{kg}$ payload. The transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station. -As shown in Figure\nbsp{}ref:fig:nass_stiff_nano_hexapod_coupling_ustation, significant coupling was observed between the nano-hexapod and micro-station dynamics. +As shown in Figure\nbsp{}ref:fig:nass_stiff_nano_hexapod_coupling_ustation, significant coupling was observed between the active platform and micro-station dynamics. This coupling introduces complex behavior that is difficult to model and predict accurately, thus corroborating the predictions of the simplified uniaxial model. -The soft nano-hexapod configuration was evaluated using a stiffness of $0.01\,N/\mu m$ with a $25\,\text{kg}$ payload. +The soft active platform configuration was evaluated using a stiffness of $0.01\,\text{N}/\mu\text{m}$ with a $25\,\text{kg}$ payload. The dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s. Figure\nbsp{}ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies. -The current approach of controlling the position in the strut frame is inadequate for soft nano-hexapods; but even shifting control to a frame matching the payload's acrlong:com would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. +The current approach of controlling the position in the strut frame is inadequate for soft active platforms; but even shifting control to a frame matching the payload's acrlong:com would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. #+name: fig:nass_soft_stiff_hexapod -#+caption: Coupling between a stiff nano-hexapod ($k_a = 100\,N/\mu m$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance ($k_a = 0.01\,N/\mu m$) nano-hexapod (\subref{fig:nass_soft_nano_hexapod_effect_Wz}) +#+caption: Coupling between a stiff active platform ($k_a = 100\,\text{N}/\mu\text{m}$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance ($k_a = 0.01\,\text{N}/\mu\text{m}$) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}) #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,N/\mu m$ - Coupling with the micro-station} +#+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\mu\text{m}$ - Coupling with the micro-station} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/nass_stiff_nano_hexapod_coupling_ustation.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,N/\mu m$ - Effect of Spindle rotation} +#+attr_latex: :caption \subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,\text{N}/\mu\text{m}$ - Effect of Spindle rotation} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -6017,7 +6017,7 @@ The current approach of controlling the position in the strut frame is inadequat **** Controller design <> -A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure\nbsp{}ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation. +A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure\nbsp{}ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at $10\,\text{Hz}$) for high performance operation. The controller structure is defined in Equation\nbsp{}eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes. \begin{equation}\label{eq:nass_robust_hac} @@ -6025,7 +6025,7 @@ The controller structure is defined in Equation\nbsp{}eqref:eq:nass_robust_hac, \end{equation} The controller performance was evaluated through two complementary analyses. -First, the decentralized loop gain shown in Figure\nbsp{}ref:fig:nass_hac_loop_gain, confirms the achievement of the desired 10Hz bandwidth. +First, the decentralized loop gain shown in Figure\nbsp{}ref:fig:nass_hac_loop_gain, confirms the achievement of the desired $10\,\text{Hz}$ bandwidth. Second, the characteristic loci analysis presented in Figure\nbsp{}ref:fig:nass_hac_loci demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope. #+name: fig:nass_hac_controller @@ -6052,16 +6052,16 @@ Second, the characteristic loci analysis presented in Figure\nbsp{}ref:fig:nass_ The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with a particular focus on tomography scanning because of its demanding performance requirements. Simulations were conducted at the maximum operational rotational velocity of $\Omega_z = 360\,\text{deg/s}$ to evaluate system performance under the most challenging conditions. -Performance metrics were established based on anticipated future beamline specifications, which specify a beam size of 200nm (horizontal) by 100nm (vertical). +Performance metrics were established based on anticipated future beamline specifications, which specify a beam size of $200\,\text{nm}$ (horizontal) by $100\,\text{nm}$ (vertical). The primary requirement stipulates that the acrshort:poi must remain within beam dimensions throughout operation. The simulation included two principal disturbance sources: ground motion and spindle vibrations. Additional noise sources, including measurement noise and electrical noise from acrfull:dac and voltage amplifiers, were not included in this analysis, as these parameters will be optimized during the detailed design phase. -Figure\nbsp{}ref:fig:nass_tomo_1kg_60rpm presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (1kg). +Figure\nbsp{}ref:fig:nass_tomo_1kg_60rpm presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration ($1\,\text{kg}$). The results demonstrate the system's capability to maintain the sample's position within the specified beam dimensions, thus validating the fundamental concept of the stabilization system. #+name: fig:nass_tomo_1kg_60rpm -#+caption: Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at $360\,\text{deg/s}$. 1kg payload is placed on top of the nano-hexapod. +#+caption: Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at $360\,\text{deg/s}$. $1\,\text{kg}$ payload is placed on top of the active platform. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_xy}XY plane} @@ -6078,7 +6078,7 @@ The results demonstrate the system's capability to maintain the sample's positio #+end_subfigure #+end_figure -The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25 and 50kg payloads, complementing the initial 1kg test case. +The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25 and $50\,\text{kg}$ payloads, complementing the initial $1\,\text{kg}$ test case. As illustrated in Figure\nbsp{}ref:fig:nass_tomography_hac_iff, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis. While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions. @@ -6089,19 +6089,19 @@ For higher mass configurations, rotational velocities are expected to be below 3 #+caption: Simulation of tomography experiments - 360deg/s. Beam size is indicated by the dashed black ellipse #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,kg$} +#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,\text{kg}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/nass_tomography_hac_iff_m1.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m25} $m = 25\,kg$} +#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m25} $m = 25\,\text{kg}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/nass_tomography_hac_iff_m25.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,kg$} +#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,\text{kg}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -6116,13 +6116,13 @@ For higher mass configurations, rotational velocities are expected to be below 3 <> The development and analysis presented in this chapter have successfully validated the Nano Active Stabilization System concept, marking the completion of the conceptual design phase. -A comprehensive control strategy has been established, effectively combining external metrology with nano-hexapod sensor measurements to achieve precise position control. +A comprehensive control strategy has been established, effectively combining external metrology with active platform sensor measurements to achieve precise position control. The control strategy implements a High Authority Control - Low Authority Control architecture - a proven approach that has been specifically adapted to meet the unique requirements of the rotating NASS. The decentralized Integral Force Feedback component has been demonstrated to provide robust active damping under various operating conditions. The addition of parallel springs to the force sensors has been shown to ensure stability during spindle rotation. The centralized High Authority Controller, operating in the frame of the struts for simplicity, has successfully achieved the desired performance objectives of maintaining a bandwidth of $10\,\text{Hz}$ while maintaining robustness against payload mass variations. -This investigation has confirmed that the moderate actuator stiffness of $1\,N/\mu m$ represents an adequate choice for the nano-hexapod, as both very stiff and very compliant configurations introduce significant performance limitations. +This investigation has confirmed that the moderate actuator stiffness of $1\,\text{N}/\mu\text{m}$ represents an adequate choice for the active platform, as both very stiff and very compliant configurations introduce significant performance limitations. Simulations of tomography experiments have been performed, with positioning accuracy requirements defined by the expected minimum beam dimensions of $200\,\text{nm}$ by $100\,\text{nm}$. The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples. @@ -6139,9 +6139,9 @@ Through a systematic progression from simplified to increasingly complex models, Using the simple uniaxial model revealed that a very stiff stabilization stage was unsuitable due to its strong coupling with the complex micro-station dynamics. Conversely, the three-degree-of-freedom rotating model demonstrated that very soft stabilization stage designs are equally problematic due to the gyroscopic effects induced by spindle rotation. -A moderate stiffness of approximately $1\,N/\mu m$ was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection. +A moderate stiffness of approximately $1\,\text{N}/\mu\text{m}$ was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection. -The multi-body modeling approach proved essential for capturing the complex dynamics of both the micro-station and the nano-hexapod. +The multi-body modeling approach proved essential for capturing the complex dynamics of both the micro-station and the active platform. This model was tuned based on extensive modal analysis and vibration measurements. The Stewart platform architecture was selected for the active platform due to its good dynamical properties, compact design, and the ability to satisfy the strict space constraints of the NASS. @@ -6149,7 +6149,7 @@ The acrshort:haclac control strategy was successfully adapted to address the uni Decentralized Integral Force Feedback with parallel springs demonstrated robust active damping capabilities across different payload masses and rotational velocities. The centralized High Authority Controller, implemented in the frame of the struts, achieved the desired $10\,\text{Hz}$ bandwidth with good robustness properties. -Simulations of tomography experiments validated the NASS concept, with positioning accuracy meeting the requirements defined by the expected minimum beam dimensions ($200\,nm \times 100\,nm$) for lightweight samples at maximum rotational velocity. +Simulations of tomography experiments validated the NASS concept, with positioning accuracy meeting the requirements defined by the expected minimum beam dimensions ($200\,\text{nm} \times 100\,\text{nm}$) for lightweight samples at maximum rotational velocity. As anticipated by the control analysis, some performance degradation was observed with heavier payloads, but the overall performance remained sufficient to validate the fundamental concept. * Detailed Design @@ -6161,14 +6161,14 @@ As anticipated by the control analysis, some performance degradation was observe :END: Following the validation of the Nano Active Stabilization System concept in the previous chapter through simulated tomography experiments, this chapter addresses the refinement of the preliminary conceptual model into an optimized implementation. -The initial validation used a nano-hexapod with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise. +The initial validation used a active platform with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise. This detailed design phase aims to optimize each component while ensuring none will limit the system's overall performance. -This chapter begins by determining the optimal geometric configuration for the nano-hexapod (Section\nbsp{}ref:sec:detail_kinematics). +This chapter begins by determining the optimal geometric configuration for the active platform (Section\nbsp{}ref:sec:detail_kinematics). To this end, a review of existing Stewart platform designs is first presented, followed by an analysis of how geometric parameters influence the system's properties—mobility, stiffness, and dynamical response—with a particular emphasis on the cubic architecture. -The chapter concludes by specifying the chosen nano-hexapod geometry and the associated actuator stroke and flexible joint angular travel requirements to achieve the desired mobility. +The chapter concludes by specifying the chosen active platform geometry and the associated actuator stroke and flexible joint angular travel requirements to achieve the desired mobility. -Section\nbsp{}ref:sec:detail_fem introduces a hybrid modeling methodology that combines acrfull:fea with multi-body dynamics to optimize critical nano-hexapod components. +Section\nbsp{}ref:sec:detail_fem introduces a hybrid modeling methodology that combines acrfull:fea with multi-body dynamics to optimize critical active platform components. This approach is first experimentally validated using an Amplified Piezoelectric Actuator, establishing confidence in the modeling technique. The methodology is then applied to two key elements: the actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and the flexible joints (Section\nbsp{}ref:sec:detail_fem_joint), enabling detailed optimization while maintaining computational efficiency for system-level simulations. @@ -6181,7 +6181,7 @@ Section\nbsp{}ref:sec:detail_instrumentation focuses on instrumentation selectio The selected instrumentation is then experimentally characterized to verify compliance with these specifications, ensuring that the combined effect of all noise sources remains within acceptable limits. # TODO - Refine this part when the corresponding section is fully written -The chapter concludes with a concise presentation of the obtained optimized nano-hexapod design in Section\nbsp{}ref:sec:detail_design, summarizing how the various optimizations contribute to a system that balances the competing requirements of precision positioning, vibration isolation, and practical implementation constraints. +The chapter concludes with a concise presentation of the obtained optimized active platform design, called the "nano-hexapod", in Section\nbsp{}ref:sec:detail_design, summarizing how the various optimizations contribute to a system that balances the competing requirements of precision positioning, vibration isolation, and practical implementation constraints. With the detailed design completed and components procured, the project advances to the experimental validation phase, which will be addressed in the subsequent chapter. ** Optimal Geometry @@ -6189,15 +6189,15 @@ With the detailed design completed and components procured, the project advances *** Introduction :ignore: The performance of a Stewart platform depends on its geometric configuration, especially the orientation of its struts and the positioning of its joints. -During the conceptual design phase of the nano-hexapod, a preliminary geometry was selected based on general principles without detailed optimization. +During the conceptual design phase of the active platform, a preliminary geometry was selected based on general principles without detailed optimization. As the project advanced to the detailed design phase, a rigorous analysis of how geometry influences system performance became essential to ensure that the final design would meet the demanding requirements of the Nano Active Stabilization System (NASS). -In this chapter, the nano-hexapod geometry is optimized through careful analysis of how design parameters influence critical performance aspects: attainable workspace, mechanical stiffness, strut-to-strut coupling for decentralized control strategies, and dynamic response in Cartesian coordinates. +In this chapter, the active platform geometry is optimized through careful analysis of how design parameters influence critical performance aspects: attainable workspace, mechanical stiffness, strut-to-strut coupling for decentralized control strategies, and dynamic response in Cartesian coordinates. The chapter begins with a comprehensive review of existing Stewart platform designs in Section\nbsp{}ref:sec:detail_kinematics_stewart_review, surveying various approaches to geometry, actuation, sensing, and joint design from the literature. Section\nbsp{}ref:sec:detail_kinematics_geometry develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process. -Section\nbsp{}ref:sec:detail_kinematics_cubic examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application. -Finally, Section\nbsp{}ref:sec:detail_kinematics_nano_hexapod presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. +Section\nbsp{}ref:sec:detail_kinematics_cubic examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the NASS applications. +Finally, Section\nbsp{}ref:sec:detail_kinematics_nano_hexapod presents the optimized active platform geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. *** Review of Stewart platforms <> @@ -6230,12 +6230,12 @@ As explained in the conceptual phase, Stewart platforms comprise the following k # TODO -\nbsp{}ref:sec:detail_fem_joint The specific geometry (i.e., position of joints and orientation of the struts) can be selected based on the application requirements, resulting in numerous designs throughout the literature. This discussion focuses primarily on Stewart platforms designed for nano-positioning and vibration control, which necessitates the use of flexible joints. -The implementation of these flexible joints, will be discussed when designing the nano-hexapod flexible joints. +The implementation of these flexible joints, will be discussed when designing the active platform flexible joints. Long stroke Stewart platforms are not addressed here as their design presents different challenges, such as singularity-free workspace and complex kinematics\nbsp{}[[cite:&merlet06_paral_robot]]. In terms of actuation, mainly two types are used: voice coil actuators and piezoelectric actuators. -Voice coil actuators, providing stroke ranges from $0.5\,mm$ to $10\,mm$, are commonly implemented in cubic architectures (as illustrated in Figures\nbsp{}ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_pph) and are mainly used for vibration isolation\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax;&thayer98_stewar;&mcinroy99_dynam;&preumont07_six_axis_singl_stage_activ]]. -For applications requiring short stroke (typically smaller than $500\,\mu m$), piezoelectric actuators present an interesting alternative, as shown in\nbsp{}[[cite:&agrawal04_algor_activ_vibrat_isolat_spacec;&furutani04_nanom_cuttin_machin_using_stewar;&yang19_dynam_model_decoup_contr_flexib]]. +Voice coil actuators, providing stroke ranges from $0.5\,\text{mm}$ to $10\,\text{mm}$, are commonly implemented in cubic architectures (as illustrated in Figures\nbsp{}ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_pph) and are mainly used for vibration isolation\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax;&thayer98_stewar;&mcinroy99_dynam;&preumont07_six_axis_singl_stage_activ]]. +For applications requiring short stroke (typically smaller than $500\,\mu\text{m}$), piezoelectric actuators present an interesting alternative, as shown in\nbsp{}[[cite:&agrawal04_algor_activ_vibrat_isolat_spacec;&furutani04_nanom_cuttin_machin_using_stewar;&yang19_dynam_model_decoup_contr_flexib]]. Examples of piezoelectric-actuated Stewart platforms are presented in Figures\nbsp{}ref:fig:detail_kinematics_ulb_pz, ref:fig:detail_kinematics_uqp and ref:fig:detail_kinematics_yang19. Although less frequently encountered, magnetostrictive actuators have been successfully implemented in\nbsp{}[[cite:&zhang11_six_dof]] (Figure\nbsp{}ref:fig:detail_kinematics_zhang11). @@ -6458,7 +6458,7 @@ Having struts further apart decreases the "lever arm" and therefore reduces the It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex. For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke. -This analysis is conducted in Section\nbsp{}ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the nano-hexapod geometry. +This analysis is conducted in Section\nbsp{}ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the active platform geometry. **** Stiffness <> @@ -6589,11 +6589,11 @@ It is also possible to implement designs with strut lengths smaller than the cub Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption\nbsp{}[[cite:&geng94_six_degree_of_freed_activ;&preumont07_six_axis_singl_stage_activ;&jafari03_orthog_gough_stewar_platf_microm]]: simplified kinematics relationships and dynamical analysis\nbsp{}[[cite:&geng94_six_degree_of_freed_activ]]; uniform stiffness in all directions\nbsp{}[[cite:&hanieh03_activ_stewar]]; uniform mobility\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt.8.5.2]]; and minimization of the cross coupling between actuators and sensors in different struts\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control\nbsp{}[[cite:&geng94_six_degree_of_freed_activ;&thayer02_six_axis_vibrat_isolat_system]]. -These properties are examined in this section to assess their relevance for the nano-hexapod. +These properties are examined in this section to assess their relevance for the active platform. The mobility and stiffness properties of the cubic configuration are analyzed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_static. Dynamical decoupling is investigated in Section\nbsp{}ref:ssec:detail_kinematics_cubic_dynamic, while decentralized control, crucial for the NASS, is examined in Section\nbsp{}ref:ssec:detail_kinematics_decentralized_control. Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_design. -The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod. +The ultimate objective is to determine the suitability of the cubic architecture for the active platform. **** Static Properties <> @@ -6782,8 +6782,8 @@ An effective strategy for improving dynamical performances involves aligning the This can be achieved by positioning the payload below the top platform, such that the acrlong:com of the moving body coincides with the cube's center (Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload). This approach was physically implemented in several studies\nbsp{}[[cite:&mcinroy99_dynam;&jafari03_orthog_gough_stewar_platf_microm]], as shown in Figure\nbsp{}ref:fig:detail_kinematics_uw_gsp. The resulting dynamics are indeed well-decoupled (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_com_cok), taking advantage from diagonal stiffness and mass matrices. -The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform. -If a design similar to Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation. +The primary limitation of this approach is that, for many applications including the NASS, the payload must be positioned above the top platform. +If a design similar to Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload were employed for the active platform, the X-ray beam would intersect with the struts during spindle rotation. #+name: fig:detail_kinematics_cubic_com_cok #+caption: Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok}) @@ -6901,7 +6901,7 @@ This section proposes modifications to the cubic architecture to enable position Three key parameters define the geometry of the cubic Stewart platform: $H$, the height of the Stewart platform (distance from fixed base to mobile platform); $H_c$, the height of the cube, as shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_schematic_full; and $H_{CoM}$, the height of the acrlong:com relative to the mobile platform (coincident with the cube's center). Depending on the cube's size $H_c$ in relation to $H$ and $H_{CoM}$, different designs emerge. -In the following examples, $H = 100\,mm$ and $H_{CoM} = 20\,mm$. +In the following examples, $H = 100\,\text{mm}$ and $H_{CoM} = 20\,\text{mm}$. ***** Small cube @@ -7041,20 +7041,20 @@ To address this limitation, modified cubic architectures have been proposed with Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform. This structural modification enables the alignment of the moving body's acrlong:com with the acrlong:cok, resulting in beneficial decoupling properties in the Cartesian frame. -*** Nano Hexapod +*** Active Platform for the NASS <> **** Introduction :ignore: -Based on previous analysis, this section aims to determine the nano-hexapod optimal geometry. -For the NASS, the chosen reference frames $\{A\}$ and $\{B\}$ coincide with the sample's acrshort:poi, which is positioned $150\,mm$ above the top platform. +Based on previous analysis, this section aims to determine the active platform optimal geometry. +For the NASS, the chosen reference frames $\{A\}$ and $\{B\}$ coincide with the sample's acrshort:poi, which is positioned $150\,\text{mm}$ above the top platform. This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused. **** Requirements <> -The design of the nano-hexapod must satisfy several constraints. -The device should fit within a cylinder with radius of $120\,mm$ and height of $95\,mm$. -Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of $\pm 50\,\mu m$. +The design of the active platform must satisfy several constraints. +The device should fit within a cylinder with radius of $120\,\text{mm}$ and height of $95\,\text{mm}$. +Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of $\pm 50\,\mu\text{m}$. At any position, the system should be capable of performing $R_x$ and $R_y$ rotations of $\pm 50\,\mu \text{rad}$. Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of $2\pi\,\text{rad/s}$ to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics. In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts. @@ -7062,23 +7062,23 @@ In terms of dynamics, the design should facilitate implementation of Integral Fo **** Obtained Geometry <> -Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads, with masses ranging from 1kg to 50kg, makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations. +Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads, with masses ranging from $1\,\text{kg}$ to $50\,\text{kg}$, makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations. -For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several considerations. +For the active platform design, the struts were oriented more vertically compared to a cubic architecture due to several considerations. First, the performance requirements in the vertical direction are more stringent than in the horizontal direction. This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise. Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes. -Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod. +Therefore, higher resonance frequencies of the active platform in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the active platform. Regarding dynamical properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that strut orientation has minimal impact on the resulting coupling characteristics. Consequently, the geometry was selected according to practical constraints. -The height between the two plates is maximized and set at $95\,mm$. -Both platforms take the maximum available size, with joints offset by $15\,mm$ from the plate surfaces and positioned along circles with radii of $120\,mm$ for the fixed joints and $110\,mm$ for the mobile joints. +The height between the two plates is maximized and set at $95\,\text{mm}$. +Both platforms take the maximum available size, with joints offset by $15\,\text{mm}$ from the plate surfaces and positioned along circles with radii of $120\,\text{mm}$ for the fixed joints and $110\,\text{mm}$ for the mobile joints. The positioning angles, as shown in Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_top, are $[255,\ 285,\ 15,\ 45,\ 135,\ 165]$ degrees for the top joints and $[220,\ 320,\ 340,\ 80,\ 100,\ 200]$ degrees for the bottom joints. #+name: fig:detail_kinematics_nano_hexapod -#+caption: Obtained architecture for the Nano Hexapod +#+caption: Obtained architecture for the active platform #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_nano_hexapod_iso}Isometric view} @@ -7101,37 +7101,37 @@ This geometry serves as the foundation for estimating required actuator stroke ( # TODO - Add link to sections Implementing a cubic architecture as proposed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_design was considered. -However, positioning the cube's center $150\,mm$ above the top platform would have resulted in platform dimensions exceeding the maximum available size. -Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the nano-hexapod, ensuring that its acrlong:com coincides with the cube's center. -Given the impracticality of consistently aligning the acrlong:com with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. +However, positioning the cube's center $150\,\text{mm}$ above the top platform would have resulted in platform dimensions exceeding the maximum available size. +Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the active platform, ensuring that its acrlong:com coincides with the cube's center. +Given the impracticality of consistently aligning the acrlong:com with the cube's center, the cubic architecture was deemed unsuitable for the NASS. **** Required Actuator stroke <> With the geometry established, the actuator stroke necessary to achieve the desired mobility can be determined. -The required mobility parameters include combined translations in the XYZ directions of $\pm 50\,\mu m$ (essentially a cubic workspace). +The required mobility parameters include combined translations in the XYZ directions of $\pm 50\,\mu\text{m}$ (essentially a cubic workspace). Additionally, at any point within this workspace, combined $R_x$ and $R_y$ rotations of $\pm 50\,\mu \text{rad}$, with $R_z$ maintained at 0, should be possible. -Calculations based on the selected geometry indicate that an actuator stroke of $\pm 94\,\mu m$ is required to achieve the desired mobility. -This specification will be used during the actuator selection process in Section ref:sec:detail_fem_actuator. +Calculations based on the selected geometry indicate that an actuator stroke of $\pm 94\,\mu\text{m}$ is required to achieve the desired mobility. +This specification will be used during the actuator selection process in Section\nbsp{}ref:sec:detail_fem_actuator. -Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of $\pm 94\,\mu m$. +Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the active platform with an actuator stroke of $\pm 94\,\mu\text{m}$. The diagram confirms that the required workspace fits within the system's capabilities. #+name: fig:detail_kinematics_nano_hexapod_mobility -#+caption: Specified translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume). +#+caption: Specified translation mobility of the active platform (grey cube) and computed Mobility (red volume). #+attr_latex: :scale 0.8 [[file:figs/detail_kinematics_nano_hexapod_mobility.png]] **** Required Joint angular stroke <> -With the nano-hexapod geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined. +With the active platform geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined. This analysis focuses solely on bending stroke, as the torsional stroke of the flexible joints is expected to be minimal given the absence of vertical rotation requirements. The required angular stroke for both fixed and mobile joints is estimated to be equal to $1\,\text{mrad}$. -This specification will guide the design of the flexible joints in Section ref:sec:detail_fem_joint. +This specification will guide the design of the flexible joints in Section\nbsp{}ref:sec:detail_fem_joint. *** Conclusion :PROPERTIES: @@ -7139,29 +7139,29 @@ This specification will guide the design of the flexible joints in Section ref:s :END: <> -This chapter has explored the optimization of the nano-hexapod geometry for the Nano Active Stabilization System (NASS). +This chapter has explored the optimization of the active platform geometry for the Nano Active Stabilization System (NASS). First, a review of existing Stewart platforms revealed two main geometric categories: cubic architectures, characterized by mutually orthogonal struts arranged along the edges of a cube, and non-cubic architectures with varied strut orientations. While cubic architectures are prevalent in the literature and attributed with beneficial properties such as simplified kinematics, uniform stiffness, and reduced cross-coupling, the performed analysis revealed that some of these advantages should be more nuanced or context-dependent than commonly described. The analytical relationships between Stewart platform geometry and its mechanical properties were established, enabling a better understanding of the trade-offs between competing requirements such as mobility and stiffness along different axes. -These insights were useful during the nano-hexapod geometry optimization. +These insights were useful during the active platform geometry optimization. For the cubic configuration, complete dynamical decoupling in the Cartesian frame can be achieved when the acrlong:com of the moving body coincides with the cube's center, but this arrangement is often impractical for real-world applications. -Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. +Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the active platform due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. -For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. +For the active platform design, a key challenge was addressing the wide range of potential payloads (1 to $50\,\text{kg}$), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies. ** Component Optimization <> *** Introduction :ignore: -During the nano-hexapod's detailed design phase, a hybrid modeling approach combining acrfull:fea with multi-body dynamics was developed. -This methodology, using reduced-order flexible bodies, was created to enable both detailed component optimization and efficient system-level simulation, addressing the impracticality of a full acrshort:fem for real-time control scenarios. +Addressing the need for both detailed component optimization and efficient system-level simulation—especially considering the limitations of full acrshort:fem for real-time control—a hybrid modeling approach was used. +This combines acrfull:fea with multi-body dynamics, employing reduced-order flexible bodies. The theoretical foundations and implementation are presented in Section\nbsp{}ref:sec:detail_fem_super_element, where experimental validation was performed using an Amplified Piezoelectric Actuator. -The framework was then applied to optimize two critical nano-hexapod elements: the actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and the flexible joints (Section\nbsp{}ref:sec:detail_fem_joint). +The framework was then applied to optimize two critical active platform elements: the actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and the flexible joints (Section\nbsp{}ref:sec:detail_fem_joint). Through this approach, system-level dynamic behavior under closed-loop control conditions could be successfully predicted while detailed component-level optimization was facilitated. *** Reduced order flexible bodies @@ -7206,7 +7206,7 @@ m = 6 \times n + p ***** Introduction :ignore: The presented modeling framework was first applied to an acrfull:apa for several reasons. -Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section\nbsp{}ref:sec:detail_fem_actuator. +Primarily, this actuator represents an excellent candidate for implementation within the active platform, as will be elaborated in Section\nbsp{}ref:sec:detail_fem_actuator. Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_picture) was available in the laboratory for experimental testing. The acrshort:apa consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure\nbsp{}ref:fig:detail_fem_apa95ml_picture) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement in the vertical direction\nbsp{}[[cite:&claeyssen07_amplif_piezoel_actuat]]. @@ -7226,11 +7226,11 @@ The specific design of the acrshort:apa allows for the simultaneous modeling of #+latex: \centering #+attr_latex: :environment tabularx :width 0.55\linewidth :placement [b] :align Xc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| *Parameter* | *Value* | -|----------------+---------------| -| Nominal Stroke | $100\,\mu m$ | -| Blocked force | $2100\,N$ | -| Stiffness | $21\,N/\mu m$ | +| *Parameter* | *Value* | +|----------------+---------------------| +| Nominal Stroke | $100\,\mu\text{m}$ | +| Blocked force | $2100\,\text{N}$ | +| Stiffness | $21\,\text{N}/\mu\text{m}$ | #+latex: \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications} #+end_minipage @@ -7243,10 +7243,10 @@ The finite element mesh, shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_mesh, #+caption: Material properties used for FEA. $E$ is the Young's modulus, $\nu$ the Poisson ratio and $\rho$ the material density #+attr_latex: :environment tabularx :width 0.55\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| | $E$ | $\nu$ | $\rho$ | -|------------------------------+-------------+--------+-----------------------| -| Stainless Steel | $190\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ | -| Piezoelectric Ceramics (PZT) | $49.5\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ | +| | $E$ | $\nu$ | $\rho$ | +|------------------------------+--------------------+--------+------------------------------| +| Stainless Steel | $190\,\text{GPa}$ | $0.31$ | $7800\,\text{kg}/\text{m}^3$ | +| Piezoelectric Ceramics (PZT) | $49.5\,\text{GPa}$ | $0.31$ | $7800\,\text{kg}/\text{m}^3$ | The definition of interface frames constitutes a critical aspect of the model preparation. Seven frames were established: one frame at the two ends of each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation. @@ -7306,32 +7306,32 @@ Yet, based on the available properties of the stacks in the data-sheet (summariz #+caption: Stack Parameters #+attr_latex: :environment tabularx :width 0.3\linewidth :align Xc #+attr_latex: :center t :booktabs t -| *Parameter* | *Value* | -|----------------+---------------------| -| Nominal Stroke | $20\,\mu m$ | -| Blocked force | $4700\,N$ | -| Stiffness | $235\,N/\mu m$ | -| Voltage Range | $-20/150\,V$ | -| Capacitance | $4.4\,\mu F$ | -| Length | $20\,mm$ | -| Stack Area | $10\times 10\,mm^2$ | +| *Parameter* | *Value* | +|----------------+-----------------------------| +| Nominal Stroke | $20\,\mu\text{m}$ | +| Blocked force | $4700\,\text{N}$ | +| Stiffness | $235\,\text{N}/\mu\text{m}$ | +| Voltage Range | $-20/150\,\text{V}$ | +| Capacitance | $4.4\,\mu\text{F}$ | +| Length | $20\,\text{mm}$ | +| Stack Area | $10\times 10\,\text{mm}^2$ | The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table\nbsp{}ref:tab:detail_fem_piezo_properties. -From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained. +From these parameters, $g_s = 5.1\,\text{V}/\mu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained. #+name: tab:detail_fem_piezo_properties #+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX #+attr_latex: :center t :booktabs t -| *Parameter* | *Value* | *Description* | -|----------------+----------------------------+--------------------------------------------------------------| -| $d_{33}$ | $680 \cdot 10^{-12}\,m/V$ | Piezoelectric constant | -| $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,F/m$ | Permittivity under constant stress | -| $s^{D}$ | $21 \cdot 10^{-12}\,m^2/N$ | Elastic compliance understand constant electric displacement | -| $c^{E}$ | $48 \cdot 10^{9}\,N/m^2$ | Young's modulus of elasticity | -| $L$ | $20\,mm$ per stack | Length of the stack | -| $A$ | $10^{-4}\,m^2$ | Area of the piezoelectric stack | -| $n$ | $160$ per stack | Number of layers in the piezoelectric stack | +| *Parameter* | *Value* | *Description* | +|----------------+------------------------------------------+--------------------------------------------------------------| +| $d_{33}$ | $680 \cdot 10^{-12}\,\text{m/V}$ | Piezoelectric constant | +| $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,\text{F/m}$ | Permittivity under constant stress | +| $s^{D}$ | $21 \cdot 10^{-12}\,\text{m}^2/\text{N}$ | Elastic compliance understand constant electric displacement | +| $c^{E}$ | $48 \cdot 10^{9}\,\text{N}/\text{m}^2$ | Young's modulus of elasticity | +| $L$ | $20\,\text{mm}$ per stack | Length of the stack | +| $A$ | $10^{-4}\,\text{m}^2$ | Area of the piezoelectric stack | +| $n$ | $160$ per stack | Number of layers in the piezoelectric stack | ***** Identification of the APA Characteristics @@ -7339,7 +7339,7 @@ Initial validation of the acrlong:fem and its integration as a reduced-order fle The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure\nbsp{}ref:fig:detail_fem_apa95ml_compliance), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames. The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML. -A value of $23\,N/\mu m$ was found which is close to the specified stiffness in the datasheet of $k = 21\,N/\mu m$. +A value of $23\,\text{N}/\mu\text{m}$ was found which is close to the specified stiffness in the datasheet of $k = 21\,\text{N}/\mu\text{m}$. The multi-body model predicted a resonant frequency under block-free conditions of $\approx 2\,\text{kHz}$ (Figure\nbsp{}ref:fig:detail_fem_apa95ml_compliance), which is in agreement with the nominal specification. @@ -7351,9 +7351,9 @@ The multi-body model predicted a resonant frequency under block-free conditions In order to estimate the stroke of the APA95ML, the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, was first determined. This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of $1.5$ was derived. -The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ relative to their length (here equal to $20\,mm$), produce an individual nominal stroke of $20\,\mu m$ (see data-sheet of the piezoelectric stacks on Table\nbsp{}ref:tab:detail_fem_stack_parameters, page\nbsp{}pageref:tab:detail_fem_stack_parameters). -As three stacks are used, the horizontal displacement is $60\,\mu m$. -Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of $90\,\mu m$ which falls within the manufacturer-specified range of $80\,\mu m$ and $120\,\mu m$. +The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ relative to their length (here equal to $20\,\text{mm}$), produce an individual nominal stroke of $20\,\mu\text{m}$ (see data-sheet of the piezoelectric stacks on Table\nbsp{}ref:tab:detail_fem_stack_parameters, page\nbsp{}pageref:tab:detail_fem_stack_parameters). +As three stacks are used, the horizontal displacement is $60\,\mu\text{m}$. +Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of $90\,\mu\text{m}$ which falls within the manufacturer-specified range of $80\,\mu\text{m}$ and $120\,\mu\text{m}$. The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include acrshort:fem into multi-body model. @@ -7364,7 +7364,7 @@ The high degree of concordance observed across multiple performance metrics prov Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation. The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element. -The test bench illustrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_bench_schematic was used, which consists of a $5.7\,kg$ granite suspended on top of the APA95ML. +The test bench illustrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_bench_schematic was used, which consists of a $5.7\,\text{kg}$ granite suspended on top of the APA95ML. The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement $y$. A acrfull:dac was used to generate the control signal $u$, which was subsequently conditioned through a voltage amplifier with a gain of $20$, ultimately yielding the effective voltage $V_a$ across the two piezoelectric stacks. Measurement of the sensor stack voltage $V_s$ was performed using an acrshort:adc. @@ -7386,7 +7386,7 @@ To improve the quality of the obtained frequency domain data, averaging and wind The obtained acrshortpl:frf from $V_a$ to $V_s$ and to $y$ are compared with the theoretical predictions derived from the multi-body model in Figure\nbsp{}ref:fig:detail_fem_apa95ml_comp_plant. -The difference in phase between the model and the measurements can be attributed to the sampling time of $0.1\,ms$ and to additional delays induced by electronic instrumentation related to the interferometer. +The difference in phase between the model and the measurements can be attributed to the sampling time of $0.1\,\text{ms}$ and to additional delays induced by electronic instrumentation related to the interferometer. The presence of a non-minimum phase zero in the measured system response (Figure\nbsp{}ref:fig:detail_fem_apa95ml_comp_plant_sensor), shall be addressed during the experimental phase. Regarding the amplitude characteristics, the constants $g_a$ and $g_s$ could be further refined through calibration against the experimental data. @@ -7449,7 +7449,7 @@ The close agreement between experimental measurements and theoretical prediction The experimental validation with an Amplified Piezoelectric Actuator confirms that this methodology accurately predicts both open-loop and closed-loop dynamic behaviors. This verification establishes its effectiveness for component design and system analysis applications. -The approach will be especially beneficial for optimizing actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and flexible joints (Section\nbsp{}ref:sec:detail_fem_joint) for the nano-hexapod. +The approach will be especially beneficial for optimizing actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and flexible joints (Section\nbsp{}ref:sec:detail_fem_joint) for the active platform. *** Actuator Selection <> @@ -7459,20 +7459,20 @@ The approach will be especially beneficial for optimizing actuators (Section\nbs The actuator selection process was driven by several critical requirements derived from previous dynamic analyses. A primary consideration is the actuator stiffness, which significantly impacts system dynamics through multiple mechanisms. The spindle rotation induces gyroscopic effects that modify plant dynamics and increase coupling, necessitating sufficient stiffness. -Conversely, the actuator stiffness must be carefully limited to ensure the nano-hexapod's suspension modes remain below the problematic modes of the micro-station to limit the coupling between the two structures. -These competing requirements suggest an optimal stiffness of approximately $1\,N/\mu m$. +Conversely, the actuator stiffness must be carefully limited to ensure the active platform's suspension modes remain below the problematic modes of the micro-station to limit the coupling between the two structures. +These competing requirements suggest an optimal stiffness of approximately $1\,\text{N}/\mu\text{m}$. Additional specifications arise from the control strategy and physical constraints. The implementation of the decentralized Integral Force Feedback (IFF) architecture necessitates force sensors to be collocated with each actuator. -The system's geometric constraints limit the actuator height to 50mm, given the nano-hexapod's maximum height of 95mm and the presence of flexible joints at each strut extremity. +The system's geometric constraints limit the actuator height to 50mm, given the active platform's maximum height of 95mm and the presence of flexible joints at each strut extremity. Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility. -An actuator stroke of $\approx 200\,\mu m$ is therefore required. +An actuator stroke of $\approx 200\,\mu\text{m}$ is therefore required. Three actuator technologies were evaluated (examples of such actuators are shown in Figure\nbsp{}ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators. Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity. #+name: fig:detail_fem_actuator_pictures -#+caption: Example of actuators considered for the nano-hexapod. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}). +#+caption: Example of actuators considered for the active platform. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_voice_coil_picture}Voice Coil} @@ -7495,13 +7495,13 @@ Variable reluctance actuators were not considered despite their superior efficie #+end_subfigure #+end_figure -Voice coil actuators (shown in Figure\nbsp{}ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,N/\mu m$), would require forces in the order of $200\,N$ to achieve the specified $200\,\mu m$ displacement. +Voice coil actuators (shown in Figure\nbsp{}ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,\text{N}/\mu\text{m}$), would require forces in the order of $200\,\text{N}$ to achieve the specified $200\,\mu\text{m}$ displacement. While these actuators offer excellent linearity and long strokes capabilities, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability. Their advantages (linearity and long stroke) were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions. Conventional piezoelectric stack actuators (shown in Figure\nbsp{}ref:fig:detail_fem_piezo_picture) present two significant limitations for the current application. -Their stroke is inherently limited to approximately $0.1\,\%$ of their length, meaning that even with the maximum allowable height of $50\,mm$, the achievable stroke would only be $50\,\mu m$, insufficient for the application. -Additionally, their extremely high stiffness, typically around $100\,N/\mu m$, exceeds the desired specifications by two orders of magnitude. +Their stroke is inherently limited to approximately $0.1\,\%$ of their length, meaning that even with the maximum allowable height of $50\,\text{mm}$, the achievable stroke would only be $50\,\mu\text{m}$, insufficient for the application. +Additionally, their extremely high stiffness, typically around $100\,\text{N}/\mu\text{m}$, exceeds the desired specifications by two orders of magnitude. Amplified Piezoelectric Actuators emerged as the optimal solution by addressing these limitations through a specific mechanical design. The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile. @@ -7514,16 +7514,16 @@ This selection was further reinforced by previous experience with acrshortpl:apa The demonstrated accuracy of the modeling approach for the APA95ML provides confidence in the reliable prediction of the APA300ML's dynamic characteristics, thereby supporting both the selection decision and subsequent dynamical analyses. #+name: tab:detail_fem_piezo_act_models -#+caption: List of some amplified piezoelectric actuators that could be used for the nano-hexapod +#+caption: List of some amplified piezoelectric actuators that could be used for the active platform #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccccc #+attr_latex: :center t :booktabs t :float t -| *Specification* | APA150M | *APA300ML* | APA400MML | FPA-0500E-P | FPA-0300E-S | -|------------------------------------+---------+------------+-----------+-------------+-------------| -| Stroke $> 200\, [\mu m]$ | 187 | 304 | 368 | 432 | 240 | -| Stiffness $\approx 1\, [N/\mu m]$ | 0.7 | 1.8 | 0.55 | 0.87 | 0.58 | -| Resolution $< 2\, [nm]$ | 2 | 3 | 4 | | | -| Blocked Force $> 100\, [N]$ | 127 | 546 | 201 | 376 | 139 | -| Height $< 50\, [mm]$ | 22 | 30 | 24 | 27 | 16 | +| *Specification* | APA150M | *APA300ML* | APA400MML | FPA-0500E-P | FPA-0300E-S | +|----------------------------------------------+---------+------------+-----------+-------------+-------------| +| Stroke $> 200\,\mu\text{m}$ | 187 | 304 | 368 | 432 | 240 | +| Stiffness $\approx 1\,\text{N}/\mu\text{m}$ | 0.7 | 1.8 | 0.55 | 0.87 | 0.58 | +| Resolution $< 2\,\text{nm}$ | 2 | 3 | 4 | | | +| Blocked Force $> 100\,\text{N}$ | 127 | 546 | 201 | 376 | 139 | +| Height $< 50\,\text{mm}$ | 22 | 30 | 24 | 27 | 16 | **** APA300ML - Reduced Order Flexible Body <> @@ -7593,16 +7593,16 @@ While higher-order modes and non-axial flexibility are not captured, the model a #+caption: Summary of the obtained parameters for the 2 DoF APA300ML model #+attr_latex: :environment tabularx :width 0.25\linewidth :align cc #+attr_latex: :center t :booktabs t -| *Parameter* | *Value* | -|-------------+-----------------| -| $k_1$ | $0.30\,N/\mu m$ | -| $k_e$ | $4.3\, N/\mu m$ | -| $k_a$ | $2.15\,N/\mu m$ | -| $c_1$ | $18\,Ns/m$ | -| $c_e$ | $0.7\,Ns/m$ | -| $c_a$ | $0.35\,Ns/m$ | -| $g_a$ | $2.7\,N/V$ | -| $g_s$ | $0.53\,V/\mu m$ | +| *Parameter* | *Value* | +|-------------+------------------------------| +| $k_1$ | $0.30\,\text{N}/\mu\text{m}$ | +| $k_e$ | $4.3\,\text{N}/\mu\text{m}$ | +| $k_a$ | $2.15\,\text{N}/\mu\text{m}$ | +| $c_1$ | $18\,\text{Ns/m}$ | +| $c_e$ | $0.7\,\text{Ns/m}$ | +| $c_a$ | $0.35\,\text{Ns/m}$ | +| $g_a$ | $2.7\,\text{N}/V$ | +| $g_s$ | $0.53\,\text{V}/\mu\text{m}$ | #+name: fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof #+caption: Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2DoF model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}) @@ -7641,10 +7641,10 @@ However, the electrical characteristics of the acrshort:apa remain crucial for i Proper consideration must be given to voltage amplifier specifications and force sensor signal conditioning requirements. These aspects will be addressed in the instrumentation chapter. -**** Validation with the Nano-Hexapod +**** Validation with the Active Platform <> -The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with acrshort:apa modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full acrshort:fem implementation. +The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with acrshort:apa modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full acrshort:fem implementation. The dynamics predicted using the flexible body model align well with the design requirements established during the conceptual phase. The dynamics from $\bm{u}$ to $\bm{V}_s$ exhibits the desired alternating pole-zero pattern (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_hac_plant), a critical characteristic for implementing robust decentralized Integral Force Feedback. @@ -7653,12 +7653,12 @@ These findings suggest that the control performance targets established during t Comparative analysis between the high-order acrshort:fem implementation and the simplified 2DoF model (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_plants) demonstrates remarkable agreement in the frequency range of interest. This validates the use of the simplified model for time-domain simulations. -The reduction in model order is substantial: while the acrshort:fem implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete nano-hexapod. +The reduction in model order is substantial: while the acrshort:fem implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete active platform. -These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the nano-hexapod. +These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the active platform. #+name: fig:detail_fem_actuator_fem_vs_perfect_plants -#+caption: Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a nano-hexapod having actuators modelled a 2DoF system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). +#+caption: Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a active platform having actuators modelled a 2DoF system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7682,7 +7682,7 @@ These results validate both the selection of the APA300ML and the effectiveness High-precision position control at the nanometer scale requires systems to be free from friction and backlash, as these nonlinear phenomena severely limit achievable positioning accuracy. This fundamental requirement prevents the use of conventional joints, necessitating instead the implementation of flexible joints that achieve motion through elastic deformation. For Stewart platforms requiring nanometric precision, numerous flexible joint designs have been developed and successfully implemented, as illustrated in Figure\nbsp{}ref:fig:detail_fem_joints_examples. -For design simplicity and component standardization, identical joints are employed at both ends of the nano-hexapod struts. +For design simplicity and component standardization, identical joints are employed at both ends of the active platform struts. #+name: fig:detail_fem_joints_examples #+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical "universal" flexible joint used in\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_wire}) "Thin" flexible joints having "notch curves" are also used\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]]. @@ -7713,20 +7713,20 @@ This section examines how these non-ideal characteristics affect system behavior The analysis of bending and axial stiffness effects enables the establishment of comprehensive specifications for the flexible joints. These specifications guide the development and optimization of a flexible joint design through acrshort:fea (Section\nbsp{}ref:ssec:detail_fem_joint_specs). -The validation process, detailed in Section\nbsp{}ref:ssec:detail_fem_joint_validation, begins with the integration of the joints as "reduced order flexible bodies" in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints. +The validation process, detailed in Section\nbsp{}ref:ssec:detail_fem_joint_validation, begins with the integration of the joints as "reduced order flexible bodies" in the active platform model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints. **** Bending and Torsional Stiffness <> The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]] and can affect system dynamics. -To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of $1\,N/\mu m$) without parallel stiffness to the force sensors. -Flexible joint bending stiffness was varied from 0 (ideal case) to $500\,Nm/\text{rad}$. +To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of $1\,\text{N}/\mu\text{m}$) without parallel stiffness to the force sensors. +Flexible joint bending stiffness was varied from 0 (ideal case) to $500\,\text{Nm}/\text{rad}$. Analysis of the plant dynamics reveals two significant effects. For the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$, bending stiffness increases low-frequency coupling, though this remains small for realistic stiffness values (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_hac_plant). In\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]], it is established that forces remain effectively aligned with the struts when the flexible joint bending stiffness is much small than the actuator stiffness multiplied by the square of the strut length. -For the nano-hexapod, this corresponds to having the bending stiffness much lower than 9000 Nm/rad. +For the active platform, this corresponds to having the bending stiffness much lower than 9000 Nm/rad. This condition is more readily satisfied with the relatively stiff actuators selected, and could be problematic for softer Stewart platforms. For the force sensor plant, bending stiffness introduces complex conjugate zeros at low frequency (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_plant). @@ -7779,10 +7779,10 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes The limited axial stiffness ($k_a$) of flexible joints introduces an additional compliance between the actuation point and the measurement point. As explained in\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapter 6]] and in\nbsp{}[[cite:&rankers98_machin]] (effect called "actuator flexibility"), such intermediate flexibility invariably degrades control performance. -Therefore, determining the minimum acceptable axial stiffness that maintains nano-hexapod performance becomes crucial. +Therefore, determining the minimum acceptable axial stiffness that maintains active platform performance becomes crucial. The analysis incorporates the strut mass (112g per APA300ML) to accurately model internal resonance effects. -A parametric study was conducted by varying the axial stiffness from $1\,N/\mu m$ (matching actuator stiffness) to $1000\,N/\mu m$ (approximating rigid behavior). +A parametric study was conducted by varying the axial stiffness from $1\,\text{N}/\mu\text{m}$ (matching actuator stiffness) to $1000\,\text{N}/\mu\text{m}$ (approximating rigid behavior). The resulting dynamics (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_plants) reveal distinct effects on system dynamics. The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both acrshortpl:frf (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_plant) and root locus analysis (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_locus). @@ -7795,7 +7795,7 @@ First, the system exhibits strong coupling between control channels, making dece Second, control authority diminishes significantly near the resonant frequencies. These effects fundamentally limit achievable control bandwidth, making high axial stiffness essential for system performance. -Based on this analysis, an axial stiffness specification of $100\,N/\mu m$ was established for the nano-hexapod joints. +Based on this analysis, an axial stiffness specification of $100\,\text{N}/\mu\text{m}$ was established for the active platform joints. #+name: fig:detail_fem_joints_axial_stiffness_plants #+caption: Effect of axial stiffness of the flexible joints on the plant dynamics. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_axial_stiffness_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_axial_stiffness_iff_plant}) @@ -7844,16 +7844,16 @@ Based on the dynamic analysis presented in previous sections, quantitative speci #+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc #+attr_latex: :center t :booktabs t :float t -| | *Specification* | *FEM* | -|-------------------------+------------------------+-------| -| Axial Stiffness $k_a$ | $> 100\,N/\mu m$ | 94 | -| Shear Stiffness $k_s$ | $> 1\,N/\mu m$ | 13 | -| Bending Stiffness $k_f$ | $< 100\,Nm/\text{rad}$ | 5 | -| Torsion Stiffness $k_t$ | $< 500\,Nm/\text{rad}$ | 260 | -| Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | +| | *Specification* | *FEM* | +|-------------------------+-------------------------------+-------| +| Axial Stiffness $k_a$ | $> 100\,\text{N}/\mu\text{m}$ | 94 | +| Shear Stiffness $k_s$ | $> 1\,\text{N}/\mu\text{m}$ | 13 | +| Bending Stiffness $k_f$ | $< 100\,\text{Nm}/\text{rad}$ | 5 | +| Torsion Stiffness $k_t$ | $< 500\,\text{Nm}/\text{rad}$ | 260 | +| Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | Among various possible flexible joint architectures, the design shown in Figure\nbsp{}ref:fig:detail_fem_joints_design was selected for three key advantages. -First, the geometry creates coincident $x$ and $y$ rotation axes, ensuring well-defined kinematic behavior, important for the precise definition of the nano-hexapod Jacobian matrix. +First, the geometry creates coincident $x$ and $y$ rotation axes, ensuring well-defined kinematic behavior, important for the precise definition of the active platform Jacobian matrix. Second, the design allows easy tuning of different directional stiffnesses through a limited number of geometric parameters. Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational acrshortpl:dof. @@ -7879,10 +7879,10 @@ The final design, featuring a neck dimension of 0.25mm, achieves mechanical prop #+end_subfigure #+end_figure -**** Validation with the Nano-Hexapod +**** Validation with the Active Platform <> -The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from acrshort:fea. +The designed flexible joint was first validated through integration into the active platform model using reduced-order flexible bodies derived from acrshort:fea. This high-fidelity representation was created by defining two interface frames (Figure\nbsp{}ref:fig:detail_fem_joints_frames) and extracting six additional modes, resulting in reduced-order mass and stiffness matrices of dimension $18 \times 18$. The computed transfer functions from actuator forces to both force sensor measurements ($\bm{f}$ to $\bm{f}_m$) and external metrology ($\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$) demonstrate dynamics consistent with predictions from earlier analyses (Figure\nbsp{}ref:fig:detail_fem_joints_fem_vs_perfect_plants), thereby validating the joint design. @@ -7891,7 +7891,7 @@ The computed transfer functions from actuator forces to both force sensor measur [[file:figs/detail_fem_joints_frames.png]] While this detailed modeling approach provides high accuracy, it results in a significant increase in system model order. -The complete nano-hexapod model incorporates 240 states: 12 for the payload (6 DOF), 12 for the 2DOF struts, and 216 for the flexible joints (18 states for each of the 12 joints). +The complete active platform model incorporates 240 states: 12 for the payload (6 DOF), 12 for the 2DOF struts, and 216 for the flexible joints (18 states for each of the 12 joints). To improve computational efficiency, a low order representation was developed using simplified joint elements with selective compliance DoF. After evaluating various configurations, a compromise was achieved by modeling bottom joints with bending and axial stiffness ($k_f$ and $k_a$), and top joints with bending, torsional, and axial stiffness ($k_f$, $k_t$ and $k_a$). @@ -7899,7 +7899,7 @@ This simplification reduces the total model order to 48 states: 12 for the paylo While additional acrshortpl:dof could potentially capture more dynamic features, the selected configuration preserves essential system characteristics while minimizing computational complexity. #+name: fig:detail_fem_joints_fem_vs_perfect_plants -#+caption: Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a nano-hexapod having bottom joint modelled by bending stiffness $k_f$ and axial stiffness $k_a$ and top joints modelled by bending stiffness $k_f$, torsion stiffness $k_t$ and axial stiffness $k_a$. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}). +#+caption: Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a active platform having bottom joint modelled by bending stiffness $k_f$ and axial stiffness $k_a$ and top joints modelled by bending stiffness $k_f$, torsion stiffness $k_t$ and axial stiffness $k_a$. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7922,7 +7922,7 @@ While additional acrshortpl:dof could potentially capture more dynamic features, :END: <> -In this chapter, the methodology of combining acrlong:fea with multi-body modeling has been demonstrated and validated, proving particularly valuable for the detailed design of nano-hexapod components. +In this chapter, the methodology of combining acrlong:fea with multi-body modeling has been demonstrated and validated, proving particularly valuable for the detailed design of active platform components. The approach was first validated using an amplified piezoelectric actuator, where predicted dynamics showed excellent agreement with experimental measurements for both open and closed-loop behavior. This validation established confidence in the method's ability to accurately predict component behavior within a larger system. @@ -7938,7 +7938,7 @@ Such model reduction, guided by detailed understanding of component behavior, pr <> *** Introduction :ignore: -Three critical elements for the control of parallel manipulators such as the Nano-Hexapod were identified: effective use and combination of multiple sensors, appropriate plant decoupling strategies, and robust controller design for the decoupled system. +Three critical elements for the control of parallel manipulators such as the active platform were identified: effective use and combination of multiple sensors, appropriate plant decoupling strategies, and robust controller design for the decoupled system. During the conceptual design phase of the NASS, pragmatic approaches were implemented for each of these elements. The acrfull:haclac architecture was selected for combining sensors. @@ -8344,13 +8344,13 @@ The inverse magnitudes of the designed weighting functions, which represent the #+begin_minipage #+attr_latex: :environment tabularx :width 0.7\linewidth :placement [b] :align ccc #+attr_latex: :booktabs t :float nil :font \footnotesize\sf -| Parameter | $W_1(s)$ | $W_2(s)$ | -|-------------+------------------+------------------| -| $G_0$ | $0.1$ | $1000$ | -| $G_{\infty}$ | $1000$ | $0.1$ | -| $\omega_c$ | $2 \pi \cdot 10$ | $2 \pi \cdot 10$ | -| $G_c$ | $0.45$ | $0.45$ | -| $n$ | $2$ | $3$ | +| Parameter | $W_1(s)$ | $W_2(s)$ | +|--------------+------------------+------------------| +| $G_0$ | $0.1$ | $1000$ | +| $G_{\infty}$ | $1000$ | $0.1$ | +| $\omega_c$ | $2 \pi \cdot 10$ | $2 \pi \cdot 10$ | +| $G_c$ | $0.45$ | $0.45$ | +| $n$ | $2$ | $3$ | #+latex: \captionof{table}{\label{tab:detail_control_sensor_weights_params}Parameters for \(W_1(s)\) and \(W_2(s)\)} #+end_minipage \hfill @@ -8545,14 +8545,14 @@ Two reference frames are defined within this model: frame $\{M\}$ with origin $O #+latex: \centering #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align cXc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| | *Description* | *Value* | -|-------+-----------------------+-------------------| -| $l_a$ | | $0.5\,m$ | -| $h_a$ | | $0.2\,m$ | -| $k$ | Actuator stiffness | $10\,N/\mu m$ | -| $c$ | Actuator damping | $200\,Ns/m$ | -| $m$ | Payload mass | $40\,\text{kg}$ | -| $I$ | Payload $R_z$ inertia | $5\,\text{kg}m^2$ | +| | *Description* | *Value* | +|-------+-----------------------+----------------------------| +| $l_a$ | | $0.5\,\text{m}$ | +| $h_a$ | | $0.2\,\text{m}$ | +| $k$ | Actuator stiffness | $10\,\text{N}/\mu\text{m}$ | +| $c$ | Actuator damping | $200\,\text{Ns/m}$ | +| $m$ | Payload mass | $40\,\text{kg}$ | +| $I$ | Payload $R_z$ inertia | $5\,\text{kgm}^2$ | #+latex: \captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters} #+end_minipage @@ -9388,7 +9388,7 @@ The Bode plot of the controller multiplied by the complementary low-pass filter, The loop gain reveals several important characteristics: - The presence of two integrators at low frequencies, enabling accurate tracking of ramp inputs - A notch at the plant resonance frequency (arising from the plant inverse) -- A lead component near the control bandwidth of approximately 20 Hz, enhancing stability margins +- A lead component near the control bandwidth of approximately $20\,\text{Hz}$, enhancing stability margins ***** Robustness and Performance analysis @@ -9522,10 +9522,10 @@ In order to derive specifications in terms of noise spectral density for each in The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met. While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications. -Based on this analysis, the obtained maximum noise levels are as follows: acrshort:dac maximum output noise acrshort:asd is established at $14\,\mu V/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise acrshort:asd at $280\,\mu V/\sqrt{\text{Hz}}$, and acrshort:adc maximum measurement noise acrshort:asd at $11\,\mu V/\sqrt{\text{Hz}}$. +Based on this analysis, the obtained maximum noise levels are as follows: acrshort:dac maximum output noise acrshort:asd is established at $14\,\mu\text{V}/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise acrshort:asd at $280\,\mu\text{V}/\sqrt{\text{Hz}}$, and acrshort:adc maximum measurement noise acrshort:asd at $11\,\mu\text{V}/\sqrt{\text{Hz}}$. In terms of RMS noise, these translate to less than $1\,\text{mV RMS}$ for the acrshort:dac, less than $20\,\text{mV RMS}$ for the voltage amplifier, and less than $0.8\,\text{mV RMS}$ for the acrshort:adc. -If the Amplitude Spectral Density of the noise of the acrshort:adc, acrshort:dac, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS. +If the Amplitude Spectral Density of the noise of the acrshort:adc, acrshort:dac, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below $15\,\text{nm RMS}$. *** Choice of Instrumentation <> @@ -9533,8 +9533,8 @@ If the Amplitude Spectral Density of the noise of the acrshort:adc, acrshort:dac ***** Introduction :ignore: Several characteristics of piezoelectric voltage amplifiers must be considered for this application. -To take advantage of the full stroke of the piezoelectric actuator, the voltage output should range between $-20$ and $150\,V$. -The amplifier should accept an analog input voltage, preferably in the range of $-10$ to $10\,V$, as this is standard for most acrshortpl:dac. +To take advantage of the full stroke of the piezoelectric actuator, the voltage output should range between $-20$ and $150\,\text{V}$. +The amplifier should accept an analog input voltage, preferably in the range of $-10$ to $10\,\text{V}$, as this is standard for most acrshortpl:dac. ***** Small signal Bandwidth and Output Impedance @@ -9554,7 +9554,7 @@ When combined with the piezoelectric load (represented as a capacitance $C_p$), [[file:figs/detail_instrumentation_amp_output_impedance.png]] Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases. -For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to $C_p = 8.8\,\mu F$. +For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to $C_p = 8.8\,\mu\text{F}$. If a small signal bandwidth of $f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}$ is desired, the voltage amplifier output impedance should be less than $R_0 = 3.6\,\Omega$. ***** Large signal Bandwidth @@ -9562,16 +9562,16 @@ If a small signal bandwidth of $f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}$ is Large signal bandwidth relates to the maximum output capabilities of the amplifier in terms of amplitude as a function of frequency. Since the primary function of the NASS is position stabilization rather than scanning, this specification is less critical than the small signal bandwidth. -However, considering potential scanning capabilities, a worst-case scenario of a constant velocity scan (triangular reference signal) with a repetition rate of $f_r = 100\,\text{Hz}$ using the full voltage range of the piezoelectric actuator ($V_{pp} = 170\,V$) is considered. +However, considering potential scanning capabilities, a worst-case scenario of a constant velocity scan (triangular reference signal) with a repetition rate of $f_r = 100\,\text{Hz}$ using the full voltage range of the piezoelectric actuator ($V_{pp} = 170\,\text{V}$) is considered. There are two limiting factors for large signal bandwidth that should be evaluated: -1. Slew rate, which should exceed $2 \cdot V_{pp} \cdot f_r = 34\,V/ms$. +1. Slew rate, which should exceed $2 \cdot V_{pp} \cdot f_r = 34\,\text{V/ms}$. This requirement is typically easily met by commercial voltage amplifiers. 2. Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies. To achieve high voltage at high frequency, the amplifier must therefore provide substantial current. - The maximum required current can be calculated as $I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,A$. + The maximum required current can be calculated as $I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,\text{A}$. -Therefore, ideally, a voltage amplifier capable of providing $0.3\,A$ of current would be interesting for scanning applications. +Therefore, ideally, a voltage amplifier capable of providing $0.3\,\text{A}$ of current would be interesting for scanning applications. ***** Output voltage noise @@ -9580,7 +9580,7 @@ As established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_bud It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure\nbsp{}ref:fig:detail_instrumentation_amp_output_impedance. Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement\nbsp{}[[cite:&spengen20_high_voltag_amplif]]. -For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\mu F$ and a bandwidth exceeding $5\,\text{kHz}$. +For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\mu\text{F}$ and a bandwidth exceeding $5\,\text{kHz}$. ***** Choice of voltage amplifier @@ -9601,19 +9601,19 @@ The PD200 from PiezoDrive was ultimately selected because it meets all the requi #+caption: Specifications for the Voltage amplifier and considered commercial products #+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcccc #+attr_latex: :center t :booktabs t :float t -| *Specifications* | PD200 | WMA-200 | LA75B | E-505 | -| | PiezoDrive | Falco | Cedrat | PI | -|--------------------------------------+--------------------+--------------------+--------------+------------| -| Input Voltage Range: $\pm 10\,V$ | $\pm 10\,V$ | $\pm8.75\,V$ | $-1/7.5\,V$ | $-2/12\,V$ | -| Output Voltage Range: $-20/150\,V$ | $-50/150\,V$ | $\pm 175\,V$ | $-20/150\,V$ | -30/130 | -| Gain $>15$ | 20 | 20 | 20 | 10 | -| Output Current $> 300\,mA$ | $900\,mA$ | $150\,mA$ | $360\,mA$ | $215\,mA$ | -| Slew Rate $> 34\,V/ms$ | $150\,V/\mu s$ | $80\,V/\mu s$ | n/a | n/a | -| Output noise $< 20\,mV\ \text{RMS}$ | $0.7\,mV$ | $0.05\,mV$ | $3.4\,mV$ | $0.6\,mV$ | -| (10uF load) | ($10\,\mu F$ load) | ($10\,\mu F$ load) | (n/a) | (n/a) | -| Small Signal Bandwidth $> 5\,kHz$ | $6.4\,kHz$ | $300\,Hz$ | $30\,kHz$ | n/a | -| ($10\,\mu F$ load) | ($10\,\mu F$ load) | ($10\,\mu F$ load) | (unloaded) | (n/a) | -| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a | +| *Specifications* | PD200 | WMA-200 | LA75B | E-505 | +| | PiezoDrive | Falco | Cedrat | PI | +|--------------------------------------------+-----------------------------+----------------------------+---------------------+---------------------| +| Input Voltage Range: $\pm 10\,\text{V}$ | $\pm 10\,\text{V}$ | $\pm8.75\,\text{V}$ | $-1/7.5\,\text{V}$ | $-2/12\,\text{V}$ | +| Output Voltage Range: $-20/150\,\text{V}$ | $-50/150\,\text{V}$ | $\pm 175\,\text{V}$ | $-20/150\,\text{V}$ | $-30/130\,\text{V}$ | +| Gain $>15$ | 20 | 20 | 20 | 10 | +| Output Current $> 300\,\text{mA}$ | $900\,\text{mA}$ | $150\,\text{mA}$ | $360\,\text{mA}$ | $215\,\text{mA}$ | +| Slew Rate $> 34\,\text{V/ms}$ | $150\,\text{V}/\mu\text{s}$ | $80\,\text{V}/\mu\text{s}$ | n/a | n/a | +| Output noise $< 20\,\text{mV RMS}$ | $0.7\,\text{mV}$ | $0.05\,\text{mV}$ | $3.4\,\text{mV}$ | $0.6\,\text{mV}$ | +| (10uF load) | ($10\,\mu\text{F}$ load) | ($10\,\mu\text{F}$ load) | (n/a) | (n/a) | +| Small Signal Bandwidth $> 5\,\text{kHz}$ | $6.4\,\text{kHz}$ | $300\,\text{Hz}$ | $30\,\text{kHz}$ | n/a | +| ($10\,\mu\text{F}$ load) | ($10\,\mu\text{F}$ load) | ($10\,\mu\text{F}$ load) | (unloaded) | (n/a) | +| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a | **** ADC and DAC ***** Introduction :ignore: @@ -9632,7 +9632,7 @@ Based on this requirement, priority was given to acrshort:adc and acrshort:dac c Several requirements that may initially appear similar are actually distinct in nature. First, the /sampling frequency/ defines the interval between two sampled points and determines the Nyquist frequency. -Then, the /bandwidth/ specifies the maximum frequency of a measured signal (typically defined as the -3dB point) and is often limited by implemented anti-aliasing filters. +Then, the /bandwidth/ specifies the maximum frequency of a measured signal (typically defined as the $-3\,\text{dB}$ point) and is often limited by implemented anti-aliasing filters. Finally, /delay/ (or /latency/) refers to the time interval between the analog signal at the input of the acrshort:adc and the digital information transferred to the control system. Sigma-Delta acrshortpl:adc can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency. @@ -9680,33 +9680,33 @@ From a specified noise amplitude spectral density $\Gamma_{\text{max}}$, the min n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right) \end{equation} -With a sampling frequency $F_s = 10\,\text{kHz}$, an input range $\Delta V = 20\,V$ and a maximum allowed acrshort:asd $\Gamma_{\text{max}} = 11\,\mu V/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 12.4$, which is readily achievable with commercial acrshortpl:adc. +With a sampling frequency $F_s = 10\,\text{kHz}$, an input range $\Delta V = 20\,\text{V}$ and a maximum allowed acrshort:asd $\Gamma_{\text{max}} = 11\,\mu\text{V}/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 12.4$, which is readily achievable with commercial acrshortpl:adc. ***** DAC Output voltage noise -Similar to the acrshort:adc requirements, the acrshort:dac output voltage noise acrshort:asd should not exceed $14\,\mu V/\sqrt{\text{Hz}}$. -This specification corresponds to a $\pm 10\,V$ acrshort:dac with 13-bit resolution, which is easily attainable with current technology. +Similar to the acrshort:adc requirements, the acrshort:dac output voltage noise acrshort:asd should not exceed $14\,\mu\text{V}/\sqrt{\text{Hz}}$. +This specification corresponds to a $\pm 10\,\text{V}$ acrshort:dac with 13-bit resolution, which is easily attainable with current technology. ***** Choice of the ADC and DAC Board Based on the preceding analysis, the selection of suitable acrshort:adc and acrshort:dac components is straightforward. For optimal synchronicity, a Speedgoat-integrated solution was chosen. -The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, $\pm 10\,V$ range, maximum sampling rate of 200kSPS (acrlong:sps), simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity. -The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, $\pm 10\,V$ range, conversion time of $10\,\mu s$, and simultaneous update capability. +The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, $\pm 10\,\text{V}$ range, maximum sampling rate of 200kSPS (acrlong:sps), simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity. +The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, $\pm 10\,\text{V}$ range, conversion time of $10\,\mu s$, and simultaneous update capability. Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements. This will be experimentally verified in Section\nbsp{}ref:sec:detail_instrumentation_characterization. **** Relative Displacement Sensors -The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below $6\,\text{nm RMS}$ (derived from the $15\,\text{nm RMS}$ vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding $100\,\mu m$. +The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below $6\,\text{nm RMS}$ (derived from the $15\,\text{nm RMS}$ vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding $100\,\mu\text{m}$. Several sensor technologies are capable of meeting these requirements\nbsp{}[[cite:&fleming13_review_nanom_resol_posit_sensor]]. These include optical encoders (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_encoder), capacitive sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_capacitive), and eddy current sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_eddy_current), each with their own advantages and implementation considerations. #+name: fig:detail_instrumentation_sensor_examples -#+caption: Relative motion sensors considered for measuring the nano-hexapod strut motion +#+caption: Relative motion sensors considered for measuring the active platform strut motion #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_sensor_encoder}Optical Linear Encoder} @@ -9761,14 +9761,14 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic #+caption: Specifications for the relative displacement sensors and considered commercial products #+attr_latex: :environment tabularx :width 0.65\linewidth :align Xccc #+attr_latex: :center t :booktabs t :float t -| *Specifications* | Renishaw Vionic | LION CPL190 | Cedrat ECP500 | -|-----------------------------+---------------------+-------------+---------------| -| Technology | Digital Encoder | Capacitive | Eddy Current | -| Bandwidth $> 5\,\text{kHz}$ | $> 500\,\text{kHz}$ | 10kHz | 20kHz | -| Noise $< 6\,nm\,\text{RMS}$ | 1.6 nm rms | 4 nm rms | 15 nm rms | -| Range $> 100\,\mu m$ | Ruler length | 250 um | 500um | -| In line measurement | | $\times$ | $\times$ | -| Digital Output | $\times$ | | | +| *Specifications* | Renishaw Vionic | LION CPL190 | Cedrat ECP500 | +|-----------------------------+----------------------+---------------------+---------------------| +| Technology | Digital Encoder | Capacitive | Eddy Current | +| Bandwidth $> 5\,\text{kHz}$ | $> 500\,\text{kHz}$ | $10\,\text{kHz}$ | $20\,\text{kHz}$ | +| Noise $< 6\,\text{nm RMS}$ | $1.6\,\text{nm RMS}$ | $4\,\text{nm RMS}$ | $15\,\text{nm RMS}$ | +| Range $> 100\,\mu\text{m}$ | Ruler length | $250\,\mu \text{m}$ | $500\,\mu \text{m}$ | +| In line measurement | | $\times$ | $\times$ | +| Digital Output | $\times$ | | | *** Characterization of Instrumentation <> @@ -9777,7 +9777,7 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic The measurement of acrshort:adc noise was performed by short-circuiting its input with a $50\,\Omega$ resistor and recording the digital values at a sampling rate of $10\,\text{kHz}$. The amplitude spectral density of the recorded values was computed and is presented in Figure\nbsp{}ref:fig:detail_instrumentation_adc_noise_measured. -The acrshort:adc noise exhibits characteristics of white noise with an amplitude spectral density of $5.6\,\mu V/\sqrt{\text{Hz}}$ (equivalent to $0.4\,\text{mV RMS}$), which satisfies the established specifications. +The acrshort:adc noise exhibits characteristics of white noise with an amplitude spectral density of $5.6\,\mu\text{V}/\sqrt{\text{Hz}}$ (equivalent to $0.4\,\text{mV RMS}$), which satisfies the established specifications. All acrshort:adc channels demonstrated similar performance, so only one channel's noise profile is shown. If necessary, oversampling can be applied to further reduce the noise\nbsp{}[[cite:&lab13_improv_adc]]. @@ -9800,18 +9800,18 @@ The voltage amplifier employed in this setup has a gain of 20. #+caption: Schematic of the setup to validate the use of the ADC for reading the force sensor volage [[file:figs/detail_instrumentation_force_sensor_adc_setup.png]] -Step signals with an amplitude of $1\,V$ were generated using the acrshort:dac, and the acrshort:adc signal was recorded. +Step signals with an amplitude of $1\,\text{V}$ were generated using the acrshort:dac, and the acrshort:adc signal was recorded. The excitation signal (steps) and the measured voltage across the sensor stack are displayed in Figure\nbsp{}ref:fig:detail_instrumentation_step_response_force_sensor. -Two notable observations were made: an offset voltage of $2.26\,V$ was present, and the measured voltage exhibited an exponential decay response to the step input. +Two notable observations were made: an offset voltage of $2.26\,\text{V}$ was present, and the measured voltage exhibited an exponential decay response to the step input. These phenomena can be explained by examining the electrical schematic shown in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc, where the acrshort:adc has an input impedance $R_i$ and an input bias current $i_n$. The input impedance $R_i$ of the acrshort:adc, in combination with the capacitance $C_p$ of the piezoelectric stack sensor, forms an RC circuit with a time constant $\tau = R_i C_p$. The charge generated by the piezoelectric effect across the stack's capacitance gradually discharges into the input resistor of the acrshort:adc. Consequently, the transfer function from the generated voltage $V_p$ to the measured voltage $V_{\text{ADC}}$ is a first-order high-pass filter with the time constant $\tau$. -An exponential curve was fitted to the experimental data, yielding a time constant $\tau = 6.5\,s$. -With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu F$, the internal impedance of the Speedgoat acrshort:adc was calculated as $R_i = \tau/C_p = 1.5\,M\Omega$, which closely aligns with the specified value of $1\,M\Omega$ found in the datasheet. +An exponential curve was fitted to the experimental data, yielding a time constant $\tau = 6.5\,\text{s}$. +With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu\text{F}$, the internal impedance of the Speedgoat acrshort:adc was calculated as $R_i = \tau/C_p = 1.5\,M\Omega$, which closely aligns with the specified value of $1\,M\Omega$ found in the datasheet. #+name: fig:detail_instrumentation_force_sensor #+caption: Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from\nbsp{}[[cite:&reza06_piezoel_trans_vibrat_contr_dampin]]. Measured voltage $V_s$ while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}). @@ -9840,7 +9840,7 @@ This modification produces two beneficial effects: a reduction of input voltage To validate this approach, a resistor $R_p \approx 82\,k\Omega$ was added in parallel with the force sensor as shown in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_R. After incorporating this resistor, the same step response tests were performed, with results displayed in Figure\nbsp{}ref:fig:detail_instrumentation_step_response_force_sensor_R. -The measurements confirmed the expected improvements, with a substantially reduced offset voltage ($V_{\text{off}} = 0.15\,V$) and a much faster time constant ($\tau = 0.45\,s$). +The measurements confirmed the expected improvements, with a substantially reduced offset voltage ($V_{\text{off}} = 0.15\,\text{V}$) and a much faster time constant ($\tau = 0.45\,\text{s}$). These results validate both the model of the acrshort:adc and the effectiveness of the added parallel resistor as a solution. #+name: fig:detail_instrumentation_force_sensor_R @@ -9863,12 +9863,12 @@ These results validate both the model of the acrshort:adc and the effectiveness **** Instrumentation Amplifier -Because the acrshort:adc noise may be too low to measure the noise of other instruments (anything below $5.6\,\mu V/\sqrt{\text{Hz}}$ cannot be distinguished from the noise of the acrshort:adc itself), a low noise instrumentation amplifier was employed. -A Femto DLPVA-101-B-S amplifier with adjustable gains from 20dB up to 80dB was selected for this purpose. +Because the acrshort:adc noise may be too low to measure the noise of other instruments (anything below $5.6\,\mu\text{V}/\sqrt{\text{Hz}}$ cannot be distinguished from the noise of the acrshort:adc itself), a low noise instrumentation amplifier was employed. +A Femto DLPVA-101-B-S amplifier with adjustable gains from $20\,text{dB}$ up to $80\,text{dB}$ was selected for this purpose. The first step was to characterize the input[fn:detail_instrumentation_1] noise of the amplifier. This was accomplished by short-circuiting its input with a $50\,\Omega$ resistor and measuring the output voltage with the acrshort:adc (Figure\nbsp{}ref:fig:detail_instrumentation_femto_meas_setup). -The maximum amplifier gain of 80dB (equivalent to 10000) was used for this measurement. +The maximum amplifier gain of $80\,text{dB}$ (equivalent to 10000) was used for this measurement. The measured voltage $n$ was then divided by 10000 to determine the equivalent noise at the input of the voltage amplifier $n_a$. In this configuration, the noise contribution from the acrshort:adc $q_{ad}$ is rendered negligible due to the high gain employed. @@ -9898,7 +9898,7 @@ The acrshort:dac was configured to output a constant voltage (zero in this case) The Amplitude Spectral Density $\Gamma_{n_{da}}(\omega)$ of the measured signal was computed, and verification was performed to confirm that the contributions of acrshort:adc noise and amplifier noise were negligible in the measurement. The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure\nbsp{}ref:fig:detail_instrumentation_dac_output_noise. -The noise profile is predominantly white with an acrshort:asd of $0.6\,\mu V/\sqrt{\text{Hz}}$. +The noise profile is predominantly white with an acrshort:asd of $0.6\,\mu\text{V}/\sqrt{\text{Hz}}$. Minor $50\,\text{Hz}$ noise is present, along with some low frequency $1/f$ noise, but these are not expected to pose issues as they are well within specifications. It should be noted that all acrshort:dac channels demonstrated similar performance, so only one channel measurement is presented. @@ -9965,11 +9965,11 @@ While the exact cause of these peaks is not fully understood, their amplitudes r The small signal dynamics of all six PD200 amplifiers were characterized through acrshort:frf measurements. -A logarithmic sweep sine excitation voltage was generated using the Speedgoat acrshort:dac with an amplitude of $0.1\,V$, spanning frequencies from $1\,\text{Hz}$ to $5\,\text{kHz}$. +A logarithmic sweep sine excitation voltage was generated using the Speedgoat acrshort:dac with an amplitude of $0.1\,\text{V}$, spanning frequencies from $1\,\text{Hz}$ to $5\,\text{kHz}$. The output voltage of the PD200 amplifier was measured via the monitor voltage output of the amplifier, while the input voltage (generated by the acrshort:dac) was measured with a separate acrshort:adc channel of the Speedgoat system. This measurement approach eliminates the influence of ADC-DAC-related time delays in the results. -All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to 500Hz, well within the specified requirements. +All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to $500\,\text{Hz}$, well within the specified requirements. The identified dynamics shown in Figure\nbsp{}ref:fig:detail_instrumentation_pd200_tf can be accurately modeled as either a first-order low-pass filter or as a simple constant gain. @@ -10025,7 +10025,7 @@ This confirms that the selected instrumentation, with its measured noise charact This section has presented a comprehensive approach to the selection and characterization of instrumentation for the nano active stabilization system. The multi-body model created earlier served as a key tool for embedding instrumentation components and their associated noise sources within the system analysis. -From the most stringent requirement (i.e. the specification on vertical sample motion limited to 15 nm RMS), detailed specifications for each noise source were methodically derived through dynamic error budgeting. +From the most stringent requirement (i.e. the specification on vertical sample motion limited to $15\,\text{nm RMS}$), detailed specifications for each noise source were methodically derived through dynamic error budgeting. Based on these specifications, appropriate instrumentation components were selected for the system. The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate noise budgeting, such as amplitude spectral densities under specific load conditions. @@ -10034,7 +10034,7 @@ Despite these challenges, suitable components were identified that theoretically The selected instrumentation (including the IO131 ADC/DAC from Speedgoat, PD200 piezoelectric voltage amplifiers from PiezoDrive, and Vionic linear encoders from Renishaw) was procured and thoroughly characterized. Initial measurements of the acrshort:adc system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit. -All components were found to meet or exceed their respective specifications. The acrshort:adc demonstrated noise levels of $5.6\,\mu V/\sqrt{\text{Hz}}$ (versus the $11\,\mu V/\sqrt{\text{Hz}}$ specification), the acrshort:dac showed $0.6\,\mu V/\sqrt{\text{Hz}}$ (versus $14\,\mu V/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $280\,\mu V/\sqrt{\text{Hz}}$ limit, and the encoders achieved $1\,\text{nm RMS}$ noise (versus the $6\,\text{nm RMS}$ specification). +All components were found to meet or exceed their respective specifications. The acrshort:adc demonstrated noise levels of $5.6\,\mu\text{V}/\sqrt{\text{Hz}}$ (versus the $11\,\mu\text{V}/\sqrt{\text{Hz}}$ specification), the acrshort:dac showed $0.6\,\mu\text{V}/\sqrt{\text{Hz}}$ (versus $14\,\mu\text{V}/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $280\,\mu\text{V}/\sqrt{\text{Hz}}$ limit, and the encoders achieved $1\,\text{nm RMS}$ noise (versus the $6\,\text{nm RMS}$ specification). Finally, the measured noise characteristics of all instrumentation components were included into the multi-body model to predict the actual system performance. The combined effect of all noise sources was estimated to induce vertical sample vibrations of only $1.5\,\text{nm RMS}$, which is substantially below the $15\,\text{nm RMS}$ requirement. @@ -10044,7 +10044,7 @@ This rigorous methodology spanning requirement formulation, component selection, <> *** Introduction :ignore: -The detailed mechanical design of the active platform, depicted in Figure\nbsp{}ref:fig:detail_design_nano_hexapod_elements, is presented in this section. +The detailed mechanical design of the active platform (also referred to as the "nano-hexapod"), depicted in Figure\nbsp{}ref:fig:detail_design_nano_hexapod_elements, is presented in this section. Several primary objectives guided the mechanical design. First, to ensure a well-defined Jacobian matrix used in the control architecture, accurate positioning of the top flexible joint rotation points and correct orientation of the struts were required. Secondly, space constraints necessitated that the entire platform fit within a cylinder with a radius of $120\,\text{mm}$ and a height of $95\,\text{mm}$. @@ -10053,7 +10053,7 @@ This objective implies that the frequencies of (un-modelled) flexible modes pote Finally, considerations for ease of mounting, alignment, and maintenance were incorporated, specifically ensuring that struts could be easily replaced in the event of failure. #+name: fig:detail_design_nano_hexapod_elements -#+caption: Obtained mechanical design of the Active platform, the "nano-hexapod" +#+caption: Obtained mechanical design of the Active platform, called the "nano-hexapod" #+attr_latex: :width 0.95\linewidth [[file:figs/detail_design_nano_hexapod_elements.png]] @@ -10125,7 +10125,7 @@ These parts serve to fix the encoder head and the associated scale (ruler) to th ***** Plates -The design of the top and bottom plates of the active platform was governed by two main requirements: maximizing the frequency of flexible modes and ensuring accurate positioning of the top flexible joints and well-defined orientation of the struts. +The design of the top and bottom plates of the nano-hexapod was governed by two main requirements: maximizing the frequency of flexible modes and ensuring accurate positioning of the top flexible joints and well-defined orientation of the struts. To maximize the natural frequencies associated with plate flexibility, a network of reinforcing ribs was incorporated into the design, as shown for the top plate in Figure\nbsp{}ref:fig:detail_design_top_plate. Although topology optimization methods were considered, the implemented ribbed design was found to provide sufficiently high natural frequencies for the flexible modes. @@ -10137,7 +10137,7 @@ Although topology optimization methods were considered, the implemented ribbed d The interfaces for the joints on the plates incorporate V-grooves (red planes in Figure\nbsp{}ref:fig:detail_design_top_plate). The cylindrical portion of each flexible joint is constrained within its corresponding V-groove through two distinct line contacts, illustrated in Figure\nbsp{}ref:fig:detail_design_fixation_flexible_joints. These grooves consequently serve to define the nominal orientation of the struts. -High machining accuracy for these features is essential to ensure that the flexible joints are in their neutral, unstressed state when the active platform is assembled. +High machining accuracy for these features is essential to ensure that the flexible joints are in their neutral, unstressed state when the nano-hexapod is assembled. #+name: fig:detail_design_fixation_flexible_joints_platform #+caption: Fixation of the flexible points to the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joint is in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joint is just oriented (\subref{fig:detail_design_location_bot_flex}). @@ -10172,12 +10172,12 @@ This characteristic is expected to permit repeated assembly and disassembly of t ***** Finite Element Analysis -A acrfull:fea of the complete active platform assembly was performed to identify modes that could potentially affect performance. +A acrfull:fea of the complete nano-hexapod assembly was performed to identify modes that could potentially affect performance. The analysis revealed that the first six modes correspond to "suspension" modes, where the top plate effectively moves as a rigid body, and motion primarily involves axial displacement of the six struts (an example is shown in Figure\nbsp{}ref:fig:detail_design_fem_rigid_body_mode). Following these suspension modes, numerous "local" modes associated with the struts themselves were observed in the frequency range between $205\,\text{Hz}$ and $420\,\text{Hz}$. One such mode is represented in Figure\nbsp{}ref:fig:detail_design_fem_strut_mode. Although these modes do not appear to induce significant motion of the top platform, they do cause relative displacement between the encoder components (head and scale) mounted on the strut. -Consequently, such modes could potentially degrade control performance if the active platform's position is regulated using these encoder measurements. +Consequently, such modes could potentially degrade control performance if the nano-hexapod's position is regulated using these encoder measurements. The extent to which these modes might be detrimental is difficult to establish at this stage, as it depends on whether they are significantly excited by the acrshort:apa actuation and their sensitivity to strut alignment. Finally, the FEA indicated that flexible modes of the top plate itself begin to appear at frequencies above $650\,\text{Hz}$, with the first such mode shown in Figure\nbsp{}ref:fig:detail_design_fem_plate_mode. @@ -10232,7 +10232,7 @@ Dedicated supports, machined from aluminum, were designed for this purpose. It was verified through FEA that the natural modes of these supports occur at frequencies sufficiently high (first mode estimated at $1120\,\text{Hz}$) to not be problematic for control. Precise positioning of these encoder supports is achieved through machined pockets in both the top and bottom plates, visible in Figure\nbsp{}ref:fig:detail_design_top_plate (indicated in green). Although the encoders in this arrangement are aligned parallel to the nominal strut axes, they no longer measure the exact relative displacement along the strut between the flexible joint centers. -This geometric discrepancy implies that if the relative motion control of the active platform is based directly on these encoder readings, the kinematic calculations may be slightly inaccurate, potentially affecting the overall positioning accuracy of the platform. +This geometric discrepancy implies that if the relative motion control of the nano-hexapod is based directly on these encoder readings, the kinematic calculations may be slightly inaccurate, potentially affecting the overall positioning accuracy of the platform. *** Multi-Body Model <> @@ -10286,7 +10286,7 @@ Therefore, a more sophisticated model of the optical encoder was necessary. The optical encoders operate based on the interaction between an encoder head and a graduated scale or ruler. The optical encoder head contains a light source that illuminates the ruler. A reference frame $\{E\}$ fixed to the scale, represents the the light position on the scale, as illustrated in Figure\nbsp{}ref:fig:detail_design_simscape_encoder_model. -The ruler features a precise grating pattern (in this case, with a $20\,\mu m$ pitch), and its position is associated with the reference frame $\{R\}$. +The ruler features a precise grating pattern (in this case, with a $20\,\mu\text{m}$ pitch), and its position is associated with the reference frame $\{R\}$. The displacement measured by the encoder corresponds to the relative position of the encoder frame $\{E\}$ (specifically, the point where the light interacts with the scale) with respect to the ruler frame $\{R\}$, projected along the measurement direction defined by the scale. An important consequence of this measurement principle is that a relative rotation between the encoder head and the ruler, as depicted conceptually in Figure\nbsp{}ref:fig:detail_design_simscape_encoder_disp, can induce a measured displacement. @@ -10311,7 +10311,7 @@ An important consequence of this measurement principle is that a relative rotati ***** Validation of the designed active platform -The refined multi-body model of the active platform was integrated into the multi-body micro-station model. +The refined multi-body model of the nano-hexapod was integrated into the multi-body micro-station model. Dynamical analysis was performed, confirming that the platform's behavior closely approximates the dynamics of the "idealized" model used during the conceptual design phase. Consequently, closed-loop performance simulations replicating tomography experiments yielded metrics highly comparable to those previously predicted (as presented in Section\nbsp{}ref:ssec:nass_hac_tomography). Given this similarity and because analogous simulations are conducted and detailed during the experimental validation phase (Section\nbsp{}ref:sec:test_id31_hac), these specific results are not reiterated here. @@ -10443,7 +10443,7 @@ The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatnes #+latex: \centering #+attr_latex: :environment tabularx :width 0.5\linewidth :align Xc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| | *Flatness* $[\mu m]$ | +| | *Flatness* $[\mu\text{m}]$ | |-------+----------------------| | APA 1 | 8.9 | | APA 2 | 3.1 | @@ -10458,16 +10458,16 @@ The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatnes **** Electrical Measurements <> -From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu F$ and $26\,\mu F$ with a nominal capacitance of $20\,\mu F$. +From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu\text{F}$ and $26\,\mu\text{F}$ with a nominal capacitance of $20\,\mu\text{F}$. The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter[fn:test_apa_1] shown in Figure\nbsp{}ref:fig:test_apa_lcr_meter. The two stacks used as the actuator and the stack used as the force sensor were measured separately. -The measured capacitance values are summarized in Table\nbsp{}ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu F$. +The measured capacitance values are summarized in Table\nbsp{}ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu\text{F}$. However, the measured capacitance of the stacks of "APA 3" is only half of the expected capacitance. This may indicate a manufacturing defect. The measured capacitance is found to be lower than the specified value. -This may be because the manufacturer measures the capacitance with large signals ($-20\,V$ to $150\,V$), whereas it was here measured with small signals\nbsp{}[[cite:&wehrsdorfer95_large_signal_measur_piezoel_stack]]. +This may be because the manufacturer measures the capacitance with large signals ($-20\,\text{V}$ to $150\,\text{V}$), whereas it was here measured with small signals\nbsp{}[[cite:&wehrsdorfer95_large_signal_measur_piezoel_stack]]. #+attr_latex: :options [b]{0.48\textwidth} #+begin_minipage @@ -10491,7 +10491,7 @@ This may be because the manufacturer measures the capacitance with large signals | APA 5 | 4.90 | 9.66 | | APA 6 | 4.99 | 9.91 | | APA 7 | 4.85 | 9.85 | -#+latex: \captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu F$} +#+latex: \captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu\text{F}$} #+end_minipage **** Stroke and Hysteresis Measurement @@ -10499,7 +10499,7 @@ This may be because the manufacturer measures the capacitance with large signals To compare the stroke of the APA300ML with the datasheet specifications, one side of the acrshort:apa is fixed to the granite, and a displacement probe[fn:test_apa_2] is located on the other side as shown in Figure\nbsp{}ref:fig:test_apa_stroke_bench. -The voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:test_apa_12] and a voltage amplifier[fn:test_apa_13]. +The voltage across the two actuator stacks is varied from $-20\,\text{V}$ to $150\,\text{V}$ using a DAC[fn:test_apa_12] and a voltage amplifier[fn:test_apa_13]. Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure\nbsp{}ref:fig:test_apa_stroke_voltage). #+name: fig:test_apa_stroke_bench @@ -10509,9 +10509,9 @@ Note that the voltage is slowly varied as the displacement probe has a very low The measured acrshort:apa displacement is shown as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis. Typical hysteresis curves for piezoelectric stack actuators can be observed. -The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks. -This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu m$, therefore $\approx 200\,\mu m$ if only two stacks are used). -For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu m$. +The measured stroke is approximately $250\,\mu\text{m}$ when using only two of the three stacks. +This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu\text{m}$, therefore $\approx 200\,\mu\text{m}$ if only two stacks are used). +For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu\text{m}$. It is clear from Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared with the other units. This confirms the abnormal electrical measurements made in Section\nbsp{}ref:ssec:test_apa_electrical_measurements. @@ -10550,19 +10550,19 @@ The flexible modes for the same condition (i.e. one mechanical interface of the #+caption: First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode (268Hz)} +#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode ($268\,\text{Hz}$)} #+attr_latex: :options {0.35\textwidth} #+begin_subfigure #+attr_latex: :height 4.3cm [[file:figs/test_apa_mode_shapes_1.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode (399Hz)} +#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode ($399\,\text{Hz}$)} #+attr_latex: :options {0.27\textwidth} #+begin_subfigure #+attr_latex: :height 4.3cm [[file:figs/test_apa_mode_shapes_2.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode (706Hz)} +#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode ($706\,\text{Hz}$)} #+attr_latex: :options {0.35\textwidth} #+begin_subfigure #+attr_latex: :height 4.3cm @@ -10604,7 +10604,7 @@ Another explanation is the shape difference between the manufactured APA300ML an <> **** Introduction :ignore: After the measurements on the acrshort:apa were performed in Section\nbsp{}ref:sec:test_apa_basic_meas, a new test bench was used to better characterize the dynamics of the APA300ML. -This test bench, depicted in Figure\nbsp{}ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a 5kg granite block that is vertically guided by an air bearing. +This test bench, depicted in Figure\nbsp{}ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a $5\,\text{kg}$ granite block that is vertically guided by an air bearing. Thus, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors. An encoder[fn:test_apa_8] is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the acrshort:apa. @@ -10642,7 +10642,7 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a <> Because the payload is vertically guided without friction, the hysteresis of the acrshort:apa can be estimated from the motion of the payload. -A quasi static[fn:test_apa_9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the acrshort:dac. +A quasi static[fn:test_apa_9] sinusoidal excitation $V_a$ with an offset of $65\,\text{V}$ (halfway between $-20\,\text{V}$ and $150\,\text{V}$) and with an amplitude varying from $4\,\text{V}$ up to $80\,\text{V}$ is generated using the acrshort:dac. For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_meas_hysteresis. This is the typical behavior expected from a acrfull:pzt stack actuator, where the hysteresis increases as a function of the applied voltage amplitude\nbsp{}[[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]]. @@ -10655,7 +10655,7 @@ This is the typical behavior expected from a acrfull:pzt stack actuator, where t <> To estimate the stiffness of the acrshort:apa, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $\Delta d_e$ is measured using the encoder. -The acrshort:apa stiffness can then be estimated from equation\nbsp{}eqref:eq:test_apa_stiffness, with $g \approx 9.8\,m/s^2$ the acceleration of gravity. +The acrshort:apa stiffness can then be estimated from equation\nbsp{}eqref:eq:test_apa_stiffness, with $g \approx 9.8\,\text{m}/\text{s}^2$ the acceleration of gravity. \begin{equation} \label{eq:test_apa_stiffness} k_{\text{apa}} = \frac{m_a g}{\Delta d_e} @@ -10666,12 +10666,12 @@ It can be seen that there are some drifts in the measured displacement (probably These two effects induce some uncertainties in the measured stiffness. The stiffnesses are computed for all acrshortpl:apa from the two displacements $d_1$ and $d_2$ (see Figure\nbsp{}ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$. -These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,N/\mu m$. +These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,\text{N}/\mu\text{m}$. #+attr_latex: :options [b]{0.57\textwidth} #+begin_minipage #+name: fig:test_apa_meas_stiffness_time -#+caption: Measured displacement when adding (at $t \approx 3\,s$) and removing (at $t \approx 13\,s$) the mass +#+caption: Measured displacement when adding (at $t \approx 3\,\text{s}$) and removing (at $t \approx 13\,\text{s}$) the mass #+attr_latex: :scale 0.8 :float nil [[file:figs/test_apa_meas_stiffness_time.png]] #+end_minipage @@ -10689,7 +10689,7 @@ These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measu | 5 | 1.7 | 1.93 | | 6 | 1.7 | 1.92 | | 8 | 1.73 | 1.98 | -#+latex: \captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $N/\mu m$} +#+latex: \captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\mu\text{m}$} #+end_minipage The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section\nbsp{}ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. @@ -10698,7 +10698,7 @@ The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_fr \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} \end{equation} -The obtained stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and using the "static deflection" method. +The obtained stiffness is $k \approx 2\,\text{N}/\mu\text{m}$ which is close to the values found in the documentation and using the "static deflection" method. It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack\nbsp{}[[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]]. @@ -10706,7 +10706,7 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the " - $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance acrshort:adc) - $k_{\text{sc}}$: piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance) -The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$. +The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,\text{N}/\mu\text{m}$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,\text{N}/\mu\text{m}$. **** Dynamics <> @@ -10715,7 +10715,7 @@ In this section, the dynamics from the excitation voltage $u$ to the encoder mea First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure\nbsp{}ref:fig:test_apa_frf_encoder. The obtained acrshortpl:frf are similar to those of a (second order) mass-spring-damper system with: -- A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$. +- A "stiffness line" indicating a static gain equal to $\approx -17\,\mu\text{m}/V$. The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the acrshort:apa - A lightly damped resonance at $95\,\text{Hz}$ - A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the acrshort:apa support. @@ -10732,7 +10732,7 @@ As illustrated by the Root Locus plot, the poles of the /closed-loop/ system con The significance of this behavior varies with the type of sensor used, as explained in\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]]. Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain. Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell. -From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$ (which is close to what is found using a acrshort:fem). +From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,\text{N}/\mu\text{m}$ (which is close to what is found using a acrshort:fem). All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure\nbsp{}ref:fig:test_apa_frf_encoder and at the force sensor in Figure\nbsp{}ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell. @@ -10788,7 +10788,7 @@ However, this is not so important here because the zero is lightly damped (i.e. **** Effect of the resistor on the IFF Plant <> -A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\mu F$). +A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\mu\text{F}$). As explained before, this is done to limit the voltage offset due to the input bias current of the acrshort:adc as well as to limit the low frequency gain. @@ -10879,9 +10879,9 @@ It can be decomposed into three components: - the shell whose axial properties are represented by $k_1$ and $c_1$ - the actuator stacks whose contribution to the axial stiffness is represented by $k_a$ and $c_a$. The force source $f$ represents the axial force induced by the force sensor stacks. - The sensitivity $g_a$ (in $N/m$) is used to convert the applied voltage $V_a$ to the axial force $f$ + The sensitivity $g_a$ (in $\text{N/m}$) is used to convert the applied voltage $V_a$ to the axial force $f$ - the sensor stack whose contribution to the axial stiffness is represented by $k_e$ and $c_e$. - A sensor measures the stack strain $d_e$ which is then converted to a voltage $V_s$ using a sensitivity $g_s$ (in $V/m$) + A sensor measures the stack strain $d_e$ which is then converted to a voltage $V_s$ using a sensitivity $g_s$ (in $\text{V/m}$) Such a simple model has some limitations: - it only represents the axial characteristics of the acrshort:apa as it is modeled as infinitely rigid in the other directions @@ -10903,8 +10903,8 @@ Such a simple model has some limitations: First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. Both methods lead to an estimated mass of $m = 5.7\,\text{kg}$. -Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section\nbsp{}ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure\nbsp{}ref:fig:test_apa_frf_force. -Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $5\,Ns/m$. +Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,\text{N}/\mu\text{m}$ in Section\nbsp{}ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure\nbsp{}ref:fig:test_apa_frf_force. +Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $5\,\text{Ns/m}$. Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:test_apa_5]. Therefore, we have $k_e = 2 k_a$ and $c_e = 2 c_a$ as the actuator stack is composed of two stacks in series. @@ -10914,14 +10914,14 @@ In this case, the total stiffness of the acrshort:apa model is described by\nbsp k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a \end{equation} -Knowing from\nbsp{}eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\text{tot}} = 2\,N/\mu m$, we get from\nbsp{}eqref:eq:test_apa_2dof_stiffness that $k_a = 2.5\,N/\mu m$ and $k_e = 5\,N/\mu m$. +Knowing from\nbsp{}eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\text{tot}} = 2\,\text{N}/\mu\text{m}$, we get from\nbsp{}eqref:eq:test_apa_2dof_stiffness that $k_a = 2.5\,\text{N}/\mu\text{m}$ and $k_e = 5\,\text{N}/\mu\text{m}$. \begin{equation}\label{eq:test_apa_tot_stiffness} -\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,N/\mu m \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s +\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,\text{N}/\mu\text{m} \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s \end{equation} Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance. -$c_a = 50\,Ns/m$ and $c_e = 100\,Ns/m$ are obtained. +$c_a = 50\,\text{Ns/m}$ and $c_e = 100\,\text{Ns/m}$ are obtained. In the last step, $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions. @@ -10931,17 +10931,17 @@ The obtained parameters of the model shown in Figure\nbsp{}ref:fig:test_apa_2dof #+caption: Summary of the obtained parameters for the 2 DoF APA300ML model #+attr_latex: :environment tabularx :width 0.25\linewidth :align cc #+attr_latex: :center t :booktabs t -| *Parameter* | *Value* | -|-------------+------------------| -| $m$ | $5.7\,\text{kg}$ | -| $k_1$ | $0.38\,N/\mu m$ | -| $k_e$ | $5.0\, N/\mu m$ | -| $k_a$ | $2.5\,N/\mu m$ | -| $c_1$ | $5\,Ns/m$ | -| $c_e$ | $100\,Ns/m$ | -| $c_a$ | $50\,Ns/m$ | -| $g_a$ | $-2.58\,N/V$ | -| $g_s$ | $0.46\,V/\mu m$ | +| *Parameter* | *Value* | +|-------------+------------------------------| +| $m$ | $5.7\,\text{kg}$ | +| $k_1$ | $0.38\,\text{N}/\mu\text{m}$ | +| $k_e$ | $5.0\,\text{N}/\mu\text{m}$ | +| $k_a$ | $2.5\,\text{N}/\mu\text{m}$ | +| $c_1$ | $5\,\text{Ns/m}$ | +| $c_e$ | $100\,\text{Ns/m}$ | +| $c_a$ | $50\,\text{Ns/m}$ | +| $g_a$ | $-2.58\,\text{N/V}$ | +| $g_s$ | $0.46\,\text{V}/\mu\text{m}$ | ***** Obtained Dynamics :ignore: @@ -10991,7 +10991,7 @@ Finally, two /remote points/ (=4= and =5=) are located across the third piezoele Once the APA300ML /super element/ is included in the multi-body model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be extracted. The gains $g_a$ and $g_s$ are then tuned such that the gains of the transfer functions match the identified ones. -By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained. +By doing so, $g_s = 4.9\,\text{V}/\mu\text{m}$ and $g_a = 23.2\,\text{N/V}$ are obtained. To ensure that the sensitivities $g_a$ and $g_s$ are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material. @@ -11008,21 +11008,21 @@ Unfortunately, the manufacturer of the stack was not willing to share the piezoe However, based on the available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table\nbsp{}ref:tab:test_apa_piezo_properties. -From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained, which are close to the constants identified using the experimentally identified transfer functions. +From these parameters, $g_s = 5.1\,\text{V}/\mu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained, which are close to the constants identified using the experimentally identified transfer functions. #+name: tab:test_apa_piezo_properties #+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX #+attr_latex: :center t :booktabs t -| *Parameter* | *Value* | *Description* | -|----------------+----------------------------+--------------------------------------------------------------| -| $d_{33}$ | $680 \cdot 10^{-12}\,m/V$ | Piezoelectric constant | -| $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,F/m$ | Permittivity under constant stress | -| $s^{D}$ | $21 \cdot 10^{-12}\,m^2/N$ | Elastic compliance understand constant electric displacement | -| $c^{E}$ | $48 \cdot 10^{9}\,N/m^2$ | Young's modulus of elasticity | -| $L$ | $20\,mm$ per stack | Length of the stack | -| $A$ | $10^{-4}\,m^2$ | Area of the piezoelectric stack | -| $n$ | $160$ per stack | Number of layers in the piezoelectric stack | +| *Parameter* | *Value* | *Description* | +|----------------+------------------------------------------+--------------------------------------------------------------| +| $d_{33}$ | $680 \cdot 10^{-12}\,\text{m/V}$ | Piezoelectric constant | +| $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,\text{F/m}$ | Permittivity under constant stress | +| $s^{D}$ | $21 \cdot 10^{-12}\,\text{m}^2/\text{N}$ | Elastic compliance understand constant electric displacement | +| $c^{E}$ | $48 \cdot 10^{9}\,\text{N}/\text{m}^2$ | Young's modulus of elasticity | +| $L$ | $20\,\text{mm}$ per stack | Length of the stack | +| $A$ | $10^{-4}\,\text{m}^2$ | Area of the piezoelectric stack | +| $n$ | $160$ per stack | Number of layers in the piezoelectric stack | ***** Comparison of the obtained dynamics @@ -11089,13 +11089,13 @@ During the detailed design phase, specifications in terms of stiffness and strok #+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc #+attr_latex: :center t :booktabs t :float t -| | *Specification* | *FEM* | -|-------------------+------------------------+-------| -| Axial Stiffness | $> 100\,N/\mu m$ | 94 | -| Shear Stiffness | $> 1\,N/\mu m$ | 13 | -| Bending Stiffness | $< 100\,Nm/\text{rad}$ | 5 | -| Torsion Stiffness | $< 500\,Nm/\text{rad}$ | 260 | -| Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | +| | *Specification* | *FEM* | +|-------------------+-------------------------------+-------| +| Axial Stiffness | $> 100\,\text{N}/\mu\text{m}$ | 94 | +| Shear Stiffness | $> 1\,\text{N}/\mu\text{m}$ | 13 | +| Bending Stiffness | $< 100\,\text{Nm}/\text{rad}$ | 5 | +| Torsion Stiffness | $< 500\,\text{Nm}/\text{rad}$ | 260 | +| Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | After optimization using a acrshort:fem, the geometry shown in Figure\nbsp{}ref:fig:test_joints_schematic has been obtained and the corresponding flexible joint characteristics are summarized in Table\nbsp{}ref:tab:test_joints_specs. This flexible joint is a monolithic piece of stainless steel[fn:test_joints_1] manufactured using wire electrical discharge machining. @@ -11167,7 +11167,7 @@ The dimensions of the flexible joint in the Y-Z plane will contribute to the X-b The setup used to measure the dimensions of the "X" flexible beam is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_setup. What is typically observed is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_image. -It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\mu m$, +It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\mu\text{m}$, #+name: fig:test_joints_profilometer #+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image}) @@ -11188,12 +11188,12 @@ It is then possible to estimate the dimension of the flexible beam with an accur #+end_figure **** Measurement Results -The specified flexible beam thickness (gap) is $250\,\mu m$. +The specified flexible beam thickness (gap) is $250\,\mu\text{m}$. Four gaps are measured for each flexible joint (2 in the $x$ direction and 2 in the $y$ direction). The "beam thickness" is then estimated as the mean between the gaps measured on opposite sides. A histogram of the measured beam thicknesses is shown in Figure\nbsp{}ref:fig:test_joints_size_hist. -The measured thickness is less than the specified value of $250\,\mu m$, but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. +The measured thickness is less than the specified value of $250\,\mu\text{m}$, but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. However, what is more important than the true value of the thickness is the consistency between all flexible joints. @@ -11273,8 +11273,8 @@ The deflection of the joint $d_x$ is measured using a displacement sensor. [[file:figs/test_joints_bench_working_principle.png]] ***** Required external applied force -The bending stiffness is foreseen to be $k_{R_y} \approx k_{R_x} \approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx \theta_{x,\text{max}}\approx 25\,mrad$. -The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 22.5\,mm$ (see Figure\nbsp{}ref:fig:test_joints_bench_working_principle). +The bending stiffness is foreseen to be $k_{R_y} \approx k_{R_x} \approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx \theta_{x,\text{max}}\approx 25\,\text{mrad}$. +The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 22.5\,\text{mm}$ (see Figure\nbsp{}ref:fig:test_joints_bench_working_principle). The bending $\theta_y$ of the flexible joint due to the force $F_x$ is given by equation\nbsp{}eqref:eq:test_joints_deflection_force. @@ -11283,20 +11283,20 @@ The bending $\theta_y$ of the flexible joint due to the force $F_x$ is given by \end{equation} Therefore, the force that must be applied to test the full range of the flexible joints is given by equation\nbsp{}eqref:eq:test_joints_max_force. -The measurement range of the force sensor should then be higher than $5.5\,N$. +The measurement range of the force sensor should then be higher than $5.5\,\text{N}$. \begin{equation}\label{eq:test_joints_max_force} - F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h} \approx 5.5\,N + F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h} \approx 5.5\,\text{N} \end{equation} ***** Required actuator stroke and sensors range -The flexible joint is designed to allow a bending motion of $\pm 25\,mrad$. +The flexible joint is designed to allow a bending motion of $\pm 25\,\text{mrad}$. The corresponding stroke at the location of the force sensor is given by\nbsp{}eqref:eq:test_joints_max_stroke. -To test the full range of the flexible joint, the means of applying a force (explained in the next section) should allow a motion of at least $0.5\,mm$. -Similarly, the measurement range of the displacement sensor should also be higher than $0.5\,mm$. +To test the full range of the flexible joint, the means of applying a force (explained in the next section) should allow a motion of at least $0.5\,\text{mm}$. +Similarly, the measurement range of the displacement sensor should also be higher than $0.5\,\text{mm}$. \begin{equation}\label{eq:test_joints_max_stroke} -d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \approx 0.5\,mm +d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \approx 0.5\,\text{mm} \end{equation} ***** Force and Displacement measurements @@ -11339,7 +11339,7 @@ The estimated bending stiffness $k_{\text{est}}$ then depends on the shear stiff k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_s h^2}}_{\epsilon_{s}} \Bigl) \end{equation} -With an estimated shear stiffness $k_s = 13\,N/\mu m$ from the acrshort:fem and an height $h=25\,mm$, the estimation errors of the bending stiffness due to shear is $\epsilon_s < 0.1\,\%$ +With an estimated shear stiffness $k_s = 13\,\text{N}/\mu\text{m}$ from the acrshort:fem and an height $h=25\,\text{mm}$, the estimation errors of the bending stiffness due to shear is $\epsilon_s < 0.1\,\%$ ***** Effect of load cell limited stiffness As explained in the previous section, because the measurement of the flexible joint deflection is indirectly performed with the encoder, errors will be made if the load cell experiences some compression. @@ -11351,7 +11351,7 @@ The estimation error of the bending stiffness due to the limited stiffness of th k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_F h^2}}_{\epsilon_f} \Bigl) \end{equation} -With an estimated load cell stiffness of $k_f \approx 1\,N/\mu m$ (from the documentation), the errors due to the load cell limited stiffness is around $\epsilon_f = 1\,\%$. +With an estimated load cell stiffness of $k_f \approx 1\,\text{N}/\mu\text{m}$ (from the documentation), the errors due to the load cell limited stiffness is around $\epsilon_f = 1\,\%$. ***** Estimation error due to height estimation error Now consider an error $\delta h$ in the estimation of the height $h$ as described by\nbsp{}eqref:eq:test_joints_est_h_error. @@ -11366,11 +11366,11 @@ The computed bending stiffness will be\nbsp{}eqref:eq:test_joints_stiffness_heig k_{R_y, \text{est}} \approx h_{\text{est}}^2 \frac{F_x}{d_x} \approx k_{R_y} \Bigl( 1 + \underbrace{2 \frac{\delta h}{h} + \frac{\delta h ^2}{h^2}}_{\epsilon_h} \Bigl) \end{equation} -The height estimation is foreseen to be accurate to within $|\delta h| < 0.4\,mm$ which corresponds to a stiffness error $\epsilon_h < 3.5\,\%$. +The height estimation is foreseen to be accurate to within $|\delta h| < 0.4\,\text{mm}$ which corresponds to a stiffness error $\epsilon_h < 3.5\,\%$. ***** Estimation error due to force and displacement sensors accuracy -An optical encoder is used to measure the displacement (see Section\nbsp{}ref:ssec:test_joints_test_bench) whose maximum non-linearity is $40\,nm$. -As the measured displacement is foreseen to be $0.5\,mm$, the error $\epsilon_d$ due to the encoder non-linearity is negligible $\epsilon_d < 0.01\,\%$. +An optical encoder is used to measure the displacement (see Section\nbsp{}ref:ssec:test_joints_test_bench) whose maximum non-linearity is $40\,\text{nm}$. +As the measured displacement is foreseen to be $0.5\,\text{mm}$, the error $\epsilon_d$ due to the encoder non-linearity is negligible $\epsilon_d < 0.01\,\%$. The accuracy of the load cell is specified at $1\,\%$ and therefore, estimation errors of the bending stiffness due to the limited load cell accuracy should be $\epsilon_F < 1\,\%$ @@ -11388,7 +11388,7 @@ An overall accuracy of $\approx 5\,\%$ can be expected with this measurement ben |----------------------+-------------------------| | Shear effect | $\epsilon_s < 0.1\,\%$ | | Load cell compliance | $\epsilon_f = 1\,\%$ | -| Height error | $\epsilon_h < 3.5\,\%$ | +| Height error | $\epsilon_h < 3.5\,\%$ | | Displacement sensor | $\epsilon_d < 0.01\,\%$ | | Force sensor | $\epsilon_F < 1\,\%$ | @@ -11459,7 +11459,7 @@ The contact between the two load cells is well defined as one has a spherical in The measured forces are compared in Figure\nbsp{}ref:fig:test_joints_force_sensor_calib_fit. The gain mismatch between the two load cells is approximately $4\,\%$ which is higher than that specified in the data sheets. -However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\,N$ and $5\,N$. +However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\,\text{N}$ and $5\,\text{N}$. #+name: fig:test_joints_force_sensor_calib #+caption: Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in (\subref{fig:test_joints_force_sensor_calib_picture}). Comparison of the two measured forces and estimated non-linearity are shown in (\subref{fig:test_joints_force_sensor_calib_fit}) @@ -11481,10 +11481,10 @@ However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\, **** Load Cell Stiffness The objective of this measurement is to estimate the stiffness $k_F$ of the force sensor. -To do so, a stiff element (much stiffer than the estimated $k_F \approx 1\,N/\mu m$) is mounted in front of the force sensor, as shown in Figure\nbsp{}ref:fig:test_joints_meas_force_sensor_stiffness_picture. +To do so, a stiff element (much stiffer than the estimated $k_F \approx 1\,\text{N}/\mu\text{m}$) is mounted in front of the force sensor, as shown in Figure\nbsp{}ref:fig:test_joints_meas_force_sensor_stiffness_picture. Then, the force sensor is pushed against this stiff element while the force sensor and the encoder displacement are measured. The measured displacement as a function of the measured force is shown in Figure\nbsp{}ref:fig:test_joints_force_sensor_stiffness_fit. -The load cell stiffness can then be estimated by computing a linear fit and is found to be $k_F \approx 0.68\,N/\mu m$. +The load cell stiffness can then be estimated by computing a linear fit and is found to be $k_F \approx 0.68\,\text{N}/\mu\text{m}$. #+name: fig:test_joints_meas_force_sensor_stiffness #+caption: Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}), and results are shown in (\subref{fig:test_joints_force_sensor_stiffness_fit}). @@ -11511,7 +11511,7 @@ The measured force and displacement as a function of time are shown in Figure\nb Three regions can be observed: first, the force sensor tip is not in contact with the flexible joint and the measured force is zero; then, the flexible joint deforms linearly; and finally, the flexible joint comes in contact with the mechanical stop. The angular motion $\theta_{y}$ computed from the displacement $d_x$ is displayed as function of the measured torque $T_{y}$ in Figure\nbsp{}ref:fig:test_joints_meas_F_d_lin_fit. -The bending stiffness of the flexible joint can be estimated by computing the slope of the curve in the linear regime (red dashed line) and is found to be $k_{R_y} = 4.4\,Nm/\text{rad}$. +The bending stiffness of the flexible joint can be estimated by computing the slope of the curve in the linear regime (red dashed line) and is found to be $k_{R_y} = 4.4\,\text{Nm}/\text{rad}$. The bending stroke can also be estimated as shown in Figure\nbsp{}ref:fig:test_joints_meas_F_d_lin_fit and is found to be $\theta_{y,\text{max}} = 20.9\,\text{mrad}$. #+name: fig:test_joints_meas_example @@ -11539,7 +11539,7 @@ The measured angular motion as a function of the applied torque is shown in Figu This gives a first idea of the dispersion of the measured bending stiffnesses (i.e. slope of the linear region) and of the angular stroke. A histogram of the measured bending stiffnesses is shown in Figure\nbsp{}ref:fig:test_joints_bend_stiff_hist. -Most of the bending stiffnesses are between $4.6\,Nm/rad$ and $5.0\,Nm/rad$. +Most of the bending stiffnesses are between $4.6\,\text{Nm/rad}$ and $5.0\,\text{Nm/rad}$. #+name: fig:test_joints_meas_bending_results #+caption: Result of measured $k_{R_x}$ and $k_{R_y}$ stiffnesses for the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). A histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist}). @@ -11571,7 +11571,7 @@ This was crucial in preventing potential complications that could have arisen fr A dedicated test bench was developed to asses the bending stiffness of the flexible joints. Through meticulous error analysis and budgeting, a satisfactory level of measurement accuracy could be guaranteed. -The measured bending stiffness values exhibited good agreement with the predictions from the acrshort:fem ($k_{R_x} = k_{R_y} = 5\,Nm/\text{rad}$). +The measured bending stiffness values exhibited good agreement with the predictions from the acrshort:fem ($k_{R_x} = k_{R_y} = 5\,\text{Nm}/\text{rad}$). These measurements are helpful for refining the model of the flexible joints, thereby enhancing the overall accuracy of the nano-hexapod model. Furthermore, the data obtained from these measurements have provided the necessary information to select the most suitable flexible joints for the nano-hexapod, ensuring optimal performance. @@ -11613,7 +11613,7 @@ A mounting bench was developed to ensure: The mounting bench is shown in Figure\nbsp{}ref:fig:test_struts_mounting_bench_first_concept. It consists of a "main frame" (Figure\nbsp{}ref:fig:test_struts_mounting_step_0) precisely machined to ensure both correct strut length and strut coaxiality. -The coaxiality is ensured by good flatness (specified at $20\,\mu m$) between surfaces A and B and between surfaces C and D. +The coaxiality is ensured by good flatness (specified at $20\,\mu\text{m}$) between surfaces A and B and between surfaces C and D. Such flatness was checked using a FARO arm[fn:test_struts_1] (see Figure\nbsp{}ref:fig:test_struts_check_dimensions_bench) and was found to comply with the requirements. The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes. @@ -11729,27 +11729,27 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle <> A Finite Element Model[fn:test_struts_3] of the struts is developed and is used to estimate the flexible modes. -The inertia of the encoder (estimated at $15\,g$) is considered. +The inertia of the encoder (estimated at $15\,\text{g}$) is considered. The two cylindrical interfaces were fixed (boundary conditions), and the first three flexible modes were computed. -The mode shapes are displayed in Figure\nbsp{}ref:fig:test_struts_mode_shapes: an "X-bending" mode at 189Hz, a "Y-bending" mode at 285Hz and a "Z-torsion" mode at 400Hz. +The mode shapes are displayed in Figure\nbsp{}ref:fig:test_struts_mode_shapes: an "X-bending" mode at $189\,\text{Hz}$, a "Y-bending" mode at $285\,\text{Hz}$ and a "Z-torsion" mode at $400\,\text{Hz}$. #+name: fig:test_struts_mode_shapes #+caption: Spurious resonances of the struts estimated from a Finite Element Model #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_1}X-bending mode (189Hz)} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_1}X-bending mode ($189\,\text{Hz}$)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :width 0.85\linewidth [[file:figs/test_struts_mode_shapes_1.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_2}Y-bending mode (285Hz)} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_2}Y-bending mode ($285\,\text{Hz}$)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :width 0.85\linewidth [[file:figs/test_struts_mode_shapes_2.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_3}Z-torsion mode (400Hz)} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_3}Z-torsion mode ($400\,\text{Hz}$)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :width 0.85\linewidth @@ -11820,9 +11820,9 @@ This validates the quality of the acrshort:fem. #+attr_latex: :center t :booktabs t :float t | *Mode* | *FEM with Encoder* | *Exp. with Encoder* | *Exp. without Encoder* | |-----------+--------------------+---------------------+------------------------| -| X-Bending | 189Hz | 198Hz | 226Hz | -| Y-Bending | 285Hz | 293Hz | 337Hz | -| Z-Torsion | 400Hz | 381Hz | 398Hz | +| X-Bending | $189\,\text{Hz}$ | $198\,\text{Hz}$ | $226\,\text{Hz}$ | +| Y-Bending | $285\,\text{Hz}$ | $293\,\text{Hz}$ | $337\,\text{Hz}$ | +| Z-Torsion | $400\,\text{Hz}$ | $381\,\text{Hz}$ | $398\,\text{Hz}$ | *** Dynamical measurements <> @@ -11914,7 +11914,7 @@ This means that the encoder should have little effect on the effectiveness of th The dynamics measured by the encoder (i.e. $d_e/u$) and interferometers (i.e. $d_a/u$) are compared in Figure\nbsp{}ref:fig:test_struts_comp_enc_int. The dynamics from the excitation voltage $u$ to the displacement measured by the encoder $d_e$ presents a behavior that is much more complex than the dynamics of the displacement measured by the interferometer (comparison made in Figure\nbsp{}ref:fig:test_struts_comp_enc_int). -Three additional resonance frequencies can be observed at 197Hz, 290Hz and 376Hz. +Three additional resonance frequencies can be observed at $197\,\text{Hz}$, $290\,\text{Hz}$ and $376\,\text{Hz}$. These resonance frequencies match the frequencies of the flexible modes studied in Section\nbsp{}ref:sec:test_struts_flexible_modes. The good news is that these resonances are not impacting the axial motion of the strut (which is what is important for the hexapod positioning). @@ -11953,7 +11953,7 @@ A very good match can be observed between the struts. The same comparison is made for the transfer function from $u$ to $d_e$ (encoder output) in Figure\nbsp{}ref:fig:test_struts_comp_enc_plants. In this study, large dynamics differences were observed between the 5 struts. -Although the same resonance frequencies were seen for all of the struts (95Hz, 200Hz, 300Hz and 400Hz), the amplitude of the peaks were not the same. +Although the same resonance frequencies were seen for all of the struts ($95\,\text{Hz}$, $200\,\text{Hz}$, $300\,\text{Hz}$ and $400\,\text{Hz}$), the amplitude of the peaks were not the same. In addition, the location or even presence of complex conjugate zeros changes from one strut to another. The reason for this variability will be studied in the next section thanks to the strut model. @@ -12018,7 +12018,7 @@ For the flexible model, it will be shown in the next section that by adding some As shown in Figure\nbsp{}ref:fig:test_struts_comp_enc_plants, the identified dynamics from DAC voltage $u$ to encoder measured displacement $d_e$ are very different from one strut to the other. In this section, it is investigated whether poor alignment of the strut (flexible joints with respect to the acrshort:apa) can explain such dynamics. For instance, consider Figure\nbsp{}ref:fig:test_struts_misalign_schematic where there is a misalignment in the $y$ direction between the two flexible joints (well aligned thanks to the mounting procedure in Section\nbsp{}ref:sec:test_struts_mounting) and the APA300ML. -In this case, the "x-bending" mode at 200Hz (see Figure\nbsp{}ref:fig:test_struts_meas_x_bending) can be expected to have greater impact on the dynamics from the actuator to the encoder. +In this case, the "x-bending" mode at $200\,\text{Hz}$ (see Figure\nbsp{}ref:fig:test_struts_meas_x_bending) can be expected to have greater impact on the dynamics from the actuator to the encoder. #+name: fig:test_struts_misalign_schematic #+caption: Mis-alignement between the joints and the APA @@ -12029,14 +12029,14 @@ To verify this assumption, the dynamics from the output DAC voltage $u$ to the m The obtained dynamics are shown in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y. The alignment of the acrshort:apa with the flexible joints has a large influence on the dynamics from actuator voltage to the measured displacement by the encoder. The misalignment in the $y$ direction mostly influences: -- the presence of the flexible mode at 200Hz (see mode shape in Figure\nbsp{}ref:fig:test_struts_mode_shapes_1) +- the presence of the flexible mode at $200\,\text{Hz}$ (see mode shape in Figure\nbsp{}ref:fig:test_struts_mode_shapes_1) - the location of the complex conjugate zero between the first two resonances: - if $d_{y} < 0$: there is no zero between the two resonances and possibly not even between the second and third resonances - if $d_{y} > 0$: there is a complex conjugate zero between the first two resonances -- the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero) +- the location of the high frequency complex conjugate zeros at $500\,\text{Hz}$ (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero) The same can be done for misalignments in the $x$ direction. -The obtained dynamics (Figure\nbsp{}ref:fig:test_struts_effect_misalignment_x) are showing that misalignment in the $x$ direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure\nbsp{}ref:fig:test_struts_mode_shapes_2). +The obtained dynamics (Figure\nbsp{}ref:fig:test_struts_effect_misalignment_x) are showing that misalignment in the $x$ direction mostly influences the presence of the flexible mode at $300\,\text{Hz}$ (see mode shape in Figure\nbsp{}ref:fig:test_struts_mode_shapes_2). A comparison of the experimental acrshortpl:frf in Figure\nbsp{}ref:fig:test_struts_comp_enc_plants with the model dynamics for several $y$ misalignments in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y indicates a clear similarity. This similarity suggests that the identified differences in dynamics are caused by misalignment. @@ -12068,15 +12068,15 @@ Therefore, large $y$ misalignments are expected. To estimate the misalignments between the two flexible joints and the acrshort:apa: - the struts were fixed horizontally on the mounting bench, as shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_3 but without the encoder - using a length gauge[fn:test_struts_2], the height difference between the flexible joints surface and the acrshort:apa shell surface was measured for both the top and bottom joints and for both sides -- as the thickness of the flexible joint is $21\,mm$ and the thickness of the acrshort:apa shell is $20\,mm$, $0.5\,mm$ of height difference should be measured if the two are perfectly aligned +- as the thickness of the flexible joint is $21\,\text{mm}$ and the thickness of the acrshort:apa shell is $20\,\text{mm}$, $0.5\,\text{mm}$ of height difference should be measured if the two are perfectly aligned Large variations in the $y$ misalignment are found from one strut to the other (results are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). -To check the validity of the measurement, it can be verified that the sum of the measured thickness difference on each side is $1\,mm$ (equal to the thickness difference between the flexible joint and the acrshort:apa). -Thickness differences for all the struts were found to be between $0.94\,mm$ and $1.00\,mm$ which indicate low errors compared to the misalignments found in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment. +To check the validity of the measurement, it can be verified that the sum of the measured thickness difference on each side is $1\,\text{mm}$ (equal to the thickness difference between the flexible joint and the acrshort:apa). +Thickness differences for all the struts were found to be between $0.94\,\text{mm}$ and $1.00\,\text{mm}$ which indicate low errors compared to the misalignments found in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment. #+name: tab:test_struts_meas_y_misalignment -#+caption: Measured $y$ misalignment at the top and bottom of the APA. Measurements are in $mm$ +#+caption: Measured $y$ misalignment at the top and bottom of the APA. Measurements are in $\text{mm}$ #+attr_latex: :environment tabularx :width 0.2\linewidth :align Xcc #+attr_latex: :center t :booktabs t | *Strut* | *Bot* | *Top* | @@ -12108,10 +12108,10 @@ After receiving the positioning pins, the struts were mounted again with the pos This should improve the alignment of the acrshort:apa with the two flexible joints. The alignment is then estimated using a length gauge, as described in the previous sections. -Measured $y$ alignments are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\mu m$ for all the struts, which is much better than before (see Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). +Measured $y$ alignments are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\mu\text{m}$ for all the struts, which is much better than before (see Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). #+name: tab:test_struts_meas_y_misalignment_with_pin -#+caption: Measured $y$ misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in $mm$. +#+caption: Measured $y$ misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in $\text{mm}$. #+attr_latex: :environment tabularx :width 0.25\linewidth :align Xcc #+attr_latex: :center t :booktabs t | *Strut* | *Bot* | *Top* | @@ -12217,20 +12217,20 @@ The main goal of this "mounting tool" is to position the flexible joint interfac The quality of the positioning can be estimated by measuring the "straightness" of the top and bottom "V" interfaces. This corresponds to the diameter of the smallest cylinder which contains all points along the measured axis. This was again done using the FARO arm, and the results for all six struts are summarized in Table\nbsp{}ref:tab:measured_straightness. -The straightness was found to be better than $15\,\mu m$ for all struts[fn:test_nhexa_4], which is sufficiently good to not induce significant stress of the flexible joint during assembly. +The straightness was found to be better than $15\,\mu\text{m}$ for all struts[fn:test_nhexa_4], which is sufficiently good to not induce significant stress of the flexible joint during assembly. #+name: tab:measured_straightness #+caption: Measured straightness between the two "V" shapes for the six struts. These measurements were performed twice for each strut. #+attr_latex: :environment tabularx :width 0.25\linewidth :align Xcc #+attr_latex: :center t :booktabs t -| *Strut* | *Meas 1* | *Meas 2* | -|---------+--------------+--------------| -| 1 | $7\,\mu m$ | $3\, \mu m$ | -| 2 | $11\, \mu m$ | $11\, \mu m$ | -| 3 | $15\, \mu m$ | $14\, \mu m$ | -| 4 | $6\, \mu m$ | $6\, \mu m$ | -| 5 | $7\, \mu m$ | $5\, \mu m$ | -| 6 | $6\, \mu m$ | $7\, \mu m$ | +| *Strut* | *Meas 1* | *Meas 2* | +|---------+--------------------+--------------------| +| 1 | $7\,\mu\text{m}$ | $3\, \mu\text{m}$ | +| 2 | $11\, \mu\text{m}$ | $11\, \mu\text{m}$ | +| 3 | $15\, \mu\text{m}$ | $14\, \mu\text{m}$ | +| 4 | $6\, \mu\text{m}$ | $6\, \mu\text{m}$ | +| 5 | $7\, \mu\text{m}$ | $5\, \mu\text{m}$ | +| 6 | $6\, \mu\text{m}$ | $7\, \mu\text{m}$ | The encoder rulers and heads were then fixed to the top and bottom plates, respectively (Figure\nbsp{}ref:fig:test_nhexa_mount_encoder), and the encoder heads were aligned to maximize the received contrast. @@ -12288,7 +12288,7 @@ Finally, the multi-body model representing the suspended table was tuned to matc The design of the suspended table is quite straightforward. First, an optical table with high frequency flexible mode was selected[fn:test_nhexa_5]. -Then, four springs[fn:test_nhexa_6] were selected with low spring rate such that the suspension modes are below 10Hz. +Then, four springs[fn:test_nhexa_6] were selected with low spring rate such that the suspension modes are below $10\,\text{Hz}$. Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure\nbsp{}ref:fig:test_nhexa_suspended_table_cad). #+name: fig:test_nhexa_suspended_table_cad @@ -12301,8 +12301,8 @@ Finally, some interface elements were designed, and mechanical lateral mechanica In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers[fn:test_nhexa_7] were fixed to the breadboard. Using an instrumented hammer, the first 9 modes could be identified and are summarized in Table\nbsp{}ref:tab:test_nhexa_suspended_table_modes. -The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below 10Hz. -The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{}ref:fig:test_nhexa_table_flexible_modes, and are located above 700Hz. +The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below $10\,\text{Hz}$. +The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{}ref:fig:test_nhexa_table_flexible_modes, and are located above $700\,\text{Hz}$. #+attr_latex: :options [t]{0.45\textwidth} #+begin_minipage @@ -12317,16 +12317,16 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{ #+latex: \centering #+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align clX #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| *Modes* | *Frequency* | *Description* | -|---------+-------------+------------------| -| 1,2 | 1.3 Hz | X-Y translations | -| 3 | 2.0 Hz | Z rotation | -| 4 | 6.9 Hz | Z translation | -| 5,6 | 9.5 Hz | X-Y rotations | -|---------+-------------+------------------| -| 7 | 701 Hz | "Membrane" Mode | -| 8 | 989 Hz | Complex mode | -| 9 | 1025 Hz | Complex mode | +| *Modes* | *Frequency* | *Description* | +|---------+-------------------+------------------| +| 1,2 | $1.3\,\text{Hz}$ | X-Y translations | +| 3 | $2.0\,\text{Hz}$ | Z rotation | +| 4 | $6.9\,\text{Hz}$ | Z translation | +| 5,6 | $9.5\,\text{Hz}$ | X-Y rotations | +|---------+-------------------+------------------| +| 7 | $701\,\text{Hz}$ | "Membrane" Mode | +| 8 | $989\,\text{Hz}$ | Complex mode | +| 9 | $1025\,\text{Hz}$ | Complex mode | #+latex: \captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table} #+end_minipage @@ -12334,19 +12334,19 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{ #+caption: Three identified flexible modes of the suspended table #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_1}Flexible mode at 701Hz} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_1}Flexible mode at $701\,\text{Hz}$} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth [[file:figs/test_nhexa_table_flexible_mode_1.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_2}Flexible mode at 989Hz} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_2}Flexible mode at $989\,\text{Hz}$} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth [[file:figs/test_nhexa_table_flexible_mode_2.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_3}Flexible mode at 1025Hz} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_3}Flexible mode at $1025\,\text{Hz}$} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth @@ -12362,18 +12362,18 @@ The 4 springs are here modeled with "bushing joints" that have stiffness and dam The model order is 12, which corresponds to the 6 suspension modes. The inertia properties of the parts were determined from the geometry and material densities. -The stiffness of the springs was initially set from the datasheet nominal value of $17.8\,N/mm$ and then reduced down to $14\,N/mm$ to better match the measured suspension modes. -The stiffness of the springs in the horizontal plane is set at $0.5\,N/mm$. +The stiffness of the springs was initially set from the datasheet nominal value of $17.8\,\text{N/mm}$ and then reduced down to $14\,\text{N/mm}$ to better match the measured suspension modes. +The stiffness of the springs in the horizontal plane is set at $0.5\,\text{N/mm}$. The obtained suspension modes of the multi-body model are compared with the measured modes in Table\nbsp{}ref:tab:test_nhexa_suspended_table_simscape_modes. #+name: tab:test_nhexa_suspended_table_simscape_modes #+caption: Comparison of suspension modes of the multi-body model and the measured ones #+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcccc #+attr_latex: :center t :booktabs t -| Directions | $D_x$, $D_y$ | $R_z$ | $D_z$ | $R_x$, $R_y$ | -|--------------+--------------+--------+--------+--------------| -| Multi-body | 1.3 Hz | 1.8 Hz | 6.8 Hz | 9.5 Hz | -| Experimental | 1.3 Hz | 2.0 Hz | 6.9 Hz | 9.5 Hz | +| Directions | $D_x$, $D_y$ | $R_z$ | $D_z$ | $R_x$, $R_y$ | +|--------------+------------------+------------------+------------------+------------------| +| Multi-body | $1.3\,\text{Hz}$ | $1.8\,\text{Hz}$ | $6.8\,\text{Hz}$ | $9.5\,\text{Hz}$ | +| Experimental | $1.3\,\text{Hz}$ | $2.0\,\text{Hz}$ | $6.9\,\text{Hz}$ | $9.5\,\text{Hz}$ | *** Nano-Hexapod Measured Dynamics <> @@ -12411,36 +12411,36 @@ Five 3-axis accelerometers were fixed on the top platform of the nano-hexapod (F #+attr_latex: :width 0.7\linewidth [[file:figs/test_nhexa_modal_analysis.jpg]] -Between 100Hz and 200Hz, 6 suspension modes (i.e. rigid body modes of the top platform) were identified. -At around 700Hz, two flexible modes of the top plate were observed (see Figure\nbsp{}ref:fig:test_nhexa_hexa_flexible_modes). +Between $100\,\text{Hz}$ and $200\,\text{Hz}$, 6 suspension modes (i.e. rigid body modes of the top platform) were identified. +At around $700\,\text{Hz}$, two flexible modes of the top plate were observed (see Figure\nbsp{}ref:fig:test_nhexa_hexa_flexible_modes). These modes are summarized in Table\nbsp{}ref:tab:test_nhexa_hexa_modal_modes_list. #+name: tab:test_nhexa_hexa_modal_modes_list #+caption: Description of the identified modes of the Nano-Hexapod #+attr_latex: :environment tabularx :width 0.6\linewidth :align ccX #+attr_latex: :center t :booktabs t -| *Mode* | *Frequency* | *Description* | -|--------+-------------+----------------------------------------------| -| 1 | 120 Hz | Suspension Mode: Y-translation | -| 2 | 120 Hz | Suspension Mode: X-translation | -| 3 | 145 Hz | Suspension Mode: Z-translation | -| 4 | 165 Hz | Suspension Mode: Y-rotation | -| 5 | 165 Hz | Suspension Mode: X-rotation | -| 6 | 190 Hz | Suspension Mode: Z-rotation | -| 7 | 692 Hz | (flexible) Membrane mode of the top platform | -| 8 | 709 Hz | Second flexible mode of the top platform | +| *Mode* | *Frequency* | *Description* | +|--------+------------------+----------------------------------------------| +| 1 | $120\,\text{Hz}$ | Suspension Mode: Y-translation | +| 2 | $120\,\text{Hz}$ | Suspension Mode: X-translation | +| 3 | $145\,\text{Hz}$ | Suspension Mode: Z-translation | +| 4 | $165\,\text{Hz}$ | Suspension Mode: Y-rotation | +| 5 | $165\,\text{Hz}$ | Suspension Mode: X-rotation | +| 6 | $190\,\text{Hz}$ | Suspension Mode: Z-rotation | +| 7 | $692\,\text{Hz}$ | (flexible) Membrane mode of the top platform | +| 8 | $709\,\text{Hz}$ | Second flexible mode of the top platform | #+name: fig:test_nhexa_hexa_flexible_modes #+caption: Two identified flexible modes of the top plate of the Nano-Hexapod #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at 692Hz} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at $692\,\text{Hz}$} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth [[file:figs/test_nhexa_hexa_flexible_mode_1.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at 709Hz} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at $709\,\text{Hz}$} #+attr_latex: :options {\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth @@ -12456,17 +12456,17 @@ The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) t The $6 \times 6$ acrshort:frf matrix from $\bm{u}$ ot $\bm{d}_e$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de. The diagonal terms are displayed using colored lines, and all the 30 off-diagonal terms are displayed by gray lines. -All six diagonal terms are well superimposed up to at least $1\,kHz$, indicating good manufacturing and mounting uniformity. +All six diagonal terms are well superimposed up to at least $1\,\text{kHz}$, indicating good manufacturing and mounting uniformity. Below the first suspension mode, good decoupling can be observed (the amplitude of all off-diagonal terms are $\approx 20$ times smaller than the diagonal terms), indicating the correct assembly of all parts. -From 10Hz up to 1kHz, around 10 resonance frequencies can be observed. -The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section\nbsp{}ref:ssec:test_nhexa_enc_struts_modal_analysis. -Three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_coupling). -Except for the mode at 237Hz, their impact on the dynamics is small. -The two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform. -Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. +From $10\,\text{Hz}$ up to $1\,\text{kHz}$, around 10 resonance frequencies can be observed. +The first 4 are suspension modes (at $122\,\text{Hz}$, $143\,\text{Hz}$, $165\,\text{Hz}$ and $191\,\text{Hz}$) which correlate the modes measured during the modal analysis in Section\nbsp{}ref:ssec:test_nhexa_enc_struts_modal_analysis. +Three modes at $237\,\text{Hz}$, $349\,\text{Hz}$ and $395\,\text{Hz}$ are attributed to the internal strut resonances (this will be checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_coupling). +Except for the mode at $237\,\text{Hz}$, their impact on the dynamics is small. +The two modes at $665\,\text{Hz}$ and $695\,\text{Hz}$ are attributed to the flexible modes of the top platform. +Other modes can be observed above $1\,\text{kHz}$, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. -Up to at least 1kHz, an alternating pole/zero pattern is observed, which makes the control easier to tune. +Up to at least $1\,\text{kHz}$, an alternating pole/zero pattern is observed, which makes the control easier to tune. This would not have occurred if the encoders were fixed to the struts. #+name: fig:test_nhexa_identified_frf_de @@ -12475,9 +12475,9 @@ This would not have occurred if the encoders were fixed to the struts. [[file:figs/test_nhexa_identified_frf_de.png]] Similarly, the $6 \times 6$ acrshort:frf matrix from $\bm{u}$ to $\bm{V}_s$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs. -Alternating poles and zeros can be observed up to at least 2kHz, which is a necessary characteristics for applying decentralized IFF. +Alternating poles and zeros can be observed up to at least $2\,\text{kHz}$, which is a necessary characteristics for applying decentralized IFF. Similar to what was observed for the encoder outputs, all the "diagonal" terms are well superimposed, indicating that the same controller can be applied to all the struts. -The first flexible mode of the struts as 235Hz has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF. +The first flexible mode of the struts as $235\,\text{Hz}$ has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF. #+name: fig:test_nhexa_identified_frf_Vs #+caption: Measured FRF for the transfer function from $\bm{u}$ to $\bm{V}_s$. The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines. @@ -12489,26 +12489,26 @@ The first flexible mode of the struts as 235Hz has large amplitude, and therefor One major challenge for controlling the NASS is the wanted robustness to a variation of payload mass; therefore, it is necessary to understand how the dynamics of the nano-hexapod changes with a change in payload mass. -To study how the dynamics changes with the payload mass, up to three "cylindrical masses" of $13\,kg$ each can be added for a total of $\approx 40\,kg$. +To study how the dynamics changes with the payload mass, up to three "cylindrical masses" of $13\,\text{kg}$ each can be added for a total of $\approx 40\,\text{kg}$. These three cylindrical masses on top of the nano-hexapod are shown in Figure\nbsp{}ref:fig:test_nhexa_table_mass_3. #+name: fig:test_nhexa_table_mass_3 -#+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,kg$ +#+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,\text{kg}$ #+attr_org: :width 800px #+attr_latex: :width 0.8\linewidth [[file:figs/test_nhexa_table_mass_3.jpg]] -The obtained acrshortpl:frf from actuator signal $u_i$ to the associated encoder $d_{ei}$ for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de_masses. +The obtained acrshortpl:frf from actuator signal $u_i$ to the associated encoder $d_{ei}$ for the four payload conditions (no mass, $13\,\text{kg}$, $26\,\text{kg}$ and $39\,\text{kg}$) are shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de_masses. As expected, the frequency of the suspension modes decreased with increasing payload mass. The low frequency gain does not change because it is linked to the stiffness property of the nano-hexapod and not to its mass property. -The frequencies of the two flexible modes of the top plate first decreased significantly when the first mass was added (from $\approx 700\,Hz$ to $\approx 400\,Hz$). +The frequencies of the two flexible modes of the top plate first decreased significantly when the first mass was added (from $\approx 700\,\text{Hz}$ to $\approx 400\,\text{Hz}$). This is because the added mass is composed of two half cylinders that are not fixed together. Therefore, it adds a lot of mass to the top plate without increasing stiffness in one direction. When more than one "mass layer" is added, the half cylinders are added at some angles such that rigidity is added in all directions (see how the three mass "layers" are positioned in Figure\nbsp{}ref:fig:test_nhexa_table_mass_3). In this case, the frequency of these flexible modes is increased. In practice, the payload should be one solid body, and no decrease in the frequency of this flexible mode should be observed. -The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass increased. +The apparent amplitude of the flexible mode of the strut at $237\,\text{Hz}$ becomes smaller as the payload mass increased. The measured acrshortpl:frf from $u_i$ to $V_{si}$ are shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs_masses. For all tested payloads, the measured acrshort:frf always have alternating poles and zeros, indicating that IFF can be applied in a robust manner. @@ -12560,8 +12560,8 @@ The $6 \times 6$ transfer function matrices from $\bm{u}$ to $\bm{d}_e$ and from First, is it evaluated how well the models matches the "direct" terms of the measured acrshort:frf matrix. To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured acrshort:frf in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag. -It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modeled. -The three resonances that were attributed to "internal" flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable because the acrshortpl:apa are here modeled as a simple uniaxial 2-DoF system. +It can be seen that the 4 suspension modes of the nano-hexapod (at $122\,\text{Hz}$, $143\,\text{Hz}$, $165\,\text{Hz}$ and $191\,\text{Hz}$) are well modeled. +The three resonances that were attributed to "internal" flexible modes of the struts (at $237\,\text{Hz}$, $349\,\text{Hz}$ and $395\,\text{Hz}$) cannot be seen in the model, which is reasonable because the acrshortpl:apa are here modeled as a simple uniaxial 2-DoF system. At higher frequencies, no resonances can be observed in the model, as the top plate and the encoder supports are modeled as rigid bodies. #+name: fig:test_nhexa_comp_simscape_diag @@ -12587,7 +12587,7 @@ At higher frequencies, no resonances can be observed in the model, as the top pl Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of acrshort:mimo systems. Instead of comparing the full 36 elements of the $6 \times 6$ acrshort:frf matrix from $\bm{u}$ to $\bm{d}_e$, only the first "column" is compared (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$. -It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at 237Hz. +It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at $237\,\text{Hz}$. Similar results are observed for all other coupling terms and for the transfer function from $\bm{u}$ to $\bm{V}_s$. #+name: fig:test_nhexa_comp_simscape_de_all @@ -12597,8 +12597,8 @@ Similar results are observed for all other coupling terms and for the transfer f The APA300ML was then modeled with a /super-element/ extracted from a FE-software. The obtained transfer functions from $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$ are compared with the measured acrshort:frf in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all_flex. -While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modeled. -Even the mode 395Hz can be observed in the model. +While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at $237\,\text{Hz}$ and $349\,\text{Hz}$ are well modeled. +Even the mode $395\,\text{Hz}$ can be observed in the model. Therefore, if the modes of the struts are to be modeled, the /super-element/ of the APA300ML can be used at the cost of obtaining a much higher order model. #+name: fig:test_nhexa_comp_simscape_de_all_flex @@ -12610,7 +12610,7 @@ Therefore, if the modes of the struts are to be modeled, the /super-element/ of <> Another important characteristic of the model is that it should represents the dynamics of the system well for all considered payloads. -The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag_masses. +The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, $13\,\text{kg}$, $26\,\text{kg}$ and $39\,\text{kg}$) in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag_masses. The observed shift of the suspension modes to lower frequencies with increased payload mass is well represented by the multi-body model. The complex conjugate zeros also well match the experiments both for the encoder outputs (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_diag_masses) and the force sensor outputs (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). @@ -12636,7 +12636,7 @@ However, as decentralized IFF will be applied, the damping is actively brought, #+end_subfigure #+end_figure -In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from $u_1$ to $d_{e1}$ to $d_{e6}$ are compared in the case of the 39kg payload in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all_high_mass. +In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from $u_1$ to $d_{e1}$ to $d_{e6}$ are compared in the case of the $39\,\text{kg}$ payload in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all_high_mass. Excellent match between experimental and model coupling is observed. Therefore, the model effectively represents the system coupling for different payloads. @@ -12658,7 +12658,7 @@ Although the dynamics of the nano-hexapod was indeed impacted by the dynamics of The dynamics of the nano-hexapod was then identified in Section\nbsp{}ref:sec:test_nhexa_dynamics. Below the first suspension mode, good decoupling could be observed for the transfer function from $\bm{u}$ to $\bm{d}_e$, which enables the design of a decentralized positioning controller based on the encoders for relative positioning purposes. -Many other modes were present above 700Hz, which will inevitably limit the achievable bandwidth. +Many other modes were present above $700\,\text{Hz}$, which will inevitably limit the achievable bandwidth. The observed effect of the payload's mass on the dynamics was quite large, which also represents a complex control challenge. The acrshortpl:frf from the six DAC voltages $\bm{u}$ to the six force sensors voltages $\bm{V}_s$ all have alternating complex conjugate poles and complex conjugate zeros for all the tested payloads (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). @@ -12682,9 +12682,9 @@ If a model of the nano-hexapod was developed in one time, it would be difficult To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure\nbsp{}ref:fig:test_id31_micro_station_nano_hexapod. This section presents a comprehensive experimental evaluation of the complete system's performance on the ID31 beamline, focusing on its ability to maintain precise sample positioning under various experimental conditions. -Initially, the project planned to develop a long-stroke ($\approx 1 \, cm^3$) 5-DoF metrology system to measure the sample position relative to the granite base. +Initially, the project planned to develop a long-stroke ($\approx 1 \, \text{cm}^3$) 5-DoF metrology system to measure the sample position relative to the granite base. However, the complexity of this development prevented its completion before the experimental testing phase on ID31. -To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\mu m^3$) metrology system was developed, which is presented in Section\nbsp{}ref:sec:test_id31_metrology. +To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\mu\text{m}^3$) metrology system was developed, which is presented in Section\nbsp{}ref:sec:test_id31_metrology. Then, several key aspects of the system validation are examined. Section\nbsp{}ref:sec:test_id31_open_loop_plant analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities. @@ -12720,7 +12720,7 @@ These include tomography scans at various speeds and with different payload mass **** Introduction :ignore: The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite. -As a long-stroke ($\approx 1 \,cm^3$) metrology system was not yet developed, a stroke stroke ($\approx 100\,\mu m^3$) was used instead to validate the nano-hexapod control. +As a long-stroke ($\approx 1 \,\text{cm}^3$) metrology system was not yet developed, a stroke stroke ($\approx 100\,\mu\text{m}^3$) was used instead to validate the nano-hexapod control. The first considered option was to use the "Spindle error analyzer" shown in Figure\nbsp{}ref:fig:test_id31_lion. This system comprises 5 capacitive sensors facing two reference spheres. @@ -12753,7 +12753,7 @@ However, as the gap between the capacitive sensors and the spheres is very small Instead of using capacitive sensors, 5 fibered interferometers were used in a similar manner (Figure\nbsp{}ref:fig:test_id31_interf). At the end of each fiber, a sensor head[fn:test_id31_2] (Figure\nbsp{}ref:fig:test_id31_interf_head) is used, which consists of a lens precisely positioned with respect to the fiber's end. The lens focuses the light on the surface of the sphere, such that the reflected light comes back into the fiber and produces an interference. -In this way, the gap between the head and the reference sphere is much larger (here around $40\,mm$), thereby removing the risk of collision. +In this way, the gap between the head and the reference sphere is much larger (here around $40\,\text{mm}$), thereby removing the risk of collision. Nevertheless, the metrology system still has a limited measurement range because of the limited angular acceptance of the fibered interferometers. Indeed, when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and above some perpendicular displacement, the reflected light does not come back into the fiber, and no interference is produced. @@ -12762,8 +12762,8 @@ Indeed, when the spheres are moving perpendicularly to the beam axis, the reflec <> The proposed short-stroke metrology system is schematized in Figure\nbsp{}ref:fig:test_id31_metrology_kinematics. -The acrshort:poi is indicated by the blue frame $\{B\}$, which is located $H = 150\,mm$ above the nano-hexapod's top platform. -The spheres have a diameter $d = 25.4\,mm$, and the indicated dimensions are $l_1 = 60\,mm$ and $l_2 = 16.2\,mm$. +The acrshort:poi is indicated by the blue frame $\{B\}$, which is located $H = 150\,\text{mm}$ above the nano-hexapod's top platform. +The spheres have a diameter $d = 25.4\,\text{mm}$, and the indicated dimensions are $l_1 = 60\,\text{mm}$ and $l_2 = 16.2\,\text{mm}$. To compute the pose of $\{B\}$ with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured (small) displacements $[d_1,\ d_2,\ d_3,\ d_4,\ d_5]$ by the interferometers are first written as a function of the (small) linear and angular motion of the $\{B\}$ frame $[D_x,\ D_y,\ D_z,\ R_x,\ R_y]$ eqref:eq:test_id31_metrology_kinematics. \begin{equation}\label{eq:test_id31_metrology_kinematics} @@ -12812,7 +12812,7 @@ To not damage the sensitive sphere surface, the probes are instead positioned on The probes are first fixed to the bottom (fixed) cylinder to align the first sphere with the spindle axis. The probes are then fixed to the top (adjustable) cylinder, and the same alignment is performed. -With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around $10\,\mu m$. +With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around $10\,\mu\text{m}$. The accuracy was probably limited by the poor coaxiality between the cylinders and the spheres. However, this first alignment should be sufficient to position the two sphere within the acceptance range of the interferometers. @@ -12829,7 +12829,7 @@ Granite is used for its good mechanical and thermal stability. The interferometer beams must be placed with respect to the two reference spheres as close as possible to the ideal case shown in Figure\nbsp{}ref:fig:test_id31_metrology_kinematics. Therefore, their positions and angles must be well adjusted with respect to the two spheres. -First, the vertical positions of the spheres is adjusted using the micro-hexapod to match the heights of the interferometers. +First, the vertical positions of the spheres is adjusted using the positioning hexapod to match the heights of the interferometers. Then, the horizontal position of the gantry is adjusted such that the intensity of the light reflected back in the fiber of the top interferometer is maximized. This is equivalent as to optimize the perpendicularity between the interferometer beam and the sphere surface (i.e. the concentricity between the top beam and the sphere center). @@ -12847,14 +12847,14 @@ Therefore, this metrology can be used to better align the axis defined by the ce The alignment process requires few iterations. First, the spindle is scanned, and alignment errors are recorded. -From the errors, the motion of the micro-hexapod to better align the spheres with the spindle axis is computed and the micro-hexapod is positioned accordingly. +From the errors, the motion of the positioning hexapod to better align the spheres with the spindle axis is computed and the positioning hexapod is positioned accordingly. Then, the spindle is scanned again, and new alignment errors are recorded. This iterative process is first performed for angular errors (Figure\nbsp{}ref:fig:test_id31_metrology_align_rx_ry) and then for lateral errors (Figure\nbsp{}ref:fig:test_id31_metrology_align_dx_dy). -The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ in $R_x$ and $R_y$ orientations, $\pm 1\,\mu m$ in $D_x$ and $D_y$ directions, and less than $0.1\,\mu m$ vertically. +The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ in $R_x$ and $R_y$ orientations, $\pm 1\,\mu\text{m}$ in $D_x$ and $D_y$ directions, and less than $0.1\,\mu\text{m}$ vertically. #+name: fig:test_id31_metrology_align -#+caption: Measured angular (\subref{fig:test_id31_metrology_align_rx_ry}) and lateral (\subref{fig:test_id31_metrology_align_dx_dy}) errors during full spindle rotation. Between two rotations, the micro-hexapod is adjusted to better align the two spheres with the rotation axis. +#+caption: Measured angular (\subref{fig:test_id31_metrology_align_rx_ry}) and lateral (\subref{fig:test_id31_metrology_align_dx_dy}) errors during full spindle rotation. Between two rotations, the positioning hexapod is adjusted to better align the two spheres with the rotation axis. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_metrology_align_rx_ry}Angular alignment} @@ -12875,22 +12875,22 @@ The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ i <> Because the interferometers point to spheres and not flat surfaces, the lateral acceptance is limited. -To estimate the metrology acceptance, the micro-hexapod was used to perform three accurate scans of $\pm 1\,mm$, respectively along the $x$, $y$ and $z$ axes. +To estimate the metrology acceptance, the positioning hexapod was used to perform three accurate scans of $\pm 1\,\text{mm}$, respectively along the $x$, $y$ and $z$ axes. During these scans, the 5 interferometers are recorded individually, and the ranges in which each interferometer had enough coupling efficiency to be able to measure the displacement were estimated. Results are summarized in Table\nbsp{}ref:tab:test_id31_metrology_acceptance. -The obtained lateral acceptance for pure displacements in any direction is estimated to be around $+/-0.5\,mm$, which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS. +The obtained lateral acceptance for pure displacements in any direction is estimated to be around $\pm0.5\,\text{mm}$, which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS. #+name: tab:test_id31_metrology_acceptance #+caption: Estimated measurement range for each interferometer, and for three different directions. #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| | $D_x$ | $D_y$ | $D_z$ | -|-----------+-------------+------------+-------| -| $d_1$ (y) | $1.0\,mm$ | $>2\,mm$ | $1.35\,mm$ | -| $d_2$ (y) | $0.8\,mm$ | $>2\,mm$ | $1.01\,mm$ | -| $d_3$ (x) | $>2\,mm$ | $1.06\,mm$ | $1.38\,mm$ | -| $d_4$ (x) | $>2\,mm$ | $0.99\,mm$ | $0.94\,mm$ | -| $d_5$ (z) | $1.33\, mm$ | $1.06\,mm$ | $>2\,mm$ | +| | $D_x$ | $D_y$ | $D_z$ | +|-----------+-------------------+-------------------+-------------------| +| $d_1$ (y) | $1.0\,\text{mm}$ | $>2\,\text{mm}$ | $1.35\,\text{mm}$ | +| $d_2$ (y) | $0.8\,\text{mm}$ | $>2\,\text{mm}$ | $1.01\,\text{mm}$ | +| $d_3$ (x) | $>2\,\text{mm}$ | $1.06\,\text{mm}$ | $1.38\,\text{mm}$ | +| $d_4$ (x) | $>2\,\text{mm}$ | $0.99\,\text{mm}$ | $0.94\,\text{mm}$ | +| $d_5$ (z) | $1.33\,\text{mm}$ | $1.06\,\text{mm}$ | $>2\,\text{mm}$ | **** Estimated measurement errors <> @@ -12901,16 +12901,16 @@ Only the bandwidth and noise characteristics of the external metrology are impor However, some elements that affect the accuracy of the metrology system are discussed here. First, the "metrology kinematics" (discussed in Section\nbsp{}ref:ssec:test_id31_metrology_kinematics) is only approximate (i.e. valid for small displacements). -This can be easily seen when performing lateral $[D_x,\,D_y]$ scans using the micro-hexapod while recording the vertical interferometer (Figure\nbsp{}ref:fig:test_id31_xy_map_sphere). +This can be easily seen when performing lateral $[D_x,\,D_y]$ scans using the positioning hexapod while recording the vertical interferometer (Figure\nbsp{}ref:fig:test_id31_xy_map_sphere). As the top interferometer points to a sphere and not to a plane, lateral motion of the sphere is seen as a vertical motion by the top interferometer. Then, the reference spheres have some deviations relative to an ideal sphere [fn:test_id31_6]. These sphere are originally intended for use with capacitive sensors that integrate shape errors over large surfaces. -When using interferometers, the size of the "light spot" on the sphere surface is a circle with a diameter approximately equal to $50\,\mu m$, and therefore the measurement is more sensitive to shape errors with small features. +When using interferometers, the size of the "light spot" on the sphere surface is a circle with a diameter approximately equal to $50\,\mu\text{m}$, and therefore the measurement is more sensitive to shape errors with small features. As the light from the interferometer travels through air (as opposed to being in vacuum), the measured distance is sensitive to any variation in the refractive index of the air. Therefore, any variation in air temperature, pressure or humidity will induce measurement errors. -For instance, for a measurement length of $40\,mm$, a temperature variation of $0.1\,{}^oC$ (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of $\approx 4\,nm$. +For instance, for a measurement length of $40\,\text{mm}$, a temperature variation of $0.1\,{}^oC$ (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of $\approx 4\,\text{nm}$. Interferometers are also affected by noise\nbsp{}[[cite:&watchi18_review_compac_inter]]. The effect of noise on the translation and rotation measurements is estimated in Figure\nbsp{}ref:fig:test_id31_interf_noise. @@ -13266,7 +13266,7 @@ The obtained acrshortpl:frf are compared with the model in Figure\nbsp{}ref:fig: The implementation of a decentralized Integral Force Feedback controller was successfully demonstrated. Using the multi-body model, the controller was designed and optimized to ensure stability across all payload conditions while providing significant damping of suspension modes. -The experimental results validated the model predictions, showing a reduction in peak amplitudes by approximately a factor of 10 across the full payload range (0-39 kg). +The experimental results validated the model predictions, showing a reduction in peak amplitudes by approximately a factor of 10 across the full payload range (0 to $39\,\text{kg}$). Although higher gains could achieve better damping performance for lighter payloads, the chosen fixed-gain configuration represents a robust compromise that maintains stability and performance under all operating conditions. The good correlation between the modeled and measured damped plants confirms the effectiveness of using the multi-body model for both controller design and performance prediction. @@ -13403,7 +13403,7 @@ The obtained closed-loop positioning accuracy was found to comply with the requi **** Robustness estimation with simulation of Tomography scans <> -To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure\nbsp{}ref:fig:test_id31_picture_masses (i.e. $0\,kg$, $13\,kg$, $26\,kg$ and $39\,kg$). +To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure\nbsp{}ref:fig:test_id31_picture_masses (i.e. $0\,\text{kg}$, $13\,\text{kg}$, $26\,\text{kg}$ and $39\,\text{kg}$). The rotational velocity was set at $6\,\text{deg/s}$, which is the typical rotational velocity for heavy samples. The closed-loop systems were stable under all payload conditions, indicating good control robustness. @@ -13425,10 +13425,10 @@ The multi-body model was first validated by comparing it with the measured frequ This validation confirmed that the model can be reliably used to tune the feedback controller and evaluate its performance. An interaction analysis using the RGA-number was then performed, which revealed that higher payload masses lead to increased coupling when implementing control in the strut reference frame. -Based on this analysis, a diagonal controller with a crossover frequency of 5 Hz was designed, incorporating an integrator, a lead compensator, and a first-order low-pass filter. +Based on this analysis, a diagonal controller with a crossover frequency of $5\,\text{Hz}$ was designed, incorporating an integrator, a lead compensator, and a first-order low-pass filter. Finally, tomography experiments were simulated to validate the acrshort:haclac architecture. -The closed-loop system remained stable under all tested payload conditions (0 to 39 kg). +The closed-loop system remained stable under all tested payload conditions (0 to $39\,\text{kg}$). With no payload at $180\,\text{deg/s}$, the NASS successfully maintained the sample acrshort:poi in the beam, which fulfilled the specifications. At $6\,\text{deg/s}$, although the positioning errors increased with the payload mass (particularly in the lateral direction), the system remained stable. These results demonstrate both the effectiveness and limitations of implementing control in the frame of the struts. @@ -13451,10 +13451,10 @@ Several scientific experiments were replicated, such as: Unless explicitly stated, all closed-loop experiments were performed using the robust (i.e. conservative) high authority controller designed in Section\nbsp{}ref:ssec:test_id31_iff_hac_controller. Higher performance controllers using complementary filters are investigated in Section\nbsp{}ref:ssec:test_id31_cf_control. -For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to $200\,nm\times 100\,nm$. -In this case, the goal is to keep the sample's acrshort:poi in the beam, and therefore the $D_y$ and $D_z$ positioning errors should be less than $200\,nm$ and $100\,nm$ peak-to-peak, respectively. +For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to $200\,\text{nm}\times 100\,\text{nm}$. +In this case, the goal is to keep the sample's acrshort:poi in the beam, and therefore the $D_y$ and $D_z$ positioning errors should be less than $200\,\text{nm}$ and $100\,\text{nm}$ peak-to-peak, respectively. The $R_y$ error should be less than $1.7\,\mu\text{rad}$ peak-to-peak. -In terms of RMS errors, this corresponds to $30\,nm$ in $D_y$, $15\,nm$ in $D_z$ and $250\,\text{nrad}$ in $R_y$ (a summary of the specifications is given in Table\nbsp{}ref:tab:test_id31_experiments_specifications). +In terms of RMS errors, this corresponds to $30\,\text{nm}$ in $D_y$, $15\,\text{nm}$ in $D_z$ and $250\,\text{nrad}$ in $R_y$ (a summary of the specifications is given in Table\nbsp{}ref:tab:test_id31_experiments_specifications). Results obtained for all experiments are summarized and compared to the specifications in Section\nbsp{}ref:ssec:test_id31_scans_conclusion. @@ -13462,10 +13462,10 @@ Results obtained for all experiments are summarized and compared to the specific #+caption: Specifications for the Nano-Active-Stabilization-System #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| | $D_y$ | $D_z$ | $R_y$ | -|-------------+-------+-------+----------------------| -| peak 2 peak | 200nm | 100nm | $1.7\,\mu\text{rad}$ | -| RMS | 30nm | 15nm | $250\,\text{nrad}$ | +| | $D_y$ | $D_z$ | $R_y$ | +|-------------+------------------+------------------+----------------------| +| peak 2 peak | $200\,\text{nm}$ | $100\,\text{nm}$ | $1.7\,\mu\text{rad}$ | +| RMS | $30\,\text{nm}$ | $15\,\text{nm}$ | $250\,\text{nrad}$ | **** Tomography Scans <> @@ -13481,7 +13481,7 @@ This idealized case was simulated by first calculating the eccentricity through While this approach likely underestimates actual open-loop errors, as perfect alignment is practically unattainable, it enables a more balanced comparison with closed-loop performance. #+name: fig:test_id31_tomo_m2_1rpm_robust_hac_iff -#+caption: Tomography experiment with a rotation velocity of $6\,\text{deg/s}$, and payload mass of 26kg. Errors in the $(x,y)$ plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}). +#+caption: Tomography experiment with a rotation velocity of $6\,\text{deg/s}$, and payload mass of $26\,\text{kg}$. Errors in the $(x,y)$ plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}Errors in $(x,y)$ plane} @@ -13573,7 +13573,7 @@ This experiment also illustrates that when needed, performance can be enhanced b <> X-ray reflectivity measurements involve scanning thin structures, particularly solid/liquid interfaces, through the beam by varying the $R_y$ angle. -In this experiment, a $R_y$ scan was executed at a rotational velocity of $100\,\mu rad/s$, and the closed-loop positioning errors were monitored (Figure\nbsp{}ref:fig:test_id31_reflectivity). +In this experiment, a $R_y$ scan was executed at a rotational velocity of $100\,\mu \text{rad/s}$, and the closed-loop positioning errors were monitored (Figure\nbsp{}ref:fig:test_id31_reflectivity). The results confirmed that the NASS successfully maintained the acrshort:poi within the specified beam parameters throughout the scanning process. #+name: fig:test_id31_reflectivity @@ -13610,31 +13610,31 @@ These vertical scans can be executed either continuously or in a step-by-step ma ***** Step by Step $D_z$ motion The vertical step motion was performed exclusively with the nano-hexapod. -Testing was conducted across step sizes ranging from $10\,nm$ to $1\,\mu m$. +Testing was conducted across step sizes ranging from $10\,\text{nm}$ to $1\,\mu\text{m}$. Results are presented in Figure\nbsp{}ref:fig:test_id31_dz_mim_steps. -The system successfully resolved 10nm steps (red curve in Figure\nbsp{}ref:fig:test_id31_dz_mim_10nm_steps) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements. +The system successfully resolved $10\,\text{nm}$ steps (red curve in Figure\nbsp{}ref:fig:test_id31_dz_mim_10nm_steps) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements. In step-by-step scanning procedures, the settling time is a critical parameter as it significantly affects the total experiment duration. -The system achieved a response time of approximately $70\,ms$ to reach the target position (within $\pm 20\,nm$), as demonstrated by the $1\,\mu m$ step response in Figure\nbsp{}ref:fig:test_id31_dz_mim_1000nm_steps. +The system achieved a response time of approximately $70\,\text{ms}$ to reach the target position (within $\pm 20\,\text{nm}$), as demonstrated by the $1\,\mu\text{m}$ step response in Figure\nbsp{}ref:fig:test_id31_dz_mim_1000nm_steps. The settling duration typically decreases for smaller step sizes. #+name: fig:test_id31_dz_mim_steps -#+caption: Vertical steps performed with the nano-hexapod. 10nm steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of $50\,ms$. 100nm steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of $\pm 20\,nm$ is $\approx 70\,ms$ as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a $1\,\mu m$ step. +#+caption: Vertical steps performed with the nano-hexapod. $10\,\text{nm}$ steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of $50\,\text{ms}$. $100\,\text{nm}$ steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of $\pm 20\,\text{nm}$ is $\approx 70\,\text{ms}$ as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a $1\,\mu\text{m}$ step. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_10nm_steps}10nm steps} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_10nm_steps}$10\,\text{nm}$ steps} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dz_mim_10nm_steps.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_100nm_steps}100nm steps} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_100nm_steps}$100\,\text{nm}$ steps} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dz_mim_100nm_steps.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu$m step} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu\text{m}$ step} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13647,10 +13647,10 @@ The settling duration typically decreases for smaller step sizes. For these and subsequent experiments, the NASS performs "ramp scans" (constant velocity scans). To eliminate tracking errors, the feedback controller incorporates two integrators, compensating for the plant's lack of integral action at low frequencies. -Initial testing at $10\,\mu m/s$ demonstrated positioning errors well within specifications (indicated by dashed lines in Figure\nbsp{}ref:fig:test_id31_dz_scan_10ums). +Initial testing at $10\,\mu\text{m/s}$ demonstrated positioning errors well within specifications (indicated by dashed lines in Figure\nbsp{}ref:fig:test_id31_dz_scan_10ums). #+name: fig:test_id31_dz_scan_10ums -#+caption: $D_z$ scan at a velocity of $10\,\mu m/s$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry}) +#+caption: $D_z$ scan at a velocity of $10\,\mu \text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dy}$D_y$} @@ -13673,12 +13673,12 @@ Initial testing at $10\,\mu m/s$ demonstrated positioning errors well within spe #+end_subfigure #+end_figure -A subsequent scan at $100\,\mu m/s$ - the maximum velocity for high-precision $D_z$ scans[fn:test_id31_8] - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure\nbsp{}ref:fig:test_id31_dz_scan_100ums). +A subsequent scan at $100\,\mu\text{m/s}$ - the maximum velocity for high-precision $D_z$ scans[fn:test_id31_8] - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure\nbsp{}ref:fig:test_id31_dz_scan_100ums). Since detectors typically operate only during the constant velocity phase, these transient deviations do not compromise the measurement quality. However, performance during acceleration phases could be enhanced through the implementation of feedforward control. #+name: fig:test_id31_dz_scan_100ums -#+caption: $D_z$ scan at a velocity of $100\,\mu m/s$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry}) +#+caption: $D_z$ scan at a velocity of $100\,\mu\text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dy}$D_y$} @@ -13708,21 +13708,21 @@ However, performance during acceleration phases could be enhanced through the im Lateral scans are executed using the $T_y$ stage. The stepper motor controller[fn:test_id31_5] generates a setpoint that is transmitted to the Speedgoat. Within the Speedgoat, the system computes the positioning error by comparing the measured $D_y$ sample position against the received setpoint, and the Nano-Hexapod compensates for positioning errors introduced during $T_y$ stage scanning. -The scanning range is constrained $\pm 100\,\mu m$ due to the limited acceptance of the metrology system. +The scanning range is constrained $\pm 100\,\mu\text{m}$ due to the limited acceptance of the metrology system. ***** Slow scan -Initial testing were made with a scanning velocity of $10\,\mu m/s$, which is typical for these experiments. +Initial testing were made with a scanning velocity of $10\,\mu\text{m/s}$, which is typical for these experiments. Figure\nbsp{}ref:fig:test_id31_dy_10ums compares the positioning errors between open-loop (without NASS) and closed-loop operation. In the scanning direction, open-loop measurements reveal periodic errors (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy) attributable to the $T_y$ stage's stepper motor. These micro-stepping errors, which are inherent to stepper motor operation, occur 200 times per motor rotation with approximately $1\,\text{mrad}$ angular error amplitude. -Given the $T_y$ stage's lead screw pitch of $2\,mm$, these errors manifest as $10\,\mu m$ periodic oscillations with $\approx 300\,nm$ amplitude, which can indeed be seen in the open-loop measurements (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy). +Given the $T_y$ stage's lead screw pitch of $2\,\text{mm}$, these errors manifest as $10\,\mu\text{m}$ periodic oscillations with $\approx 300\,\text{nm}$ amplitude, which can indeed be seen in the open-loop measurements (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy). In the vertical direction (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dz), open-loop errors likely stem from metrology measurement error because the top interferometer points at a spherical target surface (see Figure\nbsp{}ref:fig:test_id31_xy_map_sphere). Under closed-loop control, positioning errors remain within specifications in all directions. #+name: fig:test_id31_dy_10ums -#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $10\,\mu m/s$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_10ums_dy}). +#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $10\,\mu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_10ums_dy}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dy} $D_y$} @@ -13747,7 +13747,7 @@ Under closed-loop control, positioning errors remain within specifications in al ***** Fast Scan -The system performance was evaluated at an increased scanning velocity of $100\,\mu m/s$, and the results are presented in Figure\nbsp{}ref:fig:test_id31_dy_100ums. +The system performance was evaluated at an increased scanning velocity of $100\,\mu\text{m/s}$, and the results are presented in Figure\nbsp{}ref:fig:test_id31_dy_100ums. At this velocity, the micro-stepping errors generate $10\,\text{Hz}$ vibrations, which are further amplified by micro-station resonances. These vibrations exceeded the NASS feedback controller bandwidth, resulting in limited attenuation under closed-loop control. This limitation exemplifies why stepper motors are suboptimal for "long-stroke/short-stroke" systems requiring precise scanning performance\nbsp{}[[cite:&dehaeze22_fastj_uhv]]. @@ -13758,7 +13758,7 @@ Alternatively, since closed-loop errors in $D_z$ and $R_y$ directions remain wit For applications requiring small $D_y$ scans, the nano-hexapod can be used exclusively, although with limited stroke capability. #+name: fig:test_id31_dy_100ums -#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $100\,\mu m/s$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_100ums_dy}). +#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $100\,\mu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_100ums_dy}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dy} $D_y$} @@ -13786,8 +13786,8 @@ For applications requiring small $D_y$ scans, the nano-hexapod can be used exclu In diffraction tomography experiments, the micro-station performs combined motions: continuous rotation around the $R_z$ axis while performing lateral scans along $D_y$. For this validation, the spindle maintained a constant rotational velocity of $6\,\text{deg/s}$ while the nano-hexapod performs the lateral scanning motion. -To avoid high-frequency vibrations typically induced by the stepper motor, the $T_y$ stage was not used, which constrained the scanning range to approximately $\pm 100\,\mu m/s$. -The system performance was evaluated at three lateral scanning velocities: $0.1\,mm/s$, $0.5\,mm/s$, and $1\,mm/s$. Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_setpoint presents both the $D_y$ position setpoints and the corresponding measured $D_y$ positions for all tested velocities. +To avoid high-frequency vibrations typically induced by the stepper motor, the $T_y$ stage was not used, which constrained the scanning range to approximately $\pm 100\,\mu\text{m/s}$. +The system performance was evaluated at three lateral scanning velocities: $0.1\,\text{mm/s}$, $0.5\,\text{mm/s}$, and $1\,\text{mm/s}$. Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_setpoint presents both the $D_y$ position setpoints and the corresponding measured $D_y$ positions for all tested velocities. #+name: fig:test_id31_diffraction_tomo_setpoint #+caption: Dy motion for several configured velocities @@ -13797,7 +13797,7 @@ The system performance was evaluated at three lateral scanning velocities: $0.1\ The positioning errors measured along $D_y$, $D_z$, and $R_y$ directions are displayed in Figure\nbsp{}ref:fig:test_id31_diffraction_tomo. The system maintained positioning errors within specifications for both $D_z$ and $R_y$ (Figures\nbsp{}ref:fig:test_id31_diffraction_tomo_dz and ref:fig:test_id31_diffraction_tomo_ry). However, the lateral positioning errors exceeded specifications during the acceleration and deceleration phases (Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_dy). -These large errors occurred only during $\approx 20\,ms$ intervals; thus, a delay of $20\,ms$ could be implemented in the detector the avoid integrating the beam when these large errors are occurring. +These large errors occurred only during $\approx 20\,\text{ms}$ intervals; thus, a delay of $20\,\text{ms}$ could be implemented in the detector the avoid integrating the beam when these large errors are occurring. Alternatively, a feedforward controller could improve the lateral positioning accuracy during these transient phases. #+name: fig:test_id31_diffraction_tomo @@ -13905,15 +13905,15 @@ For higher values of $\omega_0$, the system became unstable in the vertical dire <> A comprehensive series of experimental validations was conducted to evaluate the NASS performance over a wide range of typical scientific experiments. -The system demonstrated robust performance in most scenarios, with positioning errors generally remaining within specified tolerances (30 nm RMS in $D_y$, 15 nm RMS in $D_z$, and 250 nrad RMS in $R_y$). +The system demonstrated robust performance in most scenarios, with positioning errors generally remaining within specified tolerances ($30\,\text{nm RMS}$ in $D_y$, $15\,\text{nm RMS}$ in $D_z$, and $250\,\text{nrad RMS}$ in $R_y$). For tomography experiments, the NASS successfully maintained good positioning accuracy at rotational velocities up to $180\,\text{deg/s}$ with light payloads, though performance degraded somewhat with heavier masses. The acrshort:haclac control architecture proved particularly effective, with the decentralized IFF providing damping of nano-hexapod suspension modes, while the high authority controller addressed low-frequency disturbances. The vertical scanning capabilities were validated in both step-by-step and continuous motion modes. -The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to $100\,\mu m/s$. +The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to $100\,\mu\text{m/s}$. -For lateral scanning, the system performed well at moderate speeds ($10\,\mu m/s$) but showed limitations at higher velocities ($100\,\mu m/s$) due to stepper motor-induced vibrations in the $T_y$ stage. +For lateral scanning, the system performed well at moderate speeds ($10\,\mu\text{m/s}$) but showed limitations at higher velocities ($100\,\mu\text{m/s}$) due to stepper motor-induced vibrations in the $T_y$ stage. The most challenging test case - diffraction tomography combining rotation and lateral scanning - demonstrated the system's ability to maintain vertical and angular stability while highlighting some limitations in lateral positioning during rapid accelerations. These limitations could be addressed through feedforward control or alternative detector triggering strategies. @@ -13925,29 +13925,29 @@ The identified limitations, primarily related to high-speed lateral scanning and #+caption: Summary of the experimental results performed using the NASS on ID31. Open-loop errors are indicated on the left of the arrows. Closed-loop errors that are outside the specifications are indicated by bold number. #+attr_latex: :environment tabularx :width 0.85\linewidth :align Xccc #+attr_latex: :center t :booktabs t -| *Experiments* | $\bm{D_y}$ *[nmRMS]* | $\bm{D_z}$ *[nmRMS]* | $\bm{R_y}$ *[nradRMS]* | -|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| Tomography ($6\,\text{deg/s}$) | $142 \Rightarrow 15$ | $32 \Rightarrow 5$ | $464 \Rightarrow 56$ | -| Tomography ($6\,\text{deg/s}$, 13kg) | $149 \Rightarrow 25$ | $26 \Rightarrow 6$ | $471 \Rightarrow 55$ | -| Tomography ($6\,\text{deg/s}$, 26kg) | $202 \Rightarrow 25$ | $36 \Rightarrow 7$ | $1737 \Rightarrow 104$ | -| Tomography ($6\,\text{deg/s}$, 39kg) | $297 \Rightarrow \bm{53}$ | $38 \Rightarrow 9$ | $1737 \Rightarrow 170$ | -|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| Tomography ($180\,\text{deg/s}$) | $143 \Rightarrow \bm{38}$ | $24 \Rightarrow 11$ | $252 \Rightarrow 130$ | -| Tomography ($180\,\text{deg/s}$, custom HAC) | $143 \Rightarrow 29$ | $24 \Rightarrow 5$ | $252 \Rightarrow 142$ | -|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| Reflectivity ($100\,\mu\text{rad}/s$) | $28$ | $6$ | $118$ | -|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| $D_z$ scan ($10\,\mu m/s$) | $25$ | $5$ | $108$ | -| $D_z$ scan ($100\,\mu m/s$) | $\bm{35}$ | $9$ | $132$ | -|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| Lateral Scan ($10\,\mu m/s$) | $585 \Rightarrow 21$ | $155 \Rightarrow 10$ | $6300 \Rightarrow 60$ | -| Lateral Scan ($100\,\mu m/s$) | $1063 \Rightarrow \bm{732}$ | $167 \Rightarrow \bm{20}$ | $6445 \Rightarrow \bm{356}$ | -|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| Diffraction tomography ($6\,\text{deg/s}$, $0.1\,mm/s$) | $\bm{36}$ | $7$ | $113$ | -| Diffraction tomography ($6\,\text{deg/s}$, $0.5\,mm/s$) | $29$ | $8$ | $81$ | -| Diffraction tomography ($6\,\text{deg/s}$, $1\,mm/s$) | $\bm{53}$ | $10$ | $135$ | -|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| *Specifications* | $30$ | $15$ | $250$ | +| *Experiments* | $\bm{D_y}$ *[nmRMS]* | $\bm{D_z}$ *[nmRMS]* | $\bm{R_y}$ *[nradRMS]* | +|----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| +| Tomography ($6\,\text{deg/s}$) | $142 \Rightarrow 15$ | $32 \Rightarrow 5$ | $464 \Rightarrow 56$ | +| Tomography ($6\,\text{deg/s}$, $13\,\text{kg}$) | $149 \Rightarrow 25$ | $26 \Rightarrow 6$ | $471 \Rightarrow 55$ | +| Tomography ($6\,\text{deg/s}$, $26\,\text{kg}$) | $202 \Rightarrow 25$ | $36 \Rightarrow 7$ | $1737 \Rightarrow 104$ | +| Tomography ($6\,\text{deg/s}$, $39\,\text{kg}$) | $297 \Rightarrow \bm{53}$ | $38 \Rightarrow 9$ | $1737 \Rightarrow 170$ | +|----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| +| Tomography ($180\,\text{deg/s}$) | $143 \Rightarrow \bm{38}$ | $24 \Rightarrow 11$ | $252 \Rightarrow 130$ | +| Tomography ($180\,\text{deg/s}$, custom HAC) | $143 \Rightarrow 29$ | $24 \Rightarrow 5$ | $252 \Rightarrow 142$ | +|----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| +| Reflectivity ($100\,\mu\text{rad}/s$) | $28$ | $6$ | $118$ | +|----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| +| $D_z$ scan ($10\,\mu\text{m/s}$) | $25$ | $5$ | $108$ | +| $D_z$ scan ($100\,\mu\text{m/s}$) | $\bm{35}$ | $9$ | $132$ | +|----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| +| Lateral Scan ($10\,\mu\text{m/s}$) | $585 \Rightarrow 21$ | $155 \Rightarrow 10$ | $6300 \Rightarrow 60$ | +| Lateral Scan ($100\,\mu\text{m/s}$) | $1063 \Rightarrow \bm{732}$ | $167 \Rightarrow \bm{20}$ | $6445 \Rightarrow \bm{356}$ | +|----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| +| Diffraction tomography ($6\,\text{deg/s}$, $0.1\,\text{mm/s}$) | $\bm{36}$ | $7$ | $113$ | +| Diffraction tomography ($6\,\text{deg/s}$, $0.5\,\text{mm/s}$) | $29$ | $8$ | $81$ | +| Diffraction tomography ($6\,\text{deg/s}$, $1\,\text{mm/s}$) | $\bm{53}$ | $10$ | $135$ | +|----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| +| *Specifications* | $30$ | $15$ | $250$ | *** Conclusion :PROPERTIES: @@ -13962,11 +13962,11 @@ The short-stroke metrology system, while designed as a temporary solution, prove The careful alignment of the fibered interferometers targeting the two reference spheres ensured reliable measurements throughout the testing campaign. The implementation of the control architecture validated the theoretical framework developed earlier in this project. -The decentralized Integral Force Feedback (IFF) controller successfully provided robust damping of suspension modes across all payload conditions (0-39 kg), reducing peak amplitudes by approximately a factor of 10. +The decentralized Integral Force Feedback (IFF) controller successfully provided robust damping of suspension modes across all payload conditions (0 to $39\,\text{kg}$), reducing peak amplitudes by approximately a factor of 10. The High Authority Controller (HAC) effectively rejects low-frequency disturbances, although its performance showed some dependency on payload mass, particularly for lateral motion control. The experimental validation covered a wide range of scientific scenarios. -The system demonstrated remarkable performance under most conditions, meeting the stringent positioning requirements (30 nm RMS in $D_y$, 15 nm RMS in $D_z$, and 250 nrad RMS in $R_y$) for the majority of test cases. +The system demonstrated remarkable performance under most conditions, meeting the stringent positioning requirements ($30\,\text{nm RMS}$ in $D_y$, $15\,\text{nm RMS}$ in $D_z$, and $250\,\text{nrad RMS}$ in $R_y$) for the majority of test cases. Some limitations were identified, particularly in handling heavy payloads during rapid motions and in managing high-speed lateral scanning with the existing stepper motor $T_y$ stage. The successful validation of the NASS demonstrates that once an accurate online metrology system is developed, it will be ready for integration into actual beamline operations. @@ -14055,7 +14055,7 @@ Such a capability would enable the rapid generation of accurate dynamic models f ***** Better addressing plant uncertainty from a change of payload For most high-performance mechatronic systems like lithography machines or atomic force microscopes, payloads inertia are often known and fixed, allowing controllers to be precisely optimized. -However, synchrotron end-stations frequently handle samples with widely varying masses and inertias – ID31 being an extreme example, but many require nanometer positioning for samples from very light masses up to 5kg. +However, synchrotron end-stations frequently handle samples with widely varying masses and inertias – ID31 being an extreme example, but many require nanometer positioning for samples from very light masses up to $5\,\text{kg}$. The conventional strategy involves implementing controllers with relatively small bandwidth to accommodate various payloads. When controllers are optimized for a specific payload, changing payloads may destabilize the feedback loops that needs to be re-tuned. @@ -14066,7 +14066,7 @@ Potential strategies to be explored include adaptive control (involving automati ***** Control based on Complementary Filters -The control architecture based on complementary filters (detailed in Section ref:sec:detail_control_cf) has been successfully implemented in several instruments at the acrlong:esrf. +The control architecture based on complementary filters (detailed in Section\nbsp{}ref:sec:detail_control_cf) has been successfully implemented in several instruments at the acrlong:esrf. This approach has proven to be straightforward to implement and offers the valuable capability of modifying closed-loop behavior in real-time, which proves advantageous for many applications. For instance, the controller can be optimized according to the scan type: constant velocity scans benefit from a $+2$ slope for the sensitivity transfer function, while ptychography may be better served by a $+1$ slope with slightly higher bandwidth to minimize point-to-point transition times. @@ -14075,7 +14075,7 @@ Nevertheless, a more rigorous analysis of this control architecture and its comp ***** Sensor Fusion While the acrshort:haclac approach demonstrated a simple and comprehensive methodology for controlling the NASS, sensor fusion represents an interesting alternative that is worth investigating. -While the synthesis method developed for complementary filters facilitates their design (Section ref:sec:detail_control_sensor), their application specifically for sensor fusion within the NASS context was not examined in detail. +While the synthesis method developed for complementary filters facilitates their design (Section\nbsp{}ref:sec:detail_control_sensor), their application specifically for sensor fusion within the NASS context was not examined in detail. One potential approach involves fusing external metrology (used at low frequencies) with force sensors (employed at high frequencies). This configuration could enhance robustness through the collocation of force sensors with actuators. @@ -14103,7 +14103,7 @@ Consequently, the underlying micro-station's own positioning accuracy has minima Nevertheless, it remains crucial that the micro-station itself does not generate detrimental high-frequency vibrations, particularly during movements, as evidenced by issues previously encountered with stepper motors. Designing a future end-station with the understanding that a functional NASS will ensure final positioning accuracy could allow for a significantly simplified long-stroke stage architecture, perhaps chosen primarily to facilitate the integration of the online metrology. -One possible configuration, illustrated in Figure ref:fig:conclusion_nass_architecture, would comprise a long-stroke Stewart platform providing the required mobility without generating high-frequency vibrations; a spindle that needs not deliver exceptional performance but should be stiff and avoid inducing high-frequency vibrations (an air-bearing spindle might not be essential); and a short-stroke Stewart platform for correcting errors from the long-stroke stage and spindle. +One possible configuration, illustrated in Figure\nbsp{}ref:fig:conclusion_nass_architecture, would comprise a long-stroke Stewart platform providing the required mobility without generating high-frequency vibrations; a spindle that needs not deliver exceptional performance but should be stiff and avoid inducing high-frequency vibrations (an air-bearing spindle might not be essential); and a short-stroke Stewart platform for correcting errors from the long-stroke stage and spindle. #+name: fig:conclusion_nass_architecture #+caption: Proposed alternative configuration for an end-station including the Nano Active Stabilization System @@ -14115,11 +14115,11 @@ With this architecture, the online metrology could be divided into two systems, ***** Development of long stroke high performance stage As an alternative to the short-stroke/long-stroke architecture, the development of a high-performance long-stroke stage seems worth investigating. -Stages based on voice coils, offering nano-positioning capabilities with $3\,mm$ stroke, have recently been reported in the literature\nbsp{}[[cite:&schropp20_ptynam;&kelly22_delta_robot_long_travel_nano]]. +Stages based on voice coils, offering nano-positioning capabilities with $3\,\text{mm}$ stroke, have recently been reported in the literature\nbsp{}[[cite:&schropp20_ptynam;&kelly22_delta_robot_long_travel_nano]]. Magnetic levitation also emerges as a particularly interesting technology to be explored, especially for microscopy\nbsp{}[[cite:&fahmy22_magnet_xy_theta_x;&heyman23_levcub]] and tomography\nbsp{}[[cite:&dyck15_magnet_levit_six_degree_freed_rotar_table;&fahmy22_magnet_xy_theta_x]] end-stations. -Two notable designs illustrating these capabilities are shown in Figure ref:fig:conclusion_maglev. -Specifically, a compact 6DoF stage known as LevCube, providing a mobility of approximately $1\,\text{cm}^3$, is depicted in Figure ref:fig:conclusion_maglev_heyman23, while a 6DoF stage featuring infinite rotation, is shown in Figure ref:fig:conclusion_maglev_dyck15. +Two notable designs illustrating these capabilities are shown in Figure\nbsp{}ref:fig:conclusion_maglev. +Specifically, a compact 6DoF stage known as LevCube, providing a mobility of approximately $1\,\text{cm}^3$, is depicted in Figure\nbsp{}ref:fig:conclusion_maglev_heyman23, while a 6DoF stage featuring infinite rotation, is shown in Figure\nbsp{}ref:fig:conclusion_maglev_dyck15. However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature. #+name: fig:conclusion_maglev @@ -14178,15 +14178,15 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro * Footnotes -[fn:uniaxial_3]DLPVA-100-B from Femto with a voltage input noise is $2.4\,nV/\sqrt{\text{Hz}}$ -[fn:uniaxial_2]Mark Product L-22D geophones are used with a sensitivity of $88\,\frac{V}{m/s}$ and a natural frequency of $\approx 2\,\text{Hz}$ -[fn:uniaxial_1]Mark Product L4-C geophones are used with a sensitivity of $171\,\frac{V}{m/s}$ and a natural frequency of $\approx 1\,\text{Hz}$ +[fn:uniaxial_3]DLPVA-100-B from Femto with a voltage input noise is $2.4\,\text{nV}/\sqrt{\text{Hz}}$ +[fn:uniaxial_2]Mark Product L-22D geophones are used with a sensitivity of $88\,\frac{V}{\text{m/s}}$ and a natural frequency of $\approx 2\,\text{Hz}$ +[fn:uniaxial_1]Mark Product L4-C geophones are used with a sensitivity of $171\,\frac{V}{\text{m/s}}$ and a natural frequency of $\approx 1\,\text{Hz}$ [fn:modal_5]As this matrix is in general non-square, the Moore–Penrose inverse can be used instead. [fn:modal_4]NVGate software from OROS company. [fn:modal_3]OROS OR36. 24bits signal-delta ADC. -[fn:modal_2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$ -[fn:modal_1]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$. +[fn:modal_2]Kistler 9722A2000. Sensitivity of $2.3\,\text{mV/N}$ and measurement range of $2\,\text{kN}$ +[fn:modal_1]PCB 356B18. Sensitivity is $1\,\text{V/g}$, measurement range is $\pm 5\,\text{g}$ and bandwidth is $0.5$ to $5\,\text{kHz}$. [fn:ustation_11]It was probably caused by rust of the linear guides along its stroke. [fn:ustation_10]Laser source is manufactured by Agilent (5519b). @@ -14212,24 +14212,24 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:detail_instrumentation_1] For variable gain amplifiers, it is usual to refer to the input noise rather than the output noise, as the input referred noise is almost independent on the chosen gain. -[fn:test_apa_13]PD200 from PiezoDrive. The gain is $20\,V/V$ -[fn:test_apa_12]The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of $\pm 10\,V$ and 16-bits resolution +[fn:test_apa_13]PD200 from PiezoDrive. The gain is $20\,\text{V/V}$ +[fn:test_apa_12]The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of $\pm 10\,\text{V}$ and 16-bits resolution [fn:test_apa_11]Ansys\textsuperscript{\textregistered} was used [fn:test_apa_10]The transfer function fitting was computed using the =vectfit3= routine, see\nbsp{}[[cite:&gustavsen99_ration_approx_frequen_domain_respon]] [fn:test_apa_9]Frequency of the sinusoidal wave is $1\,\text{Hz}$ -[fn:test_apa_8]Renishaw Vionic, resolution of $2.5\,nm$ +[fn:test_apa_8]Renishaw Vionic, resolution of $2.5\,\text{nm}$ [fn:test_apa_7]Kistler 9722A [fn:test_apa_6]Polytec controller 3001 with sensor heads OFV512 [fn:test_apa_5]Note that this is not completely correct as it was shown in Section\nbsp{}ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. [fn:test_apa_4]The Matlab =fminsearch= command is used to fit the plane -[fn:test_apa_3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu m$ -[fn:test_apa_2]Millimar 1318 probe, specified linearity better than $1\,\mu m$ +[fn:test_apa_3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu\text{m}$ +[fn:test_apa_2]Millimar 1318 probe, specified linearity better than $1\,\mu\text{m}$ [fn:test_apa_1]LCR-819 from Gwinstek, with a specified accuracy of $0.05\%$. The measured frequency is set at $1\,\text{kHz}$ -[fn:test_joints_5]XFL212R-50N from TE Connectivity. The measurement range is $50\,N$. The specified accuracy is $1\,\%$ of the full range -[fn:test_joints_4]Resolute\texttrademark{} encoder with $1\,nm$ resolution and $\pm 40\,nm$ maximum non-linearity +[fn:test_joints_5]XFL212R-50N from TE Connectivity. The measurement range is $50\,\text{N}$. The specified accuracy is $1\,\%$ of the full range +[fn:test_joints_4]Resolute\texttrademark{} encoder with $1\,\text{nm}$ resolution and $\pm 40\,\text{nm}$ maximum non-linearity [fn:test_joints_3]V-408 PIMag\textsuperscript{\textregistered} linear stage is used. Crossed rollers are used to guide the motion. -[fn:test_joints_2]The load cell is FC22 from TE Connectivity. The measurement range is $50\,N$. The specified accuracy is $1\,\%$ of the full range +[fn:test_joints_2]The load cell is FC22 from TE Connectivity. The measurement range is $50\,\text{N}$. The specified accuracy is $1\,\%$ of the full range [fn:test_joints_1]The alloy used is called /F16PH/, also refereed as "1.4542" [fn:test_struts_7] OFV-3001 controller and OFV512 sensor head from Polytec @@ -14237,22 +14237,22 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:test_struts_5] APA300ML from Cedrat Technologies [fn:test_struts_4] Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools [fn:test_struts_3] Using Ansys\textsuperscript{\textregistered}. Flexible Joints and APA Shell are made of a stainless steel allow called /17-4 PH/. Encoder and ruler support material is aluminium. -[fn:test_struts_2] Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu m$ -[fn:test_struts_1] FARO Arm Platinum 4ft, specified accuracy of $\pm 13\mu m$ +[fn:test_struts_2] Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu\text{m}$ +[fn:test_struts_1] FARO Arm Platinum 4ft, specified accuracy of $\pm 13\mu\text{m}$ -[fn:test_nhexa_7]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$. -[fn:test_nhexa_6]"SZ8005 20 x 044" from Steinel. The spring rate is specified at $17.8\,N/mm$ +[fn:test_nhexa_7]PCB 356B18. Sensitivity is $1\,\text{V/g}$, measurement range is $\pm 5\,\text{g}$ and bandwidth is $0.5$ to $5\,\text{kHz}$. +[fn:test_nhexa_6]"SZ8005 20 x 044" from Steinel. The spring rate is specified at $17.8\,\text{N/mm}$ [fn:test_nhexa_5]The 450 mm x 450 mm x 60 mm Nexus B4545A from Thorlabs. -[fn:test_nhexa_4]As the accuracy of the FARO arm is $\pm 13\,\mu m$, the true straightness is probably better than the values indicated. The limitation of the instrument is here reached. -[fn:test_nhexa_3]The height dimension is better than $40\,\mu m$. The diameter fitting of 182g6 and 24g6 with the two plates is verified. +[fn:test_nhexa_4]As the accuracy of the FARO arm is $\pm 13\,\mu\text{m}$, the true straightness is probably better than the values indicated. The limitation of the instrument is here reached. +[fn:test_nhexa_3]The height dimension is better than $40\,\mu\text{m}$. The diameter fitting of 182g6 and 24g6 with the two plates is verified. [fn:test_nhexa_2]Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked. -[fn:test_nhexa_1]FARO Arm Platinum 4ft, specified accuracy of $\pm 13\mu m$ +[fn:test_nhexa_1]FARO Arm Platinum 4ft, specified accuracy of $\pm 13\mu\text{m}$ -[fn:test_id31_8]Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and 100nm "resolution" (equal to the vertical beam size). +[fn:test_id31_8]Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and $100\,\text{nm}$ "resolution" (equal to the vertical beam size). [fn:test_id31_7]The highest rotational velocity of $360\,\text{deg/s}$ could not be tested due to an issue in the Spindle's controller. -[fn:test_id31_6]The roundness of the spheres is specified at $50\,nm$. +[fn:test_id31_6]The roundness of the spheres is specified at $50\,\text{nm}$. [fn:test_id31_5]The "IcePAP"\nbsp{}[[cite:&janvier13_icepap]] which is developed at the ESRF. [fn:test_id31_4]Note that the eccentricity of the "point of interest" with respect to the Spindle rotation axis has been tuned based on measurements. [fn:test_id31_3]The "PEPU"\nbsp{}[[cite:&hino18_posit_encod_proces_unit]] was used for digital protocol conversion between the interferometers and the Speedgoat. [fn:test_id31_2]M12/F40 model from Attocube. -[fn:test_id31_1]Depending on the measuring range, gap can range from $\approx 1\,\mu m$ to $\approx 100\,\mu m$. +[fn:test_id31_1]Depending on the measuring range, gap can range from $\approx 1\,\mu\text{m}$ to $\approx 100\,\mu\text{m}$. diff --git a/phd-thesis.tex b/phd-thesis.tex index 3e46316..82e7bec 100644 --- a/phd-thesis.tex +++ b/phd-thesis.tex @@ -1,4 +1,4 @@ -% Created 2025-04-22 Tue 18:56 +% Created 2025-04-22 Tue 23:52 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -52,7 +52,7 @@ \newacronym{vc}{VC}{Voice Coil} \newglossaryentry{ms}{name=\ensuremath{m_s},description={{Mass of the sample}}} \newglossaryentry{mn}{name=\ensuremath{m_n},description={{Mass of the nano-hexapod}}} -\newglossaryentry{mh}{name=\ensuremath{m_h},description={{Mass of the micro-hexapod}}} +\newglossaryentry{mh}{name=\ensuremath{m_h},description={{Mass of the positioning hexapod}}} \newglossaryentry{mt}{name=\ensuremath{m_t},description={{Mass of the micro-station stages}}} \newglossaryentry{mg}{name=\ensuremath{m_g},description={{Mass of the granite}}} \newglossaryentry{xf}{name=\ensuremath{x_f},description={{Floor motion}}} @@ -273,7 +273,7 @@ While this enhanced beam quality presents unprecedented scientific opportunities \begin{figure}[htbp] \centering \includegraphics[h!tbp,scale=0.9]{figs/introduction_moore_law_brillance.png} -\caption{\label{fig:introduction_moore_law_brillance}Evolution of the peak brilliance (expressed in \(\text{photons}/s/mm^2/mrad^2/0.1\%BW\)) of synchrotron radiation facilities. Note the vertical logarithmic scale.} +\caption{\label{fig:introduction_moore_law_brillance}Evolution of the peak brilliance (expressed in \(\text{photons}/s/\text{mm}^2/\text{mrad}^2/0.1\%BW\)) of synchrotron radiation facilities. Note the vertical logarithmic scale.} \end{figure} \subsubsection*{The ID31 ESRF Beamline} Each beamline begins with a ``white'' beam generated by the insertion device. @@ -306,9 +306,9 @@ Detectors are used to capture the X-rays transmitted through or scattered by the Throughout this thesis, the standard \acrshort{esrf} coordinate system is adopted, wherein the X-axis aligns with the beam direction, Y is transverse horizontal, and Z is vertical upwards against gravity. The specific end-station employed on the ID31 beamline is designated the ``micro-station''. -As depicted in Figure~\ref{fig:introduction_micro_station_dof}, it comprises a stack of positioning stages: a translation stage (in blue), a tilt stage (in red), a spindle for continuous rotation (in yellow), and a micro-hexapod (in purple). +As depicted in Figure~\ref{fig:introduction_micro_station_dof}, it comprises a stack of positioning stages: a translation stage (in blue), a tilt stage (in red), a spindle for continuous rotation (in yellow), and a positioning hexapod (in purple). The sample itself (cyan), potentially housed within complex sample environments (e.g., for high pressure or extreme temperatures), is mounted on top of this assembly. -Each stage serves distinct positioning functions; for example, the micro-hexapod enables fine static adjustments, while the \(T_y\) translation and \(R_z\) rotation stages are used for specific scanning applications. +Each stage serves distinct positioning functions; for example, the positioning hexapod enables fine static adjustments, while the \(T_y\) translation and \(R_z\) rotation stages are used for specific scanning applications. \begin{figure}[h!tbp] \begin{subfigure}{0.52\textwidth} @@ -335,7 +335,7 @@ This reconstruction depends critically on maintaining the sample's \acrfull{poi} Mapping or scanning experiments, depicted in Figure~\ref{fig:introduction_scanning_schematic}, typically use focusing optics to have a small beam size at the sample's location. The sample is then translated perpendicular to the beam (along Y and Z axes), while data is collected at each position. -An example~\cite{sanchez-cano17_synch_x_ray_fluor_nanop} of a resulting two-dimensional map, acquired with 20nm step increments, is shown in Figure~\ref{fig:introduction_scanning_results}. +An example~\cite{sanchez-cano17_synch_x_ray_fluor_nanop} of a resulting two-dimensional map, acquired with \(20\,\text{nm}\) step increments, is shown in Figure~\ref{fig:introduction_scanning_results}. The fidelity and resolution of such images are intrinsically linked to the focused beam size and the positioning precision of the sample relative to the focused beam. Positional instabilities, such as vibrations and thermal drifts, inevitably lead to blurring and distortion in the obtained image. Other advanced imaging modalities practiced on ID31 include reflectivity, diffraction tomography, and small/wide-angle X-ray scattering (SAXS/WAXS). @@ -369,7 +369,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra \end{center} \subcaption{\label{fig:introduction_scanning_results} Obtained image \cite{sanchez-cano17_synch_x_ray_fluor_nanop}} \end{subfigure} -\caption{\label{fig:introduction_scanning}Exemple of a scanning experiment. The sample is scanned in the Y-Z plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of 20nm (\subref{fig:introduction_scanning_results}).} +\caption{\label{fig:introduction_scanning}Exemple of a scanning experiment. The sample is scanned in the Y-Z plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of \(20\,\text{nm}\) (\subref{fig:introduction_scanning_results}).} \end{figure} \subsubsection*{Need of Accurate Positioning End-Stations with High Dynamics} Continuous progress in both synchrotron source technology and X-ray optics have led to the availability of smaller, more intense, and more stable X-ray beams. @@ -391,7 +391,7 @@ The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source s \caption{\label{fig:introduction_beam_3rd_4th_gen}View of the ESRF X-ray beam before the EBS upgrade (\subref{fig:introduction_beam_3rd_gen}) and after the EBS upgrade (\subref{fig:introduction_beam_4th_gen}). The brilliance is increased, whereas the horizontal size and emittance are reduced.} \end{figure} -Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of \acrshort{esrf}, where typical spot sizes were on the order of \(10\,\mu m\) \cite{riekel89_microf_works_at_esrf}. +Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of \acrshort{esrf}, where typical spot sizes were on the order of \(10\,\mu\text{m}\) \cite{riekel89_microf_works_at_esrf}. Various technologies, including zone plates, Kirkpatrick-Baez mirrors, and compound refractive lenses, have been developed and refined, each presenting unique advantages and limitations~\cite{barrett16_reflec_optic_hard_x_ray}. The historical reduction in achievable spot sizes is represented in Figure~\ref{fig:introduction_moore_law_focus}. Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines. @@ -432,7 +432,7 @@ Recent developments in detector technology have yielded sensors with improved sp Historically, detector integration times for scanning and tomography experiments were in the range of 0.1 to 1 second. This extended integration effectively filtered high-frequency vibrations in beam or sample position, resulting in apparently stable but larger beam. -With higher X-ray flux and reduced detector noise, integration times can now be shortened to approximately 1 millisecond, with frame rates exceeding 100 Hz. +With higher X-ray flux and reduced detector noise, integration times can now be shortened to approximately 1 millisecond, with frame rates exceeding \(100\,\text{Hz}\). This reduction in integration time has two major implications for positioning requirements. Firstly, for a given spatial sampling (``pixel size''), faster integration necessitates proportionally higher scanning velocities. Secondly, the shorter integration times make the measurements more susceptible to high-frequency vibrations. @@ -516,11 +516,11 @@ The PtiNAMi microscope at DESY P06 (Figure~\ref{fig:introduction_stages_schroer} For applications requiring active compensation of measured errors, particularly with nano-beams, feedback control loops are implemented. Actuation is typically achieved using piezoelectric actuators~\cite{nazaretski15_pushin_limit,holler17_omny_pin_versat_sampl_holder,holler18_omny_tomog_nano_cryo_stage,villar18_nanop_esrf_id16a_nano_imagin_beaml,nazaretski22_new_kirkp_baez_based_scann}, 3-phase linear motors~\cite{stankevic17_inter_charac_rotat_stages_x_ray_nanot,engblom18_nanop_resul}, or \acrfull{vc} actuators~\cite{kelly22_delta_robot_long_travel_nano,geraldes23_sapot_carnaub_sirius_lnls}. -While often omitted, feedback bandwidth for such stages are relatively low (around 1 Hz), primarily targeting the compensation of slow thermal drifts. -More recently, higher bandwidths (up to 100 Hz) have been demonstrated, particularly with the use of voice coil actuators~\cite{kelly22_delta_robot_long_travel_nano,geraldes23_sapot_carnaub_sirius_lnls}. +While often omitted, feedback bandwidth for such stages are relatively low (around \(1\,\text{Hz}\)), primarily targeting the compensation of slow thermal drifts. +More recently, higher bandwidths (up to \(100\,\text{Hz}\)) have been demonstrated, particularly with the use of voice coil actuators~\cite{kelly22_delta_robot_long_travel_nano,geraldes23_sapot_carnaub_sirius_lnls}. Figure~\ref{fig:introduction_active_stations} showcases two end-stations incorporating online metrology and active feedback control. -The ID16A system at \acrshort{esrf} (Figure~\ref{fig:introduction_stages_villar}) uses capacitive sensors and a piezoelectric hexapod to compensate for rotation stage errors and to perform accurate scans~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}. +The ID16A system at \acrshort{esrf} (Figure~\ref{fig:introduction_stages_villar}) uses capacitive sensors and a piezoelectric Stewart platform to compensate for rotation stage errors and to perform accurate scans~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}. Another example, shown in Figure~\ref{fig:introduction_stages_nazaretski}, employs interferometers and piezoelectric stages to compensate for thermal drifts~\cite{nazaretski15_pushin_limit,nazaretski17_desig_perfor_x_ray_scann}. A more comprehensive review of actively controlled end-stations is provided in Section~\ref{sec:nhexa_platform_review}. @@ -544,9 +544,9 @@ For tomography experiments, correcting spindle guiding errors is critical. Correction stages are typically placed either below the spindle~\cite{stankevic17_inter_charac_rotat_stages_x_ray_nanot,holler17_omny_pin_versat_sampl_holder,holler18_omny_tomog_nano_cryo_stage,villar18_nanop_esrf_id16a_nano_imagin_beaml,engblom18_nanop_resul,nazaretski22_new_kirkp_baez_based_scann,xu23_high_nsls_ii} or above it~\cite{wang12_autom_marker_full_field_hard,schroer17_ptynam,schropp20_ptynam,geraldes23_sapot_carnaub_sirius_lnls}. In most reported cases, only translation errors are actively corrected. Payload capacities for these high-precision systems are usually limited, typically handling calibrated samples on the micron scale, although capacities up to 500g have been reported~\cite{nazaretski22_new_kirkp_baez_based_scann,kelly22_delta_robot_long_travel_nano}. -The system developed in this thesis aims for payload capabilities approximately 100 times heavier (up to 50 kg) than previous stations with similar positioning requirements. +The system developed in this thesis aims for payload capabilities approximately 100 times heavier (up to \(50\,\text{kg}\)) than previous stations with similar positioning requirements. -End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few \acrshortpl{dof} with strokes around \(100\,\mu m\). +End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few \acrshortpl{dof} with strokes around \(100\,\mu\text{m}\). Recently, \acrfull{vc} actuators were used to increase the stroke up to \(3\,\text{mm}\) \cite{kelly22_delta_robot_long_travel_nano,geraldes23_sapot_carnaub_sirius_lnls} An alternative strategy involves a ``long stroke-short stroke'' architecture, illustrated conceptually in Figure~\ref{fig:introduction_two_stage_schematic}. In this configuration, a high-accuracy, high-bandwidth short-stroke stage is mounted on top of a less precise long-stroke stage. @@ -576,9 +576,9 @@ This necessitates peak-to-peak positioning errors below \(200\,\text{nm}\) in \( Additionally, the \(R_y\) tilt angle error must remain below \(0.1\,\text{mdeg}\) (\(250\,\text{nrad RMS}\)). Given the high frame rates of modern detectors, these specified positioning errors must be maintained even when considering high-frequency vibrations. -These demanding stability requirements must be achieved within the specific context of the ID31 beamline, which necessitates the integration with the existing micro-station, accommodating a wide range of experimental configurations requiring high mobility, and handling substantial payloads up to 50 kg. +These demanding stability requirements must be achieved within the specific context of the ID31 beamline, which necessitates the integration with the existing micro-station, accommodating a wide range of experimental configurations requiring high mobility, and handling substantial payloads up to \(50\,\text{kg}\). -The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately \(10\,\mu m\) and \(10\,\mu\text{rad}\) due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. +The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately \(\SI{10}{\mu\m}\) and \(\SI{10}{\mu\rad}\) due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. The primary objective of this project is therefore defined as enhancing the positioning accuracy and stability of the ID31 micro-station by roughly two orders of magnitude, to fully leverage the capabilities offered by the ESRF-EBS source and modern detectors, without compromising its existing mobility and payload capacity. \paragraph{The Nano Active Stabilization System Concept} @@ -616,7 +616,7 @@ The active stabilization platform, positioned between the micro-station top plat It needs to provide active motion compensation in 5 \acrshortpl{dof} (\(D_x\), \(D_y\), \(D_z\), \(R_x\) and \(R_y\)). It must possess excellent dynamic properties to enable high-bandwidth control capable of suppressing vibrations and tracking desired trajectories with nanometer-level precision. Consequently, it must be free from backlash and play, and its active components (e.g., actuators) should introduce minimal vibrations. -Critically, it must accommodate payloads up to 50 kg. +Critically, it must accommodate payloads up to \(50\,\text{kg}\). A suitable candidate architecture for this platform is the Stewart platform (also known as ``hexapod''), a parallel kinematic mechanism capable of 6-DoF motion. Stewart platforms are widely employed in positioning and vibration isolation applications due to their inherent stiffness and potential for high precision. @@ -649,7 +649,7 @@ Several factors complicate the design of robust feedback control for the NASS. First, the system must operate under across diverse experimental conditions, including different scan types (tomography, linear scans) and payloads' inertia. The continuous rotation of the spindle introduces gyroscopic effects that can affect the system dynamics. As actuators of the active platforms rotate relative to stationary sensors, the control kinematics to map the errors in the frame of the active platform is complex. -But perhaps the most significant challenge is the wide variation in payload mass (1 kg up to 50 kg) that the system must accommodate. +But perhaps the most significant challenge is the wide variation in payload mass (\(1\,\text{kg}\) up to \(50\,\text{kg}\)) that the system must accommodate. Designing for robustness against large payload variations typically necessitates larger stability margins, which can compromise achievable performance. Consequently, high-performance positioning stages often work with well-characterized payload, as seen in systems like wafer-scanners or atomic force microscopes. @@ -685,7 +685,7 @@ It proved invaluable for designing and optimizing components intended for integr This methodology, detailed in Section~\ref{sec:detail_fem}, is presented as a potentially useful tool for future mechatronic instrument development. \paragraph{Control Robustness by design} -The requirement for robust operation across diverse conditions—including payloads up to 50kg, complex underlying dynamics from the micro-station, and varied operational modes like different rotation speeds—presented a critical design challenge. +The requirement for robust operation across diverse conditions—including payloads up to \(50\,\text{kg}\), complex underlying dynamics from the micro-station, and varied operational modes like different rotation speeds—presented a critical design challenge. This challenge was met by embedding robustness directly into the active platform's design, rather than depending solely on complex post-design control synthesis techniques such as \(\mathcal{H}_\infty\text{-synthesis}\) and \(\mu\text{-synthesis}\). Key elements of this strategy included the model-based evaluation of active stage designs to identify architectures inherently easier to control, the incorporation of collocated actuator/sensor pairs to leverage passivity-based guaranteed stability, and the comparison of architecture to combine several sensors such as sensor fusion and High Authority Control / Low Authority Control (HAC-LAC). Furthermore, decoupling strategies for parallel manipulators were compared (Section~\ref{sec:detail_control_decoupling}), addressing a topic identified as having limited treatment in the literature. @@ -707,7 +707,7 @@ The integration of such filters into feedback control architectures can also lea \paragraph{Experimental validation of the Nano Active Stabilization System} The conclusion of this work involved the experimental implementation and validation of the complete NASS on the ID31 beamline. -Experimental results, presented in Section~\ref{sec:test_id31}, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native \(\approx 10\,\mu m\) level down to the target \(\approx 100\,nm\) range during representative scientific experiments. +Experimental results, presented in Section~\ref{sec:test_id31}, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native \(\approx 10\,\mu\text{m}\) level down to the target \(\approx 100\,\text{nm}\) range during representative scientific experiments. Crucially, robustness to variations in sample mass and diverse experimental conditions was verified. The NASS thus provides a versatile end-station solution, uniquely combining high payload capacity with nanometer-level accuracy, enabling optimal use of the advanced capabilities of the ESRF-EBS beam and associated detectors. To the author's knowledge, this represents the first demonstration of such a 5-DoF active stabilization platform being used to enhance the accuracy of a complex positioning system to this level. @@ -728,18 +728,18 @@ Dynamic error budgeting~\cite{monkhorst04_dynam_error_budget,okyay16_mechat_desi Chapter~\ref{chap:detail} focuses on translating the validated NASS concept into an optimized, implementable design. Building upon the conceptual model which used idealized components, this phase addresses the detailed specification and optimization of each subsystem. -It starts with the determination of the optimal nano-hexapod geometry (Section~\ref{sec:detail_kinematics}), analyzing the influence of geometric parameters on mobility, stiffness, and dynamics, leading to specific requirements for actuator stroke and joint mobility. +It starts with the determination of the optimal active platform geometry (Section~\ref{sec:detail_kinematics}), analyzing the influence of geometric parameters on mobility, stiffness, and dynamics, leading to specific requirements for actuator stroke and joint mobility. A hybrid multi-body/FEA modeling methodology is introduced and experimentally validated (Section~\ref{sec:detail_fem}), then applied to optimize the actuators (Section~\ref{sec:detail_fem_actuator}) and flexible joints (Section~\ref{sec:detail_fem_joint}) while maintaining system-level simulation capability. Control strategy refinement (Section~\ref{sec:detail_control}) involves optimal integration of multiple sensors in the control architecture, evaluating decoupling strategies, and discussing controller optimization for decoupled systems. Instrumentation selection (Section~\ref{sec:detail_instrumentation}) is guided by dynamic error budgeting to establish noise specifications, followed by experimental characterization. -The chapter concludes (Section~\ref{sec:detail_design}) by presenting the final, optimized nano-hexapod design, ready for procurement and assembly. +The chapter concludes (Section~\ref{sec:detail_design}) by presenting the final, optimized active platform design, ready for procurement and assembly. \paragraph{Experimental validation} Chapter~\ref{chap:test} details the experimental validation process, proceeding systematically from component-level characterization to full system evaluation on the beamline. Actuators of the active platform were characterized, models validated, and active damping tested (Section~\ref{sec:test_apa}). Flexible joints were tested on a dedicated bench to verify stiffness and stroke specifications (Section~\ref{sec:test_joints}). Assembled struts (actuators + joints) were then characterized to ensure consistency and validate multi-body models (Section~\ref{sec:test_struts}). -The complete nano-hexapod assembly was tested on an isolated table, allowing accurate dynamic identification and model validation under various payload conditions (Section~\ref{sec:test_nhexa}). +The complete active platform assembly was tested on an isolated table, allowing accurate dynamic identification and model validation under various payload conditions (Section~\ref{sec:test_nhexa}). Finally, the integrated NASS was validated on the ID31 beamline using a purpose-built short-stroke metrology system (Section~\ref{sec:test_id31}). The implemented control architecture was tested under realistic experimental scenarios, including tomography with heavy payloads, confirming the system's performance and robustness. \chapter{Conceptual Design Development} @@ -772,9 +772,9 @@ This refined model was then validated through simulations of scientific experime For the active stabilization stage, the Stewart platform architecture was selected after careful evaluation of various options. Section~\ref{sec:nhexa} examines the kinematic and dynamic properties of this parallel manipulator, exploring its control challenges and developing appropriate control strategies for implementation within the NASS. -The multi-body modeling approach facilitated the seamless integration of the nano-hexapod with the micro-station model. +The multi-body modeling approach facilitated the seamless integration of the active platform with the micro-station model. -Finally, Section~\ref{sec:nass} validates the NASS concept through closed-loop simulations of tomography experiments. +Finally, Section~\ref{sec:nass} validates the NASS concept through \acrfull{cl} simulations of tomography experiments. These simulations incorporate realistic disturbance sources, confirming the viability of the proposed design approach and control strategies. This progressive approach, beginning with easily comprehensible simplified models, proved instrumental in developing a thorough understanding of the physical phenomena at play. @@ -786,11 +786,11 @@ In this report, a uniaxial model of the \acrfull{nass} is developed and used to Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, yielding to similar conclusions. To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section~\ref{sec:uniaxial_micro_station_model}). -Then, a model of the nano-hexapod is added on top of the micro-station. +Then, a model of the active platform is added on top of the micro-station. With the added sample and sensors, this gives a uniaxial dynamical model of the \acrshort{nass} that will be used for further analysis (Section~\ref{sec:uniaxial_nano_station_model}). The disturbances affecting position stability are identified experimentally (Section~\ref{sec:uniaxial_disturbances}) and included in the model for dynamical noise budgeting (Section~\ref{sec:uniaxial_noise_budgeting}). -In all the following analysis, three nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate nano-hexapod design. +In all the following analysis, three active platform stiffnesses are considered to better understand the trade-offs and to find the most adequate active platform design. Three sample masses are also considered to verify the robustness of the applied control strategies with respect to a change of sample. To improve the position stability of the sample, an \acrfull{haclac} strategy is applied. @@ -799,17 +799,17 @@ It consists of first actively damping the plant (the \acrshort{lac} part), and t Three active damping techniques are studied (Section~\ref{sec:uniaxial_active_damping}) which are used to both reduce the effect of disturbances and make the system easier to control afterwards. Once the system is well damped, a feedback position controller is applied and the obtained performance is analyzed (Section~\ref{sec:uniaxial_position_control}). -Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section~\ref{sec:uniaxial_support_compliance}) and the presence of dynamics between the nano-hexapod and the sample's \acrshort{poi} (Section~\ref{sec:uniaxial_payload_dynamics}). +Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section~\ref{sec:uniaxial_support_compliance}) and the presence of dynamics between the active platform and the sample's \acrshort{poi} (Section~\ref{sec:uniaxial_payload_dynamics}). \subsection{Micro Station Model} \label{sec:uniaxial_micro_station_model} In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station. \subsubsection{Measured dynamics} -The measurement setup is schematically shown in Figure~\ref{fig:uniaxial_ustation_meas_dynamics_schematic} where two vertical hammer hits are performed, one on the Granite (force \(F_{g}\)) and the other on the micro-hexapod's top platform (force \(F_{h}\)). -The vertical inertial motion of the granite \(x_{g}\) and the top platform of the micro-hexapod \(x_{h}\) are measured using geophones\footnote{Mark Product L4-C geophones are used with a sensitivity of \(171\,\frac{V}{m/s}\) and a natural frequency of \(\approx 1\,\text{Hz}\)}. +The measurement setup is schematically shown in Figure~\ref{fig:uniaxial_ustation_meas_dynamics_schematic} where two vertical hammer hits are performed, one on the Granite (force \(F_{g}\)) and the other on the positioning hexapod's top platform (force \(F_{h}\)). +The vertical inertial motion of the granite \(x_{g}\) and the top platform of the positioning hexapod \(x_{h}\) are measured using geophones\footnote{Mark Product L4-C geophones are used with a sensitivity of \(171\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 1\,\text{Hz}\)}. Three \acrfullpl{frf} were computed: one from \(F_{h}\) to \(x_{h}\) (i.e., the compliance of the micro-station), one from \(F_{g}\) to \(x_{h}\) (or from \(F_{h}\) to \(x_{g}\)) and one from \(F_{g}\) to \(x_{g}\). -Due to the poor coherence at low frequencies, these \acrlongpl{frf} will only be shown between 20 and 200Hz (solid lines in Figure~\ref{fig:uniaxial_comp_frf_meas_model}). +Due to the poor coherence at low frequencies, these \acrlongpl{frf} will only be shown between 20 and \(200\,\text{Hz}\) (solid lines in Figure~\ref{fig:uniaxial_comp_frf_meas_model}). \begin{figure}[htbp] \begin{subfigure}{0.69\textwidth} @@ -831,7 +831,7 @@ The uniaxial model of the micro-station is shown in Figure~\ref{fig:uniaxial_mod It consists of a mass spring damper system with three \acrshortpl{dof}. A mass-spring-damper system represents the granite (with mass \(m_g\), stiffness \(k_g\) and damping \(c_g\)). Another mass-spring-damper system represents the different micro-station stages (the \(T_y\) stage, the \(R_y\) stage and the \(R_z\) stage) with mass \(m_t\), damping \(c_t\) and stiffness \(k_t\). -Finally, a third mass-spring-damper system represents the micro-hexapod with mass \(m_h\), damping \(c_h\) and stiffness \(k_h\). +Finally, a third mass-spring-damper system represents the positioning hexapod with mass \(m_h\), damping \(c_h\) and stiffness \(k_h\). The masses of the different stages are estimated from the 3D model, while the stiffnesses are from the data-sheet of the manufacturers. The damping coefficients were tuned to match the damping identified from the measurements. @@ -844,9 +844,9 @@ The parameters obtained are summarized in Table~\ref{tab:uniaxial_ustation_param \toprule \textbf{Stage} & \textbf{Mass} & \textbf{Stiffness} & \textbf{Damping}\\ \midrule -Micro-Hexapod & \(m_h = 15\,\text{kg}\) & \(k_h = 61\,N/\mu m\) & \(c_h = 3\,\frac{kN}{m/s}\)\\ -\(T_y\), \(R_y\), \(R_z\) & \(m_t = 1200\,\text{kg}\) & \(k_t = 520\,N/\mu m\) & \(c_t = 80\,\frac{kN}{m/s}\)\\ -Granite & \(m_g = 2500\,\text{kg}\) & \(k_g = 950\,N/\mu m\) & \(c_g = 250\,\frac{kN}{m/s}\)\\ +Hexapod & \(m_h = 15\,\text{kg}\) & \(k_h = 61\,\text{N}/\mu\text{m}\) & \(c_h = 3\,\frac{\text{kN}}{\text{m/s}}\)\\ +\(T_y\), \(R_y\), \(R_z\) & \(m_t = 1200\,\text{kg}\) & \(k_t = 520\,\text{N}/\mu\text{m}\) & \(c_t = 80\,\frac{\text{kN}}{\text{m/s}}\)\\ +Granite & \(m_g = 2500\,\text{kg}\) & \(k_g = 950\,\text{N}/\mu\text{m}\) & \(c_g = 250\,\frac{\text{kN}}{\text{m/s}}\)\\ \bottomrule \end{tabularx} \end{table} @@ -854,7 +854,7 @@ Granite & \(m_g = 2500\,\text{kg}\) & \(k_g = 950\,N/\mu m\) & \(c_g = 250\,\fra Two disturbances are considered which are shown in red: the floor motion \(x_f\) and the stage vibrations represented by \(f_t\). The hammer impacts \(F_{h}, F_{g}\) are shown in blue, whereas the measured inertial motions \(x_{h}, x_{g}\) are shown in black. \subsubsection{Comparison of model and measurements} -The transfer functions from the forces injected by the hammers to the measured inertial motion of the micro-hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure~\ref{fig:uniaxial_comp_frf_meas_model}. +The transfer functions from the forces injected by the hammers to the measured inertial motion of the positioning hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure~\ref{fig:uniaxial_comp_frf_meas_model}. Because the uniaxial model has three \acrshortpl{dof}, only three modes with frequencies at \(70\,\text{Hz}\), \(140\,\text{Hz}\) and \(320\,\text{Hz}\) are modeled. Many more modes can be observed in the measurements (see Figure~\ref{fig:uniaxial_comp_frf_meas_model}). @@ -866,13 +866,13 @@ More accurate models will be used later on. \includegraphics[scale=1,scale=0.8]{figs/uniaxial_comp_frf_meas_model.png} \caption{\label{fig:uniaxial_comp_frf_meas_model}Comparison of the measured FRF and identified ones from the uniaxial model} \end{figure} -\subsection{Nano-Hexapod Model} +\subsection{Active Platform Model} \label{sec:uniaxial_nano_station_model} -A model of the nano-hexapod and sample is now added on top of the uniaxial model of the micro-station (Figure~\ref{fig:uniaxial_model_micro_station_nass}). +A model of the active platform and sample is now added on top of the uniaxial model of the micro-station (Figure~\ref{fig:uniaxial_model_micro_station_nass}). Disturbances (shown in red) are \gls{fs} the direct forces applied to the sample (for example cable forces), \gls{ft} representing the vibrations induced when scanning the different stages and \gls{xf} the floor motion. -The control signal is the force applied by the nano-hexapod \(f\) and the measurement is the relative motion between the sample and the granite \(d\). -The sample is here considered as a rigid body and rigidly fixed to the nano-hexapod. -The effect of resonances between the sample's \acrshort{poi} and the nano-hexapod actuator will be considered in Section~\ref{sec:uniaxial_payload_dynamics}. +The control signal is the force applied by the active platform \(f\) and the measurement is the relative motion between the sample and the granite \(d\). +The sample is here considered as a rigid body and rigidly fixed to the active platform. +The effect of resonances between the sample's \acrshort{poi} and the active platform actuator will be considered in Section~\ref{sec:uniaxial_payload_dynamics}. \begin{figure}[htbp] \begin{subfigure}{0.39\textwidth} @@ -887,19 +887,19 @@ The effect of resonances between the sample's \acrshort{poi} and the nano-hexapo \end{center} \subcaption{\label{fig:uniaxial_plant_first_params}Bode Plot of the transfer function from actuator forces $f$ to measured displacement $d$ by the metrology} \end{subfigure} -\caption{\label{fig:uniaxial_model_micro_station_nass_with_tf}Uniaxial model of the NASS (\subref{fig:uniaxial_model_micro_station_nass}) with the micro-station shown in black, the nano-hexapod represented in blue and the sample represented in green. Disturbances are shown in red. Extracted transfer function from \(f\) to \(d\) (\subref{fig:uniaxial_plant_first_params}).} +\caption{\label{fig:uniaxial_model_micro_station_nass_with_tf}Uniaxial model of the NASS (\subref{fig:uniaxial_model_micro_station_nass}) with the micro-station shown in black, the active platform represented in blue and the sample represented in green. Disturbances are shown in red. Extracted transfer function from \(f\) to \(d\) (\subref{fig:uniaxial_plant_first_params}).} \end{figure} -\subsubsection{Nano-Hexapod Parameters} -The nano-hexapod is represented by a mass spring damper system (shown in blue in Figure~\ref{fig:uniaxial_model_micro_station_nass}). +\subsubsection{Active Platform Parameters} +The active platform is represented by a mass spring damper system (shown in blue in Figure~\ref{fig:uniaxial_model_micro_station_nass}). Its mass \gls{mn} is set to \(15\,\text{kg}\) while its stiffness \(k_n\) can vary depending on the chosen architecture/technology. The sample is represented by a mass \gls{ms} that can vary from \(1\,\text{kg}\) up to \(50\,\text{kg}\). -As a first example, the nano-hexapod stiffness of is set at \(k_n = 10\,N/\mu m\) and the sample mass is chosen at \(m_s = 10\,\text{kg}\). +As a first example, the active platform stiffness of is set at \(k_n = 10\,\text{N}/\mu\text{m}\) and the sample mass is chosen at \(m_s = 10\,\text{kg}\). \subsubsection{Obtained Dynamic Response} The sensitivity to disturbances (i.e., the transfer functions from \(x_f,f_t,f_s\) to \(d\)) can be extracted from the uniaxial model of Figure~\ref{fig:uniaxial_model_micro_station_nass} and are shown in Figure~\ref{fig:uniaxial_sensitivity_dist_first_params}. The \emph{plant} (i.e., the transfer function from actuator force \(f\) to measured displacement \(d\)) is shown in Figure~\ref{fig:uniaxial_plant_first_params}. -For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~\ref{fig:uniaxial_model_micro_station_nass} will be considered: three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\), \(k_n = 1\,N/\mu m\) and \(k_n = 100\,N/\mu m\)) combined with three sample's masses (\(m_s = 1\,kg\), \(m_s = 25\,kg\) and \(m_s = 50\,kg\)). +For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~\ref{fig:uniaxial_model_micro_station_nass} will be considered: three active platform stiffnesses (\(k_n = 0.01\,\text{N}/\mu\text{m}\), \(k_n = 1\,\text{N}/\mu\text{m}\) and \(k_n = 100\,\text{N}/\mu\text{m}\)) combined with three sample's masses (\(m_s = 1\,\text{kg}\), \(m_s = 25\,\text{kg}\) and \(m_s = 50\,\text{kg}\)). \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -924,8 +924,8 @@ For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~ \end{figure} \subsection{Disturbance Identification} \label{sec:uniaxial_disturbances} -To quantify disturbances (red signals in Figure~\ref{fig:uniaxial_model_micro_station_nass}), three geophones\footnote{Mark Product L-22D geophones are used with a sensitivity of \(88\,\frac{V}{m/s}\) and a natural frequency of \(\approx 2\,\text{Hz}\)} are used. -One is located on the floor, another one on the granite, and the last one on the micro-hexapod's top platform (see Figure~\ref{fig:uniaxial_ustation_meas_disturbances}). +To quantify disturbances (red signals in Figure~\ref{fig:uniaxial_model_micro_station_nass}), three geophones\footnote{Mark Product L-22D geophones are used with a sensitivity of \(88\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 2\,\text{Hz}\)} are used. +One is located on the floor, another one on the granite, and the last one on the positioning hexapod's top platform (see Figure~\ref{fig:uniaxial_ustation_meas_disturbances}). The geophone located on the floor was used to measure the floor motion \(x_f\) while the other two geophones were used to measure vibrations introduced by scanning of the \(T_y\) stage and \(R_z\) stage (see Figure~\ref{fig:uniaxial_ustation_dynamical_id_setup}). \begin{figure}[htbp] @@ -945,22 +945,22 @@ The geophone located on the floor was used to measure the floor motion \(x_f\) w \end{figure} \subsubsection{Ground Motion} To acquire the geophone signals, the measurement setup shown in Figure~\ref{fig:uniaxial_geophone_meas_chain} is used. -The voltage generated by the geophone is amplified using a low noise voltage amplifier\footnote{DLPVA-100-B from Femto with a voltage input noise is \(2.4\,nV/\sqrt{\text{Hz}}\)} with a gain of 60dB before going to the \acrfull{adc}. +The voltage generated by the geophone is amplified using a low noise voltage amplifier\footnote{DLPVA-100-B from Femto with a voltage input noise is \(2.4\,\text{nV}/\sqrt{\text{Hz}}\)} with a gain of \(60\,\text{dB}\) before going to the \acrfull{adc}. This is done to improve the signal-to-noise ratio. To reconstruct the displacement \(x_f\) from the measured voltage \(\hat{V}_{x_f}\), the transfer function of the measurement chain from \(x_f\) to \(\hat{V}_{x_f}\) needs to be estimated. -First, the transfer function \(G_{geo}\) from the floor motion \(x_{f}\) to the generated geophone voltage \(V_{x_f}\) is shown in~\eqref{eq:uniaxial_geophone_tf}, with \(T_g = 88\,\frac{V}{m/s}\) the sensitivity of the geophone, \(f_0 = \frac{\omega_0}{2\pi} = 2\,\text{Hz}\) its resonance frequency and \(\xi = 0.7\) its damping ratio. +First, the transfer function \(G_{geo}\) from the floor motion \(x_{f}\) to the generated geophone voltage \(V_{x_f}\) is shown in~\eqref{eq:uniaxial_geophone_tf}, with \(T_g = 88\,\frac{V}{\text{m/s}}\) the sensitivity of the geophone, \(f_0 = \frac{\omega_0}{2\pi} = 2\,\text{Hz}\) its resonance frequency and \(\xi = 0.7\) its damping ratio. This model of the geophone was taken from~\cite{collette12_review}. The gain of the voltage amplifier is \(V^{\prime}_{x_f}/V_{x_f} = g_0 = 1000\). \begin{equation}\label{eq:uniaxial_geophone_tf} -G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi \omega_0 s + \omega_0^2} \quad \left[ V/m \right] +G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi \omega_0 s + \omega_0^2} \quad \left[ \text{V/m} \right] \end{equation} \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/uniaxial_geophone_meas_chain.png} -\caption{\label{fig:uniaxial_geophone_meas_chain}Measurement setup for one geophone. The inertial displacement \(x\) is converted to a voltage \(V\) by the geophone. This voltage is amplified by a factor \(g_0 = 60\,dB\) using a low-noise voltage amplifier. It is then converted to a digital value \(\hat{V}_x\) using a 16bit ADC.} +\caption{\label{fig:uniaxial_geophone_meas_chain}Measurement setup for one geophone. The inertial displacement \(x\) is converted to a voltage \(V\) by the geophone. This voltage is amplified by a factor \(g_0 = 60\,\text{dB}\) using a low-noise voltage amplifier. It is then converted to a digital value \(\hat{V}_x\) using a 16bit ADC.} \end{figure} The \acrfull{asd} of the floor motion \(\Gamma_{x_f}\) can be computed from the \acrlong{asd} of measured voltage \(\Gamma_{\hat{V}_{x_f}}\) using~\eqref{eq:uniaxial_asd_floor_motion}. @@ -987,7 +987,7 @@ The estimated \acrshort{asd} \(\Gamma_{x_f}\) of the floor motion \(x_f\) is sho \end{figure} \subsubsection{Stage Vibration} To estimate the vibrations induced by scanning the micro-station stages, two geophones are used, as shown in Figure~\ref{fig:uniaxial_ustation_dynamical_id_setup}. -The vertical relative velocity between the top platform of the micro hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. +The vertical relative velocity between the top platform of the positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. The vibrations induced by the \(T_y\) stage are not considered here because they have less amplitude than the vibrations induced by the \(R_z\) stage and because the \(T_y\) stage can be scanned at lower velocities if the induced vibrations are found to be an issue. The amplitude spectral density of the relative motion with and without the Spindle rotation are compared in Figure~\ref{fig:uniaxial_asd_vibration_spindle_rotation}. @@ -997,10 +997,10 @@ The sharp peak observed at \(24\,\text{Hz}\) is believed to be induced by electr \begin{figure}[htbp] \centering \includegraphics[scale=1,scale=0.8]{figs/uniaxial_asd_vibration_spindle_rotation.png} -\caption{\label{fig:uniaxial_asd_vibration_spindle_rotation}Amplitude Spectral Density \(\Gamma_{R_z}\) of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating} +\caption{\label{fig:uniaxial_asd_vibration_spindle_rotation}Amplitude Spectral Density \(\Gamma_{R_z}\) of the relative motion measured between the granite and the positioning hexapod's top platform during Spindle rotating} \end{figure} -To compute the equivalent disturbance force \(f_t\) (Figure~\ref{fig:uniaxial_model_micro_station}) that induces such motion, the transfer function \(G_{f_t}(s)\) from \(f_t\) to the relative motion between the micro-hexapod's top platform and the granite \((x_{h} - x_{g})\) is extracted from the model. +To compute the equivalent disturbance force \(f_t\) (Figure~\ref{fig:uniaxial_model_micro_station}) that induces such motion, the transfer function \(G_{f_t}(s)\) from \(f_t\) to the relative motion between the positioning hexapod's top platform and the granite \((x_{h} - x_{g})\) is extracted from the model. The amplitude spectral density \(\Gamma_{f_{t}}\) of the disturbance force is them computed from~\eqref{eq:uniaxial_ft_asd} and is shown in Figure~\ref{fig:uniaxial_asd_disturbance_force}. \begin{equation}\label{eq:uniaxial_ft_asd} @@ -1018,18 +1018,18 @@ This is very useful to identify what is limiting the performance of the system, \label{ssec:uniaxial_noise_budget_sensitivity} From the uniaxial model of the \acrshort{nass} (Figure~\ref{fig:uniaxial_model_micro_station_nass}), the transfer function from the disturbances (\(f_s\), \(x_f\) and \(f_t\)) to the displacement \(d\) are computed. -This is done for two extreme sample masses \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\) and three nano-hexapod stiffnesses: +This is done for two extreme sample masses \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\) and three active platform stiffnesses: \begin{itemize} -\item \(k_n = 0.01\,N/\mu m\) that represents a voice coil actuator with soft flexible guiding -\item \(k_n = 1\,N/\mu m\) that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator -\item \(k_n = 100\,N/\mu m\) that represents a stiff piezoelectric stack actuator +\item \(k_n = 0.01\,\text{N}/\mu\text{m}\) that represents a voice coil actuator with soft flexible guiding +\item \(k_n = 1\,\text{N}/\mu\text{m}\) that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator +\item \(k_n = 100\,\text{N}/\mu\text{m}\) that represents a stiff piezoelectric stack actuator \end{itemize} -The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses} for the sample mass \(m_s = 1\,\text{kg}\) (the same conclusions can be drawn with \(m_s = 50\,\text{kg}\)): +The obtained sensitivity to disturbances for the three active platform stiffnesses are shown in Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses} for the sample mass \(m_s = 1\,\text{kg}\) (the same conclusions can be drawn with \(m_s = 50\,\text{kg}\)): \begin{itemize} -\item The soft nano-hexapod is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}) -\item Between the suspension mode of the nano-hexapod (here at 5Hz for the soft nano-hexapod) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}) -\item Above the suspension mode of the nano-hexapod, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to \(1\) (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf}) +\item The soft active platform is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}) +\item Between the suspension mode of the active platform (here at \(5\,\text{Hz}\)) and the first mode of the micro-station (here at \(70\,\text{Hz}\)), the disturbances induced by the stage vibrations are filtered out (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}) +\item Above the suspension mode of the active platform, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to \(1\) (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf}) \end{itemize} \begin{figure}[htbp] @@ -1055,11 +1055,11 @@ The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses \end{figure} \subsubsection{Open-Loop Dynamic Noise Budgeting} \label{ssec:uniaxial_noise_budget_result} -Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion \(d\) for each nano-hexapod and sample configuration. -The \acrfull{cas} of the relative motion \(d\) due to both floor motion \(x_f\) and stage vibrations \(f_t\) are shown in Figure~\ref{fig:uniaxial_cas_d_disturbances_stiffnesses} for the three nano-hexapod stiffnesses. -It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft nano-hexapod below \(5\,\text{Hz}\). +Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion \(d\) for each active platform and sample configuration. +The \acrfull{cas} of the relative motion \(d\) due to both floor motion \(x_f\) and stage vibrations \(f_t\) are shown in Figure~\ref{fig:uniaxial_cas_d_disturbances_stiffnesses} for the three active platform stiffnesses. +It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft active platform below \(5\,\text{Hz}\). -The total cumulative amplitude spectrum of \(d\) for the three nano-hexapod stiffnesses and for the two samples masses are shown in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}. +The total cumulative amplitude spectrum of \(d\) for the three active platform stiffnesses and for the two samples masses are shown in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}. The conclusion is that the sample mass has little effect on the cumulative amplitude spectrum of the relative motion \(d\). \begin{figure}[htbp] @@ -1073,29 +1073,29 @@ The conclusion is that the sample mass has little effect on the cumulative ampli \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_d_disturbances_payload_masses.png} \end{center} -\subcaption{\label{fig:uniaxial_cas_d_disturbances_payload_masses}Effect of nano-hexapod stiffness $k_n$ and payload mass $m_s$} +\subcaption{\label{fig:uniaxial_cas_d_disturbances_payload_masses}Effect of active platform stiffness $k_n$ and payload mass $m_s$} \end{subfigure} -\caption{\label{fig:uniaxial_cas_d_disturbances}Cumulative Amplitude Spectrum of the relative motion \(d\). The effect of \(x_f\) and \(f_t\) are shown in (\subref{fig:uniaxial_cas_d_disturbances_stiffnesses}). The effect of sample mass for the three hexapod stiffnesses is shown in (\subref{fig:uniaxial_cas_d_disturbances_payload_masses}). The control objective of having a residual error of 20 nm RMS is shown by the horizontal black dashed line.} +\caption{\label{fig:uniaxial_cas_d_disturbances}Cumulative Amplitude Spectrum of the relative motion \(d\). The effect of \(x_f\) and \(f_t\) are shown in (\subref{fig:uniaxial_cas_d_disturbances_stiffnesses}). The effect of sample mass for the three active platform stiffnesses is shown in (\subref{fig:uniaxial_cas_d_disturbances_payload_masses}). The control objective of having a residual error of \(20\,\text{nm RMS}\) is shown by the horizontal black dashed line.} \end{figure} \subsubsection{Conclusion} The open-loop residual vibrations of \(d\) can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}. -This residual vibration of \(d\) is found to be in the order of \(100\,nm\,\text{RMS}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)), \(200\,nm\,\text{RMS}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)) and \(1\,\mu m\,\text{RMS}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)). -From this analysis, it may be concluded that the stiffer the nano-hexapod the better. +This residual vibration of \(d\) is found to be in the order of \(100\,\text{nm RMS}\) for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)), \(200\,\text{nm RMS}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)) and \(1\,\mu\text{m}\,\text{RMS}\) for the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)). +From this analysis, it may be concluded that the stiffer the active platform the better. However, what is more important is the \emph{closed-loop} residual vibration of \(d\) (i.e., while the feedback controller is used). -The goal is to obtain a closed-loop residual vibration \(\epsilon_d \approx 20\,nm\,\text{RMS}\) (represented by an horizontal dashed black line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}). -The bandwidth of the feedback controller leading to a closed-loop residual vibration of \(20\,nm\,\text{RMS}\) can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}. +The goal is to obtain a closed-loop residual vibration \(\epsilon_d \approx 20\,\text{nm RMS}\) (represented by an horizontal dashed black line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}). +The bandwidth of the feedback controller leading to a closed-loop residual vibration of \(20\,\text{nm RMS}\) can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}. -A closed loop bandwidth of \(\approx 10\,\text{Hz}\) is found for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)), \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)), and \(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)). -Therefore, while the \emph{open-loop} vibration is the lowest for the stiff nano-hexapod, it requires the largest feedback bandwidth to meet the specifications. +A closed loop bandwidth of \(\approx 10\,\text{Hz}\) is found for the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)), \(\approx 50\,\text{Hz}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)), and \(\approx 100\,\text{Hz}\) for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)). +Therefore, while the \emph{open-loop} vibration is the lowest for the stiff active platform, it requires the largest feedback bandwidth to meet the specifications. -The advantage of the soft nano-hexapod can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}. +The advantage of the soft active platform can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}. \subsection{Active Damping} \label{sec:uniaxial_active_damping} -In this section, three active damping techniques are applied to the nano-hexapod (see Figure~\ref{fig:uniaxial_active_damping_strategies}): Integral Force Feedback (IFF)~\cite{preumont91_activ}, Relative Damping Control (RDC)~\cite[Chapter 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition} and Direct Velocity Feedback (DVF)~\cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}. +In this section, three active damping techniques are applied to the active platform (see Figure~\ref{fig:uniaxial_active_damping_strategies}): Integral Force Feedback (IFF)~\cite{preumont91_activ}, Relative Damping Control (RDC)~\cite[Chapter 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition} and Direct Velocity Feedback (DVF)~\cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}. -These damping strategies are first described (Section~\ref{ssec:uniaxial_active_damping_strategies}) and are then compared in terms of achievable damping of the nano-hexapod mode (Section~\ref{ssec:uniaxial_active_damping_achievable_damping}), reduction of the effect of disturbances (i.e., \(x_f\), \(f_t\) and \(f_s\)) on the displacement \(d\) (Sections~\ref{ssec:uniaxial_active_damping_sensitivity_disturbances}). +These damping strategies are first described (Section~\ref{ssec:uniaxial_active_damping_strategies}) and are then compared in terms of achievable damping of the active platform mode (Section~\ref{ssec:uniaxial_active_damping_achievable_damping}), reduction of the effect of disturbances (i.e., \(x_f\), \(f_t\) and \(f_s\)) on the displacement \(d\) (Sections~\ref{ssec:uniaxial_active_damping_sensitivity_disturbances}). \begin{figure}[htbp] \begin{subfigure}{0.37\textwidth} @@ -1200,11 +1200,11 @@ The plant dynamics for all three active damping techniques are shown in Figure~\ All have \emph{alternating poles and zeros} meaning that the phase does not vary by more than \(180\,\text{deg}\) which makes the design of a \emph{robust} damping controller very easy. This alternating poles and zeros property is guaranteed for the IFF and \acrshort{rdc} cases because the sensors are collocated with the actuator~\cite[Chapter 7]{preumont18_vibrat_contr_activ_struc_fourt_edition}. -For the \acrshort{dvf} controller, this property is not guaranteed, and may be lost if some flexibility between the nano-hexapod and the sample is considered~\cite[Chapter 8.4]{preumont18_vibrat_contr_activ_struc_fourt_edition}. +For the \acrshort{dvf} controller, this property is not guaranteed, and may be lost if some flexibility between the active platform and the sample is considered~\cite[Chapter 8.4]{preumont18_vibrat_contr_activ_struc_fourt_edition}. -When the nano-hexapod's suspension modes are at frequencies lower than the resonances of the micro-station (blue and red curves in Figure~\ref{fig:uniaxial_plant_active_damping_techniques}), the resonances of the micro-stations have little impact on the IFF and \acrshort{dvf} transfer functions. -For the stiff nano-hexapod (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure~\ref{fig:uniaxial_plant_active_damping_techniques}. -Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used. +When the active platform's suspension modes are at frequencies lower than the resonances of the micro-station (blue and red curves in Figure~\ref{fig:uniaxial_plant_active_damping_techniques}), the resonances of the micro-stations have little impact on the IFF and \acrshort{dvf} transfer functions. +For the stiff active platform (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure~\ref{fig:uniaxial_plant_active_damping_techniques}. +Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff active platform is used. \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -1225,7 +1225,7 @@ Therefore, it is expected that the micro-station dynamics might impact the achie \end{center} \subcaption{\label{fig:uniaxial_plant_active_damping_techniques_dvf}DVF} \end{subfigure} -\caption{\label{fig:uniaxial_plant_active_damping_techniques}Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), for three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\) in blue, \(k_n = 1\,N/\mu m\) in red and \(k_n = 100\,N/\mu m\) in yellow) and three sample's masses (\(m_s = 1\,kg\): solid curves, \(m_s = 25\,kg\): dot-dashed curves, and \(m_s = 50\,kg\): dashed curves).} +\caption{\label{fig:uniaxial_plant_active_damping_techniques}Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), for three active platform stiffnesses (\(k_n = 0.01\,\text{N}/\mu\text{m}\) in blue, \(k_n = 1\,\text{N}/\mu\text{m}\) in red and \(k_n = 100\,\text{N}/\mu\text{m}\) in yellow) and three sample's masses (\(m_s = 1\,\text{kg}\): solid curves, \(m_s = 25\,\text{kg}\): dot-dashed curves, and \(m_s = 50\,\text{kg}\): dashed curves).} \end{figure} \subsubsection{Achievable Damping and Damped Plants} \label{ssec:uniaxial_active_damping_achievable_damping} @@ -1239,15 +1239,15 @@ This is illustrated in Figure~\ref{fig:uniaxial_root_locus_damping_techniques_mi \xi = \sin(\phi) \end{equation} -The Root Locus for the three nano-hexapod stiffnesses and the three active damping techniques are shown in Figure~\ref{fig:uniaxial_root_locus_damping_techniques}. -All three active damping approaches can lead to \emph{critical damping} of the nano-hexapod suspension mode (angle \(\phi\) can be increased up to 90 degrees). +The Root Locus for the three active platform stiffnesses and the three active damping techniques are shown in Figure~\ref{fig:uniaxial_root_locus_damping_techniques}. +All three active damping approaches can lead to \emph{critical damping} of the active platform suspension mode (angle \(\phi\) can be increased up to 90 degrees). There is even some damping authority on micro-station modes in the following cases: \begin{description} -\item[{IFF with a stiff nano-hexapod (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_stiff})}] This can be understood from the mechanical equivalent of IFF shown in Figure~\ref{fig:uniaxial_active_damping_iff_equiv} considering an high stiffness \(k\). -The micro-station top platform is connected to an inertial mass (the nano-hexapod) through a damper, which dampens the micro-station suspension suspension mode. -\item[{DVF with a stiff nano-hexapod (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_stiff})}] In that case, the ``sky hook damper'' (see mechanical equivalent of \acrshort{dvf} in Figure~\ref{fig:uniaxial_active_damping_dvf_equiv}) is connected to the micro-station top platform through the stiff nano-hexapod. -\item[{RDC with a soft nano-hexapod (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_micro_station_mode})}] At the frequency of the micro-station mode, the nano-hexapod top mass behaves as an inertial reference because the suspension mode of the soft nano-hexapod is at much lower frequency. -The micro-station and the nano-hexapod masses are connected through a large damper induced by \acrshort{rdc} (see mechanical equivalent in Figure~\ref{fig:uniaxial_active_damping_rdc_equiv}) which allows some damping of the micro-station. +\item[{IFF with a stiff active platform (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_stiff})}] This can be understood from the mechanical equivalent of IFF shown in Figure~\ref{fig:uniaxial_active_damping_iff_equiv} considering an high stiffness \(k\). +The micro-station top platform is connected to an inertial mass (the active platform) through a damper, which dampens the micro-station suspension suspension mode. +\item[{DVF with a stiff active platform (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_stiff})}] In that case, the ``sky hook damper'' (see mechanical equivalent of \acrshort{dvf} in Figure~\ref{fig:uniaxial_active_damping_dvf_equiv}) is connected to the micro-station top platform through the stiff active platform. +\item[{RDC with a soft active platform (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_micro_station_mode})}] At the frequency of the micro-station mode, the active platform top mass behaves as an inertial reference because the suspension mode of the soft active platform is at much lower frequency. +The micro-station and the active platform masses are connected through a large damper induced by \acrshort{rdc} (see mechanical equivalent in Figure~\ref{fig:uniaxial_active_damping_rdc_equiv}) which allows some damping of the micro-station. \end{description} \begin{figure}[htbp] @@ -1255,27 +1255,27 @@ The micro-station and the nano-hexapod masses are connected through a large damp \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_soft.png} \end{center} -\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_mid.png} \end{center} -\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_stiff.png} \end{center} -\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:uniaxial_root_locus_damping_techniques}Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three nano-hexapod stiffnesses. The Root Loci are zoomed in the suspension mode of the nano-hexapod.} +\caption{\label{fig:uniaxial_root_locus_damping_techniques}Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The Root Loci are zoomed in the suspension mode of the active platform.} \end{figure} \begin{figure}[htbp] \centering \includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png} -\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root Locus for the three damping techniques applied with the soft nano-hexapod. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.} +\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root Locus for the three damping techniques applied with the soft active platform. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the positioning hexapod.} \end{figure} The transfer functions from the plant input \(f\) to the relative displacement \(d\) while active damping is implemented are shown in Figure~\ref{fig:uniaxial_damped_plant_three_active_damping_techniques}. @@ -1286,36 +1286,36 @@ All three active damping techniques yielded similar damped plants. \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_vc.png} \end{center} -\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_md.png} \end{center} -\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_pz.png} \end{center} -\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} \caption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques}Obtained damped transfer function from \(f\) to \(d\) for the three damping techniques.} \end{figure} \subsubsection{Sensitivity to disturbances and Noise Budgeting} \label{ssec:uniaxial_active_damping_sensitivity_disturbances} -Reasonable gains are chosen for the three active damping strategies such that the nano-hexapod suspension mode is well damped. +Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped. The sensitivity to disturbances (direct forces \(f_s\), stage vibrations \(f_t\) and floor motion \(x_f\)) for all three active damping techniques are compared in Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping}. -The comparison is done with the nano-hexapod having a stiffness \(k_n = 1\,N/\mu m\). +The comparison is done with the active platform having a stiffness \(k_n = 1\,\text{N}/\mu\text{m}\). Several conclusions can be drawn by comparing the obtained sensitivity transfer functions: \begin{itemize} -\item IFF degrades the sensitivity to direct forces on the sample (i.e., the compliance) below the resonance of the nano-hexapod (Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping_fs}). +\item IFF degrades the sensitivity to direct forces on the sample (i.e., the compliance) below the resonance of the active platform (Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping_fs}). This is a well-known effect of using IFF for vibration isolation~\cite{collette15_sensor_fusion_method_high_perfor}. -\item RDC degrades the sensitivity to stage vibrations around the nano-hexapod's resonance as compared to the other two methods (Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping_ft}). +\item RDC degrades the sensitivity to stage vibrations around the active platform's resonance as compared to the other two methods (Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping_ft}). This is because the equivalent damper in parallel with the actuator (see Figure~\ref{fig:uniaxial_active_damping_rdc_equiv}) increases the transmission of the micro-station vibration to the sample which is not the same for the other two active damping strategies. -\item both IFF and \acrshort{dvf} degrade the sensitivity to floor motion below the resonance of the nano-hexapod (Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping_xf}). +\item both IFF and \acrshort{dvf} degrade the sensitivity to floor motion below the resonance of the active platform (Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping_xf}). \end{itemize} \begin{figure}[htbp] @@ -1349,19 +1349,19 @@ All three active damping methods give similar results. \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_soft.png} \end{center} -\subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.31\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_mid.png} \end{center} -\subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.31\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_stiff.png} \end{center} -\subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} \caption{\label{fig:uniaxial_cas_active_damping}Comparison of the cumulative amplitude spectrum (CAS) of the distance \(d\) for all three active damping techniques (\acrshort{ol} in black, IFF in blue, RDC in red and DVF in yellow).} \end{figure} @@ -1371,7 +1371,7 @@ Three active damping strategies have been studied for the \acrfull{nass}. Equivalent mechanical representations were derived in Section~\ref{ssec:uniaxial_active_damping_strategies} which are helpful for understanding the specific effects of each strategy. The plant dynamics were then compared in Section~\ref{ssec:uniaxial_active_damping_plants} and were found to all have alternating poles and zeros, which helps in the design of the active damping controller. However, this property is not guaranteed for \acrshort{dvf}. -The achievable damping of the nano-hexapod suspension mode can be made as large as possible for all three active damping techniques (Section~\ref{ssec:uniaxial_active_damping_achievable_damping}). +The achievable damping of the active platform suspension mode can be made as large as possible for all three active damping techniques (Section~\ref{ssec:uniaxial_active_damping_achievable_damping}). Even some damping can be applied to some micro-station modes in specific cases. The obtained damped plants were found to be similar. The damping strategies were then compared in terms of disturbance reduction in Section~\ref{ssec:uniaxial_active_damping_sensitivity_disturbances}. @@ -1431,10 +1431,10 @@ This control architecture applied to the uniaxial model is shown in Figure~\ref{ \end{figure} \subsubsection{Damped Plant Dynamics} \label{ssec:uniaxial_position_control_damped_dynamics} -The damped plants obtained for the three nano-hexapod stiffnesses are shown in Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses}. -For \(k_n = 0.01\,N/\mu m\) and \(k_n = 1\,N/\mu m\), the dynamics are quite simple and can be well approximated by a second-order plant (Figures~\ref{fig:uniaxial_hac_iff_damped_plants_masses_soft} and \ref{fig:uniaxial_hac_iff_damped_plants_masses_mid}). -However, this is not the case for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)) where two modes can be seen (Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_stiff}). -This is due to the interaction between the micro-station (modeled modes at 70Hz, 140Hz and 320Hz) and the nano-hexapod. +The damped plants obtained for the three active platform stiffnesses are shown in Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses}. +For \(k_n = 0.01\,\text{N}/\mu\text{m}\) and \(k_n = 1\,\text{N}/\mu\text{m}\), the dynamics are quite simple and can be well approximated by a second-order plant (Figures~\ref{fig:uniaxial_hac_iff_damped_plants_masses_soft} and \ref{fig:uniaxial_hac_iff_damped_plants_masses_mid}). +However, this is not the case for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)) where two modes can be seen (Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_stiff}). +This is due to the interaction between the micro-station (modeled modes at \(70\,\text{Hz}\), \(140\,\text{Hz}\) and \(320\,\text{Hz}\)) and the active platform. This effect will be further explained in Section~\ref{sec:uniaxial_support_compliance}. \begin{figure}[htbp] @@ -1442,48 +1442,48 @@ This effect will be further explained in Section~\ref{sec:uniaxial_support_compl \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_soft.png} \end{center} -\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.31\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_mid.png} \end{center} -\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.31\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_stiff.png} \end{center} -\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} \caption{\label{fig:uniaxial_hac_iff_damped_plants_masses}Obtained damped plant using Integral Force Feedback for three sample masses} \end{figure} \subsubsection{Position Feedback Controller} \label{ssec:uniaxial_position_control_design} -The objective is to design high-authority feedback controllers for the three nano-hexapods. +The objective is to design high-authority feedback controllers for the three active platforms. This controller must be robust to the change of sample's mass (from \(1\,\text{kg}\) up to \(50\,\text{kg}\)). The required feedback bandwidths were estimated in Section~\ref{sec:uniaxial_noise_budgeting}: \begin{itemize} -\item \(f_b \approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)). -Near this frequency, the plants (shown in Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_soft}) are equivalent to a mass line (i.e., slope of \(-40\,dB/\text{dec}\) and a phase of -180 degrees). -The gain of this mass line can vary up to a fact \(\approx 5\) (suspended mass from \(16\,kg\) up to \(65\,kg\)). +\item \(f_b \approx 10\,\text{Hz}\) for the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)). +Near this frequency, the plants (shown in Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_soft}) are equivalent to a mass line (i.e., slope of \(-40\,\text{dB/dec}\) and a phase of -180 degrees). +The gain of this mass line can vary up to a fact \(\approx 5\) (suspended mass from \(16\,\text{kg}\) up to \(65\,\text{kg}\)). This means that the designed controller will need to have \emph{large gain margins} to be robust to the change of sample's mass. -\item \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)). -Similar to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line (Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_mid}). -It will probably be easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode. -\item \(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)). -Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_stiff}). +\item \(\approx 50\,\text{Hz}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)). +Similar to the soft active platform, the plants near the crossover frequency are equivalent to a mass line (Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_mid}). +It will probably be easier to have a little bit more bandwidth in this configuration to be further away from the active platform suspension mode. +\item \(\approx 100\,\text{Hz}\) for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)). +Contrary to the two first active platform stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_stiff}). The micro-station is not stiff enough to have a clear stiffness line at this frequency. Therefore, there is both a change of phase and gain depending on the sample mass. This makes the robust design of the controller more complicated. \end{itemize} -Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure~\ref{fig:uniaxial_nyquist_hac}). +Position feedback controllers are designed for each active platform such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure~\ref{fig:uniaxial_nyquist_hac}). An arbitrary minimum modulus margin of \(0.25\) was chosen when designing the controllers. These \acrfullpl{hac} are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency, and a \acrfull{lpf} to increase the robustness to high frequency dynamics. -The controllers used for the three nano-hexapod are shown in Equation~\eqref{eq:uniaxial_hac_formulas}, and the parameters used are summarized in Table~\ref{tab:uniaxial_feedback_controller_parameters}. +The controllers used for the three active platform are shown in Equation~\eqref{eq:uniaxial_hac_formulas}, and the parameters used are summarized in Table~\ref{tab:uniaxial_feedback_controller_parameters}. \begin{subequations} \label{eq:uniaxial_hac_formulas} \begin{align} @@ -1510,17 +1510,17 @@ K_{\text{stiff}}(s) &= g \cdot & \textbf{Soft} & \textbf{Moderately stiff} & \textbf{Stiff}\\ \midrule \textbf{Gain} & \(g = 4 \cdot 10^5\) & \(g = 3 \cdot 10^6\) & \(g = 6 \cdot 10^{12}\)\\ -\textbf{Lead} & \(a = 5\), \(\omega_c = 20\,Hz\) & \(a = 4\), \(\omega_c = 70\,Hz\) & \(a = 5\), \(\omega_c = 100\,Hz\)\\ -\textbf{Lag} & \(\omega_0 = 5\,Hz\), \(\omega_i = 0.01\,Hz\) & \(\omega_0 = 20\,Hz\), \(\omega_i = 0.01\,Hz\) & \(\omega_i = 0.01\,Hz\)\\ -\textbf{LPF} & \(\omega_l = 200\,Hz\) & \(\omega_l = 300\,Hz\) & \(\omega_l = 500\,Hz\)\\ +\textbf{Lead} & \(a = 5\), \(\omega_c = 20\,\text{Hz}\) & \(a = 4\), \(\omega_c = 70\,\text{Hz}\) & \(a = 5\), \(\omega_c = 100\,\text{Hz}\)\\ +\textbf{Lag} & \(\omega_0 = 5\,\text{Hz}\), \(\omega_i = 0.01\,\text{Hz}\) & \(\omega_0 = 20\,\text{Hz}\), \(\omega_i = 0.01\,\text{Hz}\) & \(\omega_i = 0.01\,\text{Hz}\)\\ +\textbf{LPF} & \(\omega_l = 200\,\text{Hz}\) & \(\omega_l = 300\,\text{Hz}\) & \(\omega_l = 500\,\text{Hz}\)\\ \bottomrule \end{tabularx} \end{table} -The loop gains corresponding to the designed \acrlongpl{hac} for the three nano-hexapod are shown in Figure~\ref{fig:uniaxial_loop_gain_hac}. -We can see that for the soft and moderately stiff nano-hexapod (Figures~\ref{fig:uniaxial_nyquist_hac_vc} and \ref{fig:uniaxial_nyquist_hac_md}), the crossover frequency varies significantly with the sample mass. +The loop gains corresponding to the designed \acrlongpl{hac} for the three active platform are shown in Figure~\ref{fig:uniaxial_loop_gain_hac}. +We can see that for the soft and moderately stiff active platform (Figures~\ref{fig:uniaxial_nyquist_hac_vc} and \ref{fig:uniaxial_nyquist_hac_md}), the crossover frequency varies significantly with the sample mass. This is because the crossover frequency corresponds to the mass line of the plant (whose gain is inversely proportional to the mass). -For the stiff nano-hexapod (Figure~\ref{fig:uniaxial_nyquist_hac_pz}), it was difficult to achieve the desired closed-loop bandwidth of \(\approx 100\,\text{Hz}\). +For the stiff active platform (Figure~\ref{fig:uniaxial_nyquist_hac_pz}), it was difficult to achieve the desired closed-loop bandwidth of \(\approx 100\,\text{Hz}\). A crossover frequency of \(\approx 65\,\text{Hz}\) was achieved instead. Note that these controllers were not designed using any optimization methods. @@ -1531,19 +1531,19 @@ The goal is to have a first estimation of the attainable performance. \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_vc.png} \end{center} -\subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_md.png} \end{center} -\subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_pz.png} \end{center} -\subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} \caption{\label{fig:uniaxial_nyquist_hac}Nyquist Plot for the high authority controller. The minimum modulus margin is illustrated by a black circle.} \end{figure} @@ -1553,19 +1553,19 @@ The goal is to have a first estimation of the attainable performance. \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_vc.png} \end{center} -\subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_md.png} \end{center} -\subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_pz.png} \end{center} -\subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} \caption{\label{fig:uniaxial_loop_gain_hac}Loop gains for the High Authority Controllers} \end{figure} @@ -1573,7 +1573,7 @@ The goal is to have a first estimation of the attainable performance. \label{ssec:uniaxial_position_control_cl_noise_budget} The \acrlong{hac} are then implemented and the closed-loop sensitivities to disturbances are computed. -These are compared with the open-loop and damped plants cases in Figure~\ref{fig:uniaxial_sensitivity_dist_hac_lac} for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass). +These are compared with the open-loop and damped plants cases in Figure~\ref{fig:uniaxial_sensitivity_dist_hac_lac} for just one configuration (moderately stiff active platform with \(25\,\text{kg}\) sample's mass). As expected, the sensitivity to disturbances decreased in the controller bandwidth and slightly increased outside this bandwidth. \begin{figure}[htbp] @@ -1595,75 +1595,75 @@ As expected, the sensitivity to disturbances decreased in the controller bandwid \end{center} \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_xf}Floor motion} \end{subfigure} -\caption{\label{fig:uniaxial_sensitivity_dist_hac_lac}Change of sensitivity to disturbances with \acrshort{lac} and with \acrshort{haclac}. A nano-Hexapod with \(k_n = 1\,N/\mu m\) and a sample mass of \(25\,kg\) is used. \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), \(f_t\) disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs})} +\caption{\label{fig:uniaxial_sensitivity_dist_hac_lac}Change of sensitivity to disturbances with \acrshort{lac} and with \acrshort{haclac}. An active platform with \(k_n = 1\,\text{N}/\mu\text{m}\) and a sample mass of \(25\,\text{kg}\) is used. \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), \(f_t\) disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs})} \end{figure} -The cumulative amplitude spectrum of the motion \(d\) is computed for all nano-hexapod configurations, all sample masses and in the \acrfull{ol}, damped (IFF) and position controlled (HAC-IFF) cases. +The cumulative amplitude spectrum of the motion \(d\) is computed for all active platform configurations, all sample masses and in the \acrfull{ol}, damped (IFF) and position controlled (HAC-IFF) cases. The results are shown in Figure~\ref{fig:uniaxial_cas_hac_lac}. -Obtained root mean square values of the distance \(d\) are better for the soft nano-hexapod (\(\approx 25\,nm\) to \(\approx 35\,nm\) depending on the sample's mass) than for the stiffer nano-hexapod (from \(\approx 30\,nm\) to \(\approx 70\,nm\)). +Obtained root mean square values of the distance \(d\) are better for the soft active platform (\(\approx 25\,\text{nm}\) to \(\approx 35\,\text{nm}\) depending on the sample's mass) than for the stiffer active platform (from \(\approx 30\,\text{nm}\) to \(\approx 70\,\text{nm}\)). \begin{figure}[htbp] \begin{subfigure}{0.37\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_soft.png} \end{center} -\subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.31\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_mid.png} \end{center} -\subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.31\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_stiff.png} \end{center} -\subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:uniaxial_cas_hac_lac}Cumulative Amplitude Spectrum for all three nano-hexapod stiffnesses - Comparison of OL, IFF and \acrshort{haclac} cases} +\caption{\label{fig:uniaxial_cas_hac_lac}Cumulative Amplitude Spectrum for all three active platform stiffnesses - Comparison of OL, IFF and \acrshort{haclac} cases} \end{figure} \subsubsection{Conclusion} -On the basis of the open-loop noise budgeting made in Section~\ref{sec:uniaxial_noise_budgeting}, the closed-loop bandwidth required to obtain a vibration level of \(\approx 20\,nm\,\text{RMS}\) was estimated. +On the basis of the open-loop noise budgeting made in Section~\ref{sec:uniaxial_noise_budgeting}, the closed-loop bandwidth required to obtain a vibration level of \(\approx 20\,\text{nm RMS}\) was estimated. To achieve such bandwidth, the \acrshort{haclac} strategy was followed, which consists of first using an active damping controller (studied in Section~\ref{sec:uniaxial_active_damping}) and then adding a high authority position feedback controller. In this section, feedback controllers were designed in such a way that the required closed-loop bandwidth was reached while being robust to changes in the payload mass. -The attainable vibration control performances were estimated for the three nano-hexapod stiffnesses and were found to be close to the required values. -However, the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. -A slight advantage can be given to the soft nano-hexapod as it requires less feedback bandwidth while providing better stability results. +The attainable vibration control performances were estimated for the three active platform stiffnesses and were found to be close to the required values. +However, the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. +A slight advantage can be given to the soft active platform as it requires less feedback bandwidth while providing better stability results. \subsection{Effect of limited micro-station compliance} \label{sec:uniaxial_support_compliance} In this section, the impact of the compliance of the support (i.e., the micro-station) on the dynamics of the plant to control is studied. This is a critical point because the dynamics of the micro-station is complex, depends on the considered direction (see measurements in Figure~\ref{fig:uniaxial_comp_frf_meas_model}) and may vary with position and time. It would be much better to have a plant dynamics that is not impacted by the micro-station. -Therefore, the objective of this section is to obtain some guidance for the design of a nano-hexapod that will not be impacted by the complex micro-station dynamics. +Therefore, the objective of this section is to obtain some guidance for the design of a active platform that will not be impacted by the complex micro-station dynamics. To study this, two models are used (Figure~\ref{fig:uniaxial_support_compliance_models}). -The first one consists of the nano-hexapod directly fixed on top of the granite, thus neglecting any support compliance (Figure~\ref{fig:uniaxial_support_compliance_nano_hexapod_only}). -The second one consists of the nano-hexapod fixed on top of the micro-station having some limited compliance (Figure~\ref{fig:uniaxial_support_compliance_test_system}) +The first one consists of the active platform directly fixed on top of the granite, thus neglecting any support compliance (Figure~\ref{fig:uniaxial_support_compliance_nano_hexapod_only}). +The second one consists of the active platform fixed on top of the micro-station having some limited compliance (Figure~\ref{fig:uniaxial_support_compliance_test_system}) \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} \begin{center} \includegraphics[scale=1,scale=1]{figs/uniaxial_support_compliance_nano_hexapod_only.png} \end{center} -\subcaption{\label{fig:uniaxial_support_compliance_nano_hexapod_only}Nano-Hexapod fixed directly on the Granite} +\subcaption{\label{fig:uniaxial_support_compliance_nano_hexapod_only}Active platform fixed directly on the Granite} \end{subfigure} \begin{subfigure}{0.49\textwidth} \begin{center} \includegraphics[scale=1,scale=1]{figs/uniaxial_support_compliance_test_system.png} \end{center} -\subcaption{\label{fig:uniaxial_support_compliance_test_system}Nano-Hexapod fixed on top of the Micro-Station} +\subcaption{\label{fig:uniaxial_support_compliance_test_system}Active platform fixed on top of the Micro-Station} \end{subfigure} \caption{\label{fig:uniaxial_support_compliance_models}Models used to study the effect of limited support compliance} \end{figure} \subsubsection{Neglected support compliance} The limited compliance of the micro-station is first neglected and the uniaxial model shown in Figure~\ref{fig:uniaxial_support_compliance_nano_hexapod_only} is used. -The nano-hexapod mass (including the payload) is set at \(20\,\text{kg}\) and three hexapod stiffnesses are considered, such that their resonance frequencies are at \(\omega_{n} = 10\,\text{Hz}\), \(\omega_{n} = 70\,\text{Hz}\) and \(\omega_{n} = 400\,\text{Hz}\). +The active platform mass (including the payload) is set at \(20\,\text{kg}\) and three active platform stiffnesses are considered, such that their resonance frequencies are at \(\omega_{n} = 10\,\text{Hz}\), \(\omega_{n} = 70\,\text{Hz}\) and \(\omega_{n} = 400\,\text{Hz}\). Obtained transfer functions from \(F\) to \(L^\prime\) (shown in Figure~\ref{fig:uniaxial_effect_support_compliance_neglected}) are simple second-order low-pass filters. -When neglecting the support compliance, a large feedback bandwidth can be achieved for all three nano-hexapods. +When neglecting the support compliance, a large feedback bandwidth can be achieved for all three active platforms. \begin{figure}[htbp] \begin{subfigure}{0.37\textwidth} @@ -1690,12 +1690,12 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev Some support compliance is now added and the model shown in Figure~\ref{fig:uniaxial_support_compliance_test_system} is used. The parameters of the support (i.e., \(m_{\mu}\), \(c_{\mu}\) and \(k_{\mu}\)) are chosen to match the vertical mode at \(70\,\text{Hz}\) seen on the micro-station (Figure~\ref{fig:uniaxial_comp_frf_meas_model}). -The transfer functions from \(F\) to \(L\) (i.e., control of the relative motion of the nano-hexapod) and from \(L\) to \(d\) (i.e., control of the position between the nano-hexapod and the fixed granite) can then be computed. +The transfer functions from \(F\) to \(L\) (i.e., control of the relative motion of the active platform) and from \(L\) to \(d\) (i.e., control of the position between the active platform and the fixed granite) can then be computed. -When the relative displacement of the nano-hexapod \(L\) is controlled (dynamics shown in Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics}), having a stiff nano-hexapod (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_stiff}). +When the relative displacement of the active platform \(L\) is controlled (dynamics shown in Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics}), having a stiff active platform (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_stiff}). This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages. In such a case, the control of the piezoelectric stage using its integrated metrology (typically capacitive sensors) is quite simple as the plant is not much affected by the dynamics of the support on which it is fixed. -If a soft nano-hexapod is used, the support dynamics appears in the dynamics between \(F\) and \(L\) (see Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_soft}) which will impact the control robustness and performance. +If a soft active platform is used, the support dynamics appears in the dynamics between \(F\) and \(L\) (see Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_soft}) which will impact the control robustness and performance. \begin{figure}[htbp] \begin{subfigure}{0.37\textwidth} @@ -1720,9 +1720,9 @@ If a soft nano-hexapod is used, the support dynamics appears in the dynamics bet \end{figure} \subsubsection{Effect of support compliance on \(d/F\)} -When the motion to be controlled is the relative displacement \(d\) between the granite and the nano-hexapod's top platform (which is the case for the \acrshort{nass}), the effect of the support compliance on the plant dynamics is opposite to that previously observed. -Indeed, using a ``soft'' nano-hexapod (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_d_soft}). -Conversely, if a ``stiff'' nano-hexapod is used, the support dynamics appears in the plant dynamics (Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_d_stiff}). +When the motion to be controlled is the relative displacement \(d\) between the granite and the active platform's top platform (which is the case for the \acrshort{nass}), the effect of the support compliance on the plant dynamics is opposite to that previously observed. +Indeed, using a ``soft'' active platform (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_d_soft}). +Conversely, if a ``stiff'' active platform is used, the support dynamics appears in the plant dynamics (Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_d_stiff}). \begin{figure}[htbp] \begin{subfigure}{0.37\textwidth} @@ -1748,10 +1748,10 @@ Conversely, if a ``stiff'' nano-hexapod is used, the support dynamics appears in \subsubsection{Conclusion} To study the impact of support compliance on plant dynamics, simple models shown in Figure~\ref{fig:uniaxial_support_compliance_models} were used. -Depending on the quantity to be controlled (\(L\) or \(d\) in Figure~\ref{fig:uniaxial_support_compliance_test_system}) and on the relative location of \(\omega_\nu\) (suspension mode of the nano-hexapod) with respect to \(\omega_\mu\) (modes of the support), the interaction between the support and the nano-hexapod dynamics can drastically change (observations made are summarized in Table~\ref{tab:uniaxial_effect_compliance}). +Depending on the quantity to be controlled (\(L\) or \(d\) in Figure~\ref{fig:uniaxial_support_compliance_test_system}) and on the relative location of \(\omega_\nu\) (suspension mode of the active platform) with respect to \(\omega_\mu\) (modes of the support), the interaction between the support and the active platform dynamics can drastically change (observations made are summarized in Table~\ref{tab:uniaxial_effect_compliance}). -For the \acrfull{nass}, having the suspension mode of the nano-hexapod at lower frequencies than the suspension modes of the micro-station would make the plant less dependent on the micro-station dynamics, and therefore easier to control. -Note that the observations made in this section are also affected by the ratio between the support mass \(m_{\mu}\) and the nano-hexapod mass \(m_n\) (the effect is more pronounced when the ratio \(m_n/m_{\mu}\) increases). +For the \acrfull{nass}, having the suspension mode of the active platform at lower frequencies than the suspension modes of the micro-station would make the plant less dependent on the micro-station dynamics, and therefore easier to control. +Note that the observations made in this section are also affected by the ratio between the support mass \(m_{\mu}\) and the active platform mass \(m_n\) (the effect is more pronounced when the ratio \(m_n/m_{\mu}\) increases). \begin{table}[htbp] \caption{\label{tab:uniaxial_effect_compliance}Impact of the support dynamics on the plant dynamics} @@ -1768,8 +1768,8 @@ Note that the observations made in this section are also affected by the ratio b \subsection{Effect of Payload Dynamics} \label{sec:uniaxial_payload_dynamics} -Up to this section, the sample was modeled as a mass rigidly fixed to the nano-hexapod (as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic}). -However, such a sample may present internal dynamics, and its fixation to the nano-hexapod may have limited stiffness. +Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic}). +However, such a sample may present internal dynamics, and its fixation to the active platform may have limited stiffness. To study the effect of the sample dynamics, the models shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_schematic} are used. \begin{figure}[htbp] @@ -1790,11 +1790,11 @@ To study the effect of the sample dynamics, the models shown in Figure~\ref{fig: \subsubsection{Impact on plant dynamics} \label{ssec:uniaxial_payload_dynamics_effect_dynamics} -To study the impact of the flexibility between the nano-hexapod and the payload, a first (reference) model with a rigid payload, as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic} is used. +To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic} is used. Then ``flexible'' payload whose model is shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_schematic} are considered. The resonances of the payload are set at \(\omega_s = 20\,\text{Hz}\) and at \(\omega_s = 200\,\text{Hz}\) while its mass is either \(m_s = 1\,\text{kg}\) or \(m_s = 50\,\text{kg}\). -The transfer functions from the nano-hexapod force \(f\) to the motion of the nano-hexapod top platform are computed for all the above configurations and are compared for a soft Nano-Hexapod (\(k_n = 0.01\,N/\mu m\)) in Figure~\ref{fig:uniaxial_payload_dynamics_soft_nano_hexapod}. +The transfer functions from the active platform force \(f\) to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_soft_nano_hexapod}. It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern). The frequency of the anti-resonance corresponds to the ``free'' resonance of the sample \(\omega_s = \sqrt{k_s/m_s}\). The flexibility of the sample also changes the high frequency gain (the mass line is shifted from \(\frac{1}{(m_n + m_s)s^2}\) to \(\frac{1}{m_ns^2}\)). @@ -1804,54 +1804,54 @@ The flexibility of the sample also changes the high frequency gain (the mass lin \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_light.png} \end{center} -\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,N/\mu m$, $m_s = 1\,kg$} +\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$} \end{subfigure} \begin{subfigure}{0.49\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_heavy.png} \end{center} -\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,N/\mu m$, $m_s = 50\,kg$} +\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$} \end{subfigure} -\caption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod}Effect of the payload dynamics on the soft Nano-Hexapod. Light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy})} +\caption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod}Effect of the payload dynamics on the soft active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy})} \end{figure} -The same transfer functions are now compared when using a stiff nano-hexapod (\(k_n = 100\,N/\mu m\)) in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}. -In this case, the sample's resonance \(\omega_s\) is smaller than the nano-hexapod resonance \(\omega_n\). +The same transfer functions are now compared when using a stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}. +In this case, the sample's resonance \(\omega_s\) is smaller than the active platform resonance \(\omega_n\). This changes the zero/pole pattern to a pole/zero pattern (the frequency of the zero still being equal to \(\omega_s\)). -Even though the added sample's flexibility still shifts the high frequency mass line as for the soft nano-hexapod, the dynamics below the nano-hexapod resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}). +Even though the added sample's flexibility still shifts the high frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}). \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_light.png} \end{center} -\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,N/\mu m$, $m_s = 1\,kg$} +\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$} \end{subfigure} \begin{subfigure}{0.49\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_heavy.png} \end{center} -\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,N/\mu m$, $m_s = 50\,kg$} +\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$} \end{subfigure} -\caption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy})} +\caption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}Effect of the payload dynamics on the stiff active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy})} \end{figure} \subsubsection{Impact on close loop performances} \label{ssec:uniaxial_payload_dynamics_effect_stability} -Having a flexibility between the measured position (i.e., the top platform of the nano-hexapod) and the \acrshort{poi} to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). +Having a flexibility between the measured position (i.e., the top platform of the active platform) and the \acrshort{poi} to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). To estimate whether the sample flexibility is critical for the closed-loop position stability of the sample, the model shown in Figure~\ref{fig:uniaxial_sample_flexibility_control} is used. -This is the same model that was used in Section~\ref{sec:uniaxial_position_control} but with an added flexibility between the nano-hexapod and the sample (considered sample modes are at \(\omega_s = 20\,\text{Hz}\) and \(\omega_n = 200\,\text{Hz}\)). +This is the same model that was used in Section~\ref{sec:uniaxial_position_control} but with an added flexibility between the active platform and the sample (considered sample modes are at \(\omega_s = 20\,\text{Hz}\) and \(\omega_n = 200\,\text{Hz}\)). In this case, the measured (i.e., controlled) distance \(d\) is no longer equal to the real performance index (the distance \(y\)). \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/uniaxial_sample_flexibility_control.png} -\caption{\label{fig:uniaxial_sample_flexibility_control}Uniaxial model considering some flexibility between the nano-hexapod top platform and the sample. In this case, the measured and controlled distance \(d\) is different from the distance \(y\) which is the real performance index} +\caption{\label{fig:uniaxial_sample_flexibility_control}Uniaxial model considering some flexibility between the active platform top platform and the sample. In this case, the measured and controlled distance \(d\) is different from the distance \(y\) which is the real performance index} \end{figure} The system dynamics is computed and IFF is applied using the same gains as those used in Section~\ref{sec:uniaxial_active_damping}. -Due to the collocation between the nano-hexapod and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample. +Due to the collocation between the active platform and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample. The \acrlong{hac} used in Section~\ref{sec:uniaxial_position_control} are then implemented on the damped plants. The obtained closed-loop systems are stable, indicating good robustness. @@ -1859,7 +1859,7 @@ Finally, closed-loop noise budgeting is computed for the obtained closed-loop sy The cumulative amplitude spectrum of the measured distance \(d\) (Figure~\ref{fig:uniaxial_sample_flexibility_noise_budget_d}) shows that the added flexibility at the sample location has very little effect on the control performance. However, the cumulative amplitude spectrum of the distance \(y\) (Figure~\ref{fig:uniaxial_sample_flexibility_noise_budget_y}) shows that the stability of \(y\) is degraded when the sample flexibility is considered and is degraded as \(\omega_s\) is lowered. -What happens is that above \(\omega_s\), even though the motion \(d\) can be controlled perfectly, the sample's mass is ``isolated'' from the motion of the nano-hexapod and the control on \(y\) is not effective. +What happens is that above \(\omega_s\), even though the motion \(d\) can be controlled perfectly, the sample's mass is ``isolated'' from the motion of the active platform and the control on \(y\) is not effective. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -1874,36 +1874,36 @@ What happens is that above \(\omega_s\), even though the motion \(d\) can be con \end{center} \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_y}Cumulative Amplitude Spectrum of $y$} \end{subfigure} -\caption{\label{fig:uniaxial_sample_flexibility_noise_budget}Cumulative Amplitude Spectrum of the distances \(d\) and \(y\). The effect of the sample's flexibility does not affect much \(d\) but is detrimental to the stability of \(y\). A sample mass \(m_s = 1\,\text{kg}\) and a nano-hexapod stiffness of \(100\,N/\mu m\) are used for the simulations.} +\caption{\label{fig:uniaxial_sample_flexibility_noise_budget}Cumulative Amplitude Spectrum of the distances \(d\) and \(y\). The effect of the sample's flexibility does not affect much \(d\) but is detrimental to the stability of \(y\). A sample mass \(m_s = 1\,\text{kg}\) and a active platform stiffness of \(100\,\text{N}/\mu\text{m}\) are used for the simulations.} \end{figure} \subsubsection{Conclusion} Payload dynamics is usually a major concern when designing a positioning system. -In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample \(\omega_s\) and of the nano-hexapod \(\omega_n\). +In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample \(\omega_s\) and of the active platform \(\omega_n\). The larger the sample mass, the larger the effect (i.e., change of high frequency gain, appearance of additional resonances and anti-resonances). A zero/pole pattern is observed if \(\omega_s > \omega_n\) and a pole/zero pattern if \(\omega_s > \omega_n\). Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in~\cite[Section 4.2]{rankers98_machin}. -The general conclusion is that the stiffer the nano-hexapod, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload. +The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload. This is why high-bandwidth soft positioning stages are usually restricted to constant and calibrated payloads (CD-player, lithography machines, isolation system for gravitational wave detectors, \ldots{}), whereas stiff positioning systems are usually used when the control must be robust to a change of payload's mass (stiff piezo nano-positioning stages for instance). Having some flexibility between the measurement point and the \acrshort{poi} (i.e., the sample point to be position on the x-ray) also degrades the position stability as shown in Section~\ref{ssec:uniaxial_payload_dynamics_effect_stability}. -Therefore, it is important to take special care when designing sampling environments, especially if a soft nano-hexapod is used. +Therefore, it is important to take special care when designing sampling environments, especially if a soft active platform is used. \subsection*{Conclusion} \label{sec:uniaxial_conclusion} In this study, a uniaxial model of the nano-active-stabilization-system was tuned from both dynamical measurements (Section~\ref{sec:uniaxial_micro_station_model}) and from disturbances measurements (Section~\ref{sec:uniaxial_disturbances}). -Three active damping techniques can be used to critically damp the nano-hexapod resonances (Section~\ref{sec:uniaxial_active_damping}). +Three active damping techniques can be used to critically damp the active platform resonances (Section~\ref{sec:uniaxial_active_damping}). However, this model does not allow the determination of which one is most suited to this application (a comparison of the three active damping techniques is done in Table~\ref{tab:comp_active_damping}). -Position feedback controllers have been developed for three considered nano-hexapod stiffnesses (Section~\ref{sec:uniaxial_position_control}). +Position feedback controllers have been developed for three considered active platform stiffnesses (Section~\ref{sec:uniaxial_position_control}). These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances. -Having a soft nano-hexapod makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section~\ref{sec:uniaxial_support_compliance}) and requires less position feedback bandwidth to fulfill the requirements. -The moderately stiff nano-hexapod (\(k_n = 1\,N/\mu m\)) is requiring a higher feedback bandwidth, but still gives acceptable results. -However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performance. +Having a soft active platform makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section~\ref{sec:uniaxial_support_compliance}) and requires less position feedback bandwidth to fulfill the requirements. +The moderately stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)) is requiring a higher feedback bandwidth, but still gives acceptable results. +However, the stiff active platform is the most complex to control and gives the worst positioning performance. \section{Effect of Rotation} \label{sec:rotating} -An important aspect of the \acrfull{nass} is that the nano-hexapod continuously rotates around a vertical axis, whereas the external metrology is not. +An important aspect of the \acrfull{nass} is that the active platform continuously rotates around a vertical axis, whereas the external metrology is not. Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance. To study these effects, a model of a rotating suspended platform is first presented (Section~\ref{sec:rotating_system_description}) This model is simple enough to be able to derive its dynamics analytically and to understand its behavior, while still allowing the capture of important physical effects in play. @@ -1921,9 +1921,9 @@ This study of adapting \acrshort{iff} for the damping of rotating platforms has It is then shown that \acrfull{rdc} is less affected by gyroscopic effects (Section~\ref{sec:rotating_relative_damp_control}). Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, damped plant and closed-loop compliance and transmissibility (Section~\ref{sec:rotating_comp_act_damp}). -The previous analysis was applied to three considered nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\), \(k_n = 1\,N/\mu m\) and \(k_n = 100\,N/\mu m\)) and the optimal active damping controller was obtained in each case (Section~\ref{sec:rotating_nano_hexapod}). +The previous analysis was applied to three considered active platform stiffnesses (\(k_n = 0.01\,\text{N}/\mu\text{m}\), \(k_n = 1\,\text{N}/\mu\text{m}\) and \(k_n = 100\,\text{N}/\mu\text{m}\)) and the optimal active damping controller was obtained in each case (Section~\ref{sec:rotating_nano_hexapod}). Up until this section, the study was performed on a very simplistic model that only captures the rotation aspect, and the model parameters were not tuned to correspond to the NASS. -In the last section (Section~\ref{sec:rotating_nass}), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. +In the last section (Section~\ref{sec:rotating_nass}), a model of the micro-station is added below the active platform with a rotating spindle and parameters tuned to match the NASS dynamics. The goal is to determine whether the rotation imposes performance limitation on the NASS. \subsection{System Description and Analysis} \label{sec:rotating_system_description} @@ -2533,18 +2533,18 @@ This is very well known characteristics of these common active damping technique \end{subfigure} \caption{\label{fig:rotating_comp_techniques_trans_compliance}Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques} \end{figure} -\subsection{Rotating Nano-Hexapod} +\subsection{Rotating Active Platform} \label{sec:rotating_nano_hexapod} -The previous analysis is now applied to a model representing a rotating nano-hexapod. -Three nano-hexapod stiffnesses are tested as for the uniaxial model: \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\). +The previous analysis is now applied to a model representing a rotating active platform. +Three active platform stiffnesses are tested as for the uniaxial model: \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\). Only the maximum rotating velocity is here considered (\(\Omega = \SI{60}{rpm}\)) with the light sample (\(m_s = \SI{1}{kg}\)) because this is the worst identified case scenario in terms of gyroscopic effects. \subsubsection{Nano-Active-Stabilization-System - Plant Dynamics} -For the NASS, the maximum rotating velocity is \(\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}\) for a suspended mass on top of the nano-hexapod's actuators equal to \(m_n + m_s = \SI{16}{\kilo\gram}\). +For the NASS, the maximum rotating velocity is \(\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}\) for a suspended mass on top of the active platform's actuators equal to \(m_n + m_s = \SI{16}{\kilo\gram}\). The parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \approx \SI{0.6}{\newton\per\mm}\). -The transfer functions from the nano-hexapod actuator force \(F_u\) to the displacement of the nano-hexapod in the same direction \(d_u\) as well as in the orthogonal direction \(d_v\) (coupling) are shown in Figure~\ref{fig:rotating_nano_hexapod_dynamics} for all three considered nano-hexapod stiffnesses. -The soft nano-hexapod is the most affected by rotation. -This can be seen by the large shift of the resonance frequencies, and by the induced coupling, which is larger than that for the stiffer nano-hexapods. +The transfer functions from the active platform actuator force \(F_u\) to the displacement of the active platform in the same direction \(d_u\) as well as in the orthogonal direction \(d_v\) (coupling) are shown in Figure~\ref{fig:rotating_nano_hexapod_dynamics} for all three considered active platform stiffnesses. +The soft active platform is the most affected by rotation. +This can be seen by the large shift of the resonance frequencies, and by the induced coupling, which is larger than that for the stiffer active platforms. The coupling (or interaction) in a \acrshort{mimo} \(2 \times 2\) system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). \begin{figure}[htbp] @@ -2552,24 +2552,24 @@ The coupling (or interaction) in a \acrshort{mimo} \(2 \times 2\) system can be \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_vc.png} \end{center} -\subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_md.png} \end{center} -\subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_pz.png} \end{center} -\subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity (\(\Omega = 60\,\text{rpm}\)), and shaded lines are coupling terms at maximum rotating velocity} +\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the active platform dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity (\(\Omega = 60\,\text{rpm}\)), and shaded lines are coupling terms at maximum rotating velocity} \end{figure} \subsubsection{Optimal IFF with a High-Pass Filter} -Integral Force Feedback with an added \acrlong{hpf} is applied to the three nano-hexapods. +Integral Force Feedback with an added \acrlong{hpf} is applied to the three active platforms. First, the parameters (\(\omega_i\) and \(g\)) of the IFF controller that yield the best simultaneous damping are determined from Figure~\ref{fig:rotating_iff_hpf_nass_optimal_gain}. The IFF parameters are chosen as follows: \begin{itemize} @@ -2585,19 +2585,19 @@ The obtained IFF parameters and the achievable damping are visually shown by lar \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_vc.png} \end{center} -\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_md.png} \end{center} -\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_pz.png} \end{center} -\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} \caption{\label{fig:rotating_iff_hpf_nass_optimal_gain}For each value of \(\omega_i\), the maximum damping ratio \(\xi\) is computed (blue), and the corresponding controller gain is shown (in red). The chosen controller parameters used for further analysis are indicated by the large dots.} \end{figure} @@ -2609,24 +2609,24 @@ The obtained IFF parameters and the achievable damping are visually shown by lar \toprule \(k_n\) & \(\omega_i\) & \(g\) & \(\xi_\text{opt}\)\\ \midrule -\(0.01\,N/\mu m\) & 7.3 & 51 & 0.45\\ -\(1\,N/\mu m\) & 39 & 427 & 0.93\\ -\(100\,N/\mu m\) & 500 & 3775 & 0.94\\ +\(0.01\,\text{N}/\mu\text{m}\) & 7.3 & 51 & 0.45\\ +\(1\,\text{N}/\mu\text{m}\) & 39 & 427 & 0.93\\ +\(100\,\text{N}/\mu\text{m}\) & 500 & 3775 & 0.94\\ \bottomrule \end{tabularx} \end{table} \subsubsection{Optimal IFF with Parallel Stiffness} -For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness). -To keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased (\(k_a = k_n - k_p\), with \(k_n\) the total nano-hexapod stiffness). +For each considered active platform stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total active platform stiffness). +To keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased (\(k_a = k_n - k_p\), with \(k_n\) the total active platform stiffness). A high-pass filter is also added to limit the low-frequency gain with a cut-off frequency \(\omega_i\) equal to one tenth of the system resonance (\(\omega_i = \omega_0/10\)). The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure~\ref{fig:rotating_iff_kp_nass_optimal_gain}). -It is shown that the soft nano-hexapod cannot yield good damping because the parallel stiffness cannot be sufficiently large compared to the negative stiffness induced by the rotation. +It is shown that the soft active platform cannot yield good damping because the parallel stiffness cannot be sufficiently large compared to the negative stiffness induced by the rotation. For the two stiff options, the achievable damping decreases when the parallel stiffness is too high, as explained in Section~\ref{sec:rotating_iff_parallel_stiffness}. Such behavior can be explained by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero~\cite[chapt 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}. -This distance is larger for stiff nano-hexapod because the open-loop pole will be at higher frequencies while the open-loop zero, whereas depends on the value of the parallel stiffness, can only be made large for stiff nano-hexapods. +This distance is larger for stiff active platform because the open-loop pole will be at higher frequencies while the open-loop zero, whereas depends on the value of the parallel stiffness, can only be made large for stiff active platforms. -Let's choose \(k_p = 1\,N/mm\), \(k_p = 0.01\,N/\mu m\) and \(k_p = 1\,N/\mu m\) for the three considered nano-hexapods. +Let's choose \(k_p = 1\,\text{N/mm}\), \(k_p = 0.01\,\text{N}/\mu\text{m}\) and \(k_p = 1\,\text{N}/\mu\text{m}\) for the three considered active platforms. The corresponding optimal controller gains and achievable damping are summarized in Table~\ref{tab:rotating_iff_kp_opt_iff_kp_params_nass}. \begin{minipage}[b]{0.49\linewidth} @@ -2643,15 +2643,15 @@ The corresponding optimal controller gains and achievable damping are summarized \toprule \(k_n\) & \(k_p\) & \(g\) & \(\xi_{\text{opt}}\)\\ \midrule -\(0.01\,N/\mu m\) & \(1\,N/mm\) & 47.9 & 0.44\\ -\(1\,N/\mu m\) & \(0.01\,N/\mu m\) & 465.57 & 0.97\\ -\(100\,N/\mu m\) & \(1\,N/\mu m\) & 4624.25 & 0.99\\ +\(0.01\,\text{N}/\mu\text{m}\) & \(1\,\text{N/mm}\) & 47.9 & 0.44\\ +\(1\,\text{N}/\mu\text{m}\) & \(0.01\,\text{N}/\mu\text{m}\) & 465.57 & 0.97\\ +\(100\,\text{N}/\mu\text{m}\) & \(1\,\text{N}/\mu\text{m}\) & 4624.25 & 0.99\\ \bottomrule \end{tabularx}} \captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses} \end{minipage} \subsubsection{Optimal Relative Motion Control} -For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure~\ref{fig:rotating_rdc_optimal_gain}). +For each considered active platform stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure~\ref{fig:rotating_rdc_optimal_gain}). The gain is chosen such that 99\% of modal damping is obtained (obtained gains are summarized in Table~\ref{tab:rotating_rdc_opt_params_nass}). \begin{minipage}[b]{0.49\linewidth} @@ -2668,9 +2668,9 @@ The gain is chosen such that 99\% of modal damping is obtained (obtained gains a \toprule \(k_n\) & \(g\) & \(\xi_{\text{opt}}\)\\ \midrule -\(0.01\,N/\mu m\) & 1600 & 0.99\\ -\(1\,N/\mu m\) & 8200 & 0.99\\ -\(100\,N/\mu m\) & 80000 & 0.99\\ +\(0.01\,\text{N}/\mu\text{m}\) & 1600 & 0.99\\ +\(1\,\text{N}/\mu\text{m}\) & 8200 & 0.99\\ +\(100\,\text{N}/\mu\text{m}\) & 80000 & 0.99\\ \bottomrule \end{tabularx}} \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC} @@ -2681,8 +2681,8 @@ Now that the optimal parameters for the three considered active damping techniqu Similar to what was concluded in the previous analysis: \begin{itemize} \item \acrshort{iff} adds more coupling below the resonance frequency as compared to the open-loop and \acrshort{rdc} cases -\item All three methods yield good damping, except for \acrshort{iff} applied on the soft nano-hexapod -\item Coupling is smaller for stiff nano-hexapods +\item All three methods yield good damping, except for \acrshort{iff} applied on the soft active platform +\item Coupling is smaller for stiff active platforms \end{itemize} \begin{figure}[htbp] @@ -2690,21 +2690,21 @@ Similar to what was concluded in the previous analysis: \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_vc.png} \end{center} -\subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_md.png} \end{center} -\subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_pz.png} \end{center} -\subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with \(k_p\) in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is \(\Omega = 60\,\text{rpm}\) and the suspended mass is \(m_n + m_s = \SI{16}{\kg}\).} +\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with \(k_p\) in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three active platform stiffnesses are considered. For this analysis the rotating velocity is \(\Omega = 60\,\text{rpm}\) and the suspended mass is \(m_n + m_s = \SI{16}{\kg}\).} \end{figure} \subsection{Nano-Active-Stabilization-System with rotation} \label{sec:rotating_nass} @@ -2712,16 +2712,16 @@ Until now, the model used to study gyroscopic effects consisted of an infinitely While quite simplistic, this allowed us to study the effects of rotation and the associated limitations when active damping is to be applied. In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. \subsubsection{Nano Active Stabilization System model} -To have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modeled as shown in Figure~\ref{fig:rotating_3dof_model_schematic}) is now located on top of a model of the micro-station including (see Figure~\ref{fig:rotating_nass_model} for a 3D view): +To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modeled as shown in Figure~\ref{fig:rotating_3dof_model_schematic}) is now located on top of a model of the micro-station including (see Figure~\ref{fig:rotating_nass_model} for a 3D view): \begin{itemize} \item the floor whose motion is imposed \item a 2-DoF granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}\), \(m_g = \SI{2500}{\kg}\)) \item a 2-DoF \(T_y\) stage (\(k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}\), \(m_t = \SI{600}{\kg}\)) \item a spindle (vertical rotation) stage whose rotation is imposed (\(m_s = \SI{600}{\kg}\)) -\item a 2-DoF micro-hexapod (\(k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}\), \(m_h = \SI{15}{\kg}\)) +\item a 2-DoF positioning hexapod (\(k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}\), \(m_h = \SI{15}{\kg}\)) \end{itemize} -A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the payload is measured with respect to the granite. +A payload is rigidly fixed to the active platform and the \(x,y\) motion of the payload is measured with respect to the granite. \begin{figure}[htbp] \centering @@ -2734,9 +2734,9 @@ The dynamics of the undamped and damped plants are identified using the optimal The obtained dynamics are compared in Figure~\ref{fig:rotating_nass_plant_comp_stiffness} in which the direct terms are shown by the solid curves and the coupling terms are shown by the shaded ones. It can be observed that: \begin{itemize} -\item The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod +\item The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft active platform \item Damping added using the three proposed techniques is quite high, and the obtained plant is rather easy to control -\item There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz) +\item There is some coupling between active platform and micro-station dynamics for the stiff active platform (mode at \(200\,\text{Hz}\)) \item The two proposed IFF modifications yield similar results \end{itemize} @@ -2745,21 +2745,21 @@ It can be observed that: \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_vc.png} \end{center} -\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_md.png} \end{center} -\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_pz.png} \end{center} -\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:rotating_nass_plant_comp_stiffness}Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology} +\caption{\label{fig:rotating_nass_plant_comp_stiffness}Bode plot of the transfer function from active platform actuator to measured motion by the external metrology} \end{figure} \subsubsection{Effect of disturbances} @@ -2770,12 +2770,12 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: \begin{itemize} \item Regarding the effect of floor motion and forces applied on the payload: \begin{itemize} -\item The stiffer, the better. This can be seen in Figures~\ref{fig:rotating_nass_effect_floor_motion} and \ref{fig:rotating_nass_effect_direct_forces} where the magnitudes for the stiff hexapod are lower than those for the soft one +\item The stiffer, the better. This can be seen in Figures~\ref{fig:rotating_nass_effect_floor_motion} and \ref{fig:rotating_nass_effect_direct_forces} where the magnitudes for the stiff active platform are lower than those for the soft one \item \acrshort{iff} degrades the performance at low-frequency compared to \acrshort{rdc} \end{itemize} \item Regarding the effect of micro-station vibrations: \begin{itemize} -\item Having a soft nano-hexapod allows filtering of these vibrations between the suspension modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure~\ref{fig:rotating_nass_effect_stage_vibration_vc}). +\item Having a soft active platform allows filtering of these vibrations between the suspension modes of the active platform and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure~\ref{fig:rotating_nass_effect_stage_vibration_vc}). \end{itemize} \end{itemize} @@ -2784,21 +2784,21 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_vc.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_md.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_pz.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:rotating_nass_effect_floor_motion}Effect of floor motion \(x_{f,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. IFF is shown to increase the sensitivity to floor motion at low-frequency.} +\caption{\label{fig:rotating_nass_effect_floor_motion}Effect of floor motion \(x_{f,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three active platform stiffnesses. IFF is shown to increase the sensitivity to floor motion at low-frequency.} \end{figure} \begin{figure}[htbp] @@ -2806,21 +2806,21 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_vc.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_md.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_pz.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:rotating_nass_effect_stage_vibration}Effect of micro-station vibrations \(f_{t,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft nano-hexapod suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc})} +\caption{\label{fig:rotating_nass_effect_stage_vibration}Effect of micro-station vibrations \(f_{t,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three active platform stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft active platform suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc})} \end{figure} \begin{figure}[htbp] @@ -2828,21 +2828,21 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_vc.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_md.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,\text{N}/\mu\text{m}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_pz.png} \end{center} -\subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$} +\subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,\text{N}/\mu\text{m}$} \end{subfigure} -\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces \(f_{s,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low-frequency.} +\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces \(f_{s,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three active platform stiffnesses. Integral Force Feedback degrades this compliance at low-frequency.} \end{figure} \subsection*{Conclusion} In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model. @@ -2860,14 +2860,14 @@ It was shown that if the stiffness \(k_p\) of the additional springs is larger t These two modifications were compared with \acrlong{rdc}. While having very different implementations, both proposed modifications were found to be very similar with respect to the attainable damping and the obtained closed-loop system behavior. -This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters. -As for the uniaxial model, three nano-hexapod stiffnesses values were considered. -The dynamics of the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). -In addition, the attainable damping ratio of the soft nano-hexapod when using \acrshort{iff} is limited by gyroscopic effects. +This study has been applied to a rotating platform that corresponds to the active platform parameters. +As for the uniaxial model, three active platform stiffnesses values were considered. +The dynamics of the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). +In addition, the attainable damping ratio of the soft active platform when using \acrshort{iff} is limited by gyroscopic effects. To be closer to the \acrlong{nass} dynamics, the limited compliance of the micro-station has been considered. -Results are similar to those of the uniaxial model except that come complexity is added for the soft nano-hexapod due to the spindle's rotation. -For the moderately stiff nano-hexapod (\(k_n = 1\,N/\mu m\)), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that showed better results with the uniaxial model. +Results are similar to those of the uniaxial model except that come complexity is added for the soft active platform due to the spindle's rotation. +For the moderately stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model. \section{Micro Station - Modal Analysis} \label{sec:modal} To further improve the accuracy of the performance predictions, a model that better represents the micro-station dynamics is required. @@ -2907,7 +2907,7 @@ The obtained force and acceleration signals are described in Section~\ref{ssec:m Three types of equipment are essential for a good modal analysis. First, \emph{accelerometers} are used to measure the response of the structure. -Here, 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} shown in figure~\ref{fig:modal_accelero_M393B05} are used. +Here, 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,\text{V/g}\), measurement range is \(\pm 5\,\text{g}\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} shown in figure~\ref{fig:modal_accelero_M393B05} are used. These accelerometers were glued to the micro-station using a thin layer of wax for best results~\cite[chapt. 3.5.7]{ewins00_modal}. \begin{figure}[htbp] @@ -2932,7 +2932,7 @@ These accelerometers were glued to the micro-station using a thin layer of wax f \caption{\label{fig:modal_analysis_instrumentation}Instrumentation used for the modal analysis} \end{figure} -Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,mV/N\) and measurement range of \(2\,kN\)} (figure~\ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled manner. +Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,\text{mV/N}\) and measurement range of \(2\,\text{kN}\)} (figure~\ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled manner. Tests were conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one). The softer tip was found to give best results as it injects more energy in the low-frequency range where the coherence was low, such that the overall coherence was improved. @@ -2960,9 +2960,9 @@ In this modal analysis, it is chosen to measure the response of the structure at \label{ssec:modal_accelerometers} The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured. -A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod. +A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the positioning hexapod. The positions of the accelerometers are visually shown on a 3D model in Figure~\ref{fig:modal_location_accelerometers} and their precise locations with respect to a frame located at the \acrshort{poi} are summarized in Table~\ref{tab:modal_position_accelerometers}. -Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure~\ref{fig:modal_accelerometer_pictures}. +Pictures of the accelerometers fixed to the translation stage and to the positioning hexapod are shown in Figure~\ref{fig:modal_accelerometer_pictures}. As all key stages of the micro-station are expected to behave as solid bodies, only 6 \acrshort{dof} can be considered for each solid body. However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrshort{dof}) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section~\ref{ssec:modal_solid_body_assumption}). @@ -3020,7 +3020,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs \begin{center} \includegraphics[scale=1,height=6cm]{figs/modal_accelerometers_hexapod.jpg} \end{center} -\subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod} +\subcaption{\label{fig:modal_accelerometers_hexapod} Positioning Hexapod} \end{subfigure} \caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station stages} \end{figure} @@ -3062,7 +3062,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co The ``normalized'' \acrfull{asd} of the two signals were computed and shown in Figure~\ref{fig:modal_asd_acc_force}. Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer). -These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the \(x\) direction by accelerometer \(1\) (fixed to the micro-hexapod). +These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the \(x\) direction by accelerometer \(1\) (fixed to the positioning hexapod). Similar results were obtained for all measured \acrshortpl{frf}. \begin{figure}[htbp] @@ -3122,7 +3122,7 @@ For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that li \end{bmatrix} \end{equation} -However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod. +However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the positioning hexapod. Therefore, only \(6 \times 6 = 36\) \acrshortpl{dof} are of interest. Therefore, the objective of this section is to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36. @@ -3197,7 +3197,7 @@ Top granite & \(52\,\text{mm}\) & \(258\,\text{mm}\) & \(-778\,\text{mm}\)\\ Translation stage & \(0\) & \(14\,\text{mm}\) & \(-600\,\text{mm}\)\\ Tilt Stage & \(0\) & \(-5\,\text{mm}\) & \(-628\,\text{mm}\)\\ Spindle & \(0\) & \(0\) & \(-580\,\text{mm}\)\\ -Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\ +Positioning Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\ \bottomrule \end{tabularx} \end{table} @@ -3224,7 +3224,7 @@ From the response of one solid body expressed by its 6 \acrshortpl{dof} (i.e. fr In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements \(\bm{H}\). This is what is done here to check whether the solid body assumption is correct in the frequency band of interest. -The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure~\ref{fig:modal_comp_acc_solid_body_frf}). +The comparison is made for the 4 accelerometers fixed on the positioning hexapod (Figure~\ref{fig:modal_comp_acc_solid_body_frf}). The original \acrshortpl{frf} and those computed from the CoM responses match well in the frequency range of interest. Similar results were obtained for the other solid bodies, indicating that the solid body assumption is valid and that a multi-body model can be used to represent the dynamics of the micro-station. This also validates the reduction in the number of \acrshortpl{dof} from 69 (23 accelerometers with each 3 \acrshort{dof}) to 36 (6 solid bodies with 6 \acrshort{dof}). @@ -3232,7 +3232,7 @@ This also validates the reduction in the number of \acrshortpl{dof} from 69 (23 \begin{figure}[htbp] \centering \includegraphics[scale=1,scale=0.8]{figs/modal_comp_acc_solid_body_frf.png} -\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the micro-hexapod are shown. Input is a hammer force applied on the micro-hexapod in the \(x\) direction} +\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the positioning hexapod are shown. Input is a hammer force applied on the positioning hexapod in the \(x\) direction} \end{figure} \subsection{Modal Analysis} \label{sec:modal_analysis} @@ -3316,19 +3316,19 @@ From the obtained modal parameters, the mode shapes are computed and can be disp \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/modal_mode1_animation.jpg} \end{center} -\subcaption{\label{fig:modal_mode1_animation}$1^{st}$ mode at 11.9 Hz: tilt suspension mode of the granite} +\subcaption{\label{fig:modal_mode1_animation}$1^{st}$ mode at $11.9\,\text{Hz}$: tilt suspension mode of the granite} \end{subfigure} \begin{subfigure}{\textwidth} \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/modal_mode6_animation.jpg} \end{center} -\subcaption{\label{fig:modal_mode6_animation}$6^{th}$ mode at 69.8 Hz: vertical resonance of the spindle} +\subcaption{\label{fig:modal_mode6_animation}$6^{th}$ mode at $69.8\,\text{Hz}$: vertical resonance of the spindle} \end{subfigure} \begin{subfigure}{\textwidth} \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/modal_mode13_animation.jpg} \end{center} -\subcaption{\label{fig:modal_mode13_animation}$13^{th}$ mode at 124.2 Hz: lateral micro-hexapod resonance} +\subcaption{\label{fig:modal_mode13_animation}$13^{th}$ mode at $124.2\,\text{Hz}$: lateral hexapod resonance} \end{subfigure} \caption{\label{fig:modal_mode_animations}Three obtained mode shape animations} \end{figure} @@ -3386,7 +3386,7 @@ With \(\bm{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the response A comparison between original measured \acrshortpl{frf} and synthesized ones from the modal model is presented in Figure~\ref{fig:modal_comp_acc_frf_modal}. Whether the obtained match is good or bad is quite arbitrary. However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective. -This can be seen in Figure~\ref{fig:modal_comp_acc_frf_modal_3} that shows the \acrshort{frf} from the force applied on node 11 (i.e. on the translation stage) in the \(y\) direction to the measured acceleration at node \(2\) (i.e. at the top of the micro-hexapod) in the \(x\) direction. +This can be seen in Figure~\ref{fig:modal_comp_acc_frf_modal_3} that shows the \acrshort{frf} from the force applied on node 11 (i.e. on the translation stage) in the \(y\) direction to the measured acceleration at node \(2\) (i.e. at the top of the positioning hexapod) in the \(x\) direction. \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -3426,8 +3426,8 @@ However, the measurements are useful for tuning the parameters of the micro-stat \label{sec:ustation} From the start of this work, it became increasingly clear that an accurate micro-station model was necessary. -First, during the uniaxial study, it became clear that the micro-station dynamics affects the nano-hexapod dynamics. -Then, using the 3-DoF rotating model, it was discovered that the rotation of the nano-hexapod induces gyroscopic effects that affect the system dynamics and should therefore be modeled. +First, during the uniaxial study, it became clear that the micro-station dynamics affects the active platform dynamics. +Then, using the 3-DoF rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modeled. Finally, a modal analysis of the micro-station showed how complex the dynamics of the station is. The modal analysis also confirm that each stage behaves as a rigid body in the frequency range of interest. Therefore, a multi-body model is a good candidate to accurately represent the micro-station dynamics. @@ -3501,7 +3501,7 @@ Then, a rotation stage is used for tomography experiments. It is composed of an air bearing spindle\footnote{Made by LAB Motion Systems.}, whose angular position is controlled with a 3 phase synchronous motor based on the reading of 4 optical encoders. Additional rotary unions and slip-rings are used to be able to pass electrical signals, fluids and gazes through the rotation stage. -\paragraph{Micro-Hexapod} +\paragraph{Positioning Hexapod} Finally, a Stewart platform\footnote{Modified Zonda Hexapod by Symetrie.} is used to position the sample. It includes a DC motor and an optical linear encoders in each of the six struts. @@ -3519,7 +3519,7 @@ It can also be used to precisely position the \acrfull{poi} vertically with resp \begin{minipage}[t]{0.49\linewidth} \begin{center} \includegraphics[scale=1,scale=1]{figs/ustation_hexapod_stage.png} -\captionof{figure}{\label{fig:ustation_hexapod_stage}Micro Hexapod} +\captionof{figure}{\label{fig:ustation_hexapod_stage}Positioning Hexapod} \end{center} \end{minipage} \subsubsection{Mathematical description of a rigid body motion} @@ -3719,14 +3719,14 @@ As any motion stage induces parasitic motion in all 6 DoF, the transformation ma The homogeneous transformation matrix corresponding to the micro-station \(\bm{T}_{\mu\text{-station}}\) is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation~\eqref{eq:ustation_transformation_station}. \begin{equation}\label{eq:ustation_transformation_station} -\bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}} +\bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\text{hexapod}} \end{equation} \(\bm{T}_{\mu\text{-station}}\) represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite. If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), \(\bm{T}_{\mu\text{-station}}\) then represents the pose setpoint of the sample with respect to the granite. The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation~\eqref{eq:ustation_transformation_matrices_stages}. -The setpoints are \(D_y\) for the translation stage, \(\theta_y\) for the tilt-stage, \(\theta_z\) for the spindle, \([D_{\mu x},\ D_{\mu y}, D_{\mu z}]\) for the micro-hexapod translations and \([\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]\) for the micro-hexapod rotations. +The setpoints are \(D_y\) for the translation stage, \(\theta_y\) for the tilt-stage, \(\theta_z\) for the spindle, \([D_{\mu x},\ D_{\mu y}, D_{\mu z}]\) for the positioning hexapod translations and \([\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]\) for the positioning hexapod rotations. \begin{equation}\label{eq:ustation_transformation_matrices_stages} \begin{align} @@ -3736,7 +3736,7 @@ The setpoints are \(D_y\) for the translation stage, \(\theta_y\) for the tilt-s 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad -\bm{T}_{\mu\text{-hexapod}} = +\bm{T}_{\text{hexapod}} = \left[ \begin{array}{ccc|c} & & & D_{\mu x} \\ & \bm{R}_x(\theta_{\mu x}) \bm{R}_y(\theta_{\mu y}) \bm{R}_{z}(\theta_{\mu z}) & & D_{\mu y} \\ @@ -3766,7 +3766,7 @@ The inertia of the solid bodies and the stiffness properties of the guiding mech The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section~\ref{ssec:ustation_model_comp_dynamics}). -As the dynamics of the nano-hexapod is impacted by the micro-station compliance, the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. +As the dynamics of the active platform is impacted by the micro-station compliance, the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section~\ref{ssec:ustation_model_compliance}). \subsubsection{Multi-Body Model} \label{ssec:ustation_model_simscape} @@ -3814,11 +3814,11 @@ The spring values are summarized in Table~\ref{tab:ustation_6dof_stiffness_value \toprule \textbf{Stage} & \(D_x\) & \(D_y\) & \(D_z\) & \(R_x\) & \(R_y\) & \(R_z\)\\ \midrule -Granite & \(5\,kN/\mu m\) & \(5\,kN/\mu m\) & \(5\,kN/\mu m\) & \(25\,Nm/\mu\text{rad}\) & \(25\,Nm/\mu\text{rad}\) & \(10\,Nm/\mu\text{rad}\)\\ -Translation & \(200\,N/\mu m\) & - & \(200\,N/\mu m\) & \(60\,Nm/\mu\text{rad}\) & \(90\,Nm/\mu\text{rad}\) & \(60\,Nm/\mu\text{rad}\)\\ -Tilt & \(380\,N/\mu m\) & \(400\,N/\mu m\) & \(380\,N/\mu m\) & \(120\,Nm/\mu\text{rad}\) & - & \(120\,Nm/\mu\text{rad}\)\\ -Spindle & \(700\,N/\mu m\) & \(700\,N/\mu m\) & \(2\,kN/\mu m\) & \(10\,Nm/\mu\text{rad}\) & \(10\,Nm/\mu\text{rad}\) & -\\ -Hexapod & \(10\,N/\mu m\) & \(10\,N/\mu m\) & \(100\,N/\mu m\) & \(1.5\,Nm/rad\) & \(1.5\,Nm/rad\) & \(0.27\,Nm/rad\)\\ +Granite & \(5\,\text{kN}/\mu\text{m}\) & \(5\,\text{kN}/\mu\text{m}\) & \(5\,\text{kN}/\mu\text{m}\) & \(25\,\text{Nm}/\mu\text{rad}\) & \(25\,\text{Nm}/\mu\text{rad}\) & \(10\,\text{Nm}/\mu\text{rad}\)\\ +Translation & \(200\,\text{N}/\mu\text{m}\) & - & \(200\,\text{N}/\mu\text{m}\) & \(60\,\text{Nm}/\mu\text{rad}\) & \(90\,\text{Nm}/\mu\text{rad}\) & \(60\,\text{Nm}/\mu\text{rad}\)\\ +Tilt & \(380\,\text{N}/\mu\text{m}\) & \(400\,\text{N}/\mu\text{m}\) & \(380\,\text{N}/\mu\text{m}\) & \(120\,\text{Nm}/\mu\text{rad}\) & - & \(120\,\text{Nm}/\mu\text{rad}\)\\ +Spindle & \(700\,\text{N}/\mu\text{m}\) & \(700\,\text{N}/\mu\text{m}\) & \(2\,\text{kN}/\mu\text{m}\) & \(10\,\text{Nm}/\mu\text{rad}\) & \(10\,\text{Nm}/\mu\text{rad}\) & -\\ +Hexapod & \(10\,\text{N}/\mu\text{m}\) & \(10\,\text{N}/\mu\text{m}\) & \(100\,\text{N}/\mu\text{m}\) & \(1.5\,\text{Nm/rad}\) & \(1.5\,\text{Nm/rad}\) & \(0.27\,\text{Nm/rad}\)\\ \bottomrule \end{tabularx} \end{table} @@ -3863,8 +3863,8 @@ When considering the NASS, the most important dynamical characteristics of the m Therefore, the adopted strategy is to accurately model the micro-station compliance. The micro-station compliance was experimentally measured using the setup illustrated in Figure~\ref{fig:ustation_compliance_meas}. -Four 3-axis accelerometers were fixed to the micro-hexapod top platform. -The micro-hexapod top platform was impacted at 10 different points. +Four 3-axis accelerometers were fixed to the positioning hexapod top platform. +The positioning hexapod top platform was impacted at 10 different points. For each impact position, 10 impacts were performed to average and improve the data quality. \begin{figure}[htbp] @@ -3898,7 +3898,7 @@ Then, the acceleration in the cartesian frame can be computed using~\eqref{eq:us a_{\mathcal{X}} = \bm{J}_a^{-1} \cdot a_{\mathcal{L}} \end{equation} -Similar to what is done for the accelerometers, a Jacobian matrix \(\bm{J}_F\) is computed~\eqref{eq:ustation_compliance_force_jacobian} and used to convert the individual hammer forces \(F_{\mathcal{L}}\) to force and torques \(F_{\mathcal{X}}\) applied at the center of the micro-hexapod top plate (defined by frame \(\{\mathcal{X}\}\) in Figure~\ref{fig:ustation_compliance_meas}). +Similar to what is done for the accelerometers, a Jacobian matrix \(\bm{J}_F\) is computed~\eqref{eq:ustation_compliance_force_jacobian} and used to convert the individual hammer forces \(F_{\mathcal{L}}\) to force and torques \(F_{\mathcal{X}}\) applied at the center of the positioning hexapod top plate (defined by frame \(\{\mathcal{X}\}\) in Figure~\ref{fig:ustation_compliance_meas}). \begin{equation}\label{eq:ustation_compliance_force_jacobian} \bm{J}_F = \left[\begin{smallmatrix} @@ -3961,7 +3961,7 @@ Finally, the obtained disturbance sources are compared in Section~\ref{ssec:usta In this section, ground motion is directly measured using geophones. Vibrations induced by scanning the translation stage and the spindle are also measured using dedicated setups. -The tilt stage and the micro-hexapod also have positioning errors; however, they are not modeled here because these two stages are only used for pre-positioning and not for scanning. +The tilt stage and the positioning hexapod also have positioning errors; however, they are not modeled here because these two stages are only used for pre-positioning and not for scanning. Therefore, from a control perspective, they are not important. \paragraph{Ground Motion} @@ -3994,11 +3994,11 @@ A similar setup was used to measure the horizontal deviation (i.e. in the \(x\) \caption{\label{fig:ustation_errors_ty_setup}Experimental setup to measure the straightness (vertical deviation) of the translation stage} \end{figure} -Six scans were performed between \(-4.5\,mm\) and \(4.5\,mm\). +Six scans were performed between \(-4.5\,\text{mm}\) and \(4.5\,\text{mm}\). The results for each individual scan are shown in Figure~\ref{fig:ustation_errors_dy_vertical}. The measurement axis may not be perfectly aligned with the translation stage axis; this, a linear fit is removed from the measurement. The remaining vertical displacement is shown in Figure~\ref{fig:ustation_errors_dy_vertical_remove_mean}. -A vertical error of \(\pm300\,nm\) induced by the translation stage is expected. +A vertical error of \(\pm300\,\text{nm}\) induced by the translation stage is expected. Similar result is obtained for the \(x\) lateral direction. \begin{figure}[htbp] @@ -4014,7 +4014,7 @@ Similar result is obtained for the \(x\) lateral direction. \end{center} \subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing linear fit} \end{subfigure} -\caption{\label{fig:ustation_errors_dy}Measurement of the linear (vertical) deviation of the Translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}).} +\caption{\label{fig:ustation_errors_dy}Measurement of the vertical error of the translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}).} \end{figure} \paragraph{Spindle} @@ -4045,7 +4045,7 @@ A large fraction of the radial (Figure~\ref{fig:ustation_errors_spindle_radial}) This is displayed by the dashed circle. After removing the best circular fit from the data, the vibrations induced by the Spindle may be viewed as stochastic disturbances. However, some misalignment between the \acrshort{poi} of the sample and the rotation axis will be considered because the alignment is not perfect in practice. -The vertical motion induced by scanning the spindle is in the order of \(\pm 30\,nm\) (Figure~\ref{fig:ustation_errors_spindle_axial}). +The vertical motion induced by scanning the spindle is in the order of \(\pm 30\,\text{nm}\) (Figure~\ref{fig:ustation_errors_spindle_axial}). \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -4121,7 +4121,7 @@ The obtained power spectral density of the disturbances are shown in Figure~\ref \end{center} \subcaption{\label{fig:ustation_dist_source_spindle}Spindle} \end{subfigure} -\caption{\label{fig:ustation_dist_sources}Measured spectral density of the micro-station disturbance sources. Ground motion (\subref{fig:ustation_dist_source_ground_motion}), translation stage (\subref{fig:ustation_dist_source_translation_stage}) and spindle (\subref{fig:ustation_dist_source_spindle}).} +\caption{\label{fig:ustation_dist_sources}Measured \acrshort{asd} of the micro-station disturbance sources. Ground motion (\subref{fig:ustation_dist_source_ground_motion}), translation stage (\subref{fig:ustation_dist_source_translation_stage}) and spindle (\subref{fig:ustation_dist_source_spindle}).} \end{figure} The disturbances are characterized by their power spectral densities, as shown in Figure~\ref{fig:ustation_dist_sources}. @@ -4161,7 +4161,7 @@ Second, a constant velocity scans with the translation stage was performed and a To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of 60rpm. Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section~\ref{sec:ustation_disturbances}. -A radial offset of \(\approx 1\,\mu m\) between the \acrfull{poi} and the spindle's rotation axis is introduced to represent what is experimentally observed. +A radial offset of \(\approx 1\,\mu\text{m}\) between the \acrfull{poi} and the spindle's rotation axis is introduced to represent what is experimentally observed. During the 10 second simulation (i.e. 10 spindle turns), the position of the \acrshort{poi} with respect to the granite was recorded. Results are shown in Figure~\ref{fig:ustation_errors_model_spindle}. A good correlation with the measurements is observed both for radial errors (Figure~\ref{fig:ustation_errors_model_spindle_radial}) and axial errors (Figure~\ref{fig:ustation_errors_model_spindle_axial}). @@ -4185,7 +4185,7 @@ A good correlation with the measurements is observed both for radial errors (Fig \label{sec:ustation_experiments_ty_scans} A second experiment was performed in which the translation stage was scanned at constant velocity. -The translation stage setpoint is configured to have a ``triangular'' shape with stroke of \(\pm 4.5\, mm\). +The translation stage setpoint is configured to have a ``triangular'' shape with stroke of \(\pm 4.5\,\text{mm}\). Both ground motion and translation stage vibrations were included in the simulation. Similar to what was performed for the tomography simulation, the \acrfull{poi} position with respect to the granite was recorded and compared with the experimental measurements in Figure~\ref{fig:ustation_errors_model_dy_vertical}. A similar error amplitude was observed, thus indicating that the multi-body model with the included disturbances accurately represented the micro-station behavior in typical scientific experiments. @@ -4193,7 +4193,7 @@ A similar error amplitude was observed, thus indicating that the multi-body mode \begin{figure}[htbp] \centering \includegraphics[scale=1,scale=0.8]{figs/ustation_errors_model_dy_vertical.png} -\caption{\label{fig:ustation_errors_model_dy_vertical}Vertical errors during a constant-velocity scan of the translation stage. Comparison of the measurements and simulated errors.} +\caption{\label{fig:ustation_errors_model_dy_vertical}Vertical errors during a constant-velocity scan of the translation stage.} \end{figure} \subsection*{Conclusion} \label{sec:ustation_conclusion} @@ -4205,7 +4205,7 @@ After tuning the model parameters, a good match with the measured compliance was The disturbances affecting the sample position should also be well modeled. After experimentally estimating the disturbances (Section~\ref{sec:ustation_disturbances}), the multi-body model was finally validated by performing a tomography simulation (Figure~\ref{fig:ustation_errors_model_spindle}) as well as a simulation in which the translation stage was scanned (Figure~\ref{fig:ustation_errors_model_dy_vertical}). -\section{Nano Hexapod - Multi Body Model} +\section{Active Platform - Multi Body Model} \label{sec:nhexa} Building upon the validated multi-body model of the micro-station presented in previous sections, this section focuses on the development and integration of an active vibration platform model. @@ -4240,8 +4240,8 @@ To overcome this limitation, external metrology systems have been implemented to A review of existing sample stages with active vibration control reveals various approaches to implementing such feedback systems. In many cases, sample position control is limited to translational \acrshortpl{dof}. -At NSLS-II, for instance, a system capable of \(100\,\mu m\) stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure~\ref{fig:nhexa_stages_nazaretski}). -Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately 100 Hz (Figure~\ref{fig:nhexa_stages_sapoti}). +At NSLS-II, for instance, a system capable of \(100\,\mu\text{m}\) stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure~\ref{fig:nhexa_stages_nazaretski}). +Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately \(100\,\text{Hz}\) (Figure~\ref{fig:nhexa_stages_sapoti}). \begin{figure}[h!tbp] \begin{subfigure}{0.36\textwidth} @@ -4261,8 +4261,8 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac The integration of \(R_z\) rotational capability, which is necessary for tomography experiments, introduces additional complexity. At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section~\ref{sec:nhexa_stewart_platform}) using piezoelectric actuators has been positioned below the spindle (Figure~\ref{fig:nhexa_stages_villar}). -While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to \(50\,\mu m\) due to the inherent constraints of piezoelectric actuators. -In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering \(100\,\mu m\) stroke (Figure~\ref{fig:nhexa_stages_schroer}). +While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to \(50\,\mu\text{m}\) due to the inherent constraints of piezoelectric actuators. +In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering \(100\,\mu\text{m}\) stroke (Figure~\ref{fig:nhexa_stages_schroer}). However, attempts to implement real-time feedback using YZ external metrology proved challenging, possibly due to the poor dynamical response of the serial stage configuration. \begin{figure}[h!tbp] @@ -4292,43 +4292,43 @@ Although direct performance comparisons between these systems are challenging du \textbf{Stacked Stages} & \textbf{Specifications} & \textbf{Measured DoFs} & \textbf{Bandwidth} & \textbf{Reference}\\ \midrule Sample & light & Interferometers & 3 PID, n/a & APS\\ -\textbf{XYZ stage (piezo)} & \(D_{xyz}: 0.05\,mm\) & \(D_{xyz}\) & & ~\cite{nazaretski15_pushin_limit}\\ +\textbf{XYZ stage (piezo)} & \(D_{xyz}: 0.05\,\text{mm}\) & \(D_{xyz}\) & & ~\cite{nazaretski15_pushin_limit}\\ \midrule Sample & light & Capacitive sensors & \(\approx 10\,\text{Hz}\) & ESRF\\ Spindle & \(R_z: \pm 90\,\text{deg}\) & \(D_{xyz},\ R_{xy}\) & & ID16a\\ -\textbf{Hexapod (piezo)} & \(D_{xyz}: 0.05\,mm\) & & & ~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}\\ +\textbf{Hexapod (piezo)} & \(D_{xyz}: 0.05\,\text{mm}\) & & & ~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}\\ & \(R_{xy}: 500\,\mu\text{rad}\) & & & \\ \midrule Sample & light & Interferometers & n/a & PETRA III\\ -\textbf{XYZ stage (piezo)} & \(D_{xyz}: 0.1\,mm\) & \(D_{yz}\) & & P06\\ +\textbf{XYZ stage (piezo)} & \(D_{xyz}: 0.1\,\text{mm}\) & \(D_{yz}\) & & P06\\ Spindle & \(R_z: 180\,\text{deg}\) & & & ~\cite{schroer17_ptynam,schropp20_ptynam}\\ \midrule Sample & light & Interferometers & PID, n/a & PSI\\ Spindle & \(R_z: \pm 182\,\text{deg}\) & \(D_{yz},\ R_x\) & & OMNY\\ -\textbf{Tripod (piezo)} & \(D_{xyz}: 0.4\,mm\) & & & ~\cite{holler17_omny_pin_versat_sampl_holder,holler18_omny_tomog_nano_cryo_stage}\\ +\textbf{Tripod (piezo)} & \(D_{xyz}: 0.4\,\text{mm}\) & & & ~\cite{holler17_omny_pin_versat_sampl_holder,holler18_omny_tomog_nano_cryo_stage}\\ \midrule Sample & light & Interferometers & n/a & Soleil\\ (XY stage) & & \(D_{xyz},\ R_{xy}\) & & Nanoprobe\\ Spindle & \(R_z: 360\,\text{deg}\) & & & ~\cite{stankevic17_inter_charac_rotat_stages_x_ray_nanot,engblom18_nanop_resul}\\ -\textbf{XYZ linear motors} & \(D_{xyz}: 0.4\,mm\) & & & \\ +\textbf{XYZ linear motors} & \(D_{xyz}: 0.4\,\text{mm}\) & & & \\ \midrule -Sample & up to 0.5kg & Interferometers & n/a & NSLS\\ +Sample & up to \(0.5\,\text{kg}\) & Interferometers & n/a & NSLS\\ Spindle & \(R_z: 360\,\text{deg}\) & \(D_{xyz}\) & & SRX\\ -\textbf{XYZ stage (piezo)} & \(D_{xyz}: 0.1\,mm\) & & & ~\cite{nazaretski22_new_kirkp_baez_based_scann}\\ +\textbf{XYZ stage (piezo)} & \(D_{xyz}: 0.1\,\text{mm}\) & & & ~\cite{nazaretski22_new_kirkp_baez_based_scann}\\ \midrule -Sample & up to 0.35kg & Interferometers & \(\approx 100\,\text{Hz}\) & Diamond, I14\\ -\textbf{Parallel XYZ VC} & \(D_{xyz}: 3\,mm\) & \(D_{xyz}\) & & ~\cite{kelly22_delta_robot_long_travel_nano}\\ +Sample & up to \(0.35\,\text{kg}\) & Interferometers & \(\approx 100\,\text{Hz}\) & Diamond, I14\\ +\textbf{Parallel XYZ VC} & \(D_{xyz}: 3\,\text{mm}\) & \(D_{xyz}\) & & ~\cite{kelly22_delta_robot_long_travel_nano}\\ \midrule Sample & light & Capacitive sensors & \(\approx 100\,\text{Hz}\) & LNLS\\ -\textbf{Parallel XYZ VC} & \(D_{xyz}: 3\,mm\) & and interferometers & & CARNAUBA\\ +\textbf{Parallel XYZ VC} & \(D_{xyz}: 3\,\text{mm}\) & and interferometers & & CARNAUBA\\ (Spindle) & \(R_z: \pm 110 \,\text{deg}\) & \(D_{xyz}\) & & ~\cite{geraldes23_sapot_carnaub_sirius_lnls}\\ \midrule -Sample & up to 50kg & \(D_{xyz},\ R_{xy}\) & & ESRF\\ +Sample & up to \(50\,\text{kg}\) & \(D_{xyz},\ R_{xy}\) & & ESRF\\ \textbf{Active Platform} & & & & ID31\\ -(Micro-Hexapod) & & & & ~\cite{dehaeze18_sampl_stabil_for_tomog_exper,dehaeze21_mechat_approac_devel_nano_activ_stabil_system}\\ +(Hexapod) & & & & ~\cite{dehaeze18_sampl_stabil_for_tomog_exper,dehaeze21_mechat_approac_devel_nano_activ_stabil_system}\\ Spindle & \(R_z: 360\,\text{deg}\) & & & \\ Tilt-Stage & \(R_y: \pm 3\,\text{deg}\) & & & \\ -Translation Stage & \(D_y: \pm 10\,mm\) & & & \\ +Translation Stage & \(D_y: \pm 10\,\text{mm}\) & & & \\ \bottomrule \end{tabularx} \end{table} @@ -4340,9 +4340,9 @@ In conventional systems, active platforms typically correct spindle positioning The NASS, however, faces a more complex task: it must compensate for positioning errors of the translation and tilt stages in real-time during their operation, including corrections along their primary axes of motion. For instance, when the translation stage moves along Y, the active platform must not only correct for unwanted motions in other directions but also correct the position along Y, which necessitate some synchronization between the control of the long stroke stages and the control of the active platform. -The second major distinguishing feature of the NASS is its capability to handle payload masses up to 50 kg, exceeding typical capacities in the literature by two orders of magnitude. +The second major distinguishing feature of the NASS is its capability to handle payload masses up to \(50\,\text{kg}\), exceeding typical capacities in the literature by two orders of magnitude. This substantial increase in payload mass fundamentally alters the system's dynamic behavior, as the sample mass significantly influences the overall system dynamics, in contrast to conventional systems where sample masses are negligible relative to the stage mass. -This characteristic introduces significant control challenges, as the feedback system must remain stable and maintain performance across a wide range of payload masses (from a few kilograms to 50 kg), requiring robust control strategies to handle such large plant variations. +This characteristic introduces significant control challenges, as the feedback system must remain stable and maintain performance across a wide range of payload masses (from a few kilograms to \(50\,\text{kg}\)), requiring robust control strategies to handle such large plant variations. The NASS also distinguishes itself through its high mobility and versatility, which are achieved through the use of multiple stacked stages (translation stage, tilt stage, spindle, positioning hexapod) that enable a wide range of experimental configurations. The resulting mechanical structure exhibits complex dynamics with multiple resonance modes in the low frequency range. @@ -4353,10 +4353,10 @@ The primary control requirements focus on \([D_y,\ D_z,\ R_y]\) motions; however \label{ssec:nhexa_active_platforms} The choice of the active platform architecture for the NASS requires careful consideration of several critical specifications. -The platform must provide control over five \acrshortpl{dof} (\(D_x\), \(D_y\), \(D_z\), \(R_x\), and \(R_y\)), with strokes exceeding \(100\,\mu m\) to correct for micro-station positioning errors, while fitting within a cylindrical envelope of 300 mm diameter and 95 mm height. -It must accommodate payloads up to 50 kg while maintaining high dynamical performance. -For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to 100 Hz. -However, achieving such resonance frequencies with a 50 kg payload would require unrealistic stiffness values of approximately \(2000\,N/\mu m\). +The platform must provide control over five \acrshortpl{dof} (\(D_x\), \(D_y\), \(D_z\), \(R_x\), and \(R_y\)), with strokes exceeding \(100\,\mu\text{m}\) to correct for micro-station positioning errors, while fitting within a cylindrical envelope of 300 mm diameter and 95 mm height. +It must accommodate payloads up to \(50\,\text{kg}\) while maintaining high dynamical performance. +For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to \(100\,\text{Hz}\). +However, achieving such resonance frequencies with a \(50\,\text{kg}\) payload would require unrealistic stiffness values of approximately \(2000\,\text{N}/\mu\text{m}\). This limitation necessitates alternative control approaches, and the High \acrfull{haclac} strategy is proposed to address this challenge. To this purpose, the design includes force sensors for active damping. Compliant mechanisms must also be used to eliminate friction and backlash, which would otherwise compromise the nano-positioning capabilities. @@ -4422,11 +4422,11 @@ These characteristics make the Stewart platforms particularly valuable in applic For the NASS application, the Stewart platform architecture offers three key advantages. First, as a fully parallel manipulator, all the motion errors of the micro-station can be compensated through the coordinated action of the six actuators. -Second, its compact design compared to serial manipulators makes it ideal for integration on top micro-station where only \(95\,mm\) of height is available. +Second, its compact design compared to serial manipulators makes it ideal for integration on top micro-station where only \(95\,\text{mm}\) of height is available. Third, the good dynamical properties should enable high-bandwidth positioning control. While Stewart platforms excel in precision and stiffness, they typically exhibit a relatively limited workspace compared to serial manipulators. -However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of \(10\,\mu m\). +However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of \(10\,\mu\text{m}\). This section provides a comprehensive analysis of the Stewart platform's properties, focusing on aspects crucial for precision positioning applications. The analysis encompasses the platform's kinematic relationships (Section~\ref{ssec:nhexa_stewart_platform_kinematics}), the use of the Jacobian matrix (Section~\ref{ssec:nhexa_stewart_platform_jacobian}), static behavior (Section~\ref{ssec:nhexa_stewart_platform_static}), and dynamic characteristics (Section~\ref{ssec:nhexa_stewart_platform_dynamics}). @@ -4454,7 +4454,7 @@ To facilitate the rigorous analysis of the Stewart platform, four reference fram Frames \(\{F\}\) and \(\{M\}\) serve primarily to define the joint locations. In contrast, frames \(\{A\}\) and \(\{B\}\) are used to describe the relative motion of the two platforms through the position vector \({}^A\bm{P}_B\) of frame \(\{B\}\) expressed in frame \(\{A\}\) and the rotation matrix \({}^A\bm{R}_B\) expressing the orientation of \(\{B\}\) with respect to \(\{A\}\). -For the nano-hexapod, frames \(\{A\}\) and \(\{B\}\) are chosen to be located at the theoretical focus point of the X-ray light which is \(150\,mm\) above the top platform, i.e. above \(\{M\}\). +For the active platform, frames \(\{A\}\) and \(\{B\}\) are chosen to be located at the theoretical focus point of the X-ray light which is \(150\,\text{mm}\) above the top platform, i.e. above \(\{M\}\). The location of the joints and the orientation and length of the struts are crucial for subsequent kinematic, static, and dynamic analyses of the Stewart platform. The center of rotation for the joint fixed to the base is noted \(\bm{a}_i\), while \(\bm{b}_i\) is used for the top platform joints. @@ -4495,14 +4495,14 @@ The obtained strut lengths are given by~\eqref{eq:nhexa_inverse_kinematics}. \end{equation} If the position and orientation of the platform lie in the feasible workspace, the solution is unique. -While configurations outside this workspace yield complex numbers, this only becomes relevant for large displacements that far exceed the nano-hexapod's operating range. +While configurations outside this workspace yield complex numbers, this only becomes relevant for large displacements that far exceed the active platform's operating range. \paragraph{Forward Kinematics} The forward kinematic problem seeks to determine the platform pose \(\bm{\mathcal{X}}\) given a set of strut lengths \(\bm{\mathcal{L}}\). Unlike inverse kinematics, this presents a significant challenge because it requires solving a system of nonlinear equations. Although various numerical methods exist for solving this problem, they can be computationally intensive and may not guarantee convergence to the correct solution. -For the nano-hexapod application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative. +For the active platform application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative. This approximation, which is developed in subsequent sections through the Jacobian matrix analysis, is particularly useful for real-time control applications. \subsubsection{The Jacobian Matrix} \label{ssec:nhexa_stewart_platform_jacobian} @@ -4573,16 +4573,16 @@ The accuracy of the Jacobian-based forward kinematics solution was estimated by For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation~\eqref{eq:nhexa_inverse_kinematics}. These strut lengths are then used with the Jacobian to estimate the platform pose~\eqref{eq:nhexa_forward_kinematics_approximate}, from which the error between the estimated and true poses can be calculated, both in terms of position \(\epsilon_D\) and orientation \(\epsilon_R\). -For motion strokes from \(1\,\mu m\) to \(10\,mm\), the errors are estimated for all direction of motion, and the worst case errors are shown in Figure~\ref{fig:nhexa_forward_kinematics_approximate_errors}. -The results demonstrate that for displacements up to approximately \(1\,\%\) of the hexapod's size (which corresponds to \(100\,\mu m\) as the size of the Stewart platform is here \(\approx 100\,mm\)), the Jacobian approximation provides excellent accuracy. +For motion strokes from \(1\,\mu\text{m}\) to \(10\,\text{mm}\), the errors are estimated for all direction of motion, and the worst case errors are shown in Figure~\ref{fig:nhexa_forward_kinematics_approximate_errors}. +The results demonstrate that for displacements up to approximately \(1\,\%\) of the hexapod's size (which corresponds to \(100\,\mu\text{m}\) as the size of the Stewart platform is here \(\approx 100\,\text{mm}\)), the Jacobian approximation provides excellent accuracy. -Since the maximum required stroke of the nano-hexapod (\(\approx 100\,\mu m\)) is three orders of magnitude smaller than its overall size (\(\approx 100\,mm\)), the Jacobian matrix can be considered constant throughout the workspace. +Since the maximum required stroke of the active platform (\(\approx 100\,\mu\text{m}\)) is three orders of magnitude smaller than its overall size (\(\approx 100\,\text{mm}\)), the Jacobian matrix can be considered constant throughout the workspace. It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy. \begin{figure}[htbp] \centering \includegraphics[scale=1,scale=0.8]{figs/nhexa_forward_kinematics_approximate_errors.png} -\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,mm\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true positioin and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.} +\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,\text{mm}\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true positioin and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.} \end{figure} \paragraph{Static Forces} @@ -4721,16 +4721,16 @@ Through this multi-body modeling approach, each component model (including joint The analysis is structured as follows. First, the multi-body model is developed, and the geometric parameters, inertial properties, and actuator characteristics are established (Section~\ref{ssec:nhexa_model_def}). The model is then validated through comparison with the analytical equations in a simplified configuration (Section~\ref{ssec:nhexa_model_validation}). -Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section~\ref{ssec:nhexa_model_dynamics}). +Finally, the validated model is employed to analyze the active platform dynamics, from which insights for the control system design are derived (Section~\ref{ssec:nhexa_model_dynamics}). \subsubsection{Model Definition} \label{ssec:nhexa_model_def} \paragraph{Geometry} The Stewart platform's geometry is defined by two principal coordinate frames (Figure~\ref{fig:nhexa_stewart_model_def}): a fixed base frame \(\{F\}\) and a moving platform frame \(\{M\}\). The joints connecting the actuators to these frames are located at positions \({}^F\bm{a}_i\) and \({}^M\bm{b}_i\) respectively. -The \acrshort{poi}, denoted by frame \(\{A\}\), is situated \(150\,mm\) above the moving platform frame \(\{M\}\). +The \acrshort{poi}, denoted by frame \(\{A\}\), is situated \(150\,\text{mm}\) above the moving platform frame \(\{M\}\). -The geometric parameters of the nano-hexapod are summarized in Table~\ref{tab:nhexa_stewart_model_geometry}. +The geometric parameters of the active platform are summarized in Table~\ref{tab:nhexa_stewart_model_geometry}. These parameters define the positions of all connection points in their respective coordinate frames. From these parameters, key kinematic properties can be derived: the strut orientations \(\hat{\bm{s}}_i\), strut lengths \(l_i\), and the system's Jacobian matrix \(\bm{J}\). @@ -4769,9 +4769,9 @@ From these parameters, key kinematic properties can be derived: the strut orient \paragraph{Inertia of Plates} The fixed base and moving platform were modeled as solid cylindrical bodies. -The base platform was characterized by a radius of \(120\,mm\) and thickness of \(15\,mm\), matching the dimensions of the micro-hexapod's top platform. -The moving platform was similarly modeled with a radius of \(110\,mm\) and thickness of \(15\,mm\). -Both platforms were assigned a mass of \(5\,kg\). +The base platform was characterized by a radius of \(120\,\text{mm}\) and thickness of \(15\,\text{mm}\), matching the dimensions of the positioning hexapod's top platform. +The moving platform was similarly modeled with a radius of \(110\,\text{mm}\) and thickness of \(15\,\text{mm}\). +Both platforms were assigned a mass of \(5\,\text{kg}\). \paragraph{Joints} The platform's joints play a crucial role in its dynamic behavior. @@ -4794,7 +4794,7 @@ This modular approach to actuator modeling allows for future refinements as the \begin{minipage}[b]{0.6\linewidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/nhexa_actuator_model.png} -\captionof{figure}{\label{fig:nhexa_actuator_model}Model of the nano-hexapod actuators} +\captionof{figure}{\label{fig:nhexa_actuator_model}Model of the active platform actuators} \end{center} \end{minipage} \hfill @@ -4805,9 +4805,9 @@ This modular approach to actuator modeling allows for future refinements as the \toprule & Value\\ \midrule -\(k_a\) & \(1\,N/\mu m\)\\ -\(c_a\) & \(50\,N/(m/s)\)\\ -\(k_p\) & \(0.05\,N/\mu m\)\\ +\(k_a\) & \(1\,\text{N}/\mu\text{m}\)\\ +\(c_a\) & \(50\,\text{Ns}/\text{m}\)\\ +\(k_p\) & \(0.05\,\text{N}/\mu\text{m}\)\\ \bottomrule \end{tabularx}} \captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters} @@ -4822,7 +4822,7 @@ A three-dimensional visualization of the model is presented in Figure~\ref{fig:n \begin{minipage}[b]{0.6\linewidth} \begin{center} \includegraphics[scale=1,scale=1]{figs/nhexa_stewart_model_input_outputs.png} -\captionof{figure}{\label{fig:nhexa_stewart_model_input_outputs}Nano-Hexapod plant with inputs and outputs. Frames \(\{F\}\) and \(\{M\}\) can be connected to other elements in the model.} +\captionof{figure}{\label{fig:nhexa_stewart_model_input_outputs}Active platform plant with inputs and outputs. Frames \(\{F\}\) and \(\{M\}\) can be connected to other elements in the model.} \end{center} \end{minipage} \hfill @@ -4837,7 +4837,7 @@ The validation of the multi-body model was performed using the simplest Stewart This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness \(k_a = 1\,\text{N}/\mu\text{m}\) and damping \(c_a = 10\,\text{N}/({\text{m}/\text{s}})\). The geometric parameters remain as specified in Table~\ref{tab:nhexa_actuator_parameters}. -While the moving platform itself is considered massless, a \(10\,\text{kg}\) cylindrical payload is mounted on top with a radius of \(r = 110\,mm\) and a height \(h = 300\,mm\). +While the moving platform itself is considered massless, a \(10\,\text{kg}\) cylindrical payload is mounted on top with a radius of \(r = 110\,\text{mm}\) and a height \(h = 300\,\text{mm}\). For the analytical model, the stiffness, damping, and mass matrices are defined in~\eqref{eq:nhexa_analytical_matrices}. @@ -4861,11 +4861,11 @@ The close agreement between both approaches across the frequency spectrum valida \includegraphics[scale=1,scale=0.8]{figs/nhexa_comp_multi_body_analytical.png} \caption{\label{fig:nhexa_comp_multi_body_analytical}Comparison of the analytical transfer functions and the multi-body model} \end{figure} -\subsubsection{Nano Hexapod Dynamics} +\subsubsection{Active Platform Dynamics} \label{ssec:nhexa_model_dynamics} -Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics was performed. -The model parameters were set according to the specifications outlined in Section~\ref{ssec:nhexa_model_def}, with a payload mass of \(10\,kg\). +Following the validation of the multi-body model, a detailed analysis of the active platform dynamics was performed. +The model parameters were set according to the specifications outlined in Section~\ref{ssec:nhexa_model_def}, with a payload mass of \(10\,\text{kg}\). The transfer functions from actuator forces \(\bm{f}\) to both strut displacements \(\bm{\mathcal{L}}\) and force measurements \(\bm{f}_n\) were derived from the multi-body model. The transfer functions relating actuator forces to strut displacements are presented in Figure~\ref{fig:nhexa_multi_body_plant_dL}. @@ -4895,11 +4895,11 @@ The inclusion of parallel stiffness introduces an additional complex conjugate z \end{center} \subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{n}$} \end{subfigure} -\caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed from the nano-hexapod multi-body model} +\caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed using the active platform multi-body model} \end{figure} \subsubsection{Conclusion} -The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system. +The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the active platform system. Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform. A key advantage of this modeling approach lies in its flexibility for future refinements. @@ -4931,7 +4931,7 @@ This strategy potentially enables better performance by explicitly accounting fo The choice between these approaches depends significantly on the degree of interaction between the different control channels, and also on the available sensors and actuators. For instance, when using external metrology systems that measure the platform's global position, centralized control becomes necessary because each sensor measurement depends on all actuator inputs. -In the context of the nano-hexapod, two distinct control strategies were examined during the conceptual phase: +In the context of the active platform, two distinct control strategies were examined during the conceptual phase: \begin{itemize} \item Decentralized Integral Force Feedback (IFF), which uses collocated force sensors to implement independent control loops for each strut (Section~\ref{ssec:nhexa_control_iff}) \item \acrfull{hac}, which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section~\ref{ssec:nhexa_control_hac_lac}) @@ -4987,7 +4987,7 @@ This is particularly valuable when performance requirements differ between direc However, significant coupling exists between certain \acrshortpl{dof}, particularly between rotations and translations (e.g., \(\epsilon_{R_x}/\mathcal{F}_y\) or \(\epsilon_{D_y}/\bm\mathcal{M}_x\)). -For the conceptual validation of the nano-hexapod, control in the strut space was selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies. +For the conceptual validation of the \acrshort{nass}, control in the strut space was selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies. More sophisticated control strategies will be explored during the detailed design phase. \begin{figure}[htbp] @@ -5003,7 +5003,7 @@ More sophisticated control strategies will be explored during the detailed desig \end{center} \subcaption{\label{fig:nhexa_plant_frame_cartesian}Plant in the Cartesian Frame} \end{subfigure} -\caption{\label{fig:nhexa_plant_frame}Bode plot of the transfer functions computed from the nano-hexapod multi-body model} +\caption{\label{fig:nhexa_plant_frame}Bode plot of the transfer functions computed using the active platform multi-body model} \end{figure} \subsubsection{Active Damping with Decentralized IFF} \label{ssec:nhexa_control_iff} @@ -5156,10 +5156,10 @@ This study establishes the theoretical framework necessary for the subsequent de \section{Validation of the Concept} \label{sec:nass} The previous chapters have established crucial foundational elements for the development of the Nano Active Stabilization System (NASS). -The uniaxial model study demonstrated that very stiff nano-hexapod configurations should be avoided due to their high coupling with the micro-station dynamics. -A rotating three-degree-of-freedom model revealed that soft nano-hexapod designs prove unsuitable due to gyroscopic effect induced by the spindle rotation. +The uniaxial model study demonstrated that very stiff active platform configurations should be avoided due to their high coupling with the micro-station dynamics. +A rotating three-degree-of-freedom model revealed that soft active platform designs prove unsuitable due to gyroscopic effect induced by the spindle rotation. To further improve the model accuracy, a multi-body model of the micro-station was developed, which was carefully tuned using experimental modal analysis. -Furthermore, a multi-body model of the nano-hexapod was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure~\ref{fig:nass_simscape_model}. +Furthermore, a multi-body model of the active platform was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure~\ref{fig:nass_simscape_model}. \begin{figure}[htbp] \centering @@ -5168,8 +5168,8 @@ Furthermore, a multi-body model of the nano-hexapod was created, that can then b \end{figure} Building upon these foundations, this chapter presents the validation of the NASS concept. -The investigation begins with the previously established nano-hexapod model with actuator stiffness \(k_a = 1\,N/\mu m\). -A thorough examination of the control kinematics is presented in Section~\ref{sec:nass_kinematics}, detailing how both external metrology and nano-hexapod internal sensors are used in the control architecture. +The investigation begins with the previously established active platform model with actuator stiffness \(k_a = 1\,\text{N}/\mu\text{m}\). +A thorough examination of the control kinematics is presented in Section~\ref{sec:nass_kinematics}, detailing how both external metrology and active platform internal sensors are used in the control architecture. The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section~\ref{sec:nass_active_damping}), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section~\ref{sec:nass_hac}). The robustness of the proposed control scheme was evaluated under various operational conditions. @@ -5193,7 +5193,7 @@ For the Nano Active Stabilization System, computing the positioning errors in th First, desired sample pose with respect to a fixed reference frame is computed using the micro-station kinematics as detailed in Section~\ref{ssec:nass_ustation_kinematics}. This fixed frame is located at the X-ray beam focal point, as it is where the \acrshort{poi} needs to be positioned. Second, it measures the actual sample pose relative to the same fix frame, described in Section~\ref{ssec:nass_sample_pose_error}. -Finally, it determines the sample pose error and maps these errors to the nano-hexapod struts, as explained in Section~\ref{ssec:nass_error_struts}. +Finally, it determines the sample pose error and maps these errors to the active platform struts, as explained in Section~\ref{ssec:nass_error_struts}. The complete control architecture is described in Section~\ref{ssec:nass_control_architecture}. \subsubsection{Micro Station Kinematics} @@ -5201,12 +5201,12 @@ The complete control architecture is described in Section~\ref{ssec:nass_control The micro-station kinematics enables the computation of the desired sample pose from the reference signals of each micro-station stage. These reference signals consist of the desired lateral position \(r_{D_y}\), tilt angle \(r_{R_y}\), and spindle angle \(r_{R_z}\). -The micro-hexapod pose is defined by six parameters: three translations (\(r_{D_{\mu x}}\), \(r_{D_{\mu y}}\), \(r_{D_{\mu z}}\)) and three rotations (\(r_{\theta_{\mu x}}\), \(r_{\theta_{\mu y}}\), \(r_{\theta_{\mu z}}\)). +The hexapod pose is defined by six parameters: three translations (\(r_{D_{\mu x}}\), \(r_{D_{\mu y}}\), \(r_{D_{\mu z}}\)) and three rotations (\(r_{\theta_{\mu x}}\), \(r_{\theta_{\mu y}}\), \(r_{\theta_{\mu z}}\)). Using these reference signals, the desired sample position relative to the fixed frame is expressed through the homogeneous transformation matrix \(\bm{T}_{\mu\text{-station}}\), as defined in equation~\eqref{eq:nass_sample_ref}. \begin{equation}\label{eq:nass_sample_ref} - \bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}} + \bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\text{hexapod}} \end{equation} \begin{equation}\label{eq:nass_ustation_matrices} @@ -5217,7 +5217,7 @@ Using these reference signals, the desired sample position relative to the fixed 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad - \bm{T}_{\mu\text{-hexapod}} = + \bm{T}_{\text{hexapod}} = \left[ \begin{array}{ccc|c} & & & r_{D_{\mu x}} \\ & \bm{R}_x(r_{\theta_{\mu x}}) \bm{R}_y(r_{\theta_{\mu y}}) \bm{R}_{z}(r_{\theta_{\mu z}}) & & r_{D_{\mu y}} \\ @@ -5245,8 +5245,8 @@ Using these reference signals, the desired sample position relative to the fixed The external metrology system measures the sample position relative to the fixed granite. Due to the system's symmetry, this metrology provides measurements for five \acrshortpl{dof}: three translations (\(D_x\), \(D_y\), \(D_z\)) and two rotations (\(R_x\), \(R_y\)). -The sixth \acrshort{dof} (\(R_z\)) is still required to compute the errors in the frame of the nano-hexapod struts (i.e. to compute the nano-hexapod inverse kinematics). -This \(R_z\) rotation is estimated by combining measurements from the spindle encoder and the nano-hexapod's internal metrology, which consists of relative motion sensors in each strut (note that the micro-hexapod is not used for \(R_z\) rotation, and is therefore ignored for \(R_z\) estimation). +The sixth \acrshort{dof} (\(R_z\)) is still required to compute the errors in the frame of the active platform struts (i.e. to compute the active platform inverse kinematics). +This \(R_z\) rotation is estimated by combining measurements from the spindle encoder and the active platform's internal metrology, which consists of relative motion sensors in each strut (note that the positioning hexapod is not used for \(R_z\) rotation, and is therefore ignored for \(R_z\) estimation). The measured sample pose is represented by the homogeneous transformation matrix \(\bm{T}_{\text{sample}}\), as shown in equation~\eqref{eq:nass_sample_pose}. @@ -5265,7 +5265,7 @@ The measured sample pose is represented by the homogeneous transformation matrix The homogeneous transformation formalism enables straightforward computation of the sample position error. This computation involves the previously computed homogeneous \(4 \times 4\) matrices: \(\bm{T}_{\mu\text{-station}}\) representing the desired pose, and \(\bm{T}_{\text{sample}}\) representing the measured pose. -Their combination yields \(\bm{T}_{\text{error}}\), which expresses the position error of the sample in the frame of the rotating nano-hexapod, as shown in equation~\eqref{eq:nass_transformation_error}. +Their combination yields \(\bm{T}_{\text{error}}\), which expresses the position error of the sample in the frame of the rotating active platform, as shown in equation~\eqref{eq:nass_transformation_error}. \begin{equation}\label{eq:nass_transformation_error} \bm{T}_{\text{error}} = \bm{T}_{\mu\text{-station}}^{-1} \cdot \bm{T}_{\text{sample}} @@ -5285,7 +5285,7 @@ From \(\bm{T}_{\text{error}}\), the position and orientation errors \(\bm{\epsil \end{align} \end{equation} -Finally, these errors are mapped to the strut space using the nano-hexapod Jacobian matrix~\eqref{eq:nass_inverse_kinematics}. +Finally, these errors are mapped to the strut space using the active platform Jacobian matrix~\eqref{eq:nass_inverse_kinematics}. \begin{equation}\label{eq:nass_inverse_kinematics} \bm{\epsilon}_{\mathcal{L}} = \bm{J} \cdot \bm{\epsilon}_{\mathcal{X}} @@ -5294,10 +5294,10 @@ Finally, these errors are mapped to the strut space using the nano-hexapod Jacob \label{ssec:nass_control_architecture} The complete control architecture is summarized in Figure~\ref{fig:nass_control_architecture}. -The sample pose is measured using external metrology for five \acrshortpl{dof}, while the sixth \acrshort{dof} (\(R_z\)) is estimated by combining measurements from the nano-hexapod encoders and spindle encoder. +The sample pose is measured using external metrology for five \acrshortpl{dof}, while the sixth \acrshort{dof} (\(R_z\)) is estimated by combining measurements from the active platform encoders and spindle encoder. -The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and micro-hexapod. -The position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the nano-hexapod frame. +The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and positioning hexapod. +The position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the active platform frame. Then, the Jacobian matrix \(\bm{J}\) maps these errors to individual strut coordinates. For control purposes, force sensors mounted on each strut are used in a decentralized manner for active damping, as detailed in Section~\ref{sec:nass_active_damping}. @@ -5312,13 +5312,13 @@ Then, the high authority controller uses the computed errors in the frame of the \label{sec:nass_active_damping} Building on the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the \acrshort{haclac} strategy. The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3DoF rotating model. -The objective here is to design a decentralized IFF controller that provides good damping of the nano-hexapod modes across payload masses ranging from \(1\) to \(50\,\text{kg}\) and rotational velocity up to \(360\,\text{deg/s}\). -The payloads used for validation have a cylindrical shape with 250 mm height and with masses of 1 kg, 25 kg, and 50 kg. +The objective here is to design a decentralized IFF controller that provides good damping of the active platform modes across payload masses ranging from \(1\) to \(50\,\text{kg}\) and rotational velocity up to \(360\,\text{deg/s}\). +The payloads used for validation have a cylindrical shape with 250 mm height and with masses of \(1\,\text{kg}\), \(25\,\text{kg}\), and \(50\,\text{kg}\). \subsubsection{IFF Plant} \label{ssec:nass_active_damping_plant} Transfer functions from actuator forces \(f_i\) to force sensor measurements \(f_{mi}\) are computed using the multi-body model. -Figure~\ref{fig:nass_iff_plant_effect_kp} examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity \(\Omega_z = 360\,\text{deg/s}\) and with a payload mass of 25 kg. +Figure~\ref{fig:nass_iff_plant_effect_kp} examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity \(\Omega_z = 360\,\text{deg/s}\) and with a payload mass of \(25\,\text{kg}\). Without parallel stiffness (Figure~\ref{fig:nass_iff_plant_no_kp}), the plant dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model. Adding parallel stiffness (Figure~\ref{fig:nass_iff_plant_kp}) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation. @@ -5338,10 +5338,10 @@ Although both cases show significant coupling around the resonances, stability i \end{center} \subcaption{\label{fig:nass_iff_plant_kp}with parallel stiffness} \end{subfigure} -\caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness parallel to the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and a payload mass of 25kg. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp})} +\caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness parallel to the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and a payload mass of \(25\,\text{kg}\). The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp})} \end{figure} -The effect of rotation, as shown in Figure~\ref{fig:nass_iff_plant_effect_rotation}, is negligible as the actuator stiffness (\(k_a = 1\,N/\mu m\)) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). +The effect of rotation, as shown in Figure~\ref{fig:nass_iff_plant_effect_rotation}, is negligible as the actuator stiffness (\(k_a = 1\,\text{N}/\mu\text{m}\)) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). Figure~\ref{fig:nass_iff_plant_effect_payload} illustrate the effect of payload mass on the plant dynamics. The poles and zeros shift in frequency as the payload mass varies. @@ -5416,12 +5416,12 @@ The results demonstrate that the closed-loop poles remain within the left-half p \end{figure} \subsection{Centralized Active Vibration Control} \label{sec:nass_hac} -The implementation of high-bandwidth position control for the nano-hexapod presents several technical challenges. -The plant dynamics exhibits complex behavior influenced by multiple factors, including payload mass, rotational velocity, and the mechanical coupling between the nano-hexapod and the micro-station. +The implementation of high-bandwidth position control for the active platform presents several technical challenges. +The plant dynamics exhibits complex behavior influenced by multiple factors, including payload mass, rotational velocity, and the mechanical coupling between the active platform and the micro-station. This section presents the development and validation of a centralized control strategy designed to achieve precise sample positioning during high-speed tomography experiments. First, a comprehensive analysis of the plant dynamics is presented in Section~\ref{ssec:nass_hac_plant}, examining the effects of spindle rotation, payload mass variation, and the implementation of Integral Force Feedback (IFF). -Section~\ref{ssec:nass_hac_stiffness} validates previous modeling predictions that both overly stiff and compliant nano-hexapod configurations lead to degraded performance. +Section~\ref{ssec:nass_hac_stiffness} validates previous modeling predictions that both overly stiff and compliant active platform configurations lead to degraded performance. Building upon these findings, Section~\ref{ssec:nass_hac_controller} presents the design of a robust high-authority controller that maintains stability across varying payload masses while achieving the desired control bandwidth. The performance of the developed control strategy was validated through simulations of tomography experiments in Section~\ref{ssec:nass_hac_tomography}. @@ -5432,13 +5432,13 @@ Particular attention was paid to the system's behavior under maximum rotational The plant dynamics from force inputs \(\bm{f}\) to the strut errors \(\bm{\epsilon}_{\mathcal{L}}\) were first extracted from the multi-body model without the implementation of the decentralized IFF. The influence of spindle rotation on plant dynamics was investigated, and the results are presented in Figure~\ref{fig:nass_undamped_plant_effect_Wz}. -While rotational motion introduces coupling effects at low frequencies, these effects remain minimal at operational velocities, owing to the high stiffness characteristics of the nano-hexapod assembly. +While rotational motion introduces coupling effects at low frequencies, these effects remain minimal at operational velocities, owing to the high stiffness characteristics of the active platform assembly. Payload mass emerged as a significant parameter affecting system behavior, as illustrated in Figure~\ref{fig:nass_undamped_plant_effect_mass}. As expected, increasing the payload mass decreased the resonance frequencies while amplifying coupling at low frequency. These mass-dependent dynamic changes present considerable challenges for control system design, particularly for configurations with high payload masses. -Additional operational parameters were systematically evaluated, including the \(R_y\) tilt angle, \(R_z\) spindle position, and micro-hexapod position. +Additional operational parameters were systematically evaluated, including the \(R_y\) tilt angle, \(R_z\) spindle position, and positioning hexapod position. These factors were found to exert negligible influence on the plant dynamics, which can be attributed to the effective mechanical decoupling achieved between the plant and micro-station dynamics. This decoupling characteristic ensures consistent performance across various operational configurations. This also validates the developed control strategy. @@ -5462,13 +5462,13 @@ This also validates the developed control strategy. The Decentralized Integral Force Feedback was implemented in the multi-body model, and transfer functions from force inputs \(\bm{f}^\prime\) of the damped plant to the strut errors \(\bm{\epsilon}_{\mathcal{L}}\) were extracted from this model. The effectiveness of the IFF implementation was first evaluated with a \(1\,\text{kg}\) payload, as demonstrated in Figure~\ref{fig:nass_comp_undamped_damped_plant_m1}. -The results indicate successful damping of the nano-hexapod resonance modes, although a minor increase in low-frequency coupling was observed. +The results indicate successful damping of the active platform resonance modes, although a minor increase in low-frequency coupling was observed. This trade-off was considered acceptable, given the overall improvement in system behavior. The benefits of IFF implementation were further assessed across the full range of payload configurations, and the results are presented in Figure~\ref{fig:nass_hac_plants}. -For all tested payloads (\(1\,\text{kg}\), \(25\,\text{kg}\) and \(50\,\text{kg}\)), the decentralized IFF significantly damped the nano-hexapod modes and therefore simplified the system dynamics. +For all tested payloads (\(1\,\text{kg}\), \(25\,\text{kg}\) and \(50\,\text{kg}\)), the decentralized IFF significantly damped the active platform modes and therefore simplified the system dynamics. More importantly, in the vicinity of the desired high authority control bandwidth (i.e. between \(10\,\text{Hz}\) and \(50\,\text{Hz}\)), the damped dynamics (shown in red) exhibited minimal gain and phase variations with frequency. -For the undamped plants (shown in blue), achieving robust control with bandwidth above 10Hz while maintaining stability across different payload masses would be practically impossible. +For the undamped plants (shown in blue), achieving robust control with bandwidth above \(10\,\text{Hz}\) while maintaining stability across different payload masses would be practically impossible. \begin{figure}[h!tbp] \begin{subfigure}{0.48\textwidth} @@ -5486,53 +5486,53 @@ For the undamped plants (shown in blue), achieving robust control with bandwidth \caption{\label{fig:nass_hac_plant}Effect of Decentralized Integral Force Feedback on the positioning plant for a \(1\,\text{kg}\) sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}).} \end{figure} -The coupling between the nano-hexapod and the micro-station was evaluated through a comparative analysis of plant dynamics under two mounting conditions. -In the first configuration, the nano-hexapod was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance. +The coupling between the active platform and the micro-station was evaluated through a comparative analysis of plant dynamics under two mounting conditions. +In the first configuration, the active platform was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance. As illustrated in Figure~\ref{fig:nass_effect_ustation_compliance}, the complex dynamics of the micro-station were found to have little impact on the plant dynamics. -The only observable difference manifests as additional alternating poles and zeros above 100Hz, a frequency range sufficiently beyond the control bandwidth to avoid interference with the system performance. -This result confirms effective dynamic decoupling between the nano-hexapod and the supporting micro-station structure. +The only observable difference manifests as additional alternating poles and zeros above \(100\,\text{Hz}\), a frequency range sufficiently beyond the control bandwidth to avoid interference with the system performance. +This result confirms effective dynamic decoupling between the active platform and the supporting micro-station structure. \begin{figure}[htbp] \centering \includegraphics[h!tbp,scale=0.8]{figs/nass_effect_ustation_compliance.png} \caption{\label{fig:nass_effect_ustation_compliance}Effect of the micro-station limited compliance on the plant dynamics} \end{figure} -\subsubsection{Effect of Nano-Hexapod Stiffness on System Dynamics} +\subsubsection{Effect of Active Platform Stiffness on System Dynamics} \label{ssec:nass_hac_stiffness} -The influence of nano-hexapod stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models. -These models suggest that a moderate stiffness of approximately \(1\,N/\mu m\) would provide better performance than either very stiff or very soft configurations. +The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models. +These models suggest that a moderate stiffness of approximately \(1\,\text{N}/\mu\text{m}\) would provide better performance than either very stiff or very soft configurations. -For the stiff nano-hexapod analysis, a system with an actuator stiffness of \(100\,N/\mu m\) was simulated with a \(25\,\text{kg}\) payload. +For the stiff active platform analysis, a system with an actuator stiffness of \(100\,\text{N}/\mu\text{m}\) was simulated with a \(25\,\text{kg}\) payload. The transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station. -As shown in Figure~\ref{fig:nass_stiff_nano_hexapod_coupling_ustation}, significant coupling was observed between the nano-hexapod and micro-station dynamics. +As shown in Figure~\ref{fig:nass_stiff_nano_hexapod_coupling_ustation}, significant coupling was observed between the active platform and micro-station dynamics. This coupling introduces complex behavior that is difficult to model and predict accurately, thus corroborating the predictions of the simplified uniaxial model. -The soft nano-hexapod configuration was evaluated using a stiffness of \(0.01\,N/\mu m\) with a \(25\,\text{kg}\) payload. +The soft active platform configuration was evaluated using a stiffness of \(0.01\,\text{N}/\mu\text{m}\) with a \(25\,\text{kg}\) payload. The dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s. Figure~\ref{fig:nass_soft_nano_hexapod_effect_Wz} demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies. -The current approach of controlling the position in the strut frame is inadequate for soft nano-hexapods; but even shifting control to a frame matching the payload's \acrlong{com} would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. +The current approach of controlling the position in the strut frame is inadequate for soft active platforms; but even shifting control to a frame matching the payload's \acrlong{com} would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. \begin{figure}[h!tbp] \begin{subfigure}{0.48\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/nass_stiff_nano_hexapod_coupling_ustation.png} \end{center} -\subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,N/\mu m$ - Coupling with the micro-station} +\subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\mu\text{m}$ - Coupling with the micro-station} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/nass_soft_nano_hexapod_effect_Wz.png} \end{center} -\subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,N/\mu m$ - Effect of Spindle rotation} +\subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,\text{N}/\mu\text{m}$ - Effect of Spindle rotation} \end{subfigure} -\caption{\label{fig:nass_soft_stiff_hexapod}Coupling between a stiff nano-hexapod (\(k_a = 100\,N/\mu m\)) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance (\(k_a = 0.01\,N/\mu m\)) nano-hexapod (\subref{fig:nass_soft_nano_hexapod_effect_Wz})} +\caption{\label{fig:nass_soft_stiff_hexapod}Coupling between a stiff active platform (\(k_a = 100\,\text{N}/\mu\text{m}\)) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance (\(k_a = 0.01\,\text{N}/\mu\text{m}\)) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz})} \end{figure} \subsubsection{Controller design} \label{ssec:nass_hac_controller} -A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure~\ref{fig:nass_hac_plants}), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation. +A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure~\ref{fig:nass_hac_plants}), and achievement of sufficient bandwidth (targeted at \(10\,\text{Hz}\)) for high performance operation. The controller structure is defined in Equation~\eqref{eq:nass_robust_hac}, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes. \begin{equation}\label{eq:nass_robust_hac} @@ -5540,7 +5540,7 @@ The controller structure is defined in Equation~\eqref{eq:nass_robust_hac}, inco \end{equation} The controller performance was evaluated through two complementary analyses. -First, the decentralized loop gain shown in Figure~\ref{fig:nass_hac_loop_gain}, confirms the achievement of the desired 10Hz bandwidth. +First, the decentralized loop gain shown in Figure~\ref{fig:nass_hac_loop_gain}, confirms the achievement of the desired \(10\,\text{Hz}\) bandwidth. Second, the characteristic loci analysis presented in Figure~\ref{fig:nass_hac_loci} demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope. \begin{figure}[h!tbp] @@ -5564,12 +5564,12 @@ Second, the characteristic loci analysis presented in Figure~\ref{fig:nass_hac_l The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with a particular focus on tomography scanning because of its demanding performance requirements. Simulations were conducted at the maximum operational rotational velocity of \(\Omega_z = 360\,\text{deg/s}\) to evaluate system performance under the most challenging conditions. -Performance metrics were established based on anticipated future beamline specifications, which specify a beam size of 200nm (horizontal) by 100nm (vertical). +Performance metrics were established based on anticipated future beamline specifications, which specify a beam size of \(200\,\text{nm}\) (horizontal) by \(100\,\text{nm}\) (vertical). The primary requirement stipulates that the \acrshort{poi} must remain within beam dimensions throughout operation. The simulation included two principal disturbance sources: ground motion and spindle vibrations. Additional noise sources, including measurement noise and electrical noise from \acrfull{dac} and voltage amplifiers, were not included in this analysis, as these parameters will be optimized during the detailed design phase. -Figure~\ref{fig:nass_tomo_1kg_60rpm} presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (1kg). +Figure~\ref{fig:nass_tomo_1kg_60rpm} presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (\(1\,\text{kg}\)). The results demonstrate the system's capability to maintain the sample's position within the specified beam dimensions, thus validating the fundamental concept of the stabilization system. \begin{figure}[h!tbp] @@ -5585,10 +5585,10 @@ The results demonstrate the system's capability to maintain the sample's positio \end{center} \subcaption{\label{fig:nass_tomo_1kg_60rpm_yz}YZ plane} \end{subfigure} -\caption{\label{fig:nass_tomo_1kg_60rpm}Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at \(360\,\text{deg/s}\). 1kg payload is placed on top of the nano-hexapod.} +\caption{\label{fig:nass_tomo_1kg_60rpm}Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at \(360\,\text{deg/s}\). \(1\,\text{kg}\) payload is placed on top of the active platform.} \end{figure} -The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25 and 50kg payloads, complementing the initial 1kg test case. +The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25 and \(50\,\text{kg}\) payloads, complementing the initial \(1\,\text{kg}\) test case. As illustrated in Figure~\ref{fig:nass_tomography_hac_iff}, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis. While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions. @@ -5600,19 +5600,19 @@ For higher mass configurations, rotational velocities are expected to be below 3 \begin{center} \includegraphics[scale=1,scale=0.8]{figs/nass_tomography_hac_iff_m1.png} \end{center} -\subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,kg$} +\subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,\text{kg}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/nass_tomography_hac_iff_m25.png} \end{center} -\subcaption{\label{fig:nass_tomography_hac_iff_m25} $m = 25\,kg$} +\subcaption{\label{fig:nass_tomography_hac_iff_m25} $m = 25\,\text{kg}$} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/nass_tomography_hac_iff_m50.png} \end{center} -\subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,kg$} +\subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,\text{kg}$} \end{subfigure} \caption{\label{fig:nass_tomography_hac_iff}Simulation of tomography experiments - 360deg/s. Beam size is indicated by the dashed black ellipse} \end{figure} @@ -5620,13 +5620,13 @@ For higher mass configurations, rotational velocities are expected to be below 3 \label{sec:nass_conclusion} The development and analysis presented in this chapter have successfully validated the Nano Active Stabilization System concept, marking the completion of the conceptual design phase. -A comprehensive control strategy has been established, effectively combining external metrology with nano-hexapod sensor measurements to achieve precise position control. +A comprehensive control strategy has been established, effectively combining external metrology with active platform sensor measurements to achieve precise position control. The control strategy implements a High Authority Control - Low Authority Control architecture - a proven approach that has been specifically adapted to meet the unique requirements of the rotating NASS. The decentralized Integral Force Feedback component has been demonstrated to provide robust active damping under various operating conditions. The addition of parallel springs to the force sensors has been shown to ensure stability during spindle rotation. The centralized High Authority Controller, operating in the frame of the struts for simplicity, has successfully achieved the desired performance objectives of maintaining a bandwidth of \(10\,\text{Hz}\) while maintaining robustness against payload mass variations. -This investigation has confirmed that the moderate actuator stiffness of \(1\,N/\mu m\) represents an adequate choice for the nano-hexapod, as both very stiff and very compliant configurations introduce significant performance limitations. +This investigation has confirmed that the moderate actuator stiffness of \(1\,\text{N}/\mu\text{m}\) represents an adequate choice for the active platform, as both very stiff and very compliant configurations introduce significant performance limitations. Simulations of tomography experiments have been performed, with positioning accuracy requirements defined by the expected minimum beam dimensions of \(200\,\text{nm}\) by \(100\,\text{nm}\). The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples. @@ -5639,9 +5639,9 @@ Through a systematic progression from simplified to increasingly complex models, Using the simple uniaxial model revealed that a very stiff stabilization stage was unsuitable due to its strong coupling with the complex micro-station dynamics. Conversely, the three-degree-of-freedom rotating model demonstrated that very soft stabilization stage designs are equally problematic due to the gyroscopic effects induced by spindle rotation. -A moderate stiffness of approximately \(1\,N/\mu m\) was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection. +A moderate stiffness of approximately \(1\,\text{N}/\mu\text{m}\) was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection. -The multi-body modeling approach proved essential for capturing the complex dynamics of both the micro-station and the nano-hexapod. +The multi-body modeling approach proved essential for capturing the complex dynamics of both the micro-station and the active platform. This model was tuned based on extensive modal analysis and vibration measurements. The Stewart platform architecture was selected for the active platform due to its good dynamical properties, compact design, and the ability to satisfy the strict space constraints of the NASS. @@ -5649,21 +5649,21 @@ The \acrshort{haclac} control strategy was successfully adapted to address the u Decentralized Integral Force Feedback with parallel springs demonstrated robust active damping capabilities across different payload masses and rotational velocities. The centralized High Authority Controller, implemented in the frame of the struts, achieved the desired \(10\,\text{Hz}\) bandwidth with good robustness properties. -Simulations of tomography experiments validated the NASS concept, with positioning accuracy meeting the requirements defined by the expected minimum beam dimensions (\(200\,nm \times 100\,nm\)) for lightweight samples at maximum rotational velocity. +Simulations of tomography experiments validated the NASS concept, with positioning accuracy meeting the requirements defined by the expected minimum beam dimensions (\(200\,\text{nm} \times 100\,\text{nm}\)) for lightweight samples at maximum rotational velocity. As anticipated by the control analysis, some performance degradation was observed with heavier payloads, but the overall performance remained sufficient to validate the fundamental concept. \chapter{Detailed Design} \label{chap:detail} \minitoc \subsubsection*{Abstract} Following the validation of the Nano Active Stabilization System concept in the previous chapter through simulated tomography experiments, this chapter addresses the refinement of the preliminary conceptual model into an optimized implementation. -The initial validation used a nano-hexapod with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise. +The initial validation used a active platform with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise. This detailed design phase aims to optimize each component while ensuring none will limit the system's overall performance. -This chapter begins by determining the optimal geometric configuration for the nano-hexapod (Section~\ref{sec:detail_kinematics}). +This chapter begins by determining the optimal geometric configuration for the active platform (Section~\ref{sec:detail_kinematics}). To this end, a review of existing Stewart platform designs is first presented, followed by an analysis of how geometric parameters influence the system's properties—mobility, stiffness, and dynamical response—with a particular emphasis on the cubic architecture. -The chapter concludes by specifying the chosen nano-hexapod geometry and the associated actuator stroke and flexible joint angular travel requirements to achieve the desired mobility. +The chapter concludes by specifying the chosen active platform geometry and the associated actuator stroke and flexible joint angular travel requirements to achieve the desired mobility. -Section~\ref{sec:detail_fem} introduces a hybrid modeling methodology that combines \acrfull{fea} with multi-body dynamics to optimize critical nano-hexapod components. +Section~\ref{sec:detail_fem} introduces a hybrid modeling methodology that combines \acrfull{fea} with multi-body dynamics to optimize critical active platform components. This approach is first experimentally validated using an Amplified Piezoelectric Actuator, establishing confidence in the modeling technique. The methodology is then applied to two key elements: the actuators (Section~\ref{sec:detail_fem_actuator}) and the flexible joints (Section~\ref{sec:detail_fem_joint}), enabling detailed optimization while maintaining computational efficiency for system-level simulations. @@ -5675,20 +5675,20 @@ Third, the optimization of controllers for decoupled plants is discussed, introd Section~\ref{sec:detail_instrumentation} focuses on instrumentation selection using a dynamic error budgeting approach to establish maximum acceptable noise specifications for each component. The selected instrumentation is then experimentally characterized to verify compliance with these specifications, ensuring that the combined effect of all noise sources remains within acceptable limits. -The chapter concludes with a concise presentation of the obtained optimized nano-hexapod design in Section~\ref{sec:detail_design}, summarizing how the various optimizations contribute to a system that balances the competing requirements of precision positioning, vibration isolation, and practical implementation constraints. +The chapter concludes with a concise presentation of the obtained optimized active platform design, called the ``nano-hexapod'', in Section~\ref{sec:detail_design}, summarizing how the various optimizations contribute to a system that balances the competing requirements of precision positioning, vibration isolation, and practical implementation constraints. With the detailed design completed and components procured, the project advances to the experimental validation phase, which will be addressed in the subsequent chapter. \section{Optimal Geometry} \label{sec:detail_kinematics} The performance of a Stewart platform depends on its geometric configuration, especially the orientation of its struts and the positioning of its joints. -During the conceptual design phase of the nano-hexapod, a preliminary geometry was selected based on general principles without detailed optimization. +During the conceptual design phase of the active platform, a preliminary geometry was selected based on general principles without detailed optimization. As the project advanced to the detailed design phase, a rigorous analysis of how geometry influences system performance became essential to ensure that the final design would meet the demanding requirements of the Nano Active Stabilization System (NASS). -In this chapter, the nano-hexapod geometry is optimized through careful analysis of how design parameters influence critical performance aspects: attainable workspace, mechanical stiffness, strut-to-strut coupling for decentralized control strategies, and dynamic response in Cartesian coordinates. +In this chapter, the active platform geometry is optimized through careful analysis of how design parameters influence critical performance aspects: attainable workspace, mechanical stiffness, strut-to-strut coupling for decentralized control strategies, and dynamic response in Cartesian coordinates. The chapter begins with a comprehensive review of existing Stewart platform designs in Section~\ref{sec:detail_kinematics_stewart_review}, surveying various approaches to geometry, actuation, sensing, and joint design from the literature. Section~\ref{sec:detail_kinematics_geometry} develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process. -Section~\ref{sec:detail_kinematics_cubic} examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application. -Finally, Section~\ref{sec:detail_kinematics_nano_hexapod} presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. +Section~\ref{sec:detail_kinematics_cubic} examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the NASS applications. +Finally, Section~\ref{sec:detail_kinematics_nano_hexapod} presents the optimized active platform geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. \subsection{Review of Stewart platforms} \label{sec:detail_kinematics_stewart_review} @@ -5716,12 +5716,12 @@ As explained in the conceptual phase, Stewart platforms comprise the following k The specific geometry (i.e., position of joints and orientation of the struts) can be selected based on the application requirements, resulting in numerous designs throughout the literature. This discussion focuses primarily on Stewart platforms designed for nano-positioning and vibration control, which necessitates the use of flexible joints. -The implementation of these flexible joints, will be discussed when designing the nano-hexapod flexible joints. +The implementation of these flexible joints, will be discussed when designing the active platform flexible joints. Long stroke Stewart platforms are not addressed here as their design presents different challenges, such as singularity-free workspace and complex kinematics~\cite{merlet06_paral_robot}. In terms of actuation, mainly two types are used: voice coil actuators and piezoelectric actuators. -Voice coil actuators, providing stroke ranges from \(0.5\,mm\) to \(10\,mm\), are commonly implemented in cubic architectures (as illustrated in Figures~\ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_pph}) and are mainly used for vibration isolation~\cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax,thayer98_stewar,mcinroy99_dynam,preumont07_six_axis_singl_stage_activ}. -For applications requiring short stroke (typically smaller than \(500\,\mu m\)), piezoelectric actuators present an interesting alternative, as shown in~\cite{agrawal04_algor_activ_vibrat_isolat_spacec,furutani04_nanom_cuttin_machin_using_stewar,yang19_dynam_model_decoup_contr_flexib}. +Voice coil actuators, providing stroke ranges from \(0.5\,\text{mm}\) to \(10\,\text{mm}\), are commonly implemented in cubic architectures (as illustrated in Figures~\ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_pph}) and are mainly used for vibration isolation~\cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax,thayer98_stewar,mcinroy99_dynam,preumont07_six_axis_singl_stage_activ}. +For applications requiring short stroke (typically smaller than \(500\,\mu\text{m}\)), piezoelectric actuators present an interesting alternative, as shown in~\cite{agrawal04_algor_activ_vibrat_isolat_spacec,furutani04_nanom_cuttin_machin_using_stewar,yang19_dynam_model_decoup_contr_flexib}. Examples of piezoelectric-actuated Stewart platforms are presented in Figures~\ref{fig:detail_kinematics_ulb_pz}, \ref{fig:detail_kinematics_uqp} and \ref{fig:detail_kinematics_yang19}. Although less frequently encountered, magnetostrictive actuators have been successfully implemented in~\cite{zhang11_six_dof} (Figure~\ref{fig:detail_kinematics_zhang11}). @@ -5924,7 +5924,7 @@ Having struts further apart decreases the ``lever arm'' and therefore reduces th It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex. For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke. -This analysis is conducted in Section~\ref{sec:detail_kinematics_nano_hexapod} to estimate the required actuator stroke for the nano-hexapod geometry. +This analysis is conducted in Section~\ref{sec:detail_kinematics_nano_hexapod} to estimate the required actuator stroke for the active platform geometry. \subsubsection{Stiffness} \label{ssec:detail_kinematics_geometry_stiffness} The stiffness matrix defines how the top platform of the Stewart platform (i.e. frame \(\{B\}\)) deforms with respect to its fixed base (i.e. frame \(\{A\}\)) due to static forces/torques applied between frames \(\{A\}\) and \(\{B\}\). @@ -6044,11 +6044,11 @@ It is also possible to implement designs with strut lengths smaller than the cub Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption~\cite{geng94_six_degree_of_freed_activ,preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}: simplified kinematics relationships and dynamical analysis~\cite{geng94_six_degree_of_freed_activ}; uniform stiffness in all directions~\cite{hanieh03_activ_stewar}; uniform mobility~\cite[, chapt.8.5.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}; and minimization of the cross coupling between actuators and sensors in different struts~\cite{preumont07_six_axis_singl_stage_activ}. This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control~\cite{geng94_six_degree_of_freed_activ,thayer02_six_axis_vibrat_isolat_system}. -These properties are examined in this section to assess their relevance for the nano-hexapod. +These properties are examined in this section to assess their relevance for the active platform. The mobility and stiffness properties of the cubic configuration are analyzed in Section~\ref{ssec:detail_kinematics_cubic_static}. Dynamical decoupling is investigated in Section~\ref{ssec:detail_kinematics_cubic_dynamic}, while decentralized control, crucial for the NASS, is examined in Section~\ref{ssec:detail_kinematics_decentralized_control}. Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section~\ref{ssec:detail_kinematics_cubic_design}. -The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod. +The ultimate objective is to determine the suitability of the cubic architecture for the active platform. \subsubsection{Static Properties} \label{ssec:detail_kinematics_cubic_static} \paragraph{Stiffness matrix for the Cubic architecture} @@ -6226,8 +6226,8 @@ An effective strategy for improving dynamical performances involves aligning the This can be achieved by positioning the payload below the top platform, such that the \acrlong{com} of the moving body coincides with the cube's center (Figure~\ref{fig:detail_kinematics_cubic_centered_payload}). This approach was physically implemented in several studies~\cite{mcinroy99_dynam,jafari03_orthog_gough_stewar_platf_microm}, as shown in Figure~\ref{fig:detail_kinematics_uw_gsp}. The resulting dynamics are indeed well-decoupled (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_com_cok}), taking advantage from diagonal stiffness and mass matrices. -The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform. -If a design similar to Figure~\ref{fig:detail_kinematics_cubic_centered_payload} were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation. +The primary limitation of this approach is that, for many applications including the NASS, the payload must be positioned above the top platform. +If a design similar to Figure~\ref{fig:detail_kinematics_cubic_centered_payload} were employed for the active platform, the X-ray beam would intersect with the struts during spindle rotation. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -6332,7 +6332,7 @@ This section proposes modifications to the cubic architecture to enable position Three key parameters define the geometry of the cubic Stewart platform: \(H\), the height of the Stewart platform (distance from fixed base to mobile platform); \(H_c\), the height of the cube, as shown in Figure~\ref{fig:detail_kinematics_cubic_schematic_full}; and \(H_{CoM}\), the height of the \acrlong{com} relative to the mobile platform (coincident with the cube's center). Depending on the cube's size \(H_c\) in relation to \(H\) and \(H_{CoM}\), different designs emerge. -In the following examples, \(H = 100\,mm\) and \(H_{CoM} = 20\,mm\). +In the following examples, \(H = 100\,\text{mm}\) and \(H_{CoM} = 20\,\text{mm}\). \paragraph{Small cube} When the cube size \(H_c\) is smaller than twice the height of the CoM \(H_{CoM}\) \eqref{eq:detail_kinematics_cube_small}, the resulting design is shown in Figure~\ref{fig:detail_kinematics_cubic_above_small}. @@ -6459,36 +6459,36 @@ However, this arrangement presents practical challenges, as the cube's center is To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform. Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform. This structural modification enables the alignment of the moving body's \acrlong{com} with the \acrlong{cok}, resulting in beneficial decoupling properties in the Cartesian frame. -\subsection{Nano Hexapod} +\subsection{Active Platform for the NASS} \label{sec:detail_kinematics_nano_hexapod} -Based on previous analysis, this section aims to determine the nano-hexapod optimal geometry. -For the NASS, the chosen reference frames \(\{A\}\) and \(\{B\}\) coincide with the sample's \acrshort{poi}, which is positioned \(150\,mm\) above the top platform. +Based on previous analysis, this section aims to determine the active platform optimal geometry. +For the NASS, the chosen reference frames \(\{A\}\) and \(\{B\}\) coincide with the sample's \acrshort{poi}, which is positioned \(150\,\text{mm}\) above the top platform. This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused. \subsubsection{Requirements} \label{ssec:detail_kinematics_nano_hexapod_requirements} -The design of the nano-hexapod must satisfy several constraints. -The device should fit within a cylinder with radius of \(120\,mm\) and height of \(95\,mm\). -Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of \(\pm 50\,\mu m\). +The design of the active platform must satisfy several constraints. +The device should fit within a cylinder with radius of \(120\,\text{mm}\) and height of \(95\,\text{mm}\). +Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of \(\pm 50\,\mu\text{m}\). At any position, the system should be capable of performing \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\). Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of \(2\pi\,\text{rad/s}\) to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics. In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts. \subsubsection{Obtained Geometry} \label{ssec:detail_kinematics_nano_hexapod_geometry} -Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads, with masses ranging from 1kg to 50kg, makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations. +Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads, with masses ranging from \(1\,\text{kg}\) to \(50\,\text{kg}\), makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations. -For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several considerations. +For the active platform design, the struts were oriented more vertically compared to a cubic architecture due to several considerations. First, the performance requirements in the vertical direction are more stringent than in the horizontal direction. This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise. Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes. -Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod. +Therefore, higher resonance frequencies of the active platform in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the active platform. Regarding dynamical properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that strut orientation has minimal impact on the resulting coupling characteristics. Consequently, the geometry was selected according to practical constraints. -The height between the two plates is maximized and set at \(95\,mm\). -Both platforms take the maximum available size, with joints offset by \(15\,mm\) from the plate surfaces and positioned along circles with radii of \(120\,mm\) for the fixed joints and \(110\,mm\) for the mobile joints. +The height between the two plates is maximized and set at \(95\,\text{mm}\). +Both platforms take the maximum available size, with joints offset by \(15\,\text{mm}\) from the plate surfaces and positioned along circles with radii of \(120\,\text{mm}\) for the fixed joints and \(110\,\text{mm}\) for the mobile joints. The positioning angles, as shown in Figure~\ref{fig:detail_kinematics_nano_hexapod_top}, are \([255,\ 285,\ 15,\ 45,\ 135,\ 165]\) degrees for the top joints and \([220,\ 320,\ 340,\ 80,\ 100,\ 200]\) degrees for the bottom joints. \begin{figure}[htbp] @@ -6504,66 +6504,66 @@ The positioning angles, as shown in Figure~\ref{fig:detail_kinematics_nano_hexap \end{center} \subcaption{\label{fig:detail_kinematics_nano_hexapod_top}Top view} \end{subfigure} -\caption{\label{fig:detail_kinematics_nano_hexapod}Obtained architecture for the Nano Hexapod} +\caption{\label{fig:detail_kinematics_nano_hexapod}Obtained architecture for the active platform} \end{figure} The resulting geometry is illustrated in Figure~\ref{fig:detail_kinematics_nano_hexapod}. While minor refinements may occur during detailed mechanical design to address manufacturing and assembly considerations, the fundamental geometry will remain consistent with this configuration. This geometry serves as the foundation for estimating required actuator stroke (Section~\ref{ssec:detail_kinematics_nano_hexapod_actuator_stroke}), determining flexible joint stroke requirements (Section~\ref{ssec:detail_kinematics_nano_hexapod_joint_stroke}), performing noise budgeting for instrumentation selection, and developing control strategies. Implementing a cubic architecture as proposed in Section~\ref{ssec:detail_kinematics_cubic_design} was considered. -However, positioning the cube's center \(150\,mm\) above the top platform would have resulted in platform dimensions exceeding the maximum available size. -Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the nano-hexapod, ensuring that its \acrlong{com} coincides with the cube's center. -Given the impracticality of consistently aligning the \acrlong{com} with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. +However, positioning the cube's center \(150\,\text{mm}\) above the top platform would have resulted in platform dimensions exceeding the maximum available size. +Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the active platform, ensuring that its \acrlong{com} coincides with the cube's center. +Given the impracticality of consistently aligning the \acrlong{com} with the cube's center, the cubic architecture was deemed unsuitable for the NASS. \subsubsection{Required Actuator stroke} \label{ssec:detail_kinematics_nano_hexapod_actuator_stroke} With the geometry established, the actuator stroke necessary to achieve the desired mobility can be determined. -The required mobility parameters include combined translations in the XYZ directions of \(\pm 50\,\mu m\) (essentially a cubic workspace). +The required mobility parameters include combined translations in the XYZ directions of \(\pm 50\,\mu\text{m}\) (essentially a cubic workspace). Additionally, at any point within this workspace, combined \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\), with \(R_z\) maintained at 0, should be possible. -Calculations based on the selected geometry indicate that an actuator stroke of \(\pm 94\,\mu m\) is required to achieve the desired mobility. -This specification will be used during the actuator selection process in Section \ref{sec:detail_fem_actuator}. +Calculations based on the selected geometry indicate that an actuator stroke of \(\pm 94\,\mu\text{m}\) is required to achieve the desired mobility. +This specification will be used during the actuator selection process in Section~\ref{sec:detail_fem_actuator}. -Figure~\ref{fig:detail_kinematics_nano_hexapod_mobility} illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of \(\pm 94\,\mu m\). +Figure~\ref{fig:detail_kinematics_nano_hexapod_mobility} illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the active platform with an actuator stroke of \(\pm 94\,\mu\text{m}\). The diagram confirms that the required workspace fits within the system's capabilities. \begin{figure}[htbp] \centering \includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_nano_hexapod_mobility.png} -\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Specified translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).} +\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Specified translation mobility of the active platform (grey cube) and computed Mobility (red volume).} \end{figure} \subsubsection{Required Joint angular stroke} \label{ssec:detail_kinematics_nano_hexapod_joint_stroke} -With the nano-hexapod geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined. +With the active platform geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined. This analysis focuses solely on bending stroke, as the torsional stroke of the flexible joints is expected to be minimal given the absence of vertical rotation requirements. The required angular stroke for both fixed and mobile joints is estimated to be equal to \(1\,\text{mrad}\). -This specification will guide the design of the flexible joints in Section \ref{sec:detail_fem_joint}. +This specification will guide the design of the flexible joints in Section~\ref{sec:detail_fem_joint}. \subsection*{Conclusion} \label{sec:detail_kinematics_conclusion} -This chapter has explored the optimization of the nano-hexapod geometry for the Nano Active Stabilization System (NASS). +This chapter has explored the optimization of the active platform geometry for the Nano Active Stabilization System (NASS). First, a review of existing Stewart platforms revealed two main geometric categories: cubic architectures, characterized by mutually orthogonal struts arranged along the edges of a cube, and non-cubic architectures with varied strut orientations. While cubic architectures are prevalent in the literature and attributed with beneficial properties such as simplified kinematics, uniform stiffness, and reduced cross-coupling, the performed analysis revealed that some of these advantages should be more nuanced or context-dependent than commonly described. The analytical relationships between Stewart platform geometry and its mechanical properties were established, enabling a better understanding of the trade-offs between competing requirements such as mobility and stiffness along different axes. -These insights were useful during the nano-hexapod geometry optimization. +These insights were useful during the active platform geometry optimization. For the cubic configuration, complete dynamical decoupling in the Cartesian frame can be achieved when the \acrlong{com} of the moving body coincides with the cube's center, but this arrangement is often impractical for real-world applications. -Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. +Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the active platform due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. -For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. +For the active platform design, a key challenge was addressing the wide range of potential payloads (1 to \(50\,\text{kg}\)), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies. \section{Component Optimization} \label{sec:detail_fem} -During the nano-hexapod's detailed design phase, a hybrid modeling approach combining \acrfull{fea} with multi-body dynamics was developed. -This methodology, using reduced-order flexible bodies, was created to enable both detailed component optimization and efficient system-level simulation, addressing the impracticality of a full \acrshort{fem} for real-time control scenarios. +Addressing the need for both detailed component optimization and efficient system-level simulation—especially considering the limitations of full \acrshort{fem} for real-time control—a hybrid modeling approach was used. +This combines \acrfull{fea} with multi-body dynamics, employing reduced-order flexible bodies. The theoretical foundations and implementation are presented in Section~\ref{sec:detail_fem_super_element}, where experimental validation was performed using an Amplified Piezoelectric Actuator. -The framework was then applied to optimize two critical nano-hexapod elements: the actuators (Section~\ref{sec:detail_fem_actuator}) and the flexible joints (Section~\ref{sec:detail_fem_joint}). +The framework was then applied to optimize two critical active platform elements: the actuators (Section~\ref{sec:detail_fem_actuator}) and the flexible joints (Section~\ref{sec:detail_fem_joint}). Through this approach, system-level dynamic behavior under closed-loop control conditions could be successfully predicted while detailed component-level optimization was facilitated. \subsection{Reduced order flexible bodies} \label{sec:detail_fem_super_element} @@ -6601,7 +6601,7 @@ m = 6 \times n + p \subsubsection{Example with an Amplified Piezoelectric Actuator} \label{ssec:detail_fem_super_element_example} The presented modeling framework was first applied to an \acrfull{apa} for several reasons. -Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section~\ref{sec:detail_fem_actuator}. +Primarily, this actuator represents an excellent candidate for implementation within the active platform, as will be elaborated in Section~\ref{sec:detail_fem_actuator}. Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure~\ref{fig:detail_fem_apa95ml_picture}) was available in the laboratory for experimental testing. The \acrshort{apa} consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure~\ref{fig:detail_fem_apa95ml_picture}) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement in the vertical direction~\cite{claeyssen07_amplif_piezoel_actuat}. @@ -6622,9 +6622,9 @@ The specific design of the \acrshort{apa} allows for the simultaneous modeling o \toprule \textbf{Parameter} & \textbf{Value}\\ \midrule -Nominal Stroke & \(100\,\mu m\)\\ -Blocked force & \(2100\,N\)\\ -Stiffness & \(21\,N/\mu m\)\\ +Nominal Stroke & \(100\,\mu\text{m}\)\\ +Blocked force & \(2100\,\text{N}\)\\ +Stiffness & \(21\,\text{N}/\mu\text{m}\)\\ \bottomrule \end{tabularx}} \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications} @@ -6641,8 +6641,8 @@ The finite element mesh, shown in Figure~\ref{fig:detail_fem_apa95ml_mesh}, was \toprule & \(E\) & \(\nu\) & \(\rho\)\\ \midrule -Stainless Steel & \(190\,GPa\) & \(0.31\) & \(7800\,\text{kg}/m^3\)\\ -Piezoelectric Ceramics (PZT) & \(49.5\,GPa\) & \(0.31\) & \(7800\,\text{kg}/m^3\)\\ +Stainless Steel & \(190\,\text{GPa}\) & \(0.31\) & \(7800\,\text{kg}/\text{m}^3\)\\ +Piezoelectric Ceramics (PZT) & \(49.5\,\text{GPa}\) & \(0.31\) & \(7800\,\text{kg}/\text{m}^3\)\\ \bottomrule \end{tabularx} \end{table} @@ -6704,19 +6704,19 @@ Yet, based on the available properties of the stacks in the data-sheet (summariz \toprule \textbf{Parameter} & \textbf{Value}\\ \midrule -Nominal Stroke & \(20\,\mu m\)\\ -Blocked force & \(4700\,N\)\\ -Stiffness & \(235\,N/\mu m\)\\ -Voltage Range & \(-20/150\,V\)\\ -Capacitance & \(4.4\,\mu F\)\\ -Length & \(20\,mm\)\\ -Stack Area & \(10\times 10\,mm^2\)\\ +Nominal Stroke & \(20\,\mu\text{m}\)\\ +Blocked force & \(4700\,\text{N}\)\\ +Stiffness & \(235\,\text{N}/\mu\text{m}\)\\ +Voltage Range & \(-20/150\,\text{V}\)\\ +Capacitance & \(4.4\,\mu\text{F}\)\\ +Length & \(20\,\text{mm}\)\\ +Stack Area & \(10\times 10\,\text{mm}^2\)\\ \bottomrule \end{tabularx} \end{table} The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table~\ref{tab:detail_fem_piezo_properties}. -From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained. +From these parameters, \(g_s = 5.1\,\text{V}/\mu\text{m}\) and \(g_a = 26\,\text{N/V}\) were obtained. \begin{table}[htbp] \caption{\label{tab:detail_fem_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities} @@ -6725,12 +6725,12 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine \toprule \textbf{Parameter} & \textbf{Value} & \textbf{Description}\\ \midrule -\(d_{33}\) & \(680 \cdot 10^{-12}\,m/V\) & Piezoelectric constant\\ -\(\epsilon^{T}\) & \(4.0 \cdot 10^{-8}\,F/m\) & Permittivity under constant stress\\ -\(s^{D}\) & \(21 \cdot 10^{-12}\,m^2/N\) & Elastic compliance understand constant electric displacement\\ -\(c^{E}\) & \(48 \cdot 10^{9}\,N/m^2\) & Young's modulus of elasticity\\ -\(L\) & \(20\,mm\) per stack & Length of the stack\\ -\(A\) & \(10^{-4}\,m^2\) & Area of the piezoelectric stack\\ +\(d_{33}\) & \(680 \cdot 10^{-12}\,\text{m/V}\) & Piezoelectric constant\\ +\(\epsilon^{T}\) & \(4.0 \cdot 10^{-8}\,\text{F/m}\) & Permittivity under constant stress\\ +\(s^{D}\) & \(21 \cdot 10^{-12}\,\text{m}^2/\text{N}\) & Elastic compliance understand constant electric displacement\\ +\(c^{E}\) & \(48 \cdot 10^{9}\,\text{N}/\text{m}^2\) & Young's modulus of elasticity\\ +\(L\) & \(20\,\text{mm}\) per stack & Length of the stack\\ +\(A\) & \(10^{-4}\,\text{m}^2\) & Area of the piezoelectric stack\\ \(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\ \bottomrule \end{tabularx} @@ -6741,7 +6741,7 @@ Initial validation of the \acrlong{fem} and its integration as a reduced-order f The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure~\ref{fig:detail_fem_apa95ml_compliance}), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames. The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML. -A value of \(23\,N/\mu m\) was found which is close to the specified stiffness in the datasheet of \(k = 21\,N/\mu m\). +A value of \(23\,\text{N}/\mu\text{m}\) was found which is close to the specified stiffness in the datasheet of \(k = 21\,\text{N}/\mu\text{m}\). The multi-body model predicted a resonant frequency under block-free conditions of \(\approx 2\,\text{kHz}\) (Figure~\ref{fig:detail_fem_apa95ml_compliance}), which is in agreement with the nominal specification. @@ -6754,9 +6754,9 @@ The multi-body model predicted a resonant frequency under block-free conditions In order to estimate the stroke of the APA95ML, the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, was first determined. This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of \(1.5\) was derived. -The piezoelectric stacks, exhibiting a typical strain response of \(0.1\,\%\) relative to their length (here equal to \(20\,mm\)), produce an individual nominal stroke of \(20\,\mu m\) (see data-sheet of the piezoelectric stacks on Table~\ref{tab:detail_fem_stack_parameters}, page~\pageref{tab:detail_fem_stack_parameters}). -As three stacks are used, the horizontal displacement is \(60\,\mu m\). -Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of \(90\,\mu m\) which falls within the manufacturer-specified range of \(80\,\mu m\) and \(120\,\mu m\). +The piezoelectric stacks, exhibiting a typical strain response of \(0.1\,\%\) relative to their length (here equal to \(20\,\text{mm}\)), produce an individual nominal stroke of \(20\,\mu\text{m}\) (see data-sheet of the piezoelectric stacks on Table~\ref{tab:detail_fem_stack_parameters}, page~\pageref{tab:detail_fem_stack_parameters}). +As three stacks are used, the horizontal displacement is \(60\,\mu\text{m}\). +Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of \(90\,\mu\text{m}\) which falls within the manufacturer-specified range of \(80\,\mu\text{m}\) and \(120\,\mu\text{m}\). The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include \acrshort{fem} into multi-body model. \subsubsection{Experimental Validation} @@ -6764,7 +6764,7 @@ The high degree of concordance observed across multiple performance metrics prov Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation. The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element. -The test bench illustrated in Figure~\ref{fig:detail_fem_apa95ml_bench_schematic} was used, which consists of a \(5.7\,kg\) granite suspended on top of the APA95ML. +The test bench illustrated in Figure~\ref{fig:detail_fem_apa95ml_bench_schematic} was used, which consists of a \(5.7\,\text{kg}\) granite suspended on top of the APA95ML. The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement \(y\). A \acrfull{dac} was used to generate the control signal \(u\), which was subsequently conditioned through a voltage amplifier with a gain of \(20\), ultimately yielding the effective voltage \(V_a\) across the two piezoelectric stacks. Measurement of the sensor stack voltage \(V_s\) was performed using an \acrshort{adc}. @@ -6786,7 +6786,7 @@ To improve the quality of the obtained frequency domain data, averaging and wind The obtained \acrshortpl{frf} from \(V_a\) to \(V_s\) and to \(y\) are compared with the theoretical predictions derived from the multi-body model in Figure~\ref{fig:detail_fem_apa95ml_comp_plant}. -The difference in phase between the model and the measurements can be attributed to the sampling time of \(0.1\,ms\) and to additional delays induced by electronic instrumentation related to the interferometer. +The difference in phase between the model and the measurements can be attributed to the sampling time of \(0.1\,\text{ms}\) and to additional delays induced by electronic instrumentation related to the interferometer. The presence of a non-minimum phase zero in the measured system response (Figure~\ref{fig:detail_fem_apa95ml_comp_plant_sensor}), shall be addressed during the experimental phase. Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could be further refined through calibration against the experimental data. @@ -6843,7 +6843,7 @@ The close agreement between experimental measurements and theoretical prediction The experimental validation with an Amplified Piezoelectric Actuator confirms that this methodology accurately predicts both open-loop and closed-loop dynamic behaviors. This verification establishes its effectiveness for component design and system analysis applications. -The approach will be especially beneficial for optimizing actuators (Section~\ref{sec:detail_fem_actuator}) and flexible joints (Section~\ref{sec:detail_fem_joint}) for the nano-hexapod. +The approach will be especially beneficial for optimizing actuators (Section~\ref{sec:detail_fem_actuator}) and flexible joints (Section~\ref{sec:detail_fem_joint}) for the active platform. \subsection{Actuator Selection} \label{sec:detail_fem_actuator} \subsubsection{Choice of the Actuator based on Specifications} @@ -6852,14 +6852,14 @@ The approach will be especially beneficial for optimizing actuators (Section~\re The actuator selection process was driven by several critical requirements derived from previous dynamic analyses. A primary consideration is the actuator stiffness, which significantly impacts system dynamics through multiple mechanisms. The spindle rotation induces gyroscopic effects that modify plant dynamics and increase coupling, necessitating sufficient stiffness. -Conversely, the actuator stiffness must be carefully limited to ensure the nano-hexapod's suspension modes remain below the problematic modes of the micro-station to limit the coupling between the two structures. -These competing requirements suggest an optimal stiffness of approximately \(1\,N/\mu m\). +Conversely, the actuator stiffness must be carefully limited to ensure the active platform's suspension modes remain below the problematic modes of the micro-station to limit the coupling between the two structures. +These competing requirements suggest an optimal stiffness of approximately \(1\,\text{N}/\mu\text{m}\). Additional specifications arise from the control strategy and physical constraints. The implementation of the decentralized Integral Force Feedback (IFF) architecture necessitates force sensors to be collocated with each actuator. -The system's geometric constraints limit the actuator height to 50mm, given the nano-hexapod's maximum height of 95mm and the presence of flexible joints at each strut extremity. +The system's geometric constraints limit the actuator height to 50mm, given the active platform's maximum height of 95mm and the presence of flexible joints at each strut extremity. Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility. -An actuator stroke of \(\approx 200\,\mu m\) is therefore required. +An actuator stroke of \(\approx 200\,\mu\text{m}\) is therefore required. Three actuator technologies were evaluated (examples of such actuators are shown in Figure~\ref{fig:detail_fem_actuator_pictures}): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators. Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity. @@ -6883,16 +6883,16 @@ Variable reluctance actuators were not considered despite their superior efficie \end{center} \subcaption{\label{fig:detail_fem_fpa_picture}Amplified Piezoelectric Actuator} \end{subfigure} -\caption{\label{fig:detail_fem_actuator_pictures}Example of actuators considered for the nano-hexapod. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}).} +\caption{\label{fig:detail_fem_actuator_pictures}Example of actuators considered for the active platform. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}).} \end{figure} -Voice coil actuators (shown in Figure~\ref{fig:detail_fem_voice_coil_picture}), when combined with flexure guides of wanted stiffness (\(\approx 1\,N/\mu m\)), would require forces in the order of \(200\,N\) to achieve the specified \(200\,\mu m\) displacement. +Voice coil actuators (shown in Figure~\ref{fig:detail_fem_voice_coil_picture}), when combined with flexure guides of wanted stiffness (\(\approx 1\,\text{N}/\mu\text{m}\)), would require forces in the order of \(200\,\text{N}\) to achieve the specified \(200\,\mu\text{m}\) displacement. While these actuators offer excellent linearity and long strokes capabilities, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability. Their advantages (linearity and long stroke) were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions. Conventional piezoelectric stack actuators (shown in Figure~\ref{fig:detail_fem_piezo_picture}) present two significant limitations for the current application. -Their stroke is inherently limited to approximately \(0.1\,\%\) of their length, meaning that even with the maximum allowable height of \(50\,mm\), the achievable stroke would only be \(50\,\mu m\), insufficient for the application. -Additionally, their extremely high stiffness, typically around \(100\,N/\mu m\), exceeds the desired specifications by two orders of magnitude. +Their stroke is inherently limited to approximately \(0.1\,\%\) of their length, meaning that even with the maximum allowable height of \(50\,\text{mm}\), the achievable stroke would only be \(50\,\mu\text{m}\), insufficient for the application. +Additionally, their extremely high stiffness, typically around \(100\,\text{N}/\mu\text{m}\), exceeds the desired specifications by two orders of magnitude. Amplified Piezoelectric Actuators emerged as the optimal solution by addressing these limitations through a specific mechanical design. The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile. @@ -6905,17 +6905,17 @@ This selection was further reinforced by previous experience with \acrshortpl{ap The demonstrated accuracy of the modeling approach for the APA95ML provides confidence in the reliable prediction of the APA300ML's dynamic characteristics, thereby supporting both the selection decision and subsequent dynamical analyses. \begin{table}[htbp] -\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the nano-hexapod} +\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the active platform} \centering \begin{tabularx}{0.9\linewidth}{Xccccc} \toprule \textbf{Specification} & APA150M & \textbf{APA300ML} & APA400MML & FPA-0500E-P & FPA-0300E-S\\ \midrule -Stroke \(> 200\, [\mu m]\) & 187 & 304 & 368 & 432 & 240\\ -Stiffness \(\approx 1\, [N/\mu m]\) & 0.7 & 1.8 & 0.55 & 0.87 & 0.58\\ -Resolution \(< 2\, [nm]\) & 2 & 3 & 4 & & \\ -Blocked Force \(> 100\, [N]\) & 127 & 546 & 201 & 376 & 139\\ -Height \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\ +Stroke \(> 200\,\mu\text{m}\) & 187 & 304 & 368 & 432 & 240\\ +Stiffness \(\approx 1\,\text{N}/\mu\text{m}\) & 0.7 & 1.8 & 0.55 & 0.87 & 0.58\\ +Resolution \(< 2\,\text{nm}\) & 2 & 3 & 4 & & \\ +Blocked Force \(> 100\,\text{N}\) & 127 & 546 & 201 & 376 & 139\\ +Height \(< 50\,\text{mm}\) & 22 & 30 & 24 & 27 & 16\\ \bottomrule \end{tabularx} \end{table} @@ -6989,14 +6989,14 @@ While higher-order modes and non-axial flexibility are not captured, the model a \toprule \textbf{Parameter} & \textbf{Value}\\ \midrule -\(k_1\) & \(0.30\,N/\mu m\)\\ -\(k_e\) & \(4.3\, N/\mu m\)\\ -\(k_a\) & \(2.15\,N/\mu m\)\\ -\(c_1\) & \(18\,Ns/m\)\\ -\(c_e\) & \(0.7\,Ns/m\)\\ -\(c_a\) & \(0.35\,Ns/m\)\\ -\(g_a\) & \(2.7\,N/V\)\\ -\(g_s\) & \(0.53\,V/\mu m\)\\ +\(k_1\) & \(0.30\,\text{N}/\mu\text{m}\)\\ +\(k_e\) & \(4.3\,\text{N}/\mu\text{m}\)\\ +\(k_a\) & \(2.15\,\text{N}/\mu\text{m}\)\\ +\(c_1\) & \(18\,\text{Ns/m}\)\\ +\(c_e\) & \(0.7\,\text{Ns/m}\)\\ +\(c_a\) & \(0.35\,\text{Ns/m}\)\\ +\(g_a\) & \(2.7\,\text{N}/V\)\\ +\(g_s\) & \(0.53\,\text{V}/\mu\text{m}\)\\ \bottomrule \end{tabularx} \end{table} @@ -7035,10 +7035,10 @@ The developed models of the \acrshort{apa} do not represent such behavior, but a However, the electrical characteristics of the \acrshort{apa} remain crucial for instrumentation design. Proper consideration must be given to voltage amplifier specifications and force sensor signal conditioning requirements. These aspects will be addressed in the instrumentation chapter. -\subsubsection{Validation with the Nano-Hexapod} +\subsubsection{Validation with the Active Platform} \label{ssec:detail_fem_actuator_apa300ml_validation} -The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with \acrshort{apa} modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full \acrshort{fem} implementation. +The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with \acrshort{apa} modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full \acrshort{fem} implementation. The dynamics predicted using the flexible body model align well with the design requirements established during the conceptual phase. The dynamics from \(\bm{u}\) to \(\bm{V}_s\) exhibits the desired alternating pole-zero pattern (Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}), a critical characteristic for implementing robust decentralized Integral Force Feedback. @@ -7047,9 +7047,9 @@ These findings suggest that the control performance targets established during t Comparative analysis between the high-order \acrshort{fem} implementation and the simplified 2DoF model (Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_plants}) demonstrates remarkable agreement in the frequency range of interest. This validates the use of the simplified model for time-domain simulations. -The reduction in model order is substantial: while the \acrshort{fem} implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete nano-hexapod. +The reduction in model order is substantial: while the \acrshort{fem} implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete active platform. -These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the nano-hexapod. +These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the active platform. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -7064,14 +7064,14 @@ These results validate both the selection of the APA300ML and the effectiveness \end{center} \subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$} \end{subfigure} -\caption{\label{fig:detail_fem_actuator_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a nano-hexapod having actuators modelled a 2DoF system. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}).} +\caption{\label{fig:detail_fem_actuator_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a active platform having actuators modelled a 2DoF system. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}).} \end{figure} \subsection{Flexible Joint Design} \label{sec:detail_fem_joint} High-precision position control at the nanometer scale requires systems to be free from friction and backlash, as these nonlinear phenomena severely limit achievable positioning accuracy. This fundamental requirement prevents the use of conventional joints, necessitating instead the implementation of flexible joints that achieve motion through elastic deformation. For Stewart platforms requiring nanometric precision, numerous flexible joint designs have been developed and successfully implemented, as illustrated in Figure~\ref{fig:detail_fem_joints_examples}. -For design simplicity and component standardization, identical joints are employed at both ends of the nano-hexapod struts. +For design simplicity and component standardization, identical joints are employed at both ends of the active platform struts. \begin{figure}[htbp] \begin{subfigure}{0.3\textwidth} @@ -7100,19 +7100,19 @@ This section examines how these non-ideal characteristics affect system behavior The analysis of bending and axial stiffness effects enables the establishment of comprehensive specifications for the flexible joints. These specifications guide the development and optimization of a flexible joint design through \acrshort{fea} (Section~\ref{ssec:detail_fem_joint_specs}). -The validation process, detailed in Section~\ref{ssec:detail_fem_joint_validation}, begins with the integration of the joints as ``reduced order flexible bodies'' in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints. +The validation process, detailed in Section~\ref{ssec:detail_fem_joint_validation}, begins with the integration of the joints as ``reduced order flexible bodies'' in the active platform model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints. \subsubsection{Bending and Torsional Stiffness} \label{ssec:detail_fem_joint_bending} The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction~\cite{mcinroy02_model_desig_flexur_joint_stewar} and can affect system dynamics. -To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of \(1\,N/\mu m\)) without parallel stiffness to the force sensors. -Flexible joint bending stiffness was varied from 0 (ideal case) to \(500\,Nm/\text{rad}\). +To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of \(1\,\text{N}/\mu\text{m}\)) without parallel stiffness to the force sensors. +Flexible joint bending stiffness was varied from 0 (ideal case) to \(500\,\text{Nm}/\text{rad}\). Analysis of the plant dynamics reveals two significant effects. For the transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\), bending stiffness increases low-frequency coupling, though this remains small for realistic stiffness values (Figure~\ref{fig:detail_fem_joints_bending_stiffness_hac_plant}). In~\cite{mcinroy02_model_desig_flexur_joint_stewar}, it is established that forces remain effectively aligned with the struts when the flexible joint bending stiffness is much small than the actuator stiffness multiplied by the square of the strut length. -For the nano-hexapod, this corresponds to having the bending stiffness much lower than 9000 Nm/rad. +For the active platform, this corresponds to having the bending stiffness much lower than 9000 Nm/rad. This condition is more readily satisfied with the relatively stiff actuators selected, and could be problematic for softer Stewart platforms. For the force sensor plant, bending stiffness introduces complex conjugate zeros at low frequency (Figure~\ref{fig:detail_fem_joints_bending_stiffness_iff_plant}). @@ -7160,10 +7160,10 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes The limited axial stiffness (\(k_a\)) of flexible joints introduces an additional compliance between the actuation point and the measurement point. As explained in~\cite[, chapter 6]{preumont18_vibrat_contr_activ_struc_fourt_edition} and in~\cite{rankers98_machin} (effect called ``actuator flexibility''), such intermediate flexibility invariably degrades control performance. -Therefore, determining the minimum acceptable axial stiffness that maintains nano-hexapod performance becomes crucial. +Therefore, determining the minimum acceptable axial stiffness that maintains active platform performance becomes crucial. The analysis incorporates the strut mass (112g per APA300ML) to accurately model internal resonance effects. -A parametric study was conducted by varying the axial stiffness from \(1\,N/\mu m\) (matching actuator stiffness) to \(1000\,N/\mu m\) (approximating rigid behavior). +A parametric study was conducted by varying the axial stiffness from \(1\,\text{N}/\mu\text{m}\) (matching actuator stiffness) to \(1000\,\text{N}/\mu\text{m}\) (approximating rigid behavior). The resulting dynamics (Figure~\ref{fig:detail_fem_joints_axial_stiffness_plants}) reveal distinct effects on system dynamics. The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both \acrshortpl{frf} (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_plant}) and root locus analysis (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_locus}). @@ -7176,7 +7176,7 @@ First, the system exhibits strong coupling between control channels, making dece Second, control authority diminishes significantly near the resonant frequencies. These effects fundamentally limit achievable control bandwidth, making high axial stiffness essential for system performance. -Based on this analysis, an axial stiffness specification of \(100\,N/\mu m\) was established for the nano-hexapod joints. +Based on this analysis, an axial stiffness specification of \(100\,\text{N}/\mu\text{m}\) was established for the active platform joints. \begin{figure}[h!tbp] \begin{subfigure}{0.48\textwidth} @@ -7223,17 +7223,17 @@ Based on the dynamic analysis presented in previous sections, quantitative speci \toprule & \textbf{Specification} & \textbf{FEM}\\ \midrule -Axial Stiffness \(k_a\) & \(> 100\,N/\mu m\) & 94\\ -Shear Stiffness \(k_s\) & \(> 1\,N/\mu m\) & 13\\ -Bending Stiffness \(k_f\) & \(< 100\,Nm/\text{rad}\) & 5\\ -Torsion Stiffness \(k_t\) & \(< 500\,Nm/\text{rad}\) & 260\\ +Axial Stiffness \(k_a\) & \(> 100\,\text{N}/\mu\text{m}\) & 94\\ +Shear Stiffness \(k_s\) & \(> 1\,\text{N}/\mu\text{m}\) & 13\\ +Bending Stiffness \(k_f\) & \(< 100\,\text{Nm}/\text{rad}\) & 5\\ +Torsion Stiffness \(k_t\) & \(< 500\,\text{Nm}/\text{rad}\) & 260\\ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\ \bottomrule \end{tabularx} \end{table} Among various possible flexible joint architectures, the design shown in Figure~\ref{fig:detail_fem_joints_design} was selected for three key advantages. -First, the geometry creates coincident \(x\) and \(y\) rotation axes, ensuring well-defined kinematic behavior, important for the precise definition of the nano-hexapod Jacobian matrix. +First, the geometry creates coincident \(x\) and \(y\) rotation axes, ensuring well-defined kinematic behavior, important for the precise definition of the active platform Jacobian matrix. Second, the design allows easy tuning of different directional stiffnesses through a limited number of geometric parameters. Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational \acrshortpl{dof}. @@ -7256,10 +7256,10 @@ The final design, featuring a neck dimension of 0.25mm, achieves mechanical prop \end{subfigure} \caption{\label{fig:detail_fem_joints_design}Designed flexible joints.} \end{figure} -\subsubsection{Validation with the Nano-Hexapod} +\subsubsection{Validation with the Active Platform} \label{ssec:detail_fem_joint_validation} -The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from \acrshort{fea}. +The designed flexible joint was first validated through integration into the active platform model using reduced-order flexible bodies derived from \acrshort{fea}. This high-fidelity representation was created by defining two interface frames (Figure~\ref{fig:detail_fem_joints_frames}) and extracting six additional modes, resulting in reduced-order mass and stiffness matrices of dimension \(18 \times 18\). The computed transfer functions from actuator forces to both force sensor measurements (\(\bm{f}\) to \(\bm{f}_m\)) and external metrology (\(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\)) demonstrate dynamics consistent with predictions from earlier analyses (Figure~\ref{fig:detail_fem_joints_fem_vs_perfect_plants}), thereby validating the joint design. @@ -7270,7 +7270,7 @@ The computed transfer functions from actuator forces to both force sensor measur \end{figure} While this detailed modeling approach provides high accuracy, it results in a significant increase in system model order. -The complete nano-hexapod model incorporates 240 states: 12 for the payload (6 DOF), 12 for the 2DOF struts, and 216 for the flexible joints (18 states for each of the 12 joints). +The complete active platform model incorporates 240 states: 12 for the payload (6 DOF), 12 for the 2DOF struts, and 216 for the flexible joints (18 states for each of the 12 joints). To improve computational efficiency, a low order representation was developed using simplified joint elements with selective compliance DoF. After evaluating various configurations, a compromise was achieved by modeling bottom joints with bending and axial stiffness (\(k_f\) and \(k_a\)), and top joints with bending, torsional, and axial stiffness (\(k_f\), \(k_t\) and \(k_a\)). @@ -7290,12 +7290,12 @@ While additional \acrshortpl{dof} could potentially capture more dynamic feature \end{center} \subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$} \end{subfigure} -\caption{\label{fig:detail_fem_joints_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a nano-hexapod having bottom joint modelled by bending stiffness \(k_f\) and axial stiffness \(k_a\) and top joints modelled by bending stiffness \(k_f\), torsion stiffness \(k_t\) and axial stiffness \(k_a\). Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}).} +\caption{\label{fig:detail_fem_joints_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a active platform having bottom joint modelled by bending stiffness \(k_f\) and axial stiffness \(k_a\) and top joints modelled by bending stiffness \(k_f\), torsion stiffness \(k_t\) and axial stiffness \(k_a\). Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}).} \end{figure} \subsection*{Conclusion} \label{sec:detail_fem_conclusion} -In this chapter, the methodology of combining \acrlong{fea} with multi-body modeling has been demonstrated and validated, proving particularly valuable for the detailed design of nano-hexapod components. +In this chapter, the methodology of combining \acrlong{fea} with multi-body modeling has been demonstrated and validated, proving particularly valuable for the detailed design of active platform components. The approach was first validated using an amplified piezoelectric actuator, where predicted dynamics showed excellent agreement with experimental measurements for both open and closed-loop behavior. This validation established confidence in the method's ability to accurately predict component behavior within a larger system. @@ -7308,7 +7308,7 @@ A key outcome of this work is the development of reduced-order models that maint Such model reduction, guided by detailed understanding of component behavior, provides the foundation for subsequent control system design and optimization. \section{Control Optimization} \label{sec:detail_control} -Three critical elements for the control of parallel manipulators such as the Nano-Hexapod were identified: effective use and combination of multiple sensors, appropriate plant decoupling strategies, and robust controller design for the decoupled system. +Three critical elements for the control of parallel manipulators such as the active platform were identified: effective use and combination of multiple sensors, appropriate plant decoupling strategies, and robust controller design for the decoupled system. During the conceptual design phase of the NASS, pragmatic approaches were implemented for each of these elements. The \acrfull{haclac} architecture was selected for combining sensors. @@ -7886,12 +7886,12 @@ Two reference frames are defined within this model: frame \(\{M\}\) with origin \toprule & \textbf{Description} & \textbf{Value}\\ \midrule -\(l_a\) & & \(0.5\,m\)\\ -\(h_a\) & & \(0.2\,m\)\\ -\(k\) & Actuator stiffness & \(10\,N/\mu m\)\\ -\(c\) & Actuator damping & \(200\,Ns/m\)\\ +\(l_a\) & & \(0.5\,\text{m}\)\\ +\(h_a\) & & \(0.2\,\text{m}\)\\ +\(k\) & Actuator stiffness & \(10\,\text{N}/\mu\text{m}\)\\ +\(c\) & Actuator damping & \(200\,\text{Ns/m}\)\\ \(m\) & Payload mass & \(40\,\text{kg}\)\\ -\(I\) & Payload \(R_z\) inertia & \(5\,\text{kg}m^2\)\\ +\(I\) & Payload \(R_z\) inertia & \(5\,\text{kgm}^2\)\\ \bottomrule \end{tabularx}} \captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters} @@ -8703,7 +8703,7 @@ The loop gain reveals several important characteristics: \begin{itemize} \item The presence of two integrators at low frequencies, enabling accurate tracking of ramp inputs \item A notch at the plant resonance frequency (arising from the plant inverse) -\item A lead component near the control bandwidth of approximately 20 Hz, enhancing stability margins +\item A lead component near the control bandwidth of approximately \(20\,\text{Hz}\), enhancing stability margins \end{itemize} \paragraph{Robustness and Performance analysis} @@ -8823,16 +8823,16 @@ In order to derive specifications in terms of noise spectral density for each in The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met. While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications. -Based on this analysis, the obtained maximum noise levels are as follows: \acrshort{dac} maximum output noise \acrshort{asd} is established at \(14\,\mu V/\sqrt{\text{Hz}}\), voltage amplifier maximum output voltage noise \acrshort{asd} at \(280\,\mu V/\sqrt{\text{Hz}}\), and \acrshort{adc} maximum measurement noise \acrshort{asd} at \(11\,\mu V/\sqrt{\text{Hz}}\). +Based on this analysis, the obtained maximum noise levels are as follows: \acrshort{dac} maximum output noise \acrshort{asd} is established at \(14\,\mu\text{V}/\sqrt{\text{Hz}}\), voltage amplifier maximum output voltage noise \acrshort{asd} at \(280\,\mu\text{V}/\sqrt{\text{Hz}}\), and \acrshort{adc} maximum measurement noise \acrshort{asd} at \(11\,\mu\text{V}/\sqrt{\text{Hz}}\). In terms of RMS noise, these translate to less than \(1\,\text{mV RMS}\) for the \acrshort{dac}, less than \(20\,\text{mV RMS}\) for the voltage amplifier, and less than \(0.8\,\text{mV RMS}\) for the \acrshort{adc}. -If the Amplitude Spectral Density of the noise of the \acrshort{adc}, \acrshort{dac}, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS. +If the Amplitude Spectral Density of the noise of the \acrshort{adc}, \acrshort{dac}, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below \(15\,\text{nm RMS}\). \subsection{Choice of Instrumentation} \label{sec:detail_instrumentation_choice} \subsubsection{Piezoelectric Voltage Amplifier} Several characteristics of piezoelectric voltage amplifiers must be considered for this application. -To take advantage of the full stroke of the piezoelectric actuator, the voltage output should range between \(-20\) and \(150\,V\). -The amplifier should accept an analog input voltage, preferably in the range of \(-10\) to \(10\,V\), as this is standard for most \acrshortpl{dac}. +To take advantage of the full stroke of the piezoelectric actuator, the voltage output should range between \(-20\) and \(150\,\text{V}\). +The amplifier should accept an analog input voltage, preferably in the range of \(-10\) to \(10\,\text{V}\), as this is standard for most \acrshortpl{dac}. \paragraph{Small signal Bandwidth and Output Impedance} Small signal bandwidth is particularly important for feedback applications as it can limit the overall bandwidth of the complete feedback system. @@ -8853,25 +8853,25 @@ When combined with the piezoelectric load (represented as a capacitance \(C_p\)) \end{figure} Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases. -For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to \(C_p = 8.8\,\mu F\). +For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to \(C_p = 8.8\,\mu\text{F}\). If a small signal bandwidth of \(f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}\) is desired, the voltage amplifier output impedance should be less than \(R_0 = 3.6\,\Omega\). \paragraph{Large signal Bandwidth} Large signal bandwidth relates to the maximum output capabilities of the amplifier in terms of amplitude as a function of frequency. Since the primary function of the NASS is position stabilization rather than scanning, this specification is less critical than the small signal bandwidth. -However, considering potential scanning capabilities, a worst-case scenario of a constant velocity scan (triangular reference signal) with a repetition rate of \(f_r = 100\,\text{Hz}\) using the full voltage range of the piezoelectric actuator (\(V_{pp} = 170\,V\)) is considered. +However, considering potential scanning capabilities, a worst-case scenario of a constant velocity scan (triangular reference signal) with a repetition rate of \(f_r = 100\,\text{Hz}\) using the full voltage range of the piezoelectric actuator (\(V_{pp} = 170\,\text{V}\)) is considered. There are two limiting factors for large signal bandwidth that should be evaluated: \begin{enumerate} -\item Slew rate, which should exceed \(2 \cdot V_{pp} \cdot f_r = 34\,V/ms\). +\item Slew rate, which should exceed \(2 \cdot V_{pp} \cdot f_r = 34\,\text{V/ms}\). This requirement is typically easily met by commercial voltage amplifiers. \item Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies. To achieve high voltage at high frequency, the amplifier must therefore provide substantial current. -The maximum required current can be calculated as \(I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,A\). +The maximum required current can be calculated as \(I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,\text{A}\). \end{enumerate} -Therefore, ideally, a voltage amplifier capable of providing \(0.3\,A\) of current would be interesting for scanning applications. +Therefore, ideally, a voltage amplifier capable of providing \(0.3\,\text{A}\) of current would be interesting for scanning applications. \paragraph{Output voltage noise} As established in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting}, the output noise of the voltage amplifier should be below \(20\,\text{mV RMS}\). @@ -8879,7 +8879,7 @@ As established in Section~\ref{sec:detail_instrumentation_dynamic_error_budgetin It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure~\ref{fig:detail_instrumentation_amp_output_impedance}. Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement~\cite{spengen20_high_voltag_amplif}. -For this application, the output noise must remain below \(20\,\text{mV RMS}\) with a load of \(8.8\,\mu F\) and a bandwidth exceeding \(5\,\text{kHz}\). +For this application, the output noise must remain below \(20\,\text{mV RMS}\) with a load of \(8.8\,\mu\text{F}\) and a bandwidth exceeding \(5\,\text{kHz}\). \paragraph{Choice of voltage amplifier} The specifications are summarized in Table~\ref{tab:detail_instrumentation_amp_choice}. @@ -8903,15 +8903,15 @@ The PD200 from PiezoDrive was ultimately selected because it meets all the requi \textbf{Specifications} & PD200 & WMA-200 & LA75B & E-505\\ & PiezoDrive & Falco & Cedrat & PI\\ \midrule -Input Voltage Range: \(\pm 10\,V\) & \(\pm 10\,V\) & \(\pm8.75\,V\) & \(-1/7.5\,V\) & \(-2/12\,V\)\\ -Output Voltage Range: \(-20/150\,V\) & \(-50/150\,V\) & \(\pm 175\,V\) & \(-20/150\,V\) & -30/130\\ +Input Voltage Range: \(\pm 10\,\text{V}\) & \(\pm 10\,\text{V}\) & \(\pm8.75\,\text{V}\) & \(-1/7.5\,\text{V}\) & \(-2/12\,\text{V}\)\\ +Output Voltage Range: \(-20/150\,\text{V}\) & \(-50/150\,\text{V}\) & \(\pm 175\,\text{V}\) & \(-20/150\,\text{V}\) & \(-30/130\,\text{V}\)\\ Gain \(>15\) & 20 & 20 & 20 & 10\\ -Output Current \(> 300\,mA\) & \(900\,mA\) & \(150\,mA\) & \(360\,mA\) & \(215\,mA\)\\ -Slew Rate \(> 34\,V/ms\) & \(150\,V/\mu s\) & \(80\,V/\mu s\) & n/a & n/a\\ -Output noise \(< 20\,mV\ \text{RMS}\) & \(0.7\,mV\) & \(0.05\,mV\) & \(3.4\,mV\) & \(0.6\,mV\)\\ -(10uF load) & (\(10\,\mu F\) load) & (\(10\,\mu F\) load) & (n/a) & (n/a)\\ -Small Signal Bandwidth \(> 5\,kHz\) & \(6.4\,kHz\) & \(300\,Hz\) & \(30\,kHz\) & n/a\\ -(\(10\,\mu F\) load) & (\(10\,\mu F\) load) & (\(10\,\mu F\) load) & (unloaded) & (n/a)\\ +Output Current \(> 300\,\text{mA}\) & \(900\,\text{mA}\) & \(150\,\text{mA}\) & \(360\,\text{mA}\) & \(215\,\text{mA}\)\\ +Slew Rate \(> 34\,\text{V/ms}\) & \(150\,\text{V}/\mu\text{s}\) & \(80\,\text{V}/\mu\text{s}\) & n/a & n/a\\ +Output noise \(< 20\,\text{mV RMS}\) & \(0.7\,\text{mV}\) & \(0.05\,\text{mV}\) & \(3.4\,\text{mV}\) & \(0.6\,\text{mV}\)\\ +(10uF load) & (\(10\,\mu\text{F}\) load) & (\(10\,\mu\text{F}\) load) & (n/a) & (n/a)\\ +Small Signal Bandwidth \(> 5\,\text{kHz}\) & \(6.4\,\text{kHz}\) & \(300\,\text{Hz}\) & \(30\,\text{kHz}\) & n/a\\ +(\(10\,\mu\text{F}\) load) & (\(10\,\mu\text{F}\) load) & (\(10\,\mu\text{F}\) load) & (unloaded) & (n/a)\\ Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\) & n/a & n/a\\ \bottomrule \end{tabularx} @@ -8930,7 +8930,7 @@ Based on this requirement, priority was given to \acrshort{adc} and \acrshort{da Several requirements that may initially appear similar are actually distinct in nature. First, the \emph{sampling frequency} defines the interval between two sampled points and determines the Nyquist frequency. -Then, the \emph{bandwidth} specifies the maximum frequency of a measured signal (typically defined as the -3dB point) and is often limited by implemented anti-aliasing filters. +Then, the \emph{bandwidth} specifies the maximum frequency of a measured signal (typically defined as the \(-3\,\text{dB}\) point) and is often limited by implemented anti-aliasing filters. Finally, \emph{delay} (or \emph{latency}) refers to the time interval between the analog signal at the input of the \acrshort{adc} and the digital information transferred to the control system. Sigma-Delta \acrshortpl{adc} can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency. @@ -8979,24 +8979,24 @@ From a specified noise amplitude spectral density \(\Gamma_{\text{max}}\), the m n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right) \end{equation} -With a sampling frequency \(F_s = 10\,\text{kHz}\), an input range \(\Delta V = 20\,V\) and a maximum allowed \acrshort{asd} \(\Gamma_{\text{max}} = 11\,\mu V/\sqrt{Hz}\), the minimum number of bits is \(n_{\text{min}} = 12.4\), which is readily achievable with commercial \acrshortpl{adc}. +With a sampling frequency \(F_s = 10\,\text{kHz}\), an input range \(\Delta V = 20\,\text{V}\) and a maximum allowed \acrshort{asd} \(\Gamma_{\text{max}} = 11\,\mu\text{V}/\sqrt{Hz}\), the minimum number of bits is \(n_{\text{min}} = 12.4\), which is readily achievable with commercial \acrshortpl{adc}. \paragraph{DAC Output voltage noise} -Similar to the \acrshort{adc} requirements, the \acrshort{dac} output voltage noise \acrshort{asd} should not exceed \(14\,\mu V/\sqrt{\text{Hz}}\). -This specification corresponds to a \(\pm 10\,V\) \acrshort{dac} with 13-bit resolution, which is easily attainable with current technology. +Similar to the \acrshort{adc} requirements, the \acrshort{dac} output voltage noise \acrshort{asd} should not exceed \(14\,\mu\text{V}/\sqrt{\text{Hz}}\). +This specification corresponds to a \(\pm 10\,\text{V}\) \acrshort{dac} with 13-bit resolution, which is easily attainable with current technology. \paragraph{Choice of the ADC and DAC Board} Based on the preceding analysis, the selection of suitable \acrshort{adc} and \acrshort{dac} components is straightforward. For optimal synchronicity, a Speedgoat-integrated solution was chosen. -The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, \(\pm 10\,V\) range, maximum sampling rate of 200kSPS (\acrlong{sps}), simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity. -The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, \(\pm 10\,V\) range, conversion time of \(10\,\mu s\), and simultaneous update capability. +The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, \(\pm 10\,\text{V}\) range, maximum sampling rate of 200kSPS (\acrlong{sps}), simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity. +The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, \(\pm 10\,\text{V}\) range, conversion time of \(10\,\mu s\), and simultaneous update capability. Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements. This will be experimentally verified in Section~\ref{sec:detail_instrumentation_characterization}. \subsubsection{Relative Displacement Sensors} -The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below \(6\,\text{nm RMS}\) (derived from the \(15\,\text{nm RMS}\) vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding \(100\,\mu m\). +The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below \(6\,\text{nm RMS}\) (derived from the \(15\,\text{nm RMS}\) vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding \(100\,\mu\text{m}\). Several sensor technologies are capable of meeting these requirements~\cite{fleming13_review_nanom_resol_posit_sensor}. These include optical encoders (Figure~\ref{fig:detail_instrumentation_sensor_encoder}), capacitive sensors (Figure~\ref{fig:detail_instrumentation_sensor_capacitive}), and eddy current sensors (Figure~\ref{fig:detail_instrumentation_sensor_eddy_current}), each with their own advantages and implementation considerations. @@ -9020,7 +9020,7 @@ These include optical encoders (Figure~\ref{fig:detail_instrumentation_sensor_en \end{center} \subcaption{\label{fig:detail_instrumentation_sensor_capacitive}Capacitive Sensor} \end{subfigure} -\caption{\label{fig:detail_instrumentation_sensor_examples}Relative motion sensors considered for measuring the nano-hexapod strut motion} +\caption{\label{fig:detail_instrumentation_sensor_examples}Relative motion sensors considered for measuring the active platform strut motion} \end{figure} From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the \acrshort{apa}, as illustrated in Figure~\ref{fig:detail_instrumentation_capacitive_implementation}. @@ -9057,9 +9057,9 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic \textbf{Specifications} & Renishaw Vionic & LION CPL190 & Cedrat ECP500\\ \midrule Technology & Digital Encoder & Capacitive & Eddy Current\\ -Bandwidth \(> 5\,\text{kHz}\) & \(> 500\,\text{kHz}\) & 10kHz & 20kHz\\ -Noise \(< 6\,nm\,\text{RMS}\) & 1.6 nm rms & 4 nm rms & 15 nm rms\\ -Range \(> 100\,\mu m\) & Ruler length & 250 um & 500um\\ +Bandwidth \(> 5\,\text{kHz}\) & \(> 500\,\text{kHz}\) & \(10\,\text{kHz}\) & \(20\,\text{kHz}\)\\ +Noise \(< 6\,\text{nm RMS}\) & \(1.6\,\text{nm RMS}\) & \(4\,\text{nm RMS}\) & \(15\,\text{nm RMS}\)\\ +Range \(> 100\,\mu\text{m}\) & Ruler length & \(250\,\mu \text{m}\) & \(500\,\mu \text{m}\)\\ In line measurement & & \(\times\) & \(\times\)\\ Digital Output & \(\times\) & & \\ \bottomrule @@ -9072,7 +9072,7 @@ Digital Output & \(\times\) & & \\ The measurement of \acrshort{adc} noise was performed by short-circuiting its input with a \(50\,\Omega\) resistor and recording the digital values at a sampling rate of \(10\,\text{kHz}\). The amplitude spectral density of the recorded values was computed and is presented in Figure~\ref{fig:detail_instrumentation_adc_noise_measured}. -The \acrshort{adc} noise exhibits characteristics of white noise with an amplitude spectral density of \(5.6\,\mu V/\sqrt{\text{Hz}}\) (equivalent to \(0.4\,\text{mV RMS}\)), which satisfies the established specifications. +The \acrshort{adc} noise exhibits characteristics of white noise with an amplitude spectral density of \(5.6\,\mu\text{V}/\sqrt{\text{Hz}}\) (equivalent to \(0.4\,\text{mV RMS}\)), which satisfies the established specifications. All \acrshort{adc} channels demonstrated similar performance, so only one channel's noise profile is shown. If necessary, oversampling can be applied to further reduce the noise~\cite{lab13_improv_adc}. @@ -9097,18 +9097,18 @@ The voltage amplifier employed in this setup has a gain of 20. \caption{\label{fig:detail_instrumentation_force_sensor_adc_setup}Schematic of the setup to validate the use of the ADC for reading the force sensor volage} \end{figure} -Step signals with an amplitude of \(1\,V\) were generated using the \acrshort{dac}, and the \acrshort{adc} signal was recorded. +Step signals with an amplitude of \(1\,\text{V}\) were generated using the \acrshort{dac}, and the \acrshort{adc} signal was recorded. The excitation signal (steps) and the measured voltage across the sensor stack are displayed in Figure~\ref{fig:detail_instrumentation_step_response_force_sensor}. -Two notable observations were made: an offset voltage of \(2.26\,V\) was present, and the measured voltage exhibited an exponential decay response to the step input. +Two notable observations were made: an offset voltage of \(2.26\,\text{V}\) was present, and the measured voltage exhibited an exponential decay response to the step input. These phenomena can be explained by examining the electrical schematic shown in Figure~\ref{fig:detail_instrumentation_force_sensor_adc}, where the \acrshort{adc} has an input impedance \(R_i\) and an input bias current \(i_n\). The input impedance \(R_i\) of the \acrshort{adc}, in combination with the capacitance \(C_p\) of the piezoelectric stack sensor, forms an RC circuit with a time constant \(\tau = R_i C_p\). The charge generated by the piezoelectric effect across the stack's capacitance gradually discharges into the input resistor of the \acrshort{adc}. Consequently, the transfer function from the generated voltage \(V_p\) to the measured voltage \(V_{\text{ADC}}\) is a first-order high-pass filter with the time constant \(\tau\). -An exponential curve was fitted to the experimental data, yielding a time constant \(\tau = 6.5\,s\). -With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\mu F\), the internal impedance of the Speedgoat \acrshort{adc} was calculated as \(R_i = \tau/C_p = 1.5\,M\Omega\), which closely aligns with the specified value of \(1\,M\Omega\) found in the datasheet. +An exponential curve was fitted to the experimental data, yielding a time constant \(\tau = 6.5\,\text{s}\). +With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\mu\text{F}\), the internal impedance of the Speedgoat \acrshort{adc} was calculated as \(R_i = \tau/C_p = 1.5\,M\Omega\), which closely aligns with the specified value of \(1\,M\Omega\) found in the datasheet. \begin{figure}[htbp] \begin{subfigure}{0.61\textwidth} @@ -9135,7 +9135,7 @@ This modification produces two beneficial effects: a reduction of input voltage To validate this approach, a resistor \(R_p \approx 82\,k\Omega\) was added in parallel with the force sensor as shown in Figure~\ref{fig:detail_instrumentation_force_sensor_adc_R}. After incorporating this resistor, the same step response tests were performed, with results displayed in Figure~\ref{fig:detail_instrumentation_step_response_force_sensor_R}. -The measurements confirmed the expected improvements, with a substantially reduced offset voltage (\(V_{\text{off}} = 0.15\,V\)) and a much faster time constant (\(\tau = 0.45\,s\)). +The measurements confirmed the expected improvements, with a substantially reduced offset voltage (\(V_{\text{off}} = 0.15\,\text{V}\)) and a much faster time constant (\(\tau = 0.45\,\text{s}\)). These results validate both the model of the \acrshort{adc} and the effectiveness of the added parallel resistor as a solution. \begin{figure}[htbp] @@ -9155,12 +9155,12 @@ These results validate both the model of the \acrshort{adc} and the effectivenes \end{figure} \subsubsection{Instrumentation Amplifier} -Because the \acrshort{adc} noise may be too low to measure the noise of other instruments (anything below \(5.6\,\mu V/\sqrt{\text{Hz}}\) cannot be distinguished from the noise of the \acrshort{adc} itself), a low noise instrumentation amplifier was employed. -A Femto DLPVA-101-B-S amplifier with adjustable gains from 20dB up to 80dB was selected for this purpose. +Because the \acrshort{adc} noise may be too low to measure the noise of other instruments (anything below \(5.6\,\mu\text{V}/\sqrt{\text{Hz}}\) cannot be distinguished from the noise of the \acrshort{adc} itself), a low noise instrumentation amplifier was employed. +A Femto DLPVA-101-B-S amplifier with adjustable gains from \(20\,text{dB}\) up to \(80\,text{dB}\) was selected for this purpose. The first step was to characterize the input\footnote{For variable gain amplifiers, it is usual to refer to the input noise rather than the output noise, as the input referred noise is almost independent on the chosen gain.} noise of the amplifier. This was accomplished by short-circuiting its input with a \(50\,\Omega\) resistor and measuring the output voltage with the \acrshort{adc} (Figure~\ref{fig:detail_instrumentation_femto_meas_setup}). -The maximum amplifier gain of 80dB (equivalent to 10000) was used for this measurement. +The maximum amplifier gain of \(80\,text{dB}\) (equivalent to 10000) was used for this measurement. The measured voltage \(n\) was then divided by 10000 to determine the equivalent noise at the input of the voltage amplifier \(n_a\). In this configuration, the noise contribution from the \acrshort{adc} \(q_{ad}\) is rendered negligible due to the high gain employed. @@ -9187,7 +9187,7 @@ The \acrshort{dac} was configured to output a constant voltage (zero in this cas The Amplitude Spectral Density \(\Gamma_{n_{da}}(\omega)\) of the measured signal was computed, and verification was performed to confirm that the contributions of \acrshort{adc} noise and amplifier noise were negligible in the measurement. The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure~\ref{fig:detail_instrumentation_dac_output_noise}. -The noise profile is predominantly white with an \acrshort{asd} of \(0.6\,\mu V/\sqrt{\text{Hz}}\). +The noise profile is predominantly white with an \acrshort{asd} of \(0.6\,\mu\text{V}/\sqrt{\text{Hz}}\). Minor \(50\,\text{Hz}\) noise is present, along with some low frequency \(1/f\) noise, but these are not expected to pose issues as they are well within specifications. It should be noted that all \acrshort{dac} channels demonstrated similar performance, so only one channel measurement is presented. @@ -9254,11 +9254,11 @@ While the exact cause of these peaks is not fully understood, their amplitudes r The small signal dynamics of all six PD200 amplifiers were characterized through \acrshort{frf} measurements. -A logarithmic sweep sine excitation voltage was generated using the Speedgoat \acrshort{dac} with an amplitude of \(0.1\,V\), spanning frequencies from \(1\,\text{Hz}\) to \(5\,\text{kHz}\). +A logarithmic sweep sine excitation voltage was generated using the Speedgoat \acrshort{dac} with an amplitude of \(0.1\,\text{V}\), spanning frequencies from \(1\,\text{Hz}\) to \(5\,\text{kHz}\). The output voltage of the PD200 amplifier was measured via the monitor voltage output of the amplifier, while the input voltage (generated by the \acrshort{dac}) was measured with a separate \acrshort{adc} channel of the Speedgoat system. This measurement approach eliminates the influence of ADC-DAC-related time delays in the results. -All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to 500Hz, well within the specified requirements. +All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to \(500\,\text{Hz}\), well within the specified requirements. The identified dynamics shown in Figure~\ref{fig:detail_instrumentation_pd200_tf} can be accurately modeled as either a first-order low-pass filter or as a simple constant gain. @@ -9308,7 +9308,7 @@ This confirms that the selected instrumentation, with its measured noise charact This section has presented a comprehensive approach to the selection and characterization of instrumentation for the nano active stabilization system. The multi-body model created earlier served as a key tool for embedding instrumentation components and their associated noise sources within the system analysis. -From the most stringent requirement (i.e. the specification on vertical sample motion limited to 15 nm RMS), detailed specifications for each noise source were methodically derived through dynamic error budgeting. +From the most stringent requirement (i.e. the specification on vertical sample motion limited to \(15\,\text{nm RMS}\)), detailed specifications for each noise source were methodically derived through dynamic error budgeting. Based on these specifications, appropriate instrumentation components were selected for the system. The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate noise budgeting, such as amplitude spectral densities under specific load conditions. @@ -9317,14 +9317,14 @@ Despite these challenges, suitable components were identified that theoretically The selected instrumentation (including the IO131 ADC/DAC from Speedgoat, PD200 piezoelectric voltage amplifiers from PiezoDrive, and Vionic linear encoders from Renishaw) was procured and thoroughly characterized. Initial measurements of the \acrshort{adc} system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit. -All components were found to meet or exceed their respective specifications. The \acrshort{adc} demonstrated noise levels of \(5.6\,\mu V/\sqrt{\text{Hz}}\) (versus the \(11\,\mu V/\sqrt{\text{Hz}}\) specification), the \acrshort{dac} showed \(0.6\,\mu V/\sqrt{\text{Hz}}\) (versus \(14\,\mu V/\sqrt{\text{Hz}}\) required), the voltage amplifiers exhibited noise well below the \(280\,\mu V/\sqrt{\text{Hz}}\) limit, and the encoders achieved \(1\,\text{nm RMS}\) noise (versus the \(6\,\text{nm RMS}\) specification). +All components were found to meet or exceed their respective specifications. The \acrshort{adc} demonstrated noise levels of \(5.6\,\mu\text{V}/\sqrt{\text{Hz}}\) (versus the \(11\,\mu\text{V}/\sqrt{\text{Hz}}\) specification), the \acrshort{dac} showed \(0.6\,\mu\text{V}/\sqrt{\text{Hz}}\) (versus \(14\,\mu\text{V}/\sqrt{\text{Hz}}\) required), the voltage amplifiers exhibited noise well below the \(280\,\mu\text{V}/\sqrt{\text{Hz}}\) limit, and the encoders achieved \(1\,\text{nm RMS}\) noise (versus the \(6\,\text{nm RMS}\) specification). Finally, the measured noise characteristics of all instrumentation components were included into the multi-body model to predict the actual system performance. The combined effect of all noise sources was estimated to induce vertical sample vibrations of only \(1.5\,\text{nm RMS}\), which is substantially below the \(15\,\text{nm RMS}\) requirement. This rigorous methodology spanning requirement formulation, component selection, and experimental characterization validates the instrumentation's ability to fulfill the nano active stabilization system's demanding performance specifications. \section{Obtained Design} \label{sec:detail_design} -The detailed mechanical design of the active platform, depicted in Figure~\ref{fig:detail_design_nano_hexapod_elements}, is presented in this section. +The detailed mechanical design of the active platform (also referred to as the ``nano-hexapod''), depicted in Figure~\ref{fig:detail_design_nano_hexapod_elements}, is presented in this section. Several primary objectives guided the mechanical design. First, to ensure a well-defined Jacobian matrix used in the control architecture, accurate positioning of the top flexible joint rotation points and correct orientation of the struts were required. Secondly, space constraints necessitated that the entire platform fit within a cylinder with a radius of \(120\,\text{mm}\) and a height of \(95\,\text{mm}\). @@ -9335,7 +9335,7 @@ Finally, considerations for ease of mounting, alignment, and maintenance were in \begin{figure}[htbp] \centering \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_nano_hexapod_elements.png} -\caption{\label{fig:detail_design_nano_hexapod_elements}Obtained mechanical design of the Active platform, the ``nano-hexapod''} +\caption{\label{fig:detail_design_nano_hexapod_elements}Obtained mechanical design of the Active platform, called the ``nano-hexapod''} \end{figure} \subsection{Mechanical Design} \label{sec:detail_design_mechanics} @@ -9400,7 +9400,7 @@ To achieve this, two interface parts, fabricated from aluminum, were designed. These parts serve to fix the encoder head and the associated scale (ruler) to the two flexible joints, as depicted in Figure~\ref{fig:detail_design_strut_with_enc}. \paragraph{Plates} -The design of the top and bottom plates of the active platform was governed by two main requirements: maximizing the frequency of flexible modes and ensuring accurate positioning of the top flexible joints and well-defined orientation of the struts. +The design of the top and bottom plates of the nano-hexapod was governed by two main requirements: maximizing the frequency of flexible modes and ensuring accurate positioning of the top flexible joints and well-defined orientation of the struts. To maximize the natural frequencies associated with plate flexibility, a network of reinforcing ribs was incorporated into the design, as shown for the top plate in Figure~\ref{fig:detail_design_top_plate}. Although topology optimization methods were considered, the implemented ribbed design was found to provide sufficiently high natural frequencies for the flexible modes. @@ -9413,7 +9413,7 @@ Although topology optimization methods were considered, the implemented ribbed d The interfaces for the joints on the plates incorporate V-grooves (red planes in Figure~\ref{fig:detail_design_top_plate}). The cylindrical portion of each flexible joint is constrained within its corresponding V-groove through two distinct line contacts, illustrated in Figure~\ref{fig:detail_design_fixation_flexible_joints}. These grooves consequently serve to define the nominal orientation of the struts. -High machining accuracy for these features is essential to ensure that the flexible joints are in their neutral, unstressed state when the active platform is assembled. +High machining accuracy for these features is essential to ensure that the flexible joints are in their neutral, unstressed state when the nano-hexapod is assembled. \begin{figure} \begin{subfigure}{0.33\textwidth} @@ -9446,12 +9446,12 @@ This material was selected primarily for its high hardness, which minimizes the This characteristic is expected to permit repeated assembly and disassembly of the struts, should maintenance or reconfiguration be necessary. \paragraph{Finite Element Analysis} -A \acrfull{fea} of the complete active platform assembly was performed to identify modes that could potentially affect performance. +A \acrfull{fea} of the complete nano-hexapod assembly was performed to identify modes that could potentially affect performance. The analysis revealed that the first six modes correspond to ``suspension'' modes, where the top plate effectively moves as a rigid body, and motion primarily involves axial displacement of the six struts (an example is shown in Figure~\ref{fig:detail_design_fem_rigid_body_mode}). Following these suspension modes, numerous ``local'' modes associated with the struts themselves were observed in the frequency range between \(205\,\text{Hz}\) and \(420\,\text{Hz}\). One such mode is represented in Figure~\ref{fig:detail_design_fem_strut_mode}. Although these modes do not appear to induce significant motion of the top platform, they do cause relative displacement between the encoder components (head and scale) mounted on the strut. -Consequently, such modes could potentially degrade control performance if the active platform's position is regulated using these encoder measurements. +Consequently, such modes could potentially degrade control performance if the nano-hexapod's position is regulated using these encoder measurements. The extent to which these modes might be detrimental is difficult to establish at this stage, as it depends on whether they are significantly excited by the \acrshort{apa} actuation and their sensitivity to strut alignment. Finally, the FEA indicated that flexible modes of the top plate itself begin to appear at frequencies above \(650\,\text{Hz}\), with the first such mode shown in Figure~\ref{fig:detail_design_fem_plate_mode}. @@ -9501,7 +9501,7 @@ Dedicated supports, machined from aluminum, were designed for this purpose. It was verified through FEA that the natural modes of these supports occur at frequencies sufficiently high (first mode estimated at \(1120\,\text{Hz}\)) to not be problematic for control. Precise positioning of these encoder supports is achieved through machined pockets in both the top and bottom plates, visible in Figure~\ref{fig:detail_design_top_plate} (indicated in green). Although the encoders in this arrangement are aligned parallel to the nominal strut axes, they no longer measure the exact relative displacement along the strut between the flexible joint centers. -This geometric discrepancy implies that if the relative motion control of the active platform is based directly on these encoder readings, the kinematic calculations may be slightly inaccurate, potentially affecting the overall positioning accuracy of the platform. +This geometric discrepancy implies that if the relative motion control of the nano-hexapod is based directly on these encoder readings, the kinematic calculations may be slightly inaccurate, potentially affecting the overall positioning accuracy of the platform. \subsection{Multi-Body Model} \label{sec:detail_design_model} Prior to the procurement of mechanical components, the multi-body simulation model of the active platform was refined to incorporate the finalized design geometries. @@ -9548,7 +9548,7 @@ Therefore, a more sophisticated model of the optical encoder was necessary. The optical encoders operate based on the interaction between an encoder head and a graduated scale or ruler. The optical encoder head contains a light source that illuminates the ruler. A reference frame \(\{E\}\) fixed to the scale, represents the the light position on the scale, as illustrated in Figure~\ref{fig:detail_design_simscape_encoder_model}. -The ruler features a precise grating pattern (in this case, with a \(20\,\mu m\) pitch), and its position is associated with the reference frame \(\{R\}\). +The ruler features a precise grating pattern (in this case, with a \(20\,\mu\text{m}\) pitch), and its position is associated with the reference frame \(\{R\}\). The displacement measured by the encoder corresponds to the relative position of the encoder frame \(\{E\}\) (specifically, the point where the light interacts with the scale) with respect to the ruler frame \(\{R\}\), projected along the measurement direction defined by the scale. An important consequence of this measurement principle is that a relative rotation between the encoder head and the ruler, as depicted conceptually in Figure~\ref{fig:detail_design_simscape_encoder_disp}, can induce a measured displacement. @@ -9570,7 +9570,7 @@ An important consequence of this measurement principle is that a relative rotati \end{figure} \paragraph{Validation of the designed active platform} -The refined multi-body model of the active platform was integrated into the multi-body micro-station model. +The refined multi-body model of the nano-hexapod was integrated into the multi-body micro-station model. Dynamical analysis was performed, confirming that the platform's behavior closely approximates the dynamics of the ``idealized'' model used during the conceptual design phase. Consequently, closed-loop performance simulations replicating tomography experiments yielded metrics highly comparable to those previously predicted (as presented in Section~\ref{ssec:nass_hac_tomography}). Given this similarity and because analogous simulations are conducted and detailed during the experimental validation phase (Section~\ref{sec:test_id31_hac}), these specific results are not reiterated here. @@ -9668,7 +9668,7 @@ Finally, in Section~\ref{ssec:test_apa_spurious_resonances}, the flexible modes \label{ssec:test_apa_geometrical_measurements} To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness. -As shown in Figure~\ref{fig:test_apa_flatness_setup}, the \acrshort{apa} is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu m\)} is used to measure the height of four points on each of the APA300ML interfaces. +As shown in Figure~\ref{fig:test_apa_flatness_setup}, the \acrshort{apa} is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu\text{m}\)} is used to measure the height of four points on each of the APA300ML interfaces. From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points. The measured flatness values, summarized in Table~\ref{tab:test_apa_flatness_meas}, are within the specifications. @@ -9684,7 +9684,7 @@ The measured flatness values, summarized in Table~\ref{tab:test_apa_flatness_mea {\footnotesize\sf \begin{tabularx}{0.5\linewidth}{Xc} \toprule - & \textbf{Flatness} \([\mu m]\)\\ + & \textbf{Flatness} \([\mu\text{m}]\)\\ \midrule APA 1 & 8.9\\ APA 2 & 3.1\\ @@ -9700,16 +9700,16 @@ APA 7 & 18.7\\ \subsubsection{Electrical Measurements} \label{ssec:test_apa_electrical_measurements} -From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\mu F\) and \(26\,\mu F\) with a nominal capacitance of \(20\,\mu F\). +From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\mu\text{F}\) and \(26\,\mu\text{F}\) with a nominal capacitance of \(20\,\mu\text{F}\). The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter\footnote{LCR-819 from Gwinstek, with a specified accuracy of \(0.05\%\). The measured frequency is set at \(1\,\text{kHz}\)} shown in Figure~\ref{fig:test_apa_lcr_meter}. The two stacks used as the actuator and the stack used as the force sensor were measured separately. -The measured capacitance values are summarized in Table~\ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \mu F\). +The measured capacitance values are summarized in Table~\ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \mu\text{F}\). However, the measured capacitance of the stacks of ``APA 3'' is only half of the expected capacitance. This may indicate a manufacturing defect. The measured capacitance is found to be lower than the specified value. -This may be because the manufacturer measures the capacitance with large signals (\(-20\,V\) to \(150\,V\)), whereas it was here measured with small signals~\cite{wehrsdorfer95_large_signal_measur_piezoel_stack}. +This may be because the manufacturer measures the capacitance with large signals (\(-20\,\text{V}\) to \(150\,\text{V}\)), whereas it was here measured with small signals~\cite{wehrsdorfer95_large_signal_measur_piezoel_stack}. \begin{minipage}[b]{0.48\textwidth} \begin{center} @@ -9734,14 +9734,14 @@ APA 6 & 4.99 & 9.91\\ APA 7 & 4.85 & 9.85\\ \bottomrule \end{tabularx}} -\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu F$} +\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu\text{F}$} \end{minipage} \subsubsection{Stroke and Hysteresis Measurement} \label{ssec:test_apa_stroke_measurements} -To compare the stroke of the APA300ML with the datasheet specifications, one side of the \acrshort{apa} is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure~\ref{fig:test_apa_stroke_bench}. +To compare the stroke of the APA300ML with the datasheet specifications, one side of the \acrshort{apa} is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu\text{m}\)} is located on the other side as shown in Figure~\ref{fig:test_apa_stroke_bench}. -The voltage across the two actuator stacks is varied from \(-20\,V\) to \(150\,V\) using a DAC\footnote{The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of \(\pm 10\,V\) and 16-bits resolution} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,V/V\)}. +The voltage across the two actuator stacks is varied from \(-20\,\text{V}\) to \(150\,\text{V}\) using a DAC\footnote{The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of \(\pm 10\,\text{V}\) and 16-bits resolution} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,\text{V/V}\)}. Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure~\ref{fig:test_apa_stroke_voltage}). \begin{figure}[htbp] @@ -9752,9 +9752,9 @@ Note that the voltage is slowly varied as the displacement probe has a very low The measured \acrshort{apa} displacement is shown as a function of the applied voltage in Figure~\ref{fig:test_apa_stroke_hysteresis}. Typical hysteresis curves for piezoelectric stack actuators can be observed. -The measured stroke is approximately \(250\,\mu m\) when using only two of the three stacks. -This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu m\), therefore \(\approx 200\,\mu m\) if only two stacks are used). -For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of \(10\,\mu m\). +The measured stroke is approximately \(250\,\mu\text{m}\) when using only two of the three stacks. +This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu\text{m}\), therefore \(\approx 200\,\mu\text{m}\) if only two stacks are used). +For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of \(10\,\mu\text{m}\). It is clear from Figure~\ref{fig:test_apa_stroke_hysteresis} that ``APA 3'' has an issue compared with the other units. This confirms the abnormal electrical measurements made in Section~\ref{ssec:test_apa_electrical_measurements}. @@ -9791,19 +9791,19 @@ The flexible modes for the same condition (i.e. one mechanical interface of the \begin{center} \includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_1.png} \end{center} -\subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode (268Hz)} +\subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode ($268\,\text{Hz}$)} \end{subfigure} \begin{subfigure}{0.27\textwidth} \begin{center} \includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_2.png} \end{center} -\subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode (399Hz)} +\subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode ($399\,\text{Hz}$)} \end{subfigure} \begin{subfigure}{0.35\textwidth} \begin{center} \includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_3.png} \end{center} -\subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode (706Hz)} +\subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode ($706\,\text{Hz}$)} \end{subfigure} \caption{\label{fig:test_apa_mode_shapes}First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model} \end{figure} @@ -9839,9 +9839,9 @@ Another explanation is the shape difference between the manufactured APA300ML an \subsection{Dynamical measurements} \label{sec:test_apa_dynamics} After the measurements on the \acrshort{apa} were performed in Section~\ref{sec:test_apa_basic_meas}, a new test bench was used to better characterize the dynamics of the APA300ML. -This test bench, depicted in Figure~\ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a 5kg granite block that is vertically guided by an air bearing. +This test bench, depicted in Figure~\ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a \(5\,\text{kg}\) granite block that is vertically guided by an air bearing. Thus, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors. -An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the \acrshort{apa}. +An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,\text{nm}\)} is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the \acrshort{apa}. \begin{figure}[htbp] \begin{subfigure}{0.3\textwidth} @@ -9875,7 +9875,7 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a \label{ssec:test_apa_hysteresis} Because the payload is vertically guided without friction, the hysteresis of the \acrshort{apa} can be estimated from the motion of the payload. -A quasi static\footnote{Frequency of the sinusoidal wave is \(1\,\text{Hz}\)} sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)) and with an amplitude varying from \(4\,V\) up to \(80\,V\) is generated using the \acrshort{dac}. +A quasi static\footnote{Frequency of the sinusoidal wave is \(1\,\text{Hz}\)} sinusoidal excitation \(V_a\) with an offset of \(65\,\text{V}\) (halfway between \(-20\,\text{V}\) and \(150\,\text{V}\)) and with an amplitude varying from \(4\,\text{V}\) up to \(80\,\text{V}\) is generated using the \acrshort{dac}. For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage in Figure~\ref{fig:test_apa_meas_hysteresis}. This is the typical behavior expected from a \acrfull{pzt} stack actuator, where the hysteresis increases as a function of the applied voltage amplitude~\cite[chap. 1.4]{fleming14_desig_model_contr_nanop_system}. @@ -9888,7 +9888,7 @@ This is the typical behavior expected from a \acrfull{pzt} stack actuator, where \label{ssec:test_apa_stiffness} To estimate the stiffness of the \acrshort{apa}, a weight with known mass \(m_a = 6.4\,\text{kg}\) is added on top of the suspended granite and the deflection \(\Delta d_e\) is measured using the encoder. -The \acrshort{apa} stiffness can then be estimated from equation~\eqref{eq:test_apa_stiffness}, with \(g \approx 9.8\,m/s^2\) the acceleration of gravity. +The \acrshort{apa} stiffness can then be estimated from equation~\eqref{eq:test_apa_stiffness}, with \(g \approx 9.8\,\text{m}/\text{s}^2\) the acceleration of gravity. \begin{equation} \label{eq:test_apa_stiffness} k_{\text{apa}} = \frac{m_a g}{\Delta d_e} @@ -9899,12 +9899,12 @@ It can be seen that there are some drifts in the measured displacement (probably These two effects induce some uncertainties in the measured stiffness. The stiffnesses are computed for all \acrshortpl{apa} from the two displacements \(d_1\) and \(d_2\) (see Figure~\ref{fig:test_apa_meas_stiffness_time}) leading to two stiffness estimations \(k_1\) and \(k_2\). -These estimated stiffnesses are summarized in Table~\ref{tab:test_apa_measured_stiffnesses} and are found to be close to the specified nominal stiffness of the APA300ML \(k = 1.8\,N/\mu m\). +These estimated stiffnesses are summarized in Table~\ref{tab:test_apa_measured_stiffnesses} and are found to be close to the specified nominal stiffness of the APA300ML \(k = 1.8\,\text{N}/\mu\text{m}\). \begin{minipage}[b]{0.57\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/test_apa_meas_stiffness_time.png} -\captionof{figure}{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding (at \(t \approx 3\,s\)) and removing (at \(t \approx 13\,s\)) the mass} +\captionof{figure}{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding (at \(t \approx 3\,\text{s}\)) and removing (at \(t \approx 13\,\text{s}\)) the mass} \end{center} \end{minipage} \hfill @@ -9923,7 +9923,7 @@ APA & \(k_1\) & \(k_2\)\\ 8 & 1.73 & 1.98\\ \bottomrule \end{tabularx}} -\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $N/\mu m$} +\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\mu\text{m}$} \end{minipage} The stiffness can also be computed using equation~\eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 95\,\text{Hz}\) (estimated by the dynamical measurements shown in section~\ref{ssec:test_apa_meas_dynamics}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\). @@ -9932,7 +9932,7 @@ The stiffness can also be computed using equation~\eqref{eq:test_apa_res_freq} b \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} \end{equation} -The obtained stiffness is \(k \approx 2\,N/\mu m\) which is close to the values found in the documentation and using the ``static deflection'' method. +The obtained stiffness is \(k \approx 2\,\text{N}/\mu\text{m}\) which is close to the values found in the documentation and using the ``static deflection'' method. It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack~\cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}. @@ -9942,7 +9942,7 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the ` \item \(k_{\text{sc}}\): piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance) \end{itemize} -The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\). +The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,\text{N}/\mu\text{m}\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,\text{N}/\mu\text{m}\). \subsubsection{Dynamics} \label{ssec:test_apa_meas_dynamics} @@ -9951,7 +9951,7 @@ In this section, the dynamics from the excitation voltage \(u\) to the encoder m First, the dynamics from \(u\) to \(d_e\) for the six APA300ML are compared in Figure~\ref{fig:test_apa_frf_encoder}. The obtained \acrshortpl{frf} are similar to those of a (second order) mass-spring-damper system with: \begin{itemize} -\item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu m/V\). +\item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu\text{m}/V\). The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the \acrshort{apa} \item A lightly damped resonance at \(95\,\text{Hz}\) \item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the \acrshort{apa} support. @@ -9969,7 +9969,7 @@ As illustrated by the Root Locus plot, the poles of the \emph{closed-loop} syste The significance of this behavior varies with the type of sensor used, as explained in~\cite[chap. 7.6]{preumont18_vibrat_contr_activ_struc_fourt_edition}. Considering the transfer function from \(u\) to \(V_s\), if a controller with a very high gain is applied such that the sensor stack voltage \(V_s\) is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain. Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell. -From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m\) (which is close to what is found using a \acrshort{fem}). +From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,\text{N}/\mu\text{m}\) (which is close to what is found using a \acrshort{fem}). All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure~\ref{fig:test_apa_frf_encoder} and at the force sensor in Figure~\ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell. @@ -10019,7 +10019,7 @@ However, this is not so important here because the zero is lightly damped (i.e. \subsubsection{Effect of the resistor on the IFF Plant} \label{ssec:test_apa_resistance_sensor_stack} -A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at \(\approx 5\,\mu F\)). +A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at \(\approx 5\,\mu\text{F}\)). As explained before, this is done to limit the voltage offset due to the input bias current of the \acrshort{adc} as well as to limit the low frequency gain. @@ -10110,9 +10110,9 @@ It can be decomposed into three components: \item the shell whose axial properties are represented by \(k_1\) and \(c_1\) \item the actuator stacks whose contribution to the axial stiffness is represented by \(k_a\) and \(c_a\). The force source \(f\) represents the axial force induced by the force sensor stacks. -The sensitivity \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(f\) +The sensitivity \(g_a\) (in \(\text{N/m}\)) is used to convert the applied voltage \(V_a\) to the axial force \(f\) \item the sensor stack whose contribution to the axial stiffness is represented by \(k_e\) and \(c_e\). -A sensor measures the stack strain \(d_e\) which is then converted to a voltage \(V_s\) using a sensitivity \(g_s\) (in \(V/m\)) +A sensor measures the stack strain \(d_e\) which is then converted to a voltage \(V_s\) using a sensitivity \(g_s\) (in \(\text{V/m}\)) \end{itemize} Such a simple model has some limitations: @@ -10138,8 +10138,8 @@ Such a simple model has some limitations: First, the mass \(m\) supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. Both methods lead to an estimated mass of \(m = 5.7\,\text{kg}\). -Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,N/\mu m\) in Section~\ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure~\ref{fig:test_apa_frf_force}. -Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(5\,Ns/m\). +Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,\text{N}/\mu\text{m}\) in Section~\ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure~\ref{fig:test_apa_frf_force}. +Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(5\,\text{Ns/m}\). Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not completely correct as it was shown in Section~\ref{ssec:test_apa_stiffness} that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}. Therefore, we have \(k_e = 2 k_a\) and \(c_e = 2 c_a\) as the actuator stack is composed of two stacks in series. @@ -10149,14 +10149,14 @@ In this case, the total stiffness of the \acrshort{apa} model is described by~\e k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a \end{equation} -Knowing from~\eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{\text{tot}} = 2\,N/\mu m\), we get from~\eqref{eq:test_apa_2dof_stiffness} that \(k_a = 2.5\,N/\mu m\) and \(k_e = 5\,N/\mu m\). +Knowing from~\eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{\text{tot}} = 2\,\text{N}/\mu\text{m}\), we get from~\eqref{eq:test_apa_2dof_stiffness} that \(k_a = 2.5\,\text{N}/\mu\text{m}\) and \(k_e = 5\,\text{N}/\mu\text{m}\). \begin{equation}\label{eq:test_apa_tot_stiffness} -\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,N/\mu m \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s +\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,\text{N}/\mu\text{m} \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s \end{equation} Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance. -\(c_a = 50\,Ns/m\) and \(c_e = 100\,Ns/m\) are obtained. +\(c_a = 50\,\text{Ns/m}\) and \(c_e = 100\,\text{Ns/m}\) are obtained. In the last step, \(g_s\) and \(g_a\) can be tuned to match the gain of the identified transfer functions. @@ -10170,14 +10170,14 @@ The obtained parameters of the model shown in Figure~\ref{fig:test_apa_2dof_mode \textbf{Parameter} & \textbf{Value}\\ \midrule \(m\) & \(5.7\,\text{kg}\)\\ -\(k_1\) & \(0.38\,N/\mu m\)\\ -\(k_e\) & \(5.0\, N/\mu m\)\\ -\(k_a\) & \(2.5\,N/\mu m\)\\ -\(c_1\) & \(5\,Ns/m\)\\ -\(c_e\) & \(100\,Ns/m\)\\ -\(c_a\) & \(50\,Ns/m\)\\ -\(g_a\) & \(-2.58\,N/V\)\\ -\(g_s\) & \(0.46\,V/\mu m\)\\ +\(k_1\) & \(0.38\,\text{N}/\mu\text{m}\)\\ +\(k_e\) & \(5.0\,\text{N}/\mu\text{m}\)\\ +\(k_a\) & \(2.5\,\text{N}/\mu\text{m}\)\\ +\(c_1\) & \(5\,\text{Ns/m}\)\\ +\(c_e\) & \(100\,\text{Ns/m}\)\\ +\(c_a\) & \(50\,\text{Ns/m}\)\\ +\(g_a\) & \(-2.58\,\text{N/V}\)\\ +\(g_s\) & \(0.46\,\text{V}/\mu\text{m}\)\\ \bottomrule \end{tabularx} \end{table} @@ -10223,7 +10223,7 @@ Finally, two \emph{remote points} (\texttt{4} and \texttt{5}) are located across Once the APA300ML \emph{super element} is included in the multi-body model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be extracted. The gains \(g_a\) and \(g_s\) are then tuned such that the gains of the transfer functions match the identified ones. -By doing so, \(g_s = 4.9\,V/\mu m\) and \(g_a = 23.2\,N/V\) are obtained. +By doing so, \(g_s = 4.9\,\text{V}/\mu\text{m}\) and \(g_a = 23.2\,\text{N/V}\) are obtained. To ensure that the sensitivities \(g_a\) and \(g_s\) are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material. @@ -10240,7 +10240,7 @@ Unfortunately, the manufacturer of the stack was not willing to share the piezoe However, based on the available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties. The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table~\ref{tab:test_apa_piezo_properties}. -From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained, which are close to the constants identified using the experimentally identified transfer functions. +From these parameters, \(g_s = 5.1\,\text{V}/\mu\text{m}\) and \(g_a = 26\,\text{N/V}\) were obtained, which are close to the constants identified using the experimentally identified transfer functions. \begin{table}[htbp] \caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities} @@ -10249,12 +10249,12 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine \toprule \textbf{Parameter} & \textbf{Value} & \textbf{Description}\\ \midrule -\(d_{33}\) & \(680 \cdot 10^{-12}\,m/V\) & Piezoelectric constant\\ -\(\epsilon^{T}\) & \(4.0 \cdot 10^{-8}\,F/m\) & Permittivity under constant stress\\ -\(s^{D}\) & \(21 \cdot 10^{-12}\,m^2/N\) & Elastic compliance understand constant electric displacement\\ -\(c^{E}\) & \(48 \cdot 10^{9}\,N/m^2\) & Young's modulus of elasticity\\ -\(L\) & \(20\,mm\) per stack & Length of the stack\\ -\(A\) & \(10^{-4}\,m^2\) & Area of the piezoelectric stack\\ +\(d_{33}\) & \(680 \cdot 10^{-12}\,\text{m/V}\) & Piezoelectric constant\\ +\(\epsilon^{T}\) & \(4.0 \cdot 10^{-8}\,\text{F/m}\) & Permittivity under constant stress\\ +\(s^{D}\) & \(21 \cdot 10^{-12}\,\text{m}^2/\text{N}\) & Elastic compliance understand constant electric displacement\\ +\(c^{E}\) & \(48 \cdot 10^{9}\,\text{N}/\text{m}^2\) & Young's modulus of elasticity\\ +\(L\) & \(20\,\text{mm}\) per stack & Length of the stack\\ +\(A\) & \(10^{-4}\,\text{m}^2\) & Area of the piezoelectric stack\\ \(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\ \bottomrule \end{tabularx} @@ -10318,10 +10318,10 @@ During the detailed design phase, specifications in terms of stiffness and strok \toprule & \textbf{Specification} & \textbf{FEM}\\ \midrule -Axial Stiffness & \(> 100\,N/\mu m\) & 94\\ -Shear Stiffness & \(> 1\,N/\mu m\) & 13\\ -Bending Stiffness & \(< 100\,Nm/\text{rad}\) & 5\\ -Torsion Stiffness & \(< 500\,Nm/\text{rad}\) & 260\\ +Axial Stiffness & \(> 100\,\text{N}/\mu\text{m}\) & 94\\ +Shear Stiffness & \(> 1\,\text{N}/\mu\text{m}\) & 13\\ +Bending Stiffness & \(< 100\,\text{Nm}/\text{rad}\) & 5\\ +Torsion Stiffness & \(< 500\,\text{Nm}/\text{rad}\) & 260\\ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\ \bottomrule \end{tabularx} @@ -10394,7 +10394,7 @@ The dimensions of the flexible joint in the Y-Z plane will contribute to the X-b The setup used to measure the dimensions of the ``X'' flexible beam is shown in Figure~\ref{fig:test_joints_profilometer_setup}. What is typically observed is shown in Figure~\ref{fig:test_joints_profilometer_image}. -It is then possible to estimate the dimension of the flexible beam with an accuracy of \(\approx 5\,\mu m\), +It is then possible to estimate the dimension of the flexible beam with an accuracy of \(\approx 5\,\mu\text{m}\), \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -10412,12 +10412,12 @@ It is then possible to estimate the dimension of the flexible beam with an accur \caption{\label{fig:test_joints_profilometer}Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image})} \end{figure} \subsubsection{Measurement Results} -The specified flexible beam thickness (gap) is \(250\,\mu m\). +The specified flexible beam thickness (gap) is \(250\,\mu\text{m}\). Four gaps are measured for each flexible joint (2 in the \(x\) direction and 2 in the \(y\) direction). The ``beam thickness'' is then estimated as the mean between the gaps measured on opposite sides. A histogram of the measured beam thicknesses is shown in Figure~\ref{fig:test_joints_size_hist}. -The measured thickness is less than the specified value of \(250\,\mu m\), but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. +The measured thickness is less than the specified value of \(250\,\mu\text{m}\), but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. However, what is more important than the true value of the thickness is the consistency between all flexible joints. @@ -10493,8 +10493,8 @@ The deflection of the joint \(d_x\) is measured using a displacement sensor. \caption{\label{fig:test_joints_bench_working_principle}Working principle of the test bench used to estimate the bending stiffness \(k_{R_y}\) of the flexible joints by measuring \(F_x\), \(d_x\) and \(h\)} \end{figure} \paragraph{Required external applied force} -The bending stiffness is foreseen to be \(k_{R_y} \approx k_{R_x} \approx 5\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx \theta_{x,\text{max}}\approx 25\,mrad\). -The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 22.5\,mm\) (see Figure~\ref{fig:test_joints_bench_working_principle}). +The bending stiffness is foreseen to be \(k_{R_y} \approx k_{R_x} \approx 5\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx \theta_{x,\text{max}}\approx 25\,\text{mrad}\). +The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 22.5\,\text{mm}\) (see Figure~\ref{fig:test_joints_bench_working_principle}). The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is given by equation~\eqref{eq:test_joints_deflection_force}. @@ -10503,19 +10503,19 @@ The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is given \end{equation} Therefore, the force that must be applied to test the full range of the flexible joints is given by equation~\eqref{eq:test_joints_max_force}. -The measurement range of the force sensor should then be higher than \(5.5\,N\). +The measurement range of the force sensor should then be higher than \(5.5\,\text{N}\). \begin{equation}\label{eq:test_joints_max_force} - F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h} \approx 5.5\,N + F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h} \approx 5.5\,\text{N} \end{equation} \paragraph{Required actuator stroke and sensors range} -The flexible joint is designed to allow a bending motion of \(\pm 25\,mrad\). +The flexible joint is designed to allow a bending motion of \(\pm 25\,\text{mrad}\). The corresponding stroke at the location of the force sensor is given by~\eqref{eq:test_joints_max_stroke}. -To test the full range of the flexible joint, the means of applying a force (explained in the next section) should allow a motion of at least \(0.5\,mm\). -Similarly, the measurement range of the displacement sensor should also be higher than \(0.5\,mm\). +To test the full range of the flexible joint, the means of applying a force (explained in the next section) should allow a motion of at least \(0.5\,\text{mm}\). +Similarly, the measurement range of the displacement sensor should also be higher than \(0.5\,\text{mm}\). \begin{equation}\label{eq:test_joints_max_stroke} -d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \approx 0.5\,mm +d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \approx 0.5\,\text{mm} \end{equation} \paragraph{Force and Displacement measurements} To determine the applied force, a load cell will be used in series with the mechanism that applied the force. @@ -10556,7 +10556,7 @@ The estimated bending stiffness \(k_{\text{est}}\) then depends on the shear sti k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_s h^2}}_{\epsilon_{s}} \Bigl) \end{equation} -With an estimated shear stiffness \(k_s = 13\,N/\mu m\) from the \acrshort{fem} and an height \(h=25\,mm\), the estimation errors of the bending stiffness due to shear is \(\epsilon_s < 0.1\,\%\) +With an estimated shear stiffness \(k_s = 13\,\text{N}/\mu\text{m}\) from the \acrshort{fem} and an height \(h=25\,\text{mm}\), the estimation errors of the bending stiffness due to shear is \(\epsilon_s < 0.1\,\%\) \paragraph{Effect of load cell limited stiffness} As explained in the previous section, because the measurement of the flexible joint deflection is indirectly performed with the encoder, errors will be made if the load cell experiences some compression. @@ -10567,7 +10567,7 @@ The estimation error of the bending stiffness due to the limited stiffness of th k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_F h^2}}_{\epsilon_f} \Bigl) \end{equation} -With an estimated load cell stiffness of \(k_f \approx 1\,N/\mu m\) (from the documentation), the errors due to the load cell limited stiffness is around \(\epsilon_f = 1\,\%\). +With an estimated load cell stiffness of \(k_f \approx 1\,\text{N}/\mu\text{m}\) (from the documentation), the errors due to the load cell limited stiffness is around \(\epsilon_f = 1\,\%\). \paragraph{Estimation error due to height estimation error} Now consider an error \(\delta h\) in the estimation of the height \(h\) as described by~\eqref{eq:test_joints_est_h_error}. @@ -10581,10 +10581,10 @@ The computed bending stiffness will be~\eqref{eq:test_joints_stiffness_height_er k_{R_y, \text{est}} \approx h_{\text{est}}^2 \frac{F_x}{d_x} \approx k_{R_y} \Bigl( 1 + \underbrace{2 \frac{\delta h}{h} + \frac{\delta h ^2}{h^2}}_{\epsilon_h} \Bigl) \end{equation} -The height estimation is foreseen to be accurate to within \(|\delta h| < 0.4\,mm\) which corresponds to a stiffness error \(\epsilon_h < 3.5\,\%\). +The height estimation is foreseen to be accurate to within \(|\delta h| < 0.4\,\text{mm}\) which corresponds to a stiffness error \(\epsilon_h < 3.5\,\%\). \paragraph{Estimation error due to force and displacement sensors accuracy} -An optical encoder is used to measure the displacement (see Section~\ref{ssec:test_joints_test_bench}) whose maximum non-linearity is \(40\,nm\). -As the measured displacement is foreseen to be \(0.5\,mm\), the error \(\epsilon_d\) due to the encoder non-linearity is negligible \(\epsilon_d < 0.01\,\%\). +An optical encoder is used to measure the displacement (see Section~\ref{ssec:test_joints_test_bench}) whose maximum non-linearity is \(40\,\text{nm}\). +As the measured displacement is foreseen to be \(0.5\,\text{mm}\), the error \(\epsilon_d\) due to the encoder non-linearity is negligible \(\epsilon_d < 0.01\,\%\). The accuracy of the load cell is specified at \(1\,\%\) and therefore, estimation errors of the bending stiffness due to the limited load cell accuracy should be \(\epsilon_F < 1\,\%\) \paragraph{Conclusion} @@ -10613,13 +10613,13 @@ Force sensor & \(\epsilon_F < 1\,\%\)\\ As explained in Section~\ref{ssec:test_joints_meas_principle}, the flexible joint's bending stiffness is estimated by applying a known force to the flexible joint's tip and by measuring its deflection at the same point. -The force is applied using a load cell\footnote{The load cell is FC22 from TE Connectivity. The measurement range is \(50\,N\). The specified accuracy is \(1\,\%\) of the full range} such that the applied force to the flexible joint's tip is directly measured. +The force is applied using a load cell\footnote{The load cell is FC22 from TE Connectivity. The measurement range is \(50\,\text{N}\). The specified accuracy is \(1\,\%\) of the full range} such that the applied force to the flexible joint's tip is directly measured. To control the height and direction of the applied force, a cylinder cut in half is fixed at the tip of the force sensor (pink element in Figure~\ref{fig:test_joints_bench_side}) that initially had a flat surface. Doing so, the contact between the flexible joint cylindrical tip and the force sensor is a point (intersection of two cylinders) at a precise height, and the force is applied in a known direction. To translate the load cell at a constant height, it is fixed to a translation stage\footnote{V-408 PIMag\textsuperscript{\textregistered} linear stage is used. Crossed rollers are used to guide the motion.} which is moved by hand. Instead of measuring the displacement directly at the tip of the flexible joint (with a probe or an interferometer for instance), the displacement of the load cell itself is measured. -To do so, an encoder\footnote{Resolute\texttrademark{} encoder with \(1\,nm\) resolution and \(\pm 40\,nm\) maximum non-linearity} is used, which measures the motion of a ruler. +To do so, an encoder\footnote{Resolute\texttrademark{} encoder with \(1\,\text{nm}\) resolution and \(\pm 40\,\text{nm}\) maximum non-linearity} is used, which measures the motion of a ruler. This ruler is fixed to the translation stage in line (i.e. at the same height) with the application point to reduce Abbe errors (see Figure~\ref{fig:test_joints_bench_overview}). The flexible joint can be rotated by \(90^o\) in order to measure the bending stiffness in the two directions. @@ -10661,13 +10661,13 @@ A closer view of the force sensor tip is shown in Figure~\ref{fig:test_joints_pi \caption{\label{fig:test_joints_picture_bench}Manufactured test bench for compliance measurement of the flexible joints} \end{figure} \subsubsection{Load Cell Calibration} -In order to estimate the measured errors of the load cell ``FC2231'', it is compared against another load cell\footnote{XFL212R-50N from TE Connectivity. The measurement range is \(50\,N\). The specified accuracy is \(1\,\%\) of the full range}. +In order to estimate the measured errors of the load cell ``FC2231'', it is compared against another load cell\footnote{XFL212R-50N from TE Connectivity. The measurement range is \(50\,\text{N}\). The specified accuracy is \(1\,\%\) of the full range}. The two load cells are measured simultaneously while they are pushed against each other (see Figure~\ref{fig:test_joints_force_sensor_calib_picture}). The contact between the two load cells is well defined as one has a spherical interface and the other has a flat surface. The measured forces are compared in Figure~\ref{fig:test_joints_force_sensor_calib_fit}. The gain mismatch between the two load cells is approximately \(4\,\%\) which is higher than that specified in the data sheets. -However, the estimated non-linearity is bellow \(0.2\,\%\) for forces between \(1\,N\) and \(5\,N\). +However, the estimated non-linearity is bellow \(0.2\,\%\) for forces between \(1\,\text{N}\) and \(5\,\text{N}\). \begin{figure}[h!tbp] \begin{subfigure}{0.49\textwidth} @@ -10686,10 +10686,10 @@ However, the estimated non-linearity is bellow \(0.2\,\%\) for forces between \( \end{figure} \subsubsection{Load Cell Stiffness} The objective of this measurement is to estimate the stiffness \(k_F\) of the force sensor. -To do so, a stiff element (much stiffer than the estimated \(k_F \approx 1\,N/\mu m\)) is mounted in front of the force sensor, as shown in Figure~\ref{fig:test_joints_meas_force_sensor_stiffness_picture}. +To do so, a stiff element (much stiffer than the estimated \(k_F \approx 1\,\text{N}/\mu\text{m}\)) is mounted in front of the force sensor, as shown in Figure~\ref{fig:test_joints_meas_force_sensor_stiffness_picture}. Then, the force sensor is pushed against this stiff element while the force sensor and the encoder displacement are measured. The measured displacement as a function of the measured force is shown in Figure~\ref{fig:test_joints_force_sensor_stiffness_fit}. -The load cell stiffness can then be estimated by computing a linear fit and is found to be \(k_F \approx 0.68\,N/\mu m\). +The load cell stiffness can then be estimated by computing a linear fit and is found to be \(k_F \approx 0.68\,\text{N}/\mu\text{m}\). \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -10713,7 +10713,7 @@ The measured force and displacement as a function of time are shown in Figure~\r Three regions can be observed: first, the force sensor tip is not in contact with the flexible joint and the measured force is zero; then, the flexible joint deforms linearly; and finally, the flexible joint comes in contact with the mechanical stop. The angular motion \(\theta_{y}\) computed from the displacement \(d_x\) is displayed as function of the measured torque \(T_{y}\) in Figure~\ref{fig:test_joints_meas_F_d_lin_fit}. -The bending stiffness of the flexible joint can be estimated by computing the slope of the curve in the linear regime (red dashed line) and is found to be \(k_{R_y} = 4.4\,Nm/\text{rad}\). +The bending stiffness of the flexible joint can be estimated by computing the slope of the curve in the linear regime (red dashed line) and is found to be \(k_{R_y} = 4.4\,\text{Nm}/\text{rad}\). The bending stroke can also be estimated as shown in Figure~\ref{fig:test_joints_meas_F_d_lin_fit} and is found to be \(\theta_{y,\text{max}} = 20.9\,\text{mrad}\). \begin{figure}[htbp] @@ -10738,7 +10738,7 @@ The measured angular motion as a function of the applied torque is shown in Figu This gives a first idea of the dispersion of the measured bending stiffnesses (i.e. slope of the linear region) and of the angular stroke. A histogram of the measured bending stiffnesses is shown in Figure~\ref{fig:test_joints_bend_stiff_hist}. -Most of the bending stiffnesses are between \(4.6\,Nm/rad\) and \(5.0\,Nm/rad\). +Most of the bending stiffnesses are between \(4.6\,\text{Nm/rad}\) and \(5.0\,\text{Nm/rad}\). \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -10764,7 +10764,7 @@ This was crucial in preventing potential complications that could have arisen fr A dedicated test bench was developed to asses the bending stiffness of the flexible joints. Through meticulous error analysis and budgeting, a satisfactory level of measurement accuracy could be guaranteed. -The measured bending stiffness values exhibited good agreement with the predictions from the \acrshort{fem} (\(k_{R_x} = k_{R_y} = 5\,Nm/\text{rad}\)). +The measured bending stiffness values exhibited good agreement with the predictions from the \acrshort{fem} (\(k_{R_x} = k_{R_y} = 5\,\text{Nm}/\text{rad}\)). These measurements are helpful for refining the model of the flexible joints, thereby enhancing the overall accuracy of the nano-hexapod model. Furthermore, the data obtained from these measurements have provided the necessary information to select the most suitable flexible joints for the nano-hexapod, ensuring optimal performance. \section{Struts} @@ -10805,8 +10805,8 @@ This is important not to loose to much angular stroke during their mounting into The mounting bench is shown in Figure~\ref{fig:test_struts_mounting_bench_first_concept}. It consists of a ``main frame'' (Figure~\ref{fig:test_struts_mounting_step_0}) precisely machined to ensure both correct strut length and strut coaxiality. -The coaxiality is ensured by good flatness (specified at \(20\,\mu m\)) between surfaces A and B and between surfaces C and D. -Such flatness was checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\mu m\)} (see Figure~\ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements. +The coaxiality is ensured by good flatness (specified at \(20\,\mu\text{m}\)) between surfaces A and B and between surfaces C and D. +Such flatness was checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\mu\text{m}\)} (see Figure~\ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements. The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes. \begin{figure}[htbp] @@ -10912,28 +10912,28 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle \label{sec:test_struts_flexible_modes} A Finite Element Model\footnote{Using Ansys\textsuperscript{\textregistered}. Flexible Joints and APA Shell are made of a stainless steel allow called \emph{17-4 PH}. Encoder and ruler support material is aluminium.} of the struts is developed and is used to estimate the flexible modes. -The inertia of the encoder (estimated at \(15\,g\)) is considered. +The inertia of the encoder (estimated at \(15\,\text{g}\)) is considered. The two cylindrical interfaces were fixed (boundary conditions), and the first three flexible modes were computed. -The mode shapes are displayed in Figure~\ref{fig:test_struts_mode_shapes}: an ``X-bending'' mode at 189Hz, a ``Y-bending'' mode at 285Hz and a ``Z-torsion'' mode at 400Hz. +The mode shapes are displayed in Figure~\ref{fig:test_struts_mode_shapes}: an ``X-bending'' mode at \(189\,\text{Hz}\), a ``Y-bending'' mode at \(285\,\text{Hz}\) and a ``Z-torsion'' mode at \(400\,\text{Hz}\). \begin{figure}[htbp] \begin{subfigure}{0.32\textwidth} \begin{center} \includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_mode_shapes_1.png} \end{center} -\subcaption{\label{fig:test_struts_mode_shapes_1}X-bending mode (189Hz)} +\subcaption{\label{fig:test_struts_mode_shapes_1}X-bending mode ($189\,\text{Hz}$)} \end{subfigure} \begin{subfigure}{0.32\textwidth} \begin{center} \includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_mode_shapes_2.png} \end{center} -\subcaption{\label{fig:test_struts_mode_shapes_2}Y-bending mode (285Hz)} +\subcaption{\label{fig:test_struts_mode_shapes_2}Y-bending mode ($285\,\text{Hz}$)} \end{subfigure} \begin{subfigure}{0.32\textwidth} \begin{center} \includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_mode_shapes_3.png} \end{center} -\subcaption{\label{fig:test_struts_mode_shapes_3}Z-torsion mode (400Hz)} +\subcaption{\label{fig:test_struts_mode_shapes_3}Z-torsion mode ($400\,\text{Hz}$)} \end{subfigure} \caption{\label{fig:test_struts_mode_shapes}Spurious resonances of the struts estimated from a Finite Element Model} \end{figure} @@ -10998,9 +10998,9 @@ This validates the quality of the \acrshort{fem}. \toprule \textbf{Mode} & \textbf{FEM with Encoder} & \textbf{Exp. with Encoder} & \textbf{Exp. without Encoder}\\ \midrule -X-Bending & 189Hz & 198Hz & 226Hz\\ -Y-Bending & 285Hz & 293Hz & 337Hz\\ -Z-Torsion & 400Hz & 381Hz & 398Hz\\ +X-Bending & \(189\,\text{Hz}\) & \(198\,\text{Hz}\) & \(226\,\text{Hz}\)\\ +Y-Bending & \(285\,\text{Hz}\) & \(293\,\text{Hz}\) & \(337\,\text{Hz}\)\\ +Z-Torsion & \(400\,\text{Hz}\) & \(381\,\text{Hz}\) & \(398\,\text{Hz}\)\\ \bottomrule \end{tabularx} \end{table} @@ -11084,7 +11084,7 @@ This means that the encoder should have little effect on the effectiveness of th The dynamics measured by the encoder (i.e. \(d_e/u\)) and interferometers (i.e. \(d_a/u\)) are compared in Figure~\ref{fig:test_struts_comp_enc_int}. The dynamics from the excitation voltage \(u\) to the displacement measured by the encoder \(d_e\) presents a behavior that is much more complex than the dynamics of the displacement measured by the interferometer (comparison made in Figure~\ref{fig:test_struts_comp_enc_int}). -Three additional resonance frequencies can be observed at 197Hz, 290Hz and 376Hz. +Three additional resonance frequencies can be observed at \(197\,\text{Hz}\), \(290\,\text{Hz}\) and \(376\,\text{Hz}\). These resonance frequencies match the frequencies of the flexible modes studied in Section~\ref{sec:test_struts_flexible_modes}. The good news is that these resonances are not impacting the axial motion of the strut (which is what is important for the hexapod positioning). @@ -11120,7 +11120,7 @@ A very good match can be observed between the struts. The same comparison is made for the transfer function from \(u\) to \(d_e\) (encoder output) in Figure~\ref{fig:test_struts_comp_enc_plants}. In this study, large dynamics differences were observed between the 5 struts. -Although the same resonance frequencies were seen for all of the struts (95Hz, 200Hz, 300Hz and 400Hz), the amplitude of the peaks were not the same. +Although the same resonance frequencies were seen for all of the struts (\(95\,\text{Hz}\), \(200\,\text{Hz}\), \(300\,\text{Hz}\) and \(400\,\text{Hz}\)), the amplitude of the peaks were not the same. In addition, the location or even presence of complex conjugate zeros changes from one strut to another. The reason for this variability will be studied in the next section thanks to the strut model. \subsection{Strut Model} @@ -11179,7 +11179,7 @@ For the flexible model, it will be shown in the next section that by adding some As shown in Figure~\ref{fig:test_struts_comp_enc_plants}, the identified dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\) are very different from one strut to the other. In this section, it is investigated whether poor alignment of the strut (flexible joints with respect to the \acrshort{apa}) can explain such dynamics. For instance, consider Figure~\ref{fig:test_struts_misalign_schematic} where there is a misalignment in the \(y\) direction between the two flexible joints (well aligned thanks to the mounting procedure in Section~\ref{sec:test_struts_mounting}) and the APA300ML. -In this case, the ``x-bending'' mode at 200Hz (see Figure~\ref{fig:test_struts_meas_x_bending}) can be expected to have greater impact on the dynamics from the actuator to the encoder. +In this case, the ``x-bending'' mode at \(200\,\text{Hz}\) (see Figure~\ref{fig:test_struts_meas_x_bending}) can be expected to have greater impact on the dynamics from the actuator to the encoder. \begin{figure}[htbp] \centering @@ -11192,17 +11192,17 @@ The obtained dynamics are shown in Figure~\ref{fig:test_struts_effect_misalignme The alignment of the \acrshort{apa} with the flexible joints has a large influence on the dynamics from actuator voltage to the measured displacement by the encoder. The misalignment in the \(y\) direction mostly influences: \begin{itemize} -\item the presence of the flexible mode at 200Hz (see mode shape in Figure~\ref{fig:test_struts_mode_shapes_1}) +\item the presence of the flexible mode at \(200\,\text{Hz}\) (see mode shape in Figure~\ref{fig:test_struts_mode_shapes_1}) \item the location of the complex conjugate zero between the first two resonances: \begin{itemize} \item if \(d_{y} < 0\): there is no zero between the two resonances and possibly not even between the second and third resonances \item if \(d_{y} > 0\): there is a complex conjugate zero between the first two resonances \end{itemize} -\item the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero) +\item the location of the high frequency complex conjugate zeros at \(500\,\text{Hz}\) (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero) \end{itemize} The same can be done for misalignments in the \(x\) direction. -The obtained dynamics (Figure~\ref{fig:test_struts_effect_misalignment_x}) are showing that misalignment in the \(x\) direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure~\ref{fig:test_struts_mode_shapes_2}). +The obtained dynamics (Figure~\ref{fig:test_struts_effect_misalignment_x}) are showing that misalignment in the \(x\) direction mostly influences the presence of the flexible mode at \(300\,\text{Hz}\) (see mode shape in Figure~\ref{fig:test_struts_mode_shapes_2}). A comparison of the experimental \acrshortpl{frf} in Figure~\ref{fig:test_struts_comp_enc_plants} with the model dynamics for several \(y\) misalignments in Figure~\ref{fig:test_struts_effect_misalignment_y} indicates a clear similarity. This similarity suggests that the identified differences in dynamics are caused by misalignment. @@ -11231,17 +11231,17 @@ Therefore, large \(y\) misalignments are expected. To estimate the misalignments between the two flexible joints and the \acrshort{apa}: \begin{itemize} \item the struts were fixed horizontally on the mounting bench, as shown in Figure~\ref{fig:test_struts_mounting_step_3} but without the encoder -\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu m\)}, the height difference between the flexible joints surface and the \acrshort{apa} shell surface was measured for both the top and bottom joints and for both sides -\item as the thickness of the flexible joint is \(21\,mm\) and the thickness of the \acrshort{apa} shell is \(20\,mm\), \(0.5\,mm\) of height difference should be measured if the two are perfectly aligned +\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu\text{m}\)}, the height difference between the flexible joints surface and the \acrshort{apa} shell surface was measured for both the top and bottom joints and for both sides +\item as the thickness of the flexible joint is \(21\,\text{mm}\) and the thickness of the \acrshort{apa} shell is \(20\,\text{mm}\), \(0.5\,\text{mm}\) of height difference should be measured if the two are perfectly aligned \end{itemize} Large variations in the \(y\) misalignment are found from one strut to the other (results are summarized in Table~\ref{tab:test_struts_meas_y_misalignment}). -To check the validity of the measurement, it can be verified that the sum of the measured thickness difference on each side is \(1\,mm\) (equal to the thickness difference between the flexible joint and the \acrshort{apa}). -Thickness differences for all the struts were found to be between \(0.94\,mm\) and \(1.00\,mm\) which indicate low errors compared to the misalignments found in Table~\ref{tab:test_struts_meas_y_misalignment}. +To check the validity of the measurement, it can be verified that the sum of the measured thickness difference on each side is \(1\,\text{mm}\) (equal to the thickness difference between the flexible joint and the \acrshort{apa}). +Thickness differences for all the struts were found to be between \(0.94\,\text{mm}\) and \(1.00\,\text{mm}\) which indicate low errors compared to the misalignments found in Table~\ref{tab:test_struts_meas_y_misalignment}. \begin{table}[htbp] -\caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment at the top and bottom of the APA. Measurements are in \(mm\)} +\caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment at the top and bottom of the APA. Measurements are in \(\text{mm}\)} \centering \begin{tabularx}{0.2\linewidth}{Xcc} \toprule @@ -11277,10 +11277,10 @@ After receiving the positioning pins, the struts were mounted again with the pos This should improve the alignment of the \acrshort{apa} with the two flexible joints. The alignment is then estimated using a length gauge, as described in the previous sections. -Measured \(y\) alignments are summarized in Table~\ref{tab:test_struts_meas_y_misalignment_with_pin} and are found to be bellow \(55\mu m\) for all the struts, which is much better than before (see Table~\ref{tab:test_struts_meas_y_misalignment}). +Measured \(y\) alignments are summarized in Table~\ref{tab:test_struts_meas_y_misalignment_with_pin} and are found to be bellow \(55\mu\text{m}\) for all the struts, which is much better than before (see Table~\ref{tab:test_struts_meas_y_misalignment}). \begin{table}[htbp] -\caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in \(mm\).} +\caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in \(\text{mm}\).} \centering \begin{tabularx}{0.25\linewidth}{Xcc} \toprule @@ -11351,8 +11351,8 @@ To do so, a precisely machined mounting tool (Figure~\ref{fig:test_nhexa_center_ \caption{\label{fig:test_nhexa_received_parts}Nano-Hexapod plates (\subref{fig:test_nhexa_nano_hexapod_plates}) and mounting tool used to position the two plates during assembly (\subref{fig:test_nhexa_center_part_hexapod_mounting})} \end{figure} -The mechanical tolerances of the received plates were checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\mu m\)} (Figure~\ref{fig:test_nhexa_plates_tolerances}) and were found to comply with the requirements\footnote{Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.}. -The same was done for the mounting tool\footnote{The height dimension is better than \(40\,\mu m\). The diameter fitting of 182g6 and 24g6 with the two plates is verified.}. +The mechanical tolerances of the received plates were checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\mu\text{m}\)} (Figure~\ref{fig:test_nhexa_plates_tolerances}) and were found to comply with the requirements\footnote{Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.}. +The same was done for the mounting tool\footnote{The height dimension is better than \(40\,\mu\text{m}\). The diameter fitting of 182g6 and 24g6 with the two plates is verified.}. The two plates were then fixed to the mounting tool, as shown in Figure~\ref{fig:test_nhexa_mounting_tool_hexapod_top_view}. The main goal of this ``mounting tool'' is to position the flexible joint interfaces (the ``V'' shapes) of both plates so that a cylinder can rest on the 4 flat interfaces at the same time. @@ -11375,7 +11375,7 @@ The main goal of this ``mounting tool'' is to position the flexible joint interf The quality of the positioning can be estimated by measuring the ``straightness'' of the top and bottom ``V'' interfaces. This corresponds to the diameter of the smallest cylinder which contains all points along the measured axis. This was again done using the FARO arm, and the results for all six struts are summarized in Table~\ref{tab:measured_straightness}. -The straightness was found to be better than \(15\,\mu m\) for all struts\footnote{As the accuracy of the FARO arm is \(\pm 13\,\mu m\), the true straightness is probably better than the values indicated. The limitation of the instrument is here reached.}, which is sufficiently good to not induce significant stress of the flexible joint during assembly. +The straightness was found to be better than \(15\,\mu\text{m}\) for all struts\footnote{As the accuracy of the FARO arm is \(\pm 13\,\mu\text{m}\), the true straightness is probably better than the values indicated. The limitation of the instrument is here reached.}, which is sufficiently good to not induce significant stress of the flexible joint during assembly. \begin{table}[htbp] \caption{\label{tab:measured_straightness}Measured straightness between the two ``V'' shapes for the six struts. These measurements were performed twice for each strut.} @@ -11384,12 +11384,12 @@ The straightness was found to be better than \(15\,\mu m\) for all struts\footno \toprule \textbf{Strut} & \textbf{Meas 1} & \textbf{Meas 2}\\ \midrule -1 & \(7\,\mu m\) & \(3\, \mu m\)\\ -2 & \(11\, \mu m\) & \(11\, \mu m\)\\ -3 & \(15\, \mu m\) & \(14\, \mu m\)\\ -4 & \(6\, \mu m\) & \(6\, \mu m\)\\ -5 & \(7\, \mu m\) & \(5\, \mu m\)\\ -6 & \(6\, \mu m\) & \(7\, \mu m\)\\ +1 & \(7\,\mu\text{m}\) & \(3\, \mu\text{m}\)\\ +2 & \(11\, \mu\text{m}\) & \(11\, \mu\text{m}\)\\ +3 & \(15\, \mu\text{m}\) & \(14\, \mu\text{m}\)\\ +4 & \(6\, \mu\text{m}\) & \(6\, \mu\text{m}\)\\ +5 & \(7\, \mu\text{m}\) & \(5\, \mu\text{m}\)\\ +6 & \(6\, \mu\text{m}\) & \(7\, \mu\text{m}\)\\ \bottomrule \end{tabularx} \end{table} @@ -11444,7 +11444,7 @@ Finally, the multi-body model representing the suspended table was tuned to matc The design of the suspended table is quite straightforward. First, an optical table with high frequency flexible mode was selected\footnote{The 450 mm x 450 mm x 60 mm Nexus B4545A from Thorlabs.}. -Then, four springs\footnote{``SZ8005 20 x 044'' from Steinel. The spring rate is specified at \(17.8\,N/mm\)} were selected with low spring rate such that the suspension modes are below 10Hz. +Then, four springs\footnote{``SZ8005 20 x 044'' from Steinel. The spring rate is specified at \(17.8\,\text{N/mm}\)} were selected with low spring rate such that the suspension modes are below \(10\,\text{Hz}\). Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure~\ref{fig:test_nhexa_suspended_table_cad}). \begin{figure}[htbp] @@ -11455,10 +11455,10 @@ Finally, some interface elements were designed, and mechanical lateral mechanica \subsubsection{Modal analysis of the suspended table} \label{ssec:test_nhexa_table_identification} -In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} were fixed to the breadboard. +In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,\text{V/g}\), measurement range is \(\pm 5\,\text{g}\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} were fixed to the breadboard. Using an instrumented hammer, the first 9 modes could be identified and are summarized in Table~\ref{tab:test_nhexa_suspended_table_modes}. -The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below 10Hz. -The next modes are the flexible modes of the breadboard as shown in Figure~\ref{fig:test_nhexa_table_flexible_modes}, and are located above 700Hz. +The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below \(10\,\text{Hz}\). +The next modes are the flexible modes of the breadboard as shown in Figure~\ref{fig:test_nhexa_table_flexible_modes}, and are located above \(700\,\text{Hz}\). \begin{minipage}[t]{0.45\textwidth} \begin{center} @@ -11474,14 +11474,14 @@ The next modes are the flexible modes of the breadboard as shown in Figure~\ref{ \toprule \textbf{Modes} & \textbf{Frequency} & \textbf{Description}\\ \midrule -1,2 & 1.3 Hz & X-Y translations\\ -3 & 2.0 Hz & Z rotation\\ -4 & 6.9 Hz & Z translation\\ -5,6 & 9.5 Hz & X-Y rotations\\ +1,2 & \(1.3\,\text{Hz}\) & X-Y translations\\ +3 & \(2.0\,\text{Hz}\) & Z rotation\\ +4 & \(6.9\,\text{Hz}\) & Z translation\\ +5,6 & \(9.5\,\text{Hz}\) & X-Y rotations\\ \midrule -7 & 701 Hz & ``Membrane'' Mode\\ -8 & 989 Hz & Complex mode\\ -9 & 1025 Hz & Complex mode\\ +7 & \(701\,\text{Hz}\) & ``Membrane'' Mode\\ +8 & \(989\,\text{Hz}\) & Complex mode\\ +9 & \(1025\,\text{Hz}\) & Complex mode\\ \bottomrule \end{tabularx}} \captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table} @@ -11492,19 +11492,19 @@ The next modes are the flexible modes of the breadboard as shown in Figure~\ref{ \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_table_flexible_mode_1.jpg} \end{center} -\subcaption{\label{fig:test_nhexa_table_flexible_mode_1}Flexible mode at 701Hz} +\subcaption{\label{fig:test_nhexa_table_flexible_mode_1}Flexible mode at $701\,\text{Hz}$} \end{subfigure} \begin{subfigure}{\textwidth} \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_table_flexible_mode_2.jpg} \end{center} -\subcaption{\label{fig:test_nhexa_table_flexible_mode_2}Flexible mode at 989Hz} +\subcaption{\label{fig:test_nhexa_table_flexible_mode_2}Flexible mode at $989\,\text{Hz}$} \end{subfigure} \begin{subfigure}{\textwidth} \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_table_flexible_mode_3.jpg} \end{center} -\subcaption{\label{fig:test_nhexa_table_flexible_mode_3}Flexible mode at 1025Hz} +\subcaption{\label{fig:test_nhexa_table_flexible_mode_3}Flexible mode at $1025\,\text{Hz}$} \end{subfigure} \caption{\label{fig:test_nhexa_table_flexible_modes}Three identified flexible modes of the suspended table} \end{figure} @@ -11516,8 +11516,8 @@ The 4 springs are here modeled with ``bushing joints'' that have stiffness and d The model order is 12, which corresponds to the 6 suspension modes. The inertia properties of the parts were determined from the geometry and material densities. -The stiffness of the springs was initially set from the datasheet nominal value of \(17.8\,N/mm\) and then reduced down to \(14\,N/mm\) to better match the measured suspension modes. -The stiffness of the springs in the horizontal plane is set at \(0.5\,N/mm\). +The stiffness of the springs was initially set from the datasheet nominal value of \(17.8\,\text{N/mm}\) and then reduced down to \(14\,\text{N/mm}\) to better match the measured suspension modes. +The stiffness of the springs in the horizontal plane is set at \(0.5\,\text{N/mm}\). The obtained suspension modes of the multi-body model are compared with the measured modes in Table~\ref{tab:test_nhexa_suspended_table_simscape_modes}. \begin{table}[htbp] @@ -11527,8 +11527,8 @@ The obtained suspension modes of the multi-body model are compared with the meas \toprule Directions & \(D_x\), \(D_y\) & \(R_z\) & \(D_z\) & \(R_x\), \(R_y\)\\ \midrule -Multi-body & 1.3 Hz & 1.8 Hz & 6.8 Hz & 9.5 Hz\\ -Experimental & 1.3 Hz & 2.0 Hz & 6.9 Hz & 9.5 Hz\\ +Multi-body & \(1.3\,\text{Hz}\) & \(1.8\,\text{Hz}\) & \(6.8\,\text{Hz}\) & \(9.5\,\text{Hz}\)\\ +Experimental & \(1.3\,\text{Hz}\) & \(2.0\,\text{Hz}\) & \(6.9\,\text{Hz}\) & \(9.5\,\text{Hz}\)\\ \bottomrule \end{tabularx} \end{table} @@ -11568,8 +11568,8 @@ Five 3-axis accelerometers were fixed on the top platform of the nano-hexapod (F \caption{\label{fig:test_nhexa_modal_analysis}Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis} \end{figure} -Between 100Hz and 200Hz, 6 suspension modes (i.e. rigid body modes of the top platform) were identified. -At around 700Hz, two flexible modes of the top plate were observed (see Figure~\ref{fig:test_nhexa_hexa_flexible_modes}). +Between \(100\,\text{Hz}\) and \(200\,\text{Hz}\), 6 suspension modes (i.e. rigid body modes of the top platform) were identified. +At around \(700\,\text{Hz}\), two flexible modes of the top plate were observed (see Figure~\ref{fig:test_nhexa_hexa_flexible_modes}). These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}. \begin{table}[htbp] @@ -11579,14 +11579,14 @@ These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}. \toprule \textbf{Mode} & \textbf{Frequency} & \textbf{Description}\\ \midrule -1 & 120 Hz & Suspension Mode: Y-translation\\ -2 & 120 Hz & Suspension Mode: X-translation\\ -3 & 145 Hz & Suspension Mode: Z-translation\\ -4 & 165 Hz & Suspension Mode: Y-rotation\\ -5 & 165 Hz & Suspension Mode: X-rotation\\ -6 & 190 Hz & Suspension Mode: Z-rotation\\ -7 & 692 Hz & (flexible) Membrane mode of the top platform\\ -8 & 709 Hz & Second flexible mode of the top platform\\ +1 & \(120\,\text{Hz}\) & Suspension Mode: Y-translation\\ +2 & \(120\,\text{Hz}\) & Suspension Mode: X-translation\\ +3 & \(145\,\text{Hz}\) & Suspension Mode: Z-translation\\ +4 & \(165\,\text{Hz}\) & Suspension Mode: Y-rotation\\ +5 & \(165\,\text{Hz}\) & Suspension Mode: X-rotation\\ +6 & \(190\,\text{Hz}\) & Suspension Mode: Z-rotation\\ +7 & \(692\,\text{Hz}\) & (flexible) Membrane mode of the top platform\\ +8 & \(709\,\text{Hz}\) & Second flexible mode of the top platform\\ \bottomrule \end{tabularx} \end{table} @@ -11596,13 +11596,13 @@ These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}. \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_hexa_flexible_mode_1.jpg} \end{center} -\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at 692Hz} +\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at $692\,\text{Hz}$} \end{subfigure} \begin{subfigure}{\textwidth} \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_hexa_flexible_mode_2.jpg} \end{center} -\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at 709Hz} +\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at $709\,\text{Hz}$} \end{subfigure} \caption{\label{fig:test_nhexa_hexa_flexible_modes}Two identified flexible modes of the top plate of the Nano-Hexapod} \end{figure} @@ -11614,17 +11614,17 @@ The dynamics of the nano-hexapod from the six command signals (\(u_1\) to \(u_6\ The \(6 \times 6\) \acrshort{frf} matrix from \(\bm{u}\) ot \(\bm{d}_e\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_de}. The diagonal terms are displayed using colored lines, and all the 30 off-diagonal terms are displayed by gray lines. -All six diagonal terms are well superimposed up to at least \(1\,kHz\), indicating good manufacturing and mounting uniformity. +All six diagonal terms are well superimposed up to at least \(1\,\text{kHz}\), indicating good manufacturing and mounting uniformity. Below the first suspension mode, good decoupling can be observed (the amplitude of all off-diagonal terms are \(\approx 20\) times smaller than the diagonal terms), indicating the correct assembly of all parts. -From 10Hz up to 1kHz, around 10 resonance frequencies can be observed. -The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section~\ref{ssec:test_nhexa_enc_struts_modal_analysis}. -Three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section~\ref{ssec:test_nhexa_comp_model_coupling}). -Except for the mode at 237Hz, their impact on the dynamics is small. -The two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform. -Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. +From \(10\,\text{Hz}\) up to \(1\,\text{kHz}\), around 10 resonance frequencies can be observed. +The first 4 are suspension modes (at \(122\,\text{Hz}\), \(143\,\text{Hz}\), \(165\,\text{Hz}\) and \(191\,\text{Hz}\)) which correlate the modes measured during the modal analysis in Section~\ref{ssec:test_nhexa_enc_struts_modal_analysis}. +Three modes at \(237\,\text{Hz}\), \(349\,\text{Hz}\) and \(395\,\text{Hz}\) are attributed to the internal strut resonances (this will be checked in Section~\ref{ssec:test_nhexa_comp_model_coupling}). +Except for the mode at \(237\,\text{Hz}\), their impact on the dynamics is small. +The two modes at \(665\,\text{Hz}\) and \(695\,\text{Hz}\) are attributed to the flexible modes of the top platform. +Other modes can be observed above \(1\,\text{kHz}\), which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. -Up to at least 1kHz, an alternating pole/zero pattern is observed, which makes the control easier to tune. +Up to at least \(1\,\text{kHz}\), an alternating pole/zero pattern is observed, which makes the control easier to tune. This would not have occurred if the encoders were fixed to the struts. \begin{figure}[htbp] @@ -11634,9 +11634,9 @@ This would not have occurred if the encoders were fixed to the struts. \end{figure} Similarly, the \(6 \times 6\) \acrshort{frf} matrix from \(\bm{u}\) to \(\bm{V}_s\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_Vs}. -Alternating poles and zeros can be observed up to at least 2kHz, which is a necessary characteristics for applying decentralized IFF. +Alternating poles and zeros can be observed up to at least \(2\,\text{kHz}\), which is a necessary characteristics for applying decentralized IFF. Similar to what was observed for the encoder outputs, all the ``diagonal'' terms are well superimposed, indicating that the same controller can be applied to all the struts. -The first flexible mode of the struts as 235Hz has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF. +The first flexible mode of the struts as \(235\,\text{Hz}\) has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF. \begin{figure}[htbp] \centering @@ -11648,26 +11648,26 @@ The first flexible mode of the struts as 235Hz has large amplitude, and therefor One major challenge for controlling the NASS is the wanted robustness to a variation of payload mass; therefore, it is necessary to understand how the dynamics of the nano-hexapod changes with a change in payload mass. -To study how the dynamics changes with the payload mass, up to three ``cylindrical masses'' of \(13\,kg\) each can be added for a total of \(\approx 40\,kg\). +To study how the dynamics changes with the payload mass, up to three ``cylindrical masses'' of \(13\,\text{kg}\) each can be added for a total of \(\approx 40\,\text{kg}\). These three cylindrical masses on top of the nano-hexapod are shown in Figure~\ref{fig:test_nhexa_table_mass_3}. \begin{figure}[htbp] \centering \includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_table_mass_3.jpg} -\caption{\label{fig:test_nhexa_table_mass_3}Picture of the nano-hexapod with the added three cylindrical masses for a total of \(\approx 40\,kg\)} +\caption{\label{fig:test_nhexa_table_mass_3}Picture of the nano-hexapod with the added three cylindrical masses for a total of \(\approx 40\,\text{kg}\)} \end{figure} -The obtained \acrshortpl{frf} from actuator signal \(u_i\) to the associated encoder \(d_{ei}\) for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure~\ref{fig:test_nhexa_identified_frf_de_masses}. +The obtained \acrshortpl{frf} from actuator signal \(u_i\) to the associated encoder \(d_{ei}\) for the four payload conditions (no mass, \(13\,\text{kg}\), \(26\,\text{kg}\) and \(39\,\text{kg}\)) are shown in Figure~\ref{fig:test_nhexa_identified_frf_de_masses}. As expected, the frequency of the suspension modes decreased with increasing payload mass. The low frequency gain does not change because it is linked to the stiffness property of the nano-hexapod and not to its mass property. -The frequencies of the two flexible modes of the top plate first decreased significantly when the first mass was added (from \(\approx 700\,Hz\) to \(\approx 400\,Hz\)). +The frequencies of the two flexible modes of the top plate first decreased significantly when the first mass was added (from \(\approx 700\,\text{Hz}\) to \(\approx 400\,\text{Hz}\)). This is because the added mass is composed of two half cylinders that are not fixed together. Therefore, it adds a lot of mass to the top plate without increasing stiffness in one direction. When more than one ``mass layer'' is added, the half cylinders are added at some angles such that rigidity is added in all directions (see how the three mass ``layers'' are positioned in Figure~\ref{fig:test_nhexa_table_mass_3}). In this case, the frequency of these flexible modes is increased. In practice, the payload should be one solid body, and no decrease in the frequency of this flexible mode should be observed. -The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass increased. +The apparent amplitude of the flexible mode of the strut at \(237\,\text{Hz}\) becomes smaller as the payload mass increased. The measured \acrshortpl{frf} from \(u_i\) to \(V_{si}\) are shown in Figure~\ref{fig:test_nhexa_identified_frf_Vs_masses}. For all tested payloads, the measured \acrshort{frf} always have alternating poles and zeros, indicating that IFF can be applied in a robust manner. @@ -11714,8 +11714,8 @@ The \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(\bm{d}_e\) an First, is it evaluated how well the models matches the ``direct'' terms of the measured \acrshort{frf} matrix. To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured \acrshort{frf} in Figure~\ref{fig:test_nhexa_comp_simscape_diag}. -It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modeled. -The three resonances that were attributed to ``internal'' flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable because the \acrshortpl{apa} are here modeled as a simple uniaxial 2-DoF system. +It can be seen that the 4 suspension modes of the nano-hexapod (at \(122\,\text{Hz}\), \(143\,\text{Hz}\), \(165\,\text{Hz}\) and \(191\,\text{Hz}\)) are well modeled. +The three resonances that were attributed to ``internal'' flexible modes of the struts (at \(237\,\text{Hz}\), \(349\,\text{Hz}\) and \(395\,\text{Hz}\)) cannot be seen in the model, which is reasonable because the \acrshortpl{apa} are here modeled as a simple uniaxial 2-DoF system. At higher frequencies, no resonances can be observed in the model, as the top plate and the encoder supports are modeled as rigid bodies. \begin{figure}[htbp] @@ -11738,7 +11738,7 @@ At higher frequencies, no resonances can be observed in the model, as the top pl Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of \acrshort{mimo} systems. Instead of comparing the full 36 elements of the \(6 \times 6\) \acrshort{frf} matrix from \(\bm{u}\) to \(\bm{d}_e\), only the first ``column'' is compared (Figure~\ref{fig:test_nhexa_comp_simscape_de_all}), which corresponds to the transfer function from the command \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\). -It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at 237Hz. +It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at \(237\,\text{Hz}\). Similar results are observed for all other coupling terms and for the transfer function from \(\bm{u}\) to \(\bm{V}_s\). \begin{figure}[htbp] @@ -11749,8 +11749,8 @@ Similar results are observed for all other coupling terms and for the transfer f The APA300ML was then modeled with a \emph{super-element} extracted from a FE-software. The obtained transfer functions from \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\) are compared with the measured \acrshort{frf} in Figure~\ref{fig:test_nhexa_comp_simscape_de_all_flex}. -While the damping of the suspension modes for the \emph{super-element} is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modeled. -Even the mode 395Hz can be observed in the model. +While the damping of the suspension modes for the \emph{super-element} is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at \(237\,\text{Hz}\) and \(349\,\text{Hz}\) are well modeled. +Even the mode \(395\,\text{Hz}\) can be observed in the model. Therefore, if the modes of the struts are to be modeled, the \emph{super-element} of the APA300ML can be used at the cost of obtaining a much higher order model. \begin{figure}[htbp] @@ -11762,7 +11762,7 @@ Therefore, if the modes of the struts are to be modeled, the \emph{super-element \label{ssec:test_nhexa_comp_model_masses} Another important characteristic of the model is that it should represents the dynamics of the system well for all considered payloads. -The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure~\ref{fig:test_nhexa_comp_simscape_diag_masses}. +The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, \(13\,\text{kg}\), \(26\,\text{kg}\) and \(39\,\text{kg}\)) in Figure~\ref{fig:test_nhexa_comp_simscape_diag_masses}. The observed shift of the suspension modes to lower frequencies with increased payload mass is well represented by the multi-body model. The complex conjugate zeros also well match the experiments both for the encoder outputs (Figure~\ref{fig:test_nhexa_comp_simscape_de_diag_masses}) and the force sensor outputs (Figure~\ref{fig:test_nhexa_comp_simscape_Vs_diag_masses}). @@ -11786,7 +11786,7 @@ However, as decentralized IFF will be applied, the damping is actively brought, \caption{\label{fig:test_nhexa_comp_simscape_diag_masses}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})} \end{figure} -In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from \(u_1\) to \(d_{e1}\) to \(d_{e6}\) are compared in the case of the 39kg payload in Figure~\ref{fig:test_nhexa_comp_simscape_de_all_high_mass}. +In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from \(u_1\) to \(d_{e1}\) to \(d_{e6}\) are compared in the case of the \(39\,\text{kg}\) payload in Figure~\ref{fig:test_nhexa_comp_simscape_de_all_high_mass}. Excellent match between experimental and model coupling is observed. Therefore, the model effectively represents the system coupling for different payloads. @@ -11805,7 +11805,7 @@ Although the dynamics of the nano-hexapod was indeed impacted by the dynamics of The dynamics of the nano-hexapod was then identified in Section~\ref{sec:test_nhexa_dynamics}. Below the first suspension mode, good decoupling could be observed for the transfer function from \(\bm{u}\) to \(\bm{d}_e\), which enables the design of a decentralized positioning controller based on the encoders for relative positioning purposes. -Many other modes were present above 700Hz, which will inevitably limit the achievable bandwidth. +Many other modes were present above \(700\,\text{Hz}\), which will inevitably limit the achievable bandwidth. The observed effect of the payload's mass on the dynamics was quite large, which also represents a complex control challenge. The \acrshortpl{frf} from the six DAC voltages \(\bm{u}\) to the six force sensors voltages \(\bm{V}_s\) all have alternating complex conjugate poles and complex conjugate zeros for all the tested payloads (Figure~\ref{fig:test_nhexa_comp_simscape_Vs_diag_masses}). @@ -11826,9 +11826,9 @@ If a model of the nano-hexapod was developed in one time, it would be difficult To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure~\ref{fig:test_id31_micro_station_nano_hexapod}. This section presents a comprehensive experimental evaluation of the complete system's performance on the ID31 beamline, focusing on its ability to maintain precise sample positioning under various experimental conditions. -Initially, the project planned to develop a long-stroke (\(\approx 1 \, cm^3\)) 5-DoF metrology system to measure the sample position relative to the granite base. +Initially, the project planned to develop a long-stroke (\(\approx 1 \, \text{cm}^3\)) 5-DoF metrology system to measure the sample position relative to the granite base. However, the complexity of this development prevented its completion before the experimental testing phase on ID31. -To validate the nano-hexapod and its associated control architecture, an alternative short-stroke (\(\approx 100\,\mu m^3\)) metrology system was developed, which is presented in Section~\ref{sec:test_id31_metrology}. +To validate the nano-hexapod and its associated control architecture, an alternative short-stroke (\(\approx 100\,\mu\text{m}^3\)) metrology system was developed, which is presented in Section~\ref{sec:test_id31_metrology}. Then, several key aspects of the system validation are examined. Section~\ref{sec:test_id31_open_loop_plant} analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities. @@ -11859,11 +11859,11 @@ These include tomography scans at various speeds and with different payload mass \subsection{Short Stroke Metrology System} \label{sec:test_id31_metrology} The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite. -As a long-stroke (\(\approx 1 \,cm^3\)) metrology system was not yet developed, a stroke stroke (\(\approx 100\,\mu m^3\)) was used instead to validate the nano-hexapod control. +As a long-stroke (\(\approx 1 \,\text{cm}^3\)) metrology system was not yet developed, a stroke stroke (\(\approx 100\,\mu\text{m}^3\)) was used instead to validate the nano-hexapod control. The first considered option was to use the ``Spindle error analyzer'' shown in Figure~\ref{fig:test_id31_lion}. This system comprises 5 capacitive sensors facing two reference spheres. -However, as the gap between the capacitive sensors and the spheres is very small\footnote{Depending on the measuring range, gap can range from \(\approx 1\,\mu m\) to \(\approx 100\,\mu m\).}, the risk of damaging the spheres and the capacitive sensors is too high. +However, as the gap between the capacitive sensors and the spheres is very small\footnote{Depending on the measuring range, gap can range from \(\approx 1\,\mu\text{m}\) to \(\approx 100\,\mu\text{m}\).}, the risk of damaging the spheres and the capacitive sensors is too high. \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -11890,7 +11890,7 @@ However, as the gap between the capacitive sensors and the spheres is very small Instead of using capacitive sensors, 5 fibered interferometers were used in a similar manner (Figure~\ref{fig:test_id31_interf}). At the end of each fiber, a sensor head\footnote{M12/F40 model from Attocube.} (Figure~\ref{fig:test_id31_interf_head}) is used, which consists of a lens precisely positioned with respect to the fiber's end. The lens focuses the light on the surface of the sphere, such that the reflected light comes back into the fiber and produces an interference. -In this way, the gap between the head and the reference sphere is much larger (here around \(40\,mm\)), thereby removing the risk of collision. +In this way, the gap between the head and the reference sphere is much larger (here around \(40\,\text{mm}\)), thereby removing the risk of collision. Nevertheless, the metrology system still has a limited measurement range because of the limited angular acceptance of the fibered interferometers. Indeed, when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and above some perpendicular displacement, the reflected light does not come back into the fiber, and no interference is produced. @@ -11898,8 +11898,8 @@ Indeed, when the spheres are moving perpendicularly to the beam axis, the reflec \label{ssec:test_id31_metrology_kinematics} The proposed short-stroke metrology system is schematized in Figure~\ref{fig:test_id31_metrology_kinematics}. -The \acrshort{poi} is indicated by the blue frame \(\{B\}\), which is located \(H = 150\,mm\) above the nano-hexapod's top platform. -The spheres have a diameter \(d = 25.4\,mm\), and the indicated dimensions are \(l_1 = 60\,mm\) and \(l_2 = 16.2\,mm\). +The \acrshort{poi} is indicated by the blue frame \(\{B\}\), which is located \(H = 150\,\text{mm}\) above the nano-hexapod's top platform. +The spheres have a diameter \(d = 25.4\,\text{mm}\), and the indicated dimensions are \(l_1 = 60\,\text{mm}\) and \(l_2 = 16.2\,\text{mm}\). To compute the pose of \(\{B\}\) with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured (small) displacements \([d_1,\ d_2,\ d_3,\ d_4,\ d_5]\) by the interferometers are first written as a function of the (small) linear and angular motion of the \(\{B\}\) frame \([D_x,\ D_y,\ D_z,\ R_x,\ R_y]\) \eqref{eq:test_id31_metrology_kinematics}. \begin{equation}\label{eq:test_id31_metrology_kinematics} @@ -11945,7 +11945,7 @@ To not damage the sensitive sphere surface, the probes are instead positioned on The probes are first fixed to the bottom (fixed) cylinder to align the first sphere with the spindle axis. The probes are then fixed to the top (adjustable) cylinder, and the same alignment is performed. -With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around \(10\,\mu m\). +With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around \(10\,\mu\text{m}\). The accuracy was probably limited by the poor coaxiality between the cylinders and the spheres. However, this first alignment should be sufficient to position the two sphere within the acceptance range of the interferometers. \subsubsection{Tip-Tilt adjustment of the interferometers} @@ -11962,7 +11962,7 @@ Granite is used for its good mechanical and thermal stability. The interferometer beams must be placed with respect to the two reference spheres as close as possible to the ideal case shown in Figure~\ref{fig:test_id31_metrology_kinematics}. Therefore, their positions and angles must be well adjusted with respect to the two spheres. -First, the vertical positions of the spheres is adjusted using the micro-hexapod to match the heights of the interferometers. +First, the vertical positions of the spheres is adjusted using the positioning hexapod to match the heights of the interferometers. Then, the horizontal position of the gantry is adjusted such that the intensity of the light reflected back in the fiber of the top interferometer is maximized. This is equivalent as to optimize the perpendicularity between the interferometer beam and the sphere surface (i.e. the concentricity between the top beam and the sphere center). @@ -11979,11 +11979,11 @@ Therefore, this metrology can be used to better align the axis defined by the ce The alignment process requires few iterations. First, the spindle is scanned, and alignment errors are recorded. -From the errors, the motion of the micro-hexapod to better align the spheres with the spindle axis is computed and the micro-hexapod is positioned accordingly. +From the errors, the motion of the positioning hexapod to better align the spheres with the spindle axis is computed and the positioning hexapod is positioned accordingly. Then, the spindle is scanned again, and new alignment errors are recorded. This iterative process is first performed for angular errors (Figure~\ref{fig:test_id31_metrology_align_rx_ry}) and then for lateral errors (Figure~\ref{fig:test_id31_metrology_align_dx_dy}). -The remaining errors after alignment are in the order of \(\pm5\,\mu\text{rad}\) in \(R_x\) and \(R_y\) orientations, \(\pm 1\,\mu m\) in \(D_x\) and \(D_y\) directions, and less than \(0.1\,\mu m\) vertically. +The remaining errors after alignment are in the order of \(\pm5\,\mu\text{rad}\) in \(R_x\) and \(R_y\) orientations, \(\pm 1\,\mu\text{m}\) in \(D_x\) and \(D_y\) directions, and less than \(0.1\,\mu\text{m}\) vertically. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -11998,16 +11998,16 @@ The remaining errors after alignment are in the order of \(\pm5\,\mu\text{rad}\) \end{center} \subcaption{\label{fig:test_id31_metrology_align_dx_dy}Lateral alignment} \end{subfigure} -\caption{\label{fig:test_id31_metrology_align}Measured angular (\subref{fig:test_id31_metrology_align_rx_ry}) and lateral (\subref{fig:test_id31_metrology_align_dx_dy}) errors during full spindle rotation. Between two rotations, the micro-hexapod is adjusted to better align the two spheres with the rotation axis.} +\caption{\label{fig:test_id31_metrology_align}Measured angular (\subref{fig:test_id31_metrology_align_rx_ry}) and lateral (\subref{fig:test_id31_metrology_align_dx_dy}) errors during full spindle rotation. Between two rotations, the positioning hexapod is adjusted to better align the two spheres with the rotation axis.} \end{figure} \subsubsection{Estimated measurement volume} \label{ssec:test_id31_metrology_acceptance} Because the interferometers point to spheres and not flat surfaces, the lateral acceptance is limited. -To estimate the metrology acceptance, the micro-hexapod was used to perform three accurate scans of \(\pm 1\,mm\), respectively along the \(x\), \(y\) and \(z\) axes. +To estimate the metrology acceptance, the positioning hexapod was used to perform three accurate scans of \(\pm 1\,\text{mm}\), respectively along the \(x\), \(y\) and \(z\) axes. During these scans, the 5 interferometers are recorded individually, and the ranges in which each interferometer had enough coupling efficiency to be able to measure the displacement were estimated. Results are summarized in Table~\ref{tab:test_id31_metrology_acceptance}. -The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(+/-0.5\,mm\), which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS. +The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(\pm0.5\,\text{mm}\), which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS. \begin{table}[htbp] \caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.} @@ -12016,11 +12016,11 @@ The obtained lateral acceptance for pure displacements in any direction is estim \toprule & \(D_x\) & \(D_y\) & \(D_z\)\\ \midrule -\(d_1\) (y) & \(1.0\,mm\) & \(>2\,mm\) & \(1.35\,mm\)\\ -\(d_2\) (y) & \(0.8\,mm\) & \(>2\,mm\) & \(1.01\,mm\)\\ -\(d_3\) (x) & \(>2\,mm\) & \(1.06\,mm\) & \(1.38\,mm\)\\ -\(d_4\) (x) & \(>2\,mm\) & \(0.99\,mm\) & \(0.94\,mm\)\\ -\(d_5\) (z) & \(1.33\, mm\) & \(1.06\,mm\) & \(>2\,mm\)\\ +\(d_1\) (y) & \(1.0\,\text{mm}\) & \(>2\,\text{mm}\) & \(1.35\,\text{mm}\)\\ +\(d_2\) (y) & \(0.8\,\text{mm}\) & \(>2\,\text{mm}\) & \(1.01\,\text{mm}\)\\ +\(d_3\) (x) & \(>2\,\text{mm}\) & \(1.06\,\text{mm}\) & \(1.38\,\text{mm}\)\\ +\(d_4\) (x) & \(>2\,\text{mm}\) & \(0.99\,\text{mm}\) & \(0.94\,\text{mm}\)\\ +\(d_5\) (z) & \(1.33\,\text{mm}\) & \(1.06\,\text{mm}\) & \(>2\,\text{mm}\)\\ \bottomrule \end{tabularx} \end{table} @@ -12033,16 +12033,16 @@ Only the bandwidth and noise characteristics of the external metrology are impor However, some elements that affect the accuracy of the metrology system are discussed here. First, the ``metrology kinematics'' (discussed in Section~\ref{ssec:test_id31_metrology_kinematics}) is only approximate (i.e. valid for small displacements). -This can be easily seen when performing lateral \([D_x,\,D_y]\) scans using the micro-hexapod while recording the vertical interferometer (Figure~\ref{fig:test_id31_xy_map_sphere}). +This can be easily seen when performing lateral \([D_x,\,D_y]\) scans using the positioning hexapod while recording the vertical interferometer (Figure~\ref{fig:test_id31_xy_map_sphere}). As the top interferometer points to a sphere and not to a plane, lateral motion of the sphere is seen as a vertical motion by the top interferometer. -Then, the reference spheres have some deviations relative to an ideal sphere \footnote{The roundness of the spheres is specified at \(50\,nm\).}. +Then, the reference spheres have some deviations relative to an ideal sphere \footnote{The roundness of the spheres is specified at \(50\,\text{nm}\).}. These sphere are originally intended for use with capacitive sensors that integrate shape errors over large surfaces. -When using interferometers, the size of the ``light spot'' on the sphere surface is a circle with a diameter approximately equal to \(50\,\mu m\), and therefore the measurement is more sensitive to shape errors with small features. +When using interferometers, the size of the ``light spot'' on the sphere surface is a circle with a diameter approximately equal to \(50\,\mu\text{m}\), and therefore the measurement is more sensitive to shape errors with small features. As the light from the interferometer travels through air (as opposed to being in vacuum), the measured distance is sensitive to any variation in the refractive index of the air. Therefore, any variation in air temperature, pressure or humidity will induce measurement errors. -For instance, for a measurement length of \(40\,mm\), a temperature variation of \(0.1\,{}^oC\) (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of \(\approx 4\,nm\). +For instance, for a measurement length of \(40\,\text{mm}\), a temperature variation of \(0.1\,{}^oC\) (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of \(\approx 4\,\text{nm}\). Interferometers are also affected by noise~\cite{watchi18_review_compac_inter}. The effect of noise on the translation and rotation measurements is estimated in Figure~\ref{fig:test_id31_interf_noise}. @@ -12362,7 +12362,7 @@ The obtained \acrshortpl{frf} are compared with the model in Figure~\ref{fig:tes \subsubsection*{Conclusion} The implementation of a decentralized Integral Force Feedback controller was successfully demonstrated. Using the multi-body model, the controller was designed and optimized to ensure stability across all payload conditions while providing significant damping of suspension modes. -The experimental results validated the model predictions, showing a reduction in peak amplitudes by approximately a factor of 10 across the full payload range (0-39 kg). +The experimental results validated the model predictions, showing a reduction in peak amplitudes by approximately a factor of 10 across the full payload range (0 to \(39\,\text{kg}\)). Although higher gains could achieve better damping performance for lighter payloads, the chosen fixed-gain configuration represents a robust compromise that maintains stability and performance under all operating conditions. The good correlation between the modeled and measured damped plants confirms the effectiveness of using the multi-body model for both controller design and performance prediction. \subsection{High Authority Control in the frame of the struts} @@ -12492,7 +12492,7 @@ The obtained closed-loop positioning accuracy was found to comply with the requi \subsubsection{Robustness estimation with simulation of Tomography scans} \label{ssec:test_id31_iff_hac_robustness} -To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure~\ref{fig:test_id31_picture_masses} (i.e. \(0\,kg\), \(13\,kg\), \(26\,kg\) and \(39\,kg\)). +To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure~\ref{fig:test_id31_picture_masses} (i.e. \(0\,\text{kg}\), \(13\,\text{kg}\), \(26\,\text{kg}\) and \(39\,\text{kg}\)). The rotational velocity was set at \(6\,\text{deg/s}\), which is the typical rotational velocity for heavy samples. The closed-loop systems were stable under all payload conditions, indicating good control robustness. @@ -12510,10 +12510,10 @@ The multi-body model was first validated by comparing it with the measured frequ This validation confirmed that the model can be reliably used to tune the feedback controller and evaluate its performance. An interaction analysis using the RGA-number was then performed, which revealed that higher payload masses lead to increased coupling when implementing control in the strut reference frame. -Based on this analysis, a diagonal controller with a crossover frequency of 5 Hz was designed, incorporating an integrator, a lead compensator, and a first-order low-pass filter. +Based on this analysis, a diagonal controller with a crossover frequency of \(5\,\text{Hz}\) was designed, incorporating an integrator, a lead compensator, and a first-order low-pass filter. Finally, tomography experiments were simulated to validate the \acrshort{haclac} architecture. -The closed-loop system remained stable under all tested payload conditions (0 to 39 kg). +The closed-loop system remained stable under all tested payload conditions (0 to \(39\,\text{kg}\)). With no payload at \(180\,\text{deg/s}\), the NASS successfully maintained the sample \acrshort{poi} in the beam, which fulfilled the specifications. At \(6\,\text{deg/s}\), although the positioning errors increased with the payload mass (particularly in the lateral direction), the system remained stable. These results demonstrate both the effectiveness and limitations of implementing control in the frame of the struts. @@ -12535,10 +12535,10 @@ Several scientific experiments were replicated, such as: Unless explicitly stated, all closed-loop experiments were performed using the robust (i.e. conservative) high authority controller designed in Section~\ref{ssec:test_id31_iff_hac_controller}. Higher performance controllers using complementary filters are investigated in Section~\ref{ssec:test_id31_cf_control}. -For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to \(200\,nm\times 100\,nm\). -In this case, the goal is to keep the sample's \acrshort{poi} in the beam, and therefore the \(D_y\) and \(D_z\) positioning errors should be less than \(200\,nm\) and \(100\,nm\) peak-to-peak, respectively. +For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to \(200\,\text{nm}\times 100\,\text{nm}\). +In this case, the goal is to keep the sample's \acrshort{poi} in the beam, and therefore the \(D_y\) and \(D_z\) positioning errors should be less than \(200\,\text{nm}\) and \(100\,\text{nm}\) peak-to-peak, respectively. The \(R_y\) error should be less than \(1.7\,\mu\text{rad}\) peak-to-peak. -In terms of RMS errors, this corresponds to \(30\,nm\) in \(D_y\), \(15\,nm\) in \(D_z\) and \(250\,\text{nrad}\) in \(R_y\) (a summary of the specifications is given in Table~\ref{tab:test_id31_experiments_specifications}). +In terms of RMS errors, this corresponds to \(30\,\text{nm}\) in \(D_y\), \(15\,\text{nm}\) in \(D_z\) and \(250\,\text{nrad}\) in \(R_y\) (a summary of the specifications is given in Table~\ref{tab:test_id31_experiments_specifications}). Results obtained for all experiments are summarized and compared to the specifications in Section~\ref{ssec:test_id31_scans_conclusion}. @@ -12549,8 +12549,8 @@ Results obtained for all experiments are summarized and compared to the specific \toprule & \(D_y\) & \(D_z\) & \(R_y\)\\ \midrule -peak 2 peak & 200nm & 100nm & \(1.7\,\mu\text{rad}\)\\ -RMS & 30nm & 15nm & \(250\,\text{nrad}\)\\ +peak 2 peak & \(200\,\text{nm}\) & \(100\,\text{nm}\) & \(1.7\,\mu\text{rad}\)\\ +RMS & \(30\,\text{nm}\) & \(15\,\text{nm}\) & \(250\,\text{nrad}\)\\ \bottomrule \end{tabularx} \end{table} @@ -12580,7 +12580,7 @@ While this approach likely underestimates actual open-loop errors, as perfect al \end{center} \subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}Removed eccentricity} \end{subfigure} -\caption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff}Tomography experiment with a rotation velocity of \(6\,\text{deg/s}\), and payload mass of 26kg. Errors in the \((x,y)\) plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}).} +\caption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff}Tomography experiment with a rotation velocity of \(6\,\text{deg/s}\), and payload mass of \(26\,\text{kg}\). Errors in the \((x,y)\) plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}).} \end{figure} The residual motion (i.e. after compensating for eccentricity) in the \(Y-Z\) is compared against the minimum beam size, as illustrated in Figure~\ref{fig:test_id31_tomo_Wz36_results}. @@ -12650,7 +12650,7 @@ This experiment also illustrates that when needed, performance can be enhanced b \label{ssec:test_id31_scans_reflectivity} X-ray reflectivity measurements involve scanning thin structures, particularly solid/liquid interfaces, through the beam by varying the \(R_y\) angle. -In this experiment, a \(R_y\) scan was executed at a rotational velocity of \(100\,\mu rad/s\), and the closed-loop positioning errors were monitored (Figure~\ref{fig:test_id31_reflectivity}). +In this experiment, a \(R_y\) scan was executed at a rotational velocity of \(100\,\mu \text{rad/s}\), and the closed-loop positioning errors were monitored (Figure~\ref{fig:test_id31_reflectivity}). The results confirmed that the NASS successfully maintained the \acrshort{poi} within the specified beam parameters throughout the scanning process. \begin{figure}[htbp] @@ -12681,12 +12681,12 @@ These vertical scans can be executed either continuously or in a step-by-step ma \paragraph{Step by Step \(D_z\) motion} The vertical step motion was performed exclusively with the nano-hexapod. -Testing was conducted across step sizes ranging from \(10\,nm\) to \(1\,\mu m\). +Testing was conducted across step sizes ranging from \(10\,\text{nm}\) to \(1\,\mu\text{m}\). Results are presented in Figure~\ref{fig:test_id31_dz_mim_steps}. -The system successfully resolved 10nm steps (red curve in Figure~\ref{fig:test_id31_dz_mim_10nm_steps}) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements. +The system successfully resolved \(10\,\text{nm}\) steps (red curve in Figure~\ref{fig:test_id31_dz_mim_10nm_steps}) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements. In step-by-step scanning procedures, the settling time is a critical parameter as it significantly affects the total experiment duration. -The system achieved a response time of approximately \(70\,ms\) to reach the target position (within \(\pm 20\,nm\)), as demonstrated by the \(1\,\mu m\) step response in Figure~\ref{fig:test_id31_dz_mim_1000nm_steps}. +The system achieved a response time of approximately \(70\,\text{ms}\) to reach the target position (within \(\pm 20\,\text{nm}\)), as demonstrated by the \(1\,\mu\text{m}\) step response in Figure~\ref{fig:test_id31_dz_mim_1000nm_steps}. The settling duration typically decreases for smaller step sizes. \begin{figure}[htbp] @@ -12694,28 +12694,28 @@ The settling duration typically decreases for smaller step sizes. \begin{center} \includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_mim_10nm_steps.png} \end{center} -\subcaption{\label{fig:test_id31_dz_mim_10nm_steps}10nm steps} +\subcaption{\label{fig:test_id31_dz_mim_10nm_steps}$10\,\text{nm}$ steps} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_mim_100nm_steps.png} \end{center} -\subcaption{\label{fig:test_id31_dz_mim_100nm_steps}100nm steps} +\subcaption{\label{fig:test_id31_dz_mim_100nm_steps}$100\,\text{nm}$ steps} \end{subfigure} \begin{subfigure}{0.33\textwidth} \begin{center} \includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_mim_1000nm_steps.png} \end{center} -\subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu$m step} +\subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu\text{m}$ step} \end{subfigure} -\caption{\label{fig:test_id31_dz_mim_steps}Vertical steps performed with the nano-hexapod. 10nm steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of \(50\,ms\). 100nm steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of \(\pm 20\,nm\) is \(\approx 70\,ms\) as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a \(1\,\mu m\) step.} +\caption{\label{fig:test_id31_dz_mim_steps}Vertical steps performed with the nano-hexapod. \(10\,\text{nm}\) steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of \(50\,\text{ms}\). \(100\,\text{nm}\) steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of \(\pm 20\,\text{nm}\) is \(\approx 70\,\text{ms}\) as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a \(1\,\mu\text{m}\) step.} \end{figure} \paragraph{Continuous \(D_z\) motion: Dirty Layer Scans} For these and subsequent experiments, the NASS performs ``ramp scans'' (constant velocity scans). To eliminate tracking errors, the feedback controller incorporates two integrators, compensating for the plant's lack of integral action at low frequencies. -Initial testing at \(10\,\mu m/s\) demonstrated positioning errors well within specifications (indicated by dashed lines in Figure~\ref{fig:test_id31_dz_scan_10ums}). +Initial testing at \(10\,\mu\text{m/s}\) demonstrated positioning errors well within specifications (indicated by dashed lines in Figure~\ref{fig:test_id31_dz_scan_10ums}). \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -12736,10 +12736,10 @@ Initial testing at \(10\,\mu m/s\) demonstrated positioning errors well within s \end{center} \subcaption{\label{fig:test_id31_dz_scan_10ums_ry}$R_y$} \end{subfigure} -\caption{\label{fig:test_id31_dz_scan_10ums}\(D_z\) scan at a velocity of \(10\,\mu m/s\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry})} +\caption{\label{fig:test_id31_dz_scan_10ums}\(D_z\) scan at a velocity of \(10\,\mu \text{m/s}\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry})} \end{figure} -A subsequent scan at \(100\,\mu m/s\) - the maximum velocity for high-precision \(D_z\) scans\footnote{Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and 100nm ``resolution'' (equal to the vertical beam size).} - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure~\ref{fig:test_id31_dz_scan_100ums}). +A subsequent scan at \(100\,\mu\text{m/s}\) - the maximum velocity for high-precision \(D_z\) scans\footnote{Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and \(100\,\text{nm}\) ``resolution'' (equal to the vertical beam size).} - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure~\ref{fig:test_id31_dz_scan_100ums}). Since detectors typically operate only during the constant velocity phase, these transient deviations do not compromise the measurement quality. However, performance during acceleration phases could be enhanced through the implementation of feedforward control. @@ -12762,21 +12762,21 @@ However, performance during acceleration phases could be enhanced through the im \end{center} \subcaption{\label{fig:test_id31_dz_scan_100ums_ry}$R_y$} \end{subfigure} -\caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan at a velocity of \(100\,\mu m/s\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry})} +\caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan at a velocity of \(100\,\mu\text{m/s}\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry})} \end{figure} \subsubsection{Lateral Scans} \label{ssec:test_id31_scans_dy} Lateral scans are executed using the \(T_y\) stage. The stepper motor controller\footnote{The ``IcePAP''~\cite{janvier13_icepap} which is developed at the ESRF.} generates a setpoint that is transmitted to the Speedgoat. Within the Speedgoat, the system computes the positioning error by comparing the measured \(D_y\) sample position against the received setpoint, and the Nano-Hexapod compensates for positioning errors introduced during \(T_y\) stage scanning. -The scanning range is constrained \(\pm 100\,\mu m\) due to the limited acceptance of the metrology system. +The scanning range is constrained \(\pm 100\,\mu\text{m}\) due to the limited acceptance of the metrology system. \paragraph{Slow scan} -Initial testing were made with a scanning velocity of \(10\,\mu m/s\), which is typical for these experiments. +Initial testing were made with a scanning velocity of \(10\,\mu\text{m/s}\), which is typical for these experiments. Figure~\ref{fig:test_id31_dy_10ums} compares the positioning errors between open-loop (without NASS) and closed-loop operation. In the scanning direction, open-loop measurements reveal periodic errors (Figure~\ref{fig:test_id31_dy_10ums_dy}) attributable to the \(T_y\) stage's stepper motor. These micro-stepping errors, which are inherent to stepper motor operation, occur 200 times per motor rotation with approximately \(1\,\text{mrad}\) angular error amplitude. -Given the \(T_y\) stage's lead screw pitch of \(2\,mm\), these errors manifest as \(10\,\mu m\) periodic oscillations with \(\approx 300\,nm\) amplitude, which can indeed be seen in the open-loop measurements (Figure~\ref{fig:test_id31_dy_10ums_dy}). +Given the \(T_y\) stage's lead screw pitch of \(2\,\text{mm}\), these errors manifest as \(10\,\mu\text{m}\) periodic oscillations with \(\approx 300\,\text{nm}\) amplitude, which can indeed be seen in the open-loop measurements (Figure~\ref{fig:test_id31_dy_10ums_dy}). In the vertical direction (Figure~\ref{fig:test_id31_dy_10ums_dz}), open-loop errors likely stem from metrology measurement error because the top interferometer points at a spherical target surface (see Figure~\ref{fig:test_id31_xy_map_sphere}). Under closed-loop control, positioning errors remain within specifications in all directions. @@ -12800,11 +12800,11 @@ Under closed-loop control, positioning errors remain within specifications in al \end{center} \subcaption{\label{fig:test_id31_dy_10ums_ry} $R_y$} \end{subfigure} -\caption{\label{fig:test_id31_dy_10ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(10\,\mu m/s\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_10ums_dy}).} +\caption{\label{fig:test_id31_dy_10ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(10\,\mu\text{m/s}\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_10ums_dy}).} \end{figure} \paragraph{Fast Scan} -The system performance was evaluated at an increased scanning velocity of \(100\,\mu m/s\), and the results are presented in Figure~\ref{fig:test_id31_dy_100ums}. +The system performance was evaluated at an increased scanning velocity of \(100\,\mu\text{m/s}\), and the results are presented in Figure~\ref{fig:test_id31_dy_100ums}. At this velocity, the micro-stepping errors generate \(10\,\text{Hz}\) vibrations, which are further amplified by micro-station resonances. These vibrations exceeded the NASS feedback controller bandwidth, resulting in limited attenuation under closed-loop control. This limitation exemplifies why stepper motors are suboptimal for ``long-stroke/short-stroke'' systems requiring precise scanning performance~\cite{dehaeze22_fastj_uhv}. @@ -12833,15 +12833,15 @@ For applications requiring small \(D_y\) scans, the nano-hexapod can be used exc \end{center} \subcaption{\label{fig:test_id31_dy_100ums_ry} $R_y$} \end{subfigure} -\caption{\label{fig:test_id31_dy_100ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(100\,\mu m/s\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_100ums_dy}).} +\caption{\label{fig:test_id31_dy_100ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(100\,\mu\text{m/s}\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_100ums_dy}).} \end{figure} \subsubsection{Diffraction Tomography} \label{ssec:test_id31_scans_diffraction_tomo} In diffraction tomography experiments, the micro-station performs combined motions: continuous rotation around the \(R_z\) axis while performing lateral scans along \(D_y\). For this validation, the spindle maintained a constant rotational velocity of \(6\,\text{deg/s}\) while the nano-hexapod performs the lateral scanning motion. -To avoid high-frequency vibrations typically induced by the stepper motor, the \(T_y\) stage was not used, which constrained the scanning range to approximately \(\pm 100\,\mu m/s\). -The system performance was evaluated at three lateral scanning velocities: \(0.1\,mm/s\), \(0.5\,mm/s\), and \(1\,mm/s\). Figure~\ref{fig:test_id31_diffraction_tomo_setpoint} presents both the \(D_y\) position setpoints and the corresponding measured \(D_y\) positions for all tested velocities. +To avoid high-frequency vibrations typically induced by the stepper motor, the \(T_y\) stage was not used, which constrained the scanning range to approximately \(\pm 100\,\mu\text{m/s}\). +The system performance was evaluated at three lateral scanning velocities: \(0.1\,\text{mm/s}\), \(0.5\,\text{mm/s}\), and \(1\,\text{mm/s}\). Figure~\ref{fig:test_id31_diffraction_tomo_setpoint} presents both the \(D_y\) position setpoints and the corresponding measured \(D_y\) positions for all tested velocities. \begin{figure}[htbp] \centering @@ -12852,7 +12852,7 @@ The system performance was evaluated at three lateral scanning velocities: \(0.1 The positioning errors measured along \(D_y\), \(D_z\), and \(R_y\) directions are displayed in Figure~\ref{fig:test_id31_diffraction_tomo}. The system maintained positioning errors within specifications for both \(D_z\) and \(R_y\) (Figures~\ref{fig:test_id31_diffraction_tomo_dz} and \ref{fig:test_id31_diffraction_tomo_ry}). However, the lateral positioning errors exceeded specifications during the acceleration and deceleration phases (Figure~\ref{fig:test_id31_diffraction_tomo_dy}). -These large errors occurred only during \(\approx 20\,ms\) intervals; thus, a delay of \(20\,ms\) could be implemented in the detector the avoid integrating the beam when these large errors are occurring. +These large errors occurred only during \(\approx 20\,\text{ms}\) intervals; thus, a delay of \(20\,\text{ms}\) could be implemented in the detector the avoid integrating the beam when these large errors are occurring. Alternatively, a feedforward controller could improve the lateral positioning accuracy during these transient phases. \begin{figure}[htbp] @@ -12947,15 +12947,15 @@ For higher values of \(\omega_0\), the system became unstable in the vertical di \label{ssec:test_id31_scans_conclusion} A comprehensive series of experimental validations was conducted to evaluate the NASS performance over a wide range of typical scientific experiments. -The system demonstrated robust performance in most scenarios, with positioning errors generally remaining within specified tolerances (30 nm RMS in \(D_y\), 15 nm RMS in \(D_z\), and 250 nrad RMS in \(R_y\)). +The system demonstrated robust performance in most scenarios, with positioning errors generally remaining within specified tolerances (\(30\,\text{nm RMS}\) in \(D_y\), \(15\,\text{nm RMS}\) in \(D_z\), and \(250\,\text{nrad RMS}\) in \(R_y\)). For tomography experiments, the NASS successfully maintained good positioning accuracy at rotational velocities up to \(180\,\text{deg/s}\) with light payloads, though performance degraded somewhat with heavier masses. The \acrshort{haclac} control architecture proved particularly effective, with the decentralized IFF providing damping of nano-hexapod suspension modes, while the high authority controller addressed low-frequency disturbances. The vertical scanning capabilities were validated in both step-by-step and continuous motion modes. -The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to \(100\,\mu m/s\). +The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to \(100\,\mu\text{m/s}\). -For lateral scanning, the system performed well at moderate speeds (\(10\,\mu m/s\)) but showed limitations at higher velocities (\(100\,\mu m/s\)) due to stepper motor-induced vibrations in the \(T_y\) stage. +For lateral scanning, the system performed well at moderate speeds (\(10\,\mu\text{m/s}\)) but showed limitations at higher velocities (\(100\,\mu\text{m/s}\)) due to stepper motor-induced vibrations in the \(T_y\) stage. The most challenging test case - diffraction tomography combining rotation and lateral scanning - demonstrated the system's ability to maintain vertical and angular stability while highlighting some limitations in lateral positioning during rapid accelerations. These limitations could be addressed through feedforward control or alternative detector triggering strategies. @@ -12971,24 +12971,24 @@ The identified limitations, primarily related to high-speed lateral scanning and \textbf{Experiments} & \(\bm{D_y}\) \textbf{{[}nmRMS]} & \(\bm{D_z}\) \textbf{{[}nmRMS]} & \(\bm{R_y}\) \textbf{{[}nradRMS]}\\ \midrule Tomography (\(6\,\text{deg/s}\)) & \(142 \Rightarrow 15\) & \(32 \Rightarrow 5\) & \(464 \Rightarrow 56\)\\ -Tomography (\(6\,\text{deg/s}\), 13kg) & \(149 \Rightarrow 25\) & \(26 \Rightarrow 6\) & \(471 \Rightarrow 55\)\\ -Tomography (\(6\,\text{deg/s}\), 26kg) & \(202 \Rightarrow 25\) & \(36 \Rightarrow 7\) & \(1737 \Rightarrow 104\)\\ -Tomography (\(6\,\text{deg/s}\), 39kg) & \(297 \Rightarrow \bm{53}\) & \(38 \Rightarrow 9\) & \(1737 \Rightarrow 170\)\\ +Tomography (\(6\,\text{deg/s}\), \(13\,\text{kg}\)) & \(149 \Rightarrow 25\) & \(26 \Rightarrow 6\) & \(471 \Rightarrow 55\)\\ +Tomography (\(6\,\text{deg/s}\), \(26\,\text{kg}\)) & \(202 \Rightarrow 25\) & \(36 \Rightarrow 7\) & \(1737 \Rightarrow 104\)\\ +Tomography (\(6\,\text{deg/s}\), \(39\,\text{kg}\)) & \(297 \Rightarrow \bm{53}\) & \(38 \Rightarrow 9\) & \(1737 \Rightarrow 170\)\\ \midrule Tomography (\(180\,\text{deg/s}\)) & \(143 \Rightarrow \bm{38}\) & \(24 \Rightarrow 11\) & \(252 \Rightarrow 130\)\\ Tomography (\(180\,\text{deg/s}\), custom HAC) & \(143 \Rightarrow 29\) & \(24 \Rightarrow 5\) & \(252 \Rightarrow 142\)\\ \midrule Reflectivity (\(100\,\mu\text{rad}/s\)) & \(28\) & \(6\) & \(118\)\\ \midrule -\(D_z\) scan (\(10\,\mu m/s\)) & \(25\) & \(5\) & \(108\)\\ -\(D_z\) scan (\(100\,\mu m/s\)) & \(\bm{35}\) & \(9\) & \(132\)\\ +\(D_z\) scan (\(10\,\mu\text{m/s}\)) & \(25\) & \(5\) & \(108\)\\ +\(D_z\) scan (\(100\,\mu\text{m/s}\)) & \(\bm{35}\) & \(9\) & \(132\)\\ \midrule -Lateral Scan (\(10\,\mu m/s\)) & \(585 \Rightarrow 21\) & \(155 \Rightarrow 10\) & \(6300 \Rightarrow 60\)\\ -Lateral Scan (\(100\,\mu m/s\)) & \(1063 \Rightarrow \bm{732}\) & \(167 \Rightarrow \bm{20}\) & \(6445 \Rightarrow \bm{356}\)\\ +Lateral Scan (\(10\,\mu\text{m/s}\)) & \(585 \Rightarrow 21\) & \(155 \Rightarrow 10\) & \(6300 \Rightarrow 60\)\\ +Lateral Scan (\(100\,\mu\text{m/s}\)) & \(1063 \Rightarrow \bm{732}\) & \(167 \Rightarrow \bm{20}\) & \(6445 \Rightarrow \bm{356}\)\\ \midrule -Diffraction tomography (\(6\,\text{deg/s}\), \(0.1\,mm/s\)) & \(\bm{36}\) & \(7\) & \(113\)\\ -Diffraction tomography (\(6\,\text{deg/s}\), \(0.5\,mm/s\)) & \(29\) & \(8\) & \(81\)\\ -Diffraction tomography (\(6\,\text{deg/s}\), \(1\,mm/s\)) & \(\bm{53}\) & \(10\) & \(135\)\\ +Diffraction tomography (\(6\,\text{deg/s}\), \(0.1\,\text{mm/s}\)) & \(\bm{36}\) & \(7\) & \(113\)\\ +Diffraction tomography (\(6\,\text{deg/s}\), \(0.5\,\text{mm/s}\)) & \(29\) & \(8\) & \(81\)\\ +Diffraction tomography (\(6\,\text{deg/s}\), \(1\,\text{mm/s}\)) & \(\bm{53}\) & \(10\) & \(135\)\\ \midrule \textbf{Specifications} & \(30\) & \(15\) & \(250\)\\ \bottomrule @@ -13004,11 +13004,11 @@ The short-stroke metrology system, while designed as a temporary solution, prove The careful alignment of the fibered interferometers targeting the two reference spheres ensured reliable measurements throughout the testing campaign. The implementation of the control architecture validated the theoretical framework developed earlier in this project. -The decentralized Integral Force Feedback (IFF) controller successfully provided robust damping of suspension modes across all payload conditions (0-39 kg), reducing peak amplitudes by approximately a factor of 10. +The decentralized Integral Force Feedback (IFF) controller successfully provided robust damping of suspension modes across all payload conditions (0 to \(39\,\text{kg}\)), reducing peak amplitudes by approximately a factor of 10. The High Authority Controller (HAC) effectively rejects low-frequency disturbances, although its performance showed some dependency on payload mass, particularly for lateral motion control. The experimental validation covered a wide range of scientific scenarios. -The system demonstrated remarkable performance under most conditions, meeting the stringent positioning requirements (30 nm RMS in \(D_y\), 15 nm RMS in \(D_z\), and 250 nrad RMS in \(R_y\)) for the majority of test cases. +The system demonstrated remarkable performance under most conditions, meeting the stringent positioning requirements (\(30\,\text{nm RMS}\) in \(D_y\), \(15\,\text{nm RMS}\) in \(D_z\), and \(250\,\text{nrad RMS}\) in \(R_y\)) for the majority of test cases. Some limitations were identified, particularly in handling heavy payloads during rapid motions and in managing high-speed lateral scanning with the existing stepper motor \(T_y\) stage. The successful validation of the NASS demonstrates that once an accurate online metrology system is developed, it will be ready for integration into actual beamline operations. @@ -13089,7 +13089,7 @@ Such a capability would enable the rapid generation of accurate dynamic models f \paragraph{Better addressing plant uncertainty from a change of payload} For most high-performance mechatronic systems like lithography machines or atomic force microscopes, payloads inertia are often known and fixed, allowing controllers to be precisely optimized. -However, synchrotron end-stations frequently handle samples with widely varying masses and inertias – ID31 being an extreme example, but many require nanometer positioning for samples from very light masses up to 5kg. +However, synchrotron end-stations frequently handle samples with widely varying masses and inertias – ID31 being an extreme example, but many require nanometer positioning for samples from very light masses up to \(5\,\text{kg}\). The conventional strategy involves implementing controllers with relatively small bandwidth to accommodate various payloads. When controllers are optimized for a specific payload, changing payloads may destabilize the feedback loops that needs to be re-tuned. @@ -13099,7 +13099,7 @@ Therefore, a key objective for future work is to enhance the management of paylo Potential strategies to be explored include adaptive control (involving automatic plant identification and controller tuning after a change of payload) and robust control techniques such as \(\mu\text{-synthesis}\) (allowing the controller to be synthesized while explicitly considering a specified range of payload masses). \paragraph{Control based on Complementary Filters} -The control architecture based on complementary filters (detailed in Section \ref{sec:detail_control_cf}) has been successfully implemented in several instruments at the \acrlong{esrf}. +The control architecture based on complementary filters (detailed in Section~\ref{sec:detail_control_cf}) has been successfully implemented in several instruments at the \acrlong{esrf}. This approach has proven to be straightforward to implement and offers the valuable capability of modifying closed-loop behavior in real-time, which proves advantageous for many applications. For instance, the controller can be optimized according to the scan type: constant velocity scans benefit from a \(+2\) slope for the sensitivity transfer function, while ptychography may be better served by a \(+1\) slope with slightly higher bandwidth to minimize point-to-point transition times. @@ -13107,7 +13107,7 @@ Nevertheless, a more rigorous analysis of this control architecture and its comp \paragraph{Sensor Fusion} While the \acrshort{haclac} approach demonstrated a simple and comprehensive methodology for controlling the NASS, sensor fusion represents an interesting alternative that is worth investigating. -While the synthesis method developed for complementary filters facilitates their design (Section \ref{sec:detail_control_sensor}), their application specifically for sensor fusion within the NASS context was not examined in detail. +While the synthesis method developed for complementary filters facilitates their design (Section~\ref{sec:detail_control_sensor}), their application specifically for sensor fusion within the NASS context was not examined in detail. One potential approach involves fusing external metrology (used at low frequencies) with force sensors (employed at high frequencies). This configuration could enhance robustness through the collocation of force sensors with actuators. @@ -13133,7 +13133,7 @@ Consequently, the underlying micro-station's own positioning accuracy has minima Nevertheless, it remains crucial that the micro-station itself does not generate detrimental high-frequency vibrations, particularly during movements, as evidenced by issues previously encountered with stepper motors. Designing a future end-station with the understanding that a functional NASS will ensure final positioning accuracy could allow for a significantly simplified long-stroke stage architecture, perhaps chosen primarily to facilitate the integration of the online metrology. -One possible configuration, illustrated in Figure \ref{fig:conclusion_nass_architecture}, would comprise a long-stroke Stewart platform providing the required mobility without generating high-frequency vibrations; a spindle that needs not deliver exceptional performance but should be stiff and avoid inducing high-frequency vibrations (an air-bearing spindle might not be essential); and a short-stroke Stewart platform for correcting errors from the long-stroke stage and spindle. +One possible configuration, illustrated in Figure~\ref{fig:conclusion_nass_architecture}, would comprise a long-stroke Stewart platform providing the required mobility without generating high-frequency vibrations; a spindle that needs not deliver exceptional performance but should be stiff and avoid inducing high-frequency vibrations (an air-bearing spindle might not be essential); and a short-stroke Stewart platform for correcting errors from the long-stroke stage and spindle. \begin{figure}[htbp] \centering @@ -13145,11 +13145,11 @@ With this architecture, the online metrology could be divided into two systems, \paragraph{Development of long stroke high performance stage} As an alternative to the short-stroke/long-stroke architecture, the development of a high-performance long-stroke stage seems worth investigating. -Stages based on voice coils, offering nano-positioning capabilities with \(3\,mm\) stroke, have recently been reported in the literature~\cite{schropp20_ptynam,kelly22_delta_robot_long_travel_nano}. +Stages based on voice coils, offering nano-positioning capabilities with \(3\,\text{mm}\) stroke, have recently been reported in the literature~\cite{schropp20_ptynam,kelly22_delta_robot_long_travel_nano}. Magnetic levitation also emerges as a particularly interesting technology to be explored, especially for microscopy~\cite{fahmy22_magnet_xy_theta_x,heyman23_levcub} and tomography~\cite{dyck15_magnet_levit_six_degree_freed_rotar_table,fahmy22_magnet_xy_theta_x} end-stations. -Two notable designs illustrating these capabilities are shown in Figure \ref{fig:conclusion_maglev}. -Specifically, a compact 6DoF stage known as LevCube, providing a mobility of approximately \(1\,\text{cm}^3\), is depicted in Figure \ref{fig:conclusion_maglev_heyman23}, while a 6DoF stage featuring infinite rotation, is shown in Figure \ref{fig:conclusion_maglev_dyck15}. +Two notable designs illustrating these capabilities are shown in Figure~\ref{fig:conclusion_maglev}. +Specifically, a compact 6DoF stage known as LevCube, providing a mobility of approximately \(1\,\text{cm}^3\), is depicted in Figure~\ref{fig:conclusion_maglev_heyman23}, while a 6DoF stage featuring infinite rotation, is shown in Figure~\ref{fig:conclusion_maglev_dyck15}. However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature. \begin{figure}[htbp] diff --git a/setup.org b/setup.org index 2d05752..fed9e38 100644 --- a/setup.org +++ b/setup.org @@ -20,6 +20,7 @@ \DeclareSIUnit\px{px} \DeclareSIUnit\rms{rms} +\DeclareSIUnit\rad{rad} #+end_src ** Mathematics @@ -56,6 +57,10 @@ Use these with the proper bracket in order to ensure that they scale automatical \setabbreviationstyle[acronym]{long-short} \setglossarystyle{long-name-desc} + +% https://tex.stackexchange.com/questions/318694/glossaries-acronyms-avoid-additional-entries-in-number-list-with-indexonlyfirs +% Used to avoid many warnings +\renewcommand*{\acrshort}[1][]{\glsxtrshort[noindex,#1]} #+end_src * Config Extra