From 76b1f435ce8016a1a31ce3f2951fc9c317c58560 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 22 Apr 2025 20:58:33 +0200 Subject: [PATCH] Rework acronyms --- phd-thesis.bib | 33 +- phd-thesis.org | 1037 ++++++++++++++++++++++++------------------------ phd-thesis.tex | 1027 ++++++++++++++++++++++++----------------------- 3 files changed, 1070 insertions(+), 1027 deletions(-) diff --git a/phd-thesis.bib b/phd-thesis.bib index 705f685..bd46ecd 100644 --- a/phd-thesis.bib +++ b/phd-thesis.bib @@ -14,22 +14,6 @@ keywords = {esrf}, } -@article{raimondi21_commis_hybrid_multib_achrom_lattic, - author = {Raimondi, P. and Carmignani, N. and Carver, L. R. and - Chavanne, J. and Farvacque, L. and Le Bec, G. and Martin, D. - and Liuzzo, S. M. and Perron, T. and White, S.}, - title = {Commissioning of the Hybrid Multibend Achromat Lattice At - the European Synchrotron Radiation Facility}, - journal = {Physical Review Accelerators and Beams}, - volume = 24, - number = 11, - pages = 110701, - year = 2021, - doi = {10.1103/physrevaccelbeams.24.110701}, - url = {http://dx.doi.org/10.1103/physrevaccelbeams.24.110701}, - keywords = {esrf}, -} - @article{schoeppler17_shapin_highl_regul_glass_archit, author = {Schoeppler, V. and Reich, E. and Vacelet, J. and Rosenthal, M. and Pacureanu, A. and Rack, A. @@ -1655,9 +1639,9 @@ author = {Abbas, H. and Hai, H.}, title = {Vibration isolation concepts for non-cubic Stewart Platform using modal control}, - booktitle = {Proceedings of 2014 11th International Bhurban Conference + booktitle = {Proceedings of 11th International Bhurban Conference on Applied Sciences \& Technology (IBCAST) Islamabad, - Pakistan, 14th - 18th January, 2014}, + Pakistan}, year = 2014, doi = {10.1109/ibcast.2014.6778139}, url = {https://doi.org/10.1109/ibcast.2014.6778139}, @@ -1766,7 +1750,7 @@ @techreport{bibel92_guidel_h, author = {Bibel, J. E. and Malyevac, D. S.}, - institution = {NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA}, + institution = {Naval Surface Warfare Center Dahlgren div va}, title = {Guidelines for the selection of weighting functions for H-infinity control}, year = 1992, @@ -1950,3 +1934,14 @@ {http://jacow.org/icalepcs2017/doi/JACoW-ICALEPCS2017-THPHA072.html}, } +@misc{dehaeze25_nano_activ_stabil_zenodo, + author = {Dehaeze, T.}, + doi = {10.5281/zenodo.15254389}, + month = 5, + publisher = {Zenodo}, + title = {Nano Active Stabilization of samples for tomography + experiments: A mechatronic design approach (data, scripts and + models)}, + url = {https://doi.org/10.5281/zenodo.15254389}, + year = 2025, +} diff --git a/phd-thesis.org b/phd-thesis.org index 4eea1bb..08944fa 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -108,26 +108,53 @@ #+name: acronyms | key | abbreviation | full form | |--------+--------------+------------------------------------------------| -| haclac | HAC-LAC | High Authority Control - Low Authority Control | +| adc | ADC | Analog to Digital Converter | +| apa | APA | Amplified Piezoelectric Actuator | +| asd | ASD | Amplitude Spectrum Density | +| cas | CAS | Cumulative Amplitude Spectrum | +| cl | CL | Closed Loop | +| cmif | CMIF | Complex Modal Indication Function | +| cok | CoK | Center of Stiffness | +| com | CoM | Center of Mass | +| dac | DAC | Digital to Analog Converter | +| dvf | DVF | Direct Velocity Feedback | +| dof | DoF | Degrees of Freedom | +| ebs | EBS | Extremely Brilliant Source | +| edm | EDM | Electrical Discharge Machining | +| esrf | ESRF | European Synchrotron Radiation Facility | +| fea | FEA | Finite Element Analysis | +| fem | FEM | Finite Element Model | +| frf | FRF | Frequency Response Function | | hac | HAC | High Authority Control | | lac | LAC | Low Authority Control | -| nass | NASS | Nano Active Stabilization System | -| asd | ASD | Amplitude Spectral Density | -| psd | PSD | Power Spectral Density | -| cps | CPS | Cumulative Power Spectrum | -| cas | CAS | Cumulative Amplitude Spectrum | -| frf | FRF | Frequency Response Function | +| haclac | HAC-LAC | High Authority Control / Low Authority Control | +| hpf | HPF | High Pass Filter | | iff | IFF | Integral Force Feedback | +| lpf | LPF | Low Pass Filter | +| lqg | LQG | Linear Quadratic Gaussian | +| lsb | LSB | Least Significant Bit | +| lti | LTI | Linear Time Invariant | +| lvdt | LVDT | Linear Variable Differential Transformer | +| mif | MIF | Modal Indication Function | +| mimo | MIMO | Multi Inputs Multi Outputs | +| nass | NASS | Nano Active Stabilization System | +| np | NP | Nominal Performance | +| ns | NS | Nominal Stability | +| ol | OL | Open Loop | +| pi | PI | Proportional Integral | +| pid | PID | Proportional Integral Derivative | +| psd | PSD | Power Spectral Density | +| pzt | PZT | Lead Zirconate Titanate | +| poi | PoI | Point of Interest | | rdc | RDC | Relative Damping Control | | rga | RGA | Relative Gain Array | -| hpf | HPF | high-pass filter | -| lpf | LPF | low-pass filter | -| dof | DoF | Degree of freedom | +| rms | RMS | Root Mean Square | +| rp | RP | Robust Performance | +| rs | RS | Robust Stability | +| siso | SISO | Single Input Single Output | +| sps | SPS | Samples per Second | | svd | SVD | Singular Value Decomposition | -| mif | MIF | Mode Indicator Functions | -| dac | DAC | Digital to Analog Converter | -| fem | FEM | Finite Element Model | -| apa | APA | Amplified Piezoelectric Actuator | +| vc | VC | Voice Coil | * Title Page :ignore: @@ -272,7 +299,7 @@ The fundamental objective has been to ensure that anyone should be capable of re To achieve this goal of reproducibility, comprehensive sharing of all elements has been implemented. This includes the mathematical models developed, raw experimental data collected, and scripts used to generate the figures. -For those wishing to engage with the reproducible aspects of this work, all data and code are freely accessible [[cite:&dehaeze25_nano_activ_stabil_zenodo]]. +For those wishing to engage with the reproducible aspects of this work, all data and code are freely accessible\nbsp{}[[cite:&dehaeze25_nano_activ_stabil_zenodo]]. The organization of the code mirrors that of the manuscript, with corresponding chapters and sections. All materials have been made available under the MIT License, permitting free reuse. @@ -324,8 +351,8 @@ This global distribution of such facilities underscores the significant utility These facilities fundamentally comprise two main parts: the accelerator and storage ring, where electron acceleration and light generation occur, and the beamlines, where the intense X-ray beams are conditioned and directed for experimental use. -The European Synchrotron Radiation Facility (ESRF), shown in Figure\nbsp{}ref:fig:introduction_esrf_picture, is a joint research institution supported by 19 member countries. -The ESRF started user operations in 1994 as the world's first third-generation synchrotron. +The acrfull:esrf, shown in Figure\nbsp{}ref:fig:introduction_esrf_picture, is a joint research institution supported by 19 member countries. +The acrshort:esrf started user operations in 1994 as the world's first third-generation synchrotron. Its accelerator complex, schematically depicted in Figure\nbsp{}ref:fig:introduction_esrf_schematic, includes a linear accelerator where electrons are initially generated and accelerated, a booster synchrotron to further accelerate the electrons, and an 844-meter circumference storage ring where electrons are maintained in a stable orbit. Synchrotron light are emitted in more than 40 beamlines surrounding the storage ring, each having specialized experimental stations. @@ -349,11 +376,11 @@ These beamlines host diverse instrumentation that enables a wide spectrum of sci #+end_subfigure #+end_figure -In August 2020, following an extensive 20-month upgrade period, the ESRF inaugurated its Extremely Brilliant Source (EBS), establishing it as the world's premier fourth-generation synchrotron\nbsp{}[[cite:&raimondi21_commis_hybrid_multib_achrom_lattic]]. +In August 2020, following an extensive 20-month upgrade period, the acrshort:esrf inaugurated its acrfull:ebs, establishing it as the world's premier fourth-generation synchrotron\nbsp{}[[cite:&raimondi21_commis_hybrid_multib_achrom_lattic]]. This upgrade implemented a novel storage ring concept that substantially increases the brilliance and coherence of the X-ray beams. Brilliance, a measure of the photon flux, is a key figure of merit for synchrotron facilities. -It experienced an approximate 100-fold increase with the implementation of EBS, as shown in the historical evolution depicted in Figure\nbsp{}ref:fig:introduction_moore_law_brillance. +It experienced an approximate 100-fold increase with the implementation of acrshort:ebs, as shown in the historical evolution depicted in Figure\nbsp{}ref:fig:introduction_moore_law_brillance. 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 @@ -397,7 +424,7 @@ These components are housed in multiple Optical Hutches, as depicted in Figure\n Following the optical hutches, the conditioned beam enters the Experimental Hutch (Figure\nbsp{}ref:fig:introduction_id31_cad), where, for experiments pertinent to this work, focusing optics are used. The sample is mounted on a positioning stage, referred to as the "end-station", that enables precise alignment relative to the X-ray beam. Detectors are used to capture the X-rays transmitted through or scattered by the sample. -Throughout this thesis, the standard 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. +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). @@ -405,7 +432,7 @@ The sample itself (cyan), potentially housed within complex sample environments 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. #+name: fig:introduction_micro_station -#+caption: CAD 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. CAD view of the micro-station with associated degrees of freedom (\subref{fig:introduction_micro_station_dof}). +#+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}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_id31_cad} Experimental Hutch} @@ -427,7 +454,7 @@ Two illustrative examples are provided. Tomography experiments, schematically represented in Figure\nbsp{}ref:fig:introduction_tomography_schematic, involve placing a sample in the X-ray beam path while controlling its vertical rotation angle using a dedicated stage. Detector images are captured at numerous rotation angles, allowing the reconstruction of three-dimensional sample structure (Figure\nbsp{}ref:fig:introduction_tomography_results)\nbsp{}[[cite:&schoeppler17_shapin_highl_regul_glass_archit]]. -This reconstruction depends critically on maintaining the sample's point of interest within the beam throughout the rotation process. +This reconstruction depends critically on maintaining the sample's acrfull:poi within the beam throughout the rotation process. 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. @@ -498,7 +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 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 $10\,\mu m$ [[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. @@ -556,7 +583,7 @@ To contextualize the system developed within this thesis, a brief overview of ex The aim is to identify the specific characteristics that distinguish the proposed system from current state-of-the-art implementations. Positioning systems can be broadly categorized based on their kinematic architecture, typically serial or parallel, as exemplified by the 3-Degree-of-Freedom (DoF) platforms in Figure\nbsp{}ref:fig:introduction_kinematics. -Serial kinematics (Figure\nbsp{}ref:fig:introduction_serial_kinematics) is composed of stacked stages where each degree of freedom is controlled by a dedicated actuator. +Serial kinematics (Figure\nbsp{}ref:fig:introduction_serial_kinematics) is composed of stacked stages where each acrshort:dof is controlled by a dedicated actuator. This configuration offers great mobility, but positioning errors (e.g., guiding inaccuracies, thermal expansion) accumulate through the stack, compromising overall accuracy. Similarly, the overall dynamic performance (stiffness, resonant frequencies) is limited by the softest component in the stack, often resulting in poor dynamic behavior when many stages are combined. @@ -579,7 +606,7 @@ Similarly, the overall dynamic performance (stiffness, resonant frequencies) is #+end_figure Conversely, parallel kinematic architectures (Figure\nbsp{}ref:fig:introduction_parallel_kinematics) involve the coordinated motion of multiple actuators to achieve the desired end-effector motion. -While theoretically offering the same controlled degrees of freedom as stacked stages, parallel systems generally provide limited stroke but significantly enhanced stiffness and superior dynamic performance. +While theoretically offering the same controlled acrshortpl:dof as stacked stages, parallel systems generally provide limited stroke but significantly enhanced stiffness and superior dynamic performance. Most end stations, particularly those requiring extensive mobility, employ stacked stages. Their positioning performance consequently depends entirely on the accuracy of individual components. @@ -606,7 +633,7 @@ However, when a large number of DoFs are required, the cumulative errors and lim #+end_figure The concept of using an external metrology to measure and potentially correct for positioning errors is increasing used for nano-positioning end-stations. -Ideally, the relative position between the sample's point of interest and the X-ray beam focus would be measured directly. +Ideally, the relative position between the sample's acrfull:poi and the X-ray beam focus would be measured directly. In practice, direct measurement is often impossible; instead, the sample position is typically measured relative to a reference frame associated with the focusing optics, providing an indirect measurement. This measured position can be used in several ways: for post-processing correction of acquired data; for calibration routines to compensate for repeatable errors; or, most relevantly here, for real-time feedback control. @@ -635,12 +662,12 @@ The PtiNAMi microscope at DESY P06 (Figure\nbsp{}ref:fig:introduction_stages_sch #+end_figure 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 voice coil actuators\nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]]. +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]]. Figure\nbsp{}ref:fig:introduction_active_stations showcases two end-stations incorporating online metrology and active feedback control. -The ID16A system at 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 hexapod 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. @@ -668,8 +695,8 @@ 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. -End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few degrees of freedom with strokes around $100\,\mu m$. -Recently, voice coil actuators were used to increase the stroke up to $3\,\text{mm}$ [[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]] +End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few acrshortpl:dof with strokes around $100\,\mu m$. +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. The short-stroke stage actively compensates for errors based on metrology feedback, while the long-stroke stage performs the larger movements. @@ -698,8 +725,8 @@ This approach allows combining extended travel with high precision and good dyna The advent of fourth-generation light sources, coupled with advancements in focusing optics and detector technology, imposes stringent new requirements on sample positioning systems. -With ID31's anticipated minimum beam dimensions of approximately $200\,\text{nm}\times 100\,\text{nm}$, the primary experimental objective is maintaining the sample's point of interest within this beam. -This necessitates peak-to-peak positioning errors below $200\,\text{nm}$ in $D_y$ and $200\,\text{nm}$ in $D_z$, corresponding to RMS errors of $30\,\text{nm}$ and $15\,\text{nm}$, respectively. +With ID31's anticipated minimum beam dimensions of approximately $200\,\text{nm}\times 100\,\text{nm}$, the primary experimental objective is maintaining the sample's acrshort:poi within this beam. +This necessitates peak-to-peak positioning errors below $200\,\text{nm}$ in $D_y$ and $200\,\text{nm}$ in $D_z$, corresponding to acrfull:rms errors of $30\,\text{nm}$ and $15\,\text{nm}$, respectively. 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. @@ -711,8 +738,8 @@ The primary objective of this project is therefore defined as enhancing the posi ***** The Nano Active Stabilization System Concept -To address these challenges, the concept of a Nano Active Stabilization System (NASS) is proposed. -As schematically illustrated in Figure\nbsp{}ref:fig:introduction_nass_concept_schematic, the NASS comprises four principal components integrated with the existing micro-station (yellow): a 5-DoF online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple). +To address these challenges, the concept of a acrfull:nass is proposed. +As schematically illustrated in Figure\nbsp{}ref:fig:introduction_nass_concept_schematic, the acrshort:nass comprises four principal components integrated with the existing micro-station (yellow): a 5-DoF online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple). This system essentially functions as a high-performance, multi-axis vibration isolation and error correction platform situated between the micro-station and the sample. It actively compensates for positioning errors measured by the external metrology system. @@ -723,7 +750,7 @@ It actively compensates for positioning errors measured by the external metrolog ***** Online Metrology system -The performance of the NASS is fundamentally reliant on the accuracy and bandwidth of its online metrology system, as the active control is based directly on these measurements. +The performance of the acrshort:nass is fundamentally reliant on the accuracy and bandwidth of its online metrology system, as the active control is based directly on these measurements. This metrology system must fulfill several criteria: measure the sample position in 5 DoF (excluding rotation about the vertical Z-axis); possess a measurement range compatible with the micro-station's extensive mobility and continuous spindle rotation; achieve an accuracy compatible with the sub-100 nm positioning target; and offer high bandwidth for real-time control. #+name: fig:introduction_nass_metrology @@ -740,7 +767,7 @@ For the work presented herein, the metrology system is assumed to provide accura ***** Active Stabilization Platform Design The active stabilization platform, positioned between the micro-station top plate and the sample, must satisfy several demanding requirements. -It needs to provide active motion compensation in 5 degrees of freedom ($D_x$, $D_y$, $D_z$, $R_x$ and $R_y$). +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. @@ -772,7 +799,7 @@ A more detailed review of Stewart platform and its main components will be given ***** Robust Control The control system must compute the position measurements from the online metrology system and computes the reference positions derived from each micro-station desired movement. -It then commands the active platform in real time to stabilize the sample and compensate for all error sources, including stage imperfections, thermal drifts, and vibrations. +It then commands the active platform in real-time to stabilize the sample and compensate for all error sources, including stage imperfections, thermal drifts, and vibrations. Ensuring the stability and robustness of these feedback loops is crucial, especially within the demanding operational context of a synchrotron beamline, which requires reliable 24/7 operation with minimal intervention. Several factors complicate the design of robust feedback control for the NASS. @@ -784,7 +811,7 @@ Designing for robustness against large payload variations typically necessitates Consequently, high-performance positioning stages often work with well-characterized payload, as seen in systems like wafer-scanners or atomic force microscopes. Furthermore, unlike many systems where the active stage and sample are significantly lighter than the underlying coarse stages, the NASS payload mass can be substantially greater than the mass of the micro-station's top stage. -This leads to strong dynamic coupling between the active platform and the micro-station structure, resulting in a more complex multi-inputs multi-outputs (MIMO) system with significant cross-talk between axes. +This leads to strong dynamic coupling between the active platform and the micro-station structure, resulting in a more complex acrfull:mimo system with significant cross-talk between axes. These variations in operating conditions and payload translate into significant uncertainty or changes in the plant dynamics that the controller must handle. Therefore, the feedback controller must be designed to be robust against this plant uncertainty while still delivering the required nanometer-level performance. @@ -816,7 +843,7 @@ While the resulting system is highly specific, the documented effectiveness of t ***** Experimental validation of multi-body simulations with reduced order flexible bodies obtained by FEA -A key tool employed extensively in this work was a combined multi-body simulation and Finite Element Analysis technique, specifically using Component Mode Synthesis to represent flexible bodies within the multi-body framework\nbsp{}[[cite:&brumund21_multib_simul_reduc_order_flexib_bodies_fea]]. +A key tool employed extensively in this work was a combined multi-body simulation and acrfull:fea technique, specifically using Component Mode Synthesis to represent flexible bodies within the multi-body framework\nbsp{}[[cite:&brumund21_multib_simul_reduc_order_flexib_bodies_fea]]. This hybrid approach, while established, was experimentally validated in this work for components critical to the NASS, namely amplified piezoelectric actuators and flexible joints. It proved invaluable for designing and optimizing components intended for integration into a larger, complex dynamic system. This methodology, detailed in Section\nbsp{}ref:sec:detail_fem, is presented as a potentially useful tool for future mechatronic instrument development. @@ -831,7 +858,7 @@ Consequently, the specified performance targets were met using controllers which ***** Active Damping of rotating mechanical systems using Integral Force Feedback -During conceptual design, it was found that the guaranteed stability property of the established active damping technique known as Integral Force Feedback (IFF) is compromised when applied to rotating platforms like the NASS. +During conceptual design, it was found that the guaranteed stability property of the established active damping technique known as acrfull:iff is compromised when applied to rotating platforms like the NASS. To address this instability issue, two modifications to the classical IFF control scheme were proposed and analyzed. The first involves a minor adjustment to the control law itself, while the second incorporates physical springs in parallel with the force sensors. Stability conditions and optimal parameter tuning guidelines were derived for both modified schemes. @@ -874,19 +901,19 @@ Dynamic error budgeting\nbsp{}[[cite:&monkhorst04_dynam_error_budget;&okyay16_me Chapter\nbsp{}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\nbsp{}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\nbsp{}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\nbsp{}ref:sec:detail_fem), then applied to optimize the actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and flexible joints (Section\nbsp{}ref:sec:detail_fem_joint) while maintaining system-level simulation capability. Control strategy refinement (Section\nbsp{}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\nbsp{}ref:sec:detail_instrumentation) is guided by dynamic error budgeting to establish noise specifications, followed by experimental characterization. -The chapter concludes (Section\nbsp{}ref:sec:detail_design) by presenting the final, optimized nano-hexapod design, ready for procurement and assembly. +The chapter concludes (Section\nbsp{}ref:sec:detail_design) by presenting the final, optimized active platform design, ready for procurement and assembly. ***** Experimental validation Chapter\nbsp{}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 (IFF) tested (Section\nbsp{}ref:sec:test_apa). +Actuators of the active platform were characterized, models validated, and active damping tested (Section\nbsp{}ref:sec:test_apa). Flexible joints were tested on a dedicated bench to verify stiffness and stroke specifications (Section\nbsp{}ref:sec:test_joints). Assembled struts (actuators + joints) were then characterized to ensure consistency and validate multi-body models (Section\nbsp{}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\nbsp{}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\nbsp{}ref:sec:test_nhexa). Finally, the integrated NASS was validated on the ID31 beamline using a purpose-built short-stroke metrology system (Section\nbsp{}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. @@ -955,7 +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 point of interest (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 nano-hexapod and the sample's acrshort:poi (Section\nbsp{}ref:sec:uniaxial_payload_dynamics). *** Micro Station Model <> @@ -967,9 +994,9 @@ In this section, a uniaxial model of the micro-station is tuned to match measure 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]. -Three frequency response functions 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 $x_{g}$. +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 frequency response functions 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 200Hz (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. @@ -991,7 +1018,7 @@ Due to the poor coherence at low frequencies, these frequency response functions **** Uniaxial Model The uniaxial model of the micro-station is shown in Figure\nbsp{}ref:fig:uniaxial_model_micro_station. -It consists of a mass spring damper system with three degrees of freedom. +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 $k_h$. @@ -1016,7 +1043,7 @@ The hammer impacts $F_{h}, F_{g}$ are shown in blue, whereas the measured inerti **** 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. -Because the uniaxial model has three degrees of freedom, only three modes with frequencies at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$ are modeled. +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). However, the goal is not to have a perfect match with the measurement (this would require a much more complex model), but to have a first approximation. More accurate models will be used later on. @@ -1034,7 +1061,7 @@ A model of the nano-hexapod and sample is now added on top of the uniaxial model 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 point of interest and the nano-hexapod actuator will be considered in Section\nbsp{}ref:sec:uniaxial_payload_dynamics. +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. #+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}). @@ -1091,8 +1118,6 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nb #+end_subfigure #+end_figure -**** Identification of all combination of stiffnesses / masses :noexport: - *** Disturbance Identification <> **** Introduction :ignore: @@ -1120,7 +1145,7 @@ 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 ADC. +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. 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. @@ -1136,8 +1161,8 @@ G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi #+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. [[file:figs/uniaxial_geophone_meas_chain.png]] -The amplitude spectral density of the floor motion $\Gamma_{x_f}$ can be computed from the amplitude spectral density of measured voltage $\Gamma_{\hat{V}_{x_f}}$ using\nbsp{}eqref:eq:uniaxial_asd_floor_motion. -The estimated amplitude spectral density $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in Figure\nbsp{}ref:fig:uniaxial_asd_floor_motion_id31. +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. +The estimated acrshort:asd $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in Figure\nbsp{}ref:fig:uniaxial_asd_floor_motion_id31. \begin{equation}\label{eq:uniaxial_asd_floor_motion} \Gamma_{x_f}(\omega) = \frac{\Gamma_{\hat{V}_{x_f}}(\omega)}{|G_{geo}(j\omega)| \cdot g_0} \quad \left[ m/\sqrt{\text{Hz}} \right] @@ -1233,7 +1258,7 @@ 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 Cumulative Amplitude Spectrum 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. +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 $5\,\text{Hz}$. 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. @@ -1275,7 +1300,7 @@ The advantage of the soft nano-hexapod can be explained by its natural isolation *** 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) cite:preumont91_activ, Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] 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 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]]. 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). @@ -1312,7 +1337,7 @@ The Integral Force Feedback strategy consists of using a force sensor in series \boxed{K_{\text{IFF}}(s) = \frac{g}{s}} \end{equation} -The mechanical equivalent of this IFF strategy is a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain $k/g$ (see Figure\nbsp{}ref:fig:uniaxial_active_damping_iff_equiv). +The mechanical equivalent of this acrshort:iff strategy is a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain $k/g$ (see Figure\nbsp{}ref:fig:uniaxial_active_damping_iff_equiv). #+name: fig:uniaxial_active_damping_iff #+caption: Integral Force Feedback (\subref{fig:uniaxial_active_damping_iff_schematic}) is equivalent to a damper in series with the actuator stiffness (\subref{fig:uniaxial_active_damping_iff_equiv}) @@ -1393,10 +1418,10 @@ This is usually referred to as "/sky hook damper/". The plant dynamics for all three active damping techniques are shown in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques. 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 RDC cases because the sensors are collocated with the actuator\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7]]. -For the 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]]. +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]]. -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 DVF transfer functions. +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. @@ -1443,10 +1468,10 @@ There is even some damping authority on micro-station modes in the following cas 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 DVF in Figure\nbsp{}ref:fig:uniaxial_active_damping_dvf_equiv) is connected to the micro-station top platform through the stiff nano-hexapod. + 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 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 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. #+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. @@ -1516,7 +1541,7 @@ Several conclusions can be drawn by comparing the obtained sensitivity transfer 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). 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 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 nano-hexapod (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}) @@ -1547,7 +1572,7 @@ The cumulative amplitude spectrum of the distance $d$ with all three active damp All three active damping methods give similar results. #+name: fig:uniaxial_cas_active_damping -#+caption: Comparison of the cumulative amplitude spectrum (CAS) of the distance $d$ for all three active damping techniques (OL in black, IFF in blue, RDC in red and DVF in yellow). +#+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}$k_n = 0.01\,N/\mu m$} @@ -1575,7 +1600,7 @@ All three active damping methods give similar results. 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 DVF. +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). Even some damping can be applied to some micro-station modes in specific cases. The obtained damped plants were found to be similar. @@ -1687,7 +1712,7 @@ The required feedback bandwidths were estimated in Section\nbsp{}ref:sec:uniaxia 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). An arbitrary minimum modulus margin of $0.25$ was chosen when designing the controllers. -These high authority controllers 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 low pass filter to increase the robustness to high frequency dynamics. +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. \begin{subequations} \label{eq:uniaxial_hac_formulas} @@ -1718,7 +1743,7 @@ K_{\text{stiff}}(s) &= g \cdot | *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* | $\omega_l = 200\,Hz$ | $\omega_l = 300\,Hz$ | $\omega_l = 500\,Hz$ | -The loop gains corresponding to the designed high authority controllers for the three nano-hexapod are shown in Figure\nbsp{}ref:fig:uniaxial_loop_gain_hac. +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. 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 $\approx 100\,\text{Hz}$. @@ -1752,7 +1777,7 @@ The goal is to have a first estimation of the attainable performance. #+end_figure #+name: fig:uniaxial_loop_gain_hac -#+caption: Loop gain for the High Authority Controller +#+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}$k_n = 0.01\,N/\mu m$} @@ -1778,12 +1803,12 @@ The goal is to have a first estimation of the attainable performance. **** Closed-Loop Noise Budgeting <> -The high authority position feedback controllers are then implemented and the closed-loop sensitivities to disturbances are computed. +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). 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 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. 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}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_fs}Direct forces} @@ -1806,7 +1831,7 @@ 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 open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases. +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 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 $\approx 70\,nm$). @@ -2073,7 +2098,7 @@ 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 point-of-interest 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 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). 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 $\omega_n = 200\,\text{Hz}$). @@ -2085,7 +2110,7 @@ In this case, the measured (i.e., controlled) distance $d$ is no longer equal to 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. -The High Authority Controllers used in Section\nbsp{}ref:sec:uniaxial_position_control are then implemented on the damped plants. +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. Finally, closed-loop noise budgeting is computed for the obtained closed-loop system, and the cumulative amplitude spectrum of $d$ and $y$ are shown in Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_y. @@ -2122,7 +2147,7 @@ Such additional dynamics can induce stability issues depending on their position 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. 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 point of interest (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. +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. *** Conclusion @@ -2155,9 +2180,9 @@ This model is simple enough to be able to derive its dynamics analytically and t acrfull:iff is then applied to the rotating platform, and it is shown that the unconditional stability of acrshort:iff is lost due to the gyroscopic effects induced by the rotation (Section\nbsp{}ref:sec:rotating_iff_pure_int). Two modifications of the Integral Force Feedback are then proposed. -The first modification involves adding a high-pass filter to the acrshort:iff controller (Section\nbsp{}ref:sec:rotating_iff_pseudo_int). +The first modification involves adding a acrfull:hpf to the acrshort:iff controller (Section\nbsp{}ref:sec:rotating_iff_pseudo_int). It is shown that the acrshort:iff controller is stable for some gain values, and that damping can be added to the suspension modes. -The optimal high-pass filter cut-off frequency is computed. +The optimal acrlong:hpf cut-off frequency is computed. The second modification consists of adding a stiffness in parallel to the force sensors (Section\nbsp{}ref:sec:rotating_iff_parallel_stiffness). Under certain conditions, the unconditional stability of the IFF controller is regained. The optimal parallel stiffness is then computed. @@ -2175,7 +2200,7 @@ The goal is to determine whether the rotation imposes performance limitation on <> **** Introduction :ignore: -The system used to study gyroscopic effects consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure\nbsp{}ref:fig:rotating_3dof_model_schematic). +The system used to study gyroscopic effects consists of a 2-acrshortpl:dof translation stage on top of a rotating stage (Figure\nbsp{}ref:fig:rotating_3dof_model_schematic). The rotating stage is supposed to be ideal, meaning it induces a perfect rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\s}$. The suspended platform consists of two orthogonal actuators, each represented by three elements in parallel: a spring with a stiffness $k$ in $\si{\newton\per\meter}$, a dashpot with a damping coefficient $c$ in $\si{\newton\per(\meter\per\second)}$ and an ideal force source $F_u, F_v$. A payload with a mass $m$ in $\si{\kilo\gram}$, is mounted on the (rotating) suspended platform. @@ -2217,7 +2242,7 @@ Substituting equations\nbsp{}eqref:eq:rotating_energy_functions_lagrange into eq The uniform rotation of the system induces two /gyroscopic effects/ as shown in equation\nbsp{}eqref:eq:rotating_eom_coupled: - /Centrifugal forces/: that can be seen as an added /negative stiffness/ $- m \Omega^2$ along $\vec{i}_u$ and $\vec{i}_v$ - /Coriolis forces/: that adds /coupling/ between the two orthogonal directions. -One can verify that without rotation ($\Omega = 0$), the system becomes equivalent to two /uncoupled/ one degree of freedom mass-spring-damper systems. +One can verify that without rotation ($\Omega = 0$), the system becomes equivalent to two /uncoupled/ one acrshort:dof mass-spring-damper systems. To study the dynamics of the system, the two differential equations of motions\nbsp{}eqref:eq:rotating_eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\bm{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation\nbsp{}eqref:eq:rotating_Gd_mimo_tf is obtained. The four transfer functions in $\bm{G}_d$ are shown in equation\nbsp{}eqref:eq:rotating_Gd_indiv_el. @@ -2285,8 +2310,6 @@ Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forc #+end_subfigure #+end_figure -**** Identify Generic Dynamics :noexport: - **** System Dynamics: Effect of rotation The system dynamics from actuator forces $[F_u, F_v]$ to the relative motion $[d_u, d_v]$ is identified for several rotating velocities. Looking at the transfer function matrix $\bm{G}_d$ in equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. @@ -2317,17 +2340,17 @@ For $\Omega > \omega_0$, the low-frequency pair of complex conjugate poles $p_{- **** Introduction :ignore: The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation. -As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances cite:collette11_review_activ_vibrat_isolat_strat and to make the plant easier to control for the high authority controller. +As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances\nbsp{}[[cite:&collette11_review_activ_vibrat_isolat_strat]] and to make the plant easier to control for the high authority controller. -Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF) cite:lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc, Integral Force Feedback (IFF) cite:preumont91_activ 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. +Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF)\nbsp{}[[cite:&lin06_distur_atten_precis_hexap_point;&fanson90_posit_posit_feedb_contr_large_space_struc]], Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]] 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\nbsp{}[[cite:&preumont91_activ]], the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system. -When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros, which guarantees the stability of the closed-loop system cite:preumont02_force_feedb_versus_accel_feedb. -It was later shown that this property holds for multiple collated actuator/sensor pairs cite:preumont08_trans_zeros_struc_contr_with. +When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros, which guarantees the stability of the closed-loop system\nbsp{}[[cite:&preumont02_force_feedb_versus_accel_feedb]]. +It was later shown that this property holds for multiple collated actuator/sensor pairs\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with]]. -The main advantages of IFF over other active damping techniques are the guaranteed stability even in the presence of flexible dynamics, good performance, and robustness properties cite:preumont02_force_feedb_versus_accel_feedb. +The main advantages of IFF over other active damping techniques are the guaranteed stability even in the presence of flexible dynamics, good performance, and robustness properties\nbsp{}[[cite:&preumont02_force_feedb_versus_accel_feedb]]. -Several improvements to the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping cite:teo15_optim_integ_force_feedb_activ_vibrat_contr or adding a high-pass filter to recover the loss of compliance at low-frequency cite:chesne16_enhan_dampin_flexib_struc_using_force_feedb. -Recently, an $\mathcal{H}_\infty$ optimization criterion has been used to derive optimal gains for the IFF controller cite:zhao19_optim_integ_force_feedb_contr. \par +Several improvements to the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping\nbsp{}[[cite:&teo15_optim_integ_force_feedb_activ_vibrat_contr]] or adding a acrshort:hpf to recover the loss of compliance at low-frequency\nbsp{}[[cite:&chesne16_enhan_dampin_flexib_struc_using_force_feedb]]. +Recently, an $\mathcal{H}_\infty$ optimization criterion has been used to derive optimal gains for the IFF controller\nbsp{}[[cite:&zhao19_optim_integ_force_feedb_contr]]. However, none of these studies have been applied to rotating systems. In this section, the acrshort:iff strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alter the system dynamics and that IFF cannot be applied as is. @@ -2444,22 +2467,22 @@ The decentralized acrshort:iff controller $\bm{K}_F$ corresponds to a diagonal c \end{equation} To determine how the acrshort:iff controller affects the poles of the closed-loop system, a Root Locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_{F}$ simultaneously. -As explained in cite:preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr, the closed-loop poles start at the open-loop poles (shown by crosses) for $g = 0$ and coincide with the transmission zeros (shown by circles) as $g \to \infty$. +As explained in\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with;&skogestad07_multiv_feedb_contr]], the closed-loop poles start at the open-loop poles (shown by crosses) for $g = 0$ and coincide with the transmission zeros (shown by circles) as $g \to \infty$. -Whereas collocated IFF is usually associated with unconditional stability cite:preumont91_activ, this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null. +Whereas collocated IFF is usually associated with unconditional stability\nbsp{}[[cite:&preumont91_activ]], this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null. This can be seen in the Root Locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure\nbsp{}ref:fig:rotating_iff_bode_plot_effect_rot). The control system is thus cancels the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. -*** Integral Force Feedback with a High-Pass Filter +*** Integral Force Feedback with a High-Pass Filter <> **** Introduction :ignore: As explained in the previous section, the instability of the IFF controller applied to the rotating system is due to the high gain of the integrator at low-frequency. To limit the low-frequency controller gain, a acrfull:hpf can be added to the controller, as shown in equation\nbsp{}eqref:eq:rotating_iff_lhf. This is equivalent to slightly shifting the controller pole to the left along the real axis. -This modification of the IFF controller is typically performed to avoid saturation associated with the pure integrator cite:preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans. -This is however not the reason why this high-pass filter is added here. +This modification of the IFF controller is typically performed to avoid saturation associated with the pure integrator\nbsp{}[[cite:&preumont91_activ;&marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans]]. +This is however not the reason why this acrlong:hpf is added here. \begin{equation}\label{eq:rotating_iff_lhf} \boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} @@ -2479,7 +2502,7 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the controll \end{equation} #+name: fig:rotating_iff_modified_loop_gain_root_locus -#+caption: Comparison of the IFF with pure integrator and modified IFF with added high-pass filter ($\Omega = 0.1\omega_0$). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with $\omega_i = 0.1 \omega_0$ and $g = 2$. The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large}) +#+caption: Comparison of the IFF with pure integrator and modified IFF with added high-pass filter ($\Omega = 0.1\omega_0$). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with $\omega_i = 0.1 \omega_0$ and $g = 2$. The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_modified_loop_gain}Loop gain} @@ -2509,7 +2532,7 @@ For small values of $\omega_i$, the added damping is limited by the maximum allo For larger values of $\omega_i$, the attainable damping ratio decreases as a function of $\omega_i$ as was predicted from the root locus plot of Figure\nbsp{}ref:fig:rotating_iff_root_locus_hpf_large. #+name: fig:rotating_iff_modified_effect_wi -#+caption: Root Locus for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as $\omega_i$ increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain}) +#+caption: Root Locus for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as $\omega_i$ increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus} @@ -2630,7 +2653,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif **** Effect of $k_p$ on the attainable damping Even though the parallel stiffness $k_p$ has no impact on the open-loop poles (as the overall stiffness $k$ is kept constant), it has a large impact on the transmission zeros. -Moreover, as the attainable damping is generally proportional to the distance between poles and zeros cite:preumont18_vibrat_contr_activ_struc_fourt_edition, the parallel stiffness $k_p$ is expected to have some impact on the attainable damping. +Moreover, as the attainable damping is generally proportional to the distance between poles and zeros\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition]], the parallel stiffness $k_p$ is expected to have some impact on the attainable damping. To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus_effect_kp. The frequencies of the transmission zeros of the system increase with an increase in the parallel stiffness $k_p$ (thus getting closer to the poles), and the associated attainable damping is reduced. Therefore, even though the parallel stiffness $k_p$ should be larger than $m \Omega^2$ for stability reasons, it should not be taken too large as this would limit the attainable damping. @@ -2660,7 +2683,7 @@ The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in F Even though the two resonances are well damped, the IFF changes the low-frequency behavior of the plant, which is usually not desired. This is because "pure" integrators are used which are inducing large low-frequency loop gains. -To lower the low-frequency gain, a high-pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): +To lower the low-frequency gain, a acrshort:hpf is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): \begin{equation} K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} 1 & 0 \\ @@ -2668,14 +2691,14 @@ To lower the low-frequency gain, a high-pass filter is added to the IFF control \end{bmatrix} \end{equation} -To determine how the high-pass filter impacts the attainable damping, the controller gain $g$ is kept constant while $\omega_i$ is changed, and the minimum damping ratio of the damped plant is computed. +To determine how the acrshort:hpf impacts the attainable damping, the controller gain $g$ is kept constant while $\omega_i$ is changed, and the minimum damping ratio of the damped plant is computed. The obtained damping ratio as a function of $\omega_i/\omega_0$ (where $\omega_0$ is the resonance of the system without rotation) is shown in Figure\nbsp{}ref:fig:rotating_iff_kp_added_hpf_effect_damping. It is shown that the attainable damping ratio reduces as $\omega_i$ is increased (same conclusion than in Section\nbsp{}ref:sec:rotating_iff_pseudo_int). Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compare the obtained damped plant again with the undamped and with the "pure" IFF in Figure\nbsp{}ref:fig:rotating_iff_kp_added_hpf_damped_plant. -The added high-pass filter gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior. +The added acrshort:hpf gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior. #+name: fig:rotating_iff_optimal_hpf -#+caption:Effect of high-pass filter cut-off frequency on the obtained damping +#+caption:Effect of high-pass filter cut-off frequency on the obtained damping #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$} @@ -2698,7 +2721,7 @@ The added high-pass filter gives almost the same damping properties to the suspe **** Introduction :ignore: To apply a "Relative Damping Control" strategy, relative motion sensors are added in parallel with the actuators as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_rdc. Two controllers $K_d$ are used to feed back the relative motion to the actuator. -These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter\nbsp{}eqref:eq:rotating_rdc_controller. +These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter\nbsp{}eqref:eq:rotating_rdc_controller. \begin{equation}\label{eq:rotating_rdc_controller} K_d(s) = g \cdot \frac{s}{s + \omega_d} @@ -2837,8 +2860,6 @@ The previous analysis is now applied to a model representing a rotating nano-hex 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 $k_n = \SI{100}{\N\per\mu\m}$. 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. -**** Identify NASS dynamics :noexport: - **** 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 $m_n + m_s = \SI{16}{\kilo\gram}$. The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \approx \SI{0.6}{\newton\per\mm}$. @@ -2846,7 +2867,7 @@ The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \a 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 coupling (or interaction) in a MIMO $2 \times 2$ system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). +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 @@ -2872,8 +2893,8 @@ The coupling (or interaction) in a MIMO $2 \times 2$ system can be visually esti #+end_subfigure #+end_figure -**** Optimal IFF with a High-Pass Filter -Integral Force Feedback with an added high-pass filter is applied to the three nano-hexapods. +**** Optimal IFF with a High-Pass Filter +Integral Force Feedback with an added acrlong:hpf is applied to the three nano-hexapods. 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. @@ -2907,7 +2928,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar #+end_figure #+name: tab:rotating_iff_hpf_opt_iff_hpf_params_nass -#+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. +#+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 | $k_n$ | $\omega_i$ | $g$ | $\xi_\text{opt}$ | @@ -2919,7 +2940,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar **** 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). -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 ($\omega_i = \omega_0/10$). +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. @@ -3157,7 +3178,7 @@ In this study, the gyroscopic effects induced by the spindle's rotation have bee Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms. Two modifications of the classical acrshort:iff control have been proposed to overcome this issue. -The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators. +The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators. This is equivalent to moving the controller pole to the left along the real axis. This allows the closed-loop system to be stable up to some value of the controller gain. @@ -3188,7 +3209,7 @@ Although the inertia of each solid body can easily be estimated from its geometr Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station. The tuning approach for the multi-body model based on measurements is illustrated in Figure\nbsp{}ref:fig:modal_vibration_analysis_procedure. -First, a /response model/ is obtained, which corresponds to a set of frequency response functions computed from experimental measurements. +First, a /response model/ is obtained, which corresponds to a set of acrshortpl:frf computed from experimental measurements. From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes. This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies. @@ -3199,8 +3220,8 @@ This modal model can then be used to tune the spatial model (i.e. the multi-body The measurement setup used to obtain the response model is described in Section\nbsp{}ref:sec:modal_meas_setup. This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals. -In Section\nbsp{}ref:sec:modal_frf_processing, the obtained frequency response functions between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed. -These measurements are projected at the center of mass of each considered solid body to facilitate the further use of the results. +In Section\nbsp{}ref:sec:modal_frf_processing, the obtained acrshortpl:frf between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed. +These measurements are projected at the acrfull:com of each considered solid body to facilitate the further use of the results. The solid body assumption is then verified, validating the use of the multi-body model. Finally, the modal analysis is performed in Section\nbsp{}ref:sec:modal_analysis. @@ -3263,11 +3284,11 @@ If these local feedback controls were turned off, this would have resulted in ve The top part representing the active stabilization stage was disassembled as the active stabilization stage will be added in the multi-body model afterwards. -To perform the modal analysis from the measured responses, the $n \times n$ frequency response function matrix $\bm{H}$ needs to be measured, where $n$ is the considered number of degrees of freedom. -The $H_{jk}$ element of this acrfull:frf matrix corresponds to the frequency response function from a force $F_k$ applied at acrfull:dof $k$ to the displacement of the structure $X_j$ at acrshort:dof $j$. +To perform the modal analysis from the measured responses, the $n \times n$ acrshort:frf matrix $\bm{H}$ needs to be measured, where $n$ is the considered number of acrshortpl:dof. +The $H_{jk}$ element of this acrfull:frf matrix corresponds to the acrshort:frf from a force $F_k$ applied at acrfull:dof $k$ to the displacement of the structure $X_j$ at acrshort:dof $j$. Measuring this acrshort:frf matrix is time consuming as it requires to make $n \times n$ measurements. However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\bm{H}$ [[cite:&ewins00_modal chapt. 5.2]]. -Therefore, a minimum set of $n$ frequency response functions is required. +Therefore, a minimum set of $n$ acrshortpl:frf is required. This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof. It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode). @@ -3278,7 +3299,7 @@ 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. -The positions of the accelerometers are visually shown on a CAD model in Figure\nbsp{}ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table\nbsp{}ref:tab:modal_position_accelerometers. +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. 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. @@ -3385,7 +3406,7 @@ 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). -Similar results were obtained for all measured frequency response functions. +Similar results were obtained for all measured acrshortpl:frf. #+name: fig:modal_raw_meas_asd #+caption: Raw measurement of the accelerometer 1 in the $x$ direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the $z$ direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force}) @@ -3405,7 +3426,7 @@ Similar results were obtained for all measured frequency response functions. #+end_subfigure #+end_figure -The frequency response function from the applied force to the measured acceleration is then computed and shown Figure\nbsp{}ref:fig:modal_frf_acc_force. +The acrshort:frf from the applied force to the measured acceleration is then computed and shown Figure\nbsp{}ref:fig:modal_frf_acc_force. The quality of the obtained data can be estimated using the /coherence/ function (Figure\nbsp{}ref:fig:modal_coh_acc_force). Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest. @@ -3450,11 +3471,11 @@ For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that link \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. -Therefore, only $6 \times 6 = 36$ degrees of freedom are of interest. +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. The coordinate transformation from accelerometers acrshort:dof to the solid body 6 acrshortpl:dof (three translations and three rotations) is performed in Section\nbsp{}ref:ssec:modal_acc_to_solid_dof. -The $69 \times 3 \times 801$ frequency response matrix is then reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to its center of mass. +The $69 \times 3 \times 801$ frequency response matrix is then reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to its acrlong:com. To validate this reduction of acrshort:dof and the solid body assumption, the frequency response function at the accelerometer location are "reconstructed" from the reduced frequency response matrix and are compared with the initial measurements in Section\nbsp{}ref:ssec:modal_solid_body_assumption. @@ -3508,8 +3529,8 @@ The motion of the solid body expressed in a chosen frame $\{O\}$ can be determin Note that this matrix inversion is equivalent to resolving a mean square problem. Therefore, having more accelerometers permits better approximation of the motion of a solid body. -From the CAD model, the position of the center of mass of each solid body is computed (see Table\nbsp{}ref:tab:modal_com_solid_bodies). -The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined. +From the 3D model, the position of the acrlong:com of each solid body is computed (see Table\nbsp{}ref:tab:modal_com_solid_bodies). +The position of each accelerometer with respect to the acrlong:com of the corresponding solid body can easily be determined. #+name: tab:modal_com_solid_bodies #+caption: Center of mass of considered solid bodies with respect to the "point of interest" @@ -3524,7 +3545,7 @@ The position of each accelerometer with respect to the center of mass of the cor | Spindle | $0$ | $0$ | $-580\,\text{mm}$ | | 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 center of mass 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 $\bm{H}$. +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 $\bm{H}$. \begin{equation}\label{eq:modal_frf_matrix_com} \bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix} @@ -3548,9 +3569,9 @@ In particular, the responses at the locations of the four accelerometers can be 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 original frequency response functions and those computed from the CoM responses match well in the frequency range of interest. +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 degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof). +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 @@ -3565,7 +3586,7 @@ The goal here is to extract the modal parameters describing the modes of the mic This is performed from the acrshort:frf matrix previously extracted from the measurements. In order to perform the modal parameter extraction, the order of the modal model has to be estimated (i.e. the number of modes in the frequency band of interest). -This is achived using the acrfull:mif in section\nbsp{}ref:ssec:modal_number_of_modes. +This is achieved using the acrfull:mif in section\nbsp{}ref:ssec:modal_number_of_modes. In section\nbsp{}ref:ssec:modal_parameter_extraction, the modal parameter extraction is performed. The graphical display of the mode shapes can be computed from the modal model, which is quite useful for physical interpretation of the modes. @@ -3576,7 +3597,7 @@ To validate the quality of the modal model, the full acrshort:frf matrix is comp <> The acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$). -The complex modal indication function is defined in equation\nbsp{}eqref:eq:modal_cmif where the diagonal matrix $\Sigma$ is obtained from a acrlong:svd of the acrshort:frf matrix as shown in equation\nbsp{}eqref:eq:modal_svd. +The acrfull:cmif is defined in equation\nbsp{}eqref:eq:modal_cmif where the diagonal matrix $\Sigma$ is obtained from a acrfull:svd of the acrshort:frf matrix as shown in equation\nbsp{}eqref:eq:modal_svd. \begin{equation} \label{eq:modal_cmif} [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^{\intercal} [\Sigma(\omega)]_{n\times p} \end{equation} @@ -3664,7 +3685,7 @@ From the obtained modal parameters, the mode shapes are computed and can be disp These animations are useful for visually obtaining a better understanding of the system's dynamic behavior. For instance, the mode shape of the first mode at $11\,\text{Hz}$ (figure\nbsp{}ref:fig:modal_mode1_animation) indicates an issue with the lower granite. It turns out that four /Airloc Levelers/ are used to level the lower granite (figure\nbsp{}ref:fig:modal_airloc). -These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it has a low frequency "tilt mode". +These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it has a low frequency "tilt mode". The levelers were then better adjusted. #+name: fig:modal_airloc @@ -3711,10 +3732,10 @@ With $\bm{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response o \bm{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m} \end{equation} -A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal. +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 frequency response function 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 micro-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. @@ -3747,11 +3768,11 @@ This can be seen in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal_3 that shows t <> In this study, a modal analysis of the micro-station was performed. -Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained. -The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture. +Thanks to an adequate choice of instrumentation and proper set of measurements, high quality acrshortpl:frf can be obtained. +The obtained acrshortpl:frf indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture. It shows a lot of coupling between stages and different directions, and many modes. -By measuring 12 degrees of freedom on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body. +By measuring 12 acrshortpl:dof on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body. This confirms that a multi-body model can be used to properly model the micro-station. Although a lot of effort was put into this experimental modal analysis of the micro-station, it was difficult to obtain an accurate modal model. @@ -3790,7 +3811,7 @@ From bottom to top, the stacked stages are the translation stage $D_y$, the tilt Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex. #+name: fig:ustation_cad_view -#+caption: CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple). +#+caption: 3D view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple). #+attr_latex: :width \linewidth [[file:figs/ustation_cad_view.png]] @@ -3822,7 +3843,7 @@ Each linear guide is very stiff in radial directions such that the only DoF with This stage is mainly used in /reflectivity/ experiments where the sample $R_y$ angle is scanned. This stage can also be used to tilt the rotation axis of the Spindle. -To precisely control the $R_y$ angle, a stepper motor and two optical encoders are used in a PID feedback loop. +To precisely control the $R_y$ angle, a stepper motor and two optical encoders are used in a acrfull:pid feedback loop. #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage @@ -3852,8 +3873,8 @@ Additional rotary unions and slip-rings are used to be able to pass electrical s 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. -This stage is used to position the point of interest of the sample with respect to the spindle rotation axis. -It can also be used to precisely position the PoI vertically with respect to the x-ray. +This stage is used to position the acrshort:poi of the sample with respect to the spindle rotation axis. +It can also be used to precisely position the acrfull:poi vertically with respect to the x-ray. #+attr_latex: :options [t]{0.49\linewidth} #+begin_minipage @@ -4043,7 +4064,7 @@ Another key advantage of homogeneous transformation is the easy inverse transfor <> Each stage is described by two frames; one is attached to the fixed platform $\{A\}$ while the other is fixed to the mobile platform $\{B\}$. -At "rest" position, the two have the same pose and coincide with the point of interest ($O_A = O_B$). +At "rest" position, the two have the same pose and coincide with the acrshort:poi ($O_A = O_B$). An example of the tilt stage is shown in Figure\nbsp{}ref:fig:ustation_stage_motion. The mobile frame of the translation stage is equal to the fixed frame of the tilt stage: $\{B_{D_y}\} = \{A_{R_y}\}$. Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the spindle: $\{B_{R_y}\} = \{A_{R_z}\}$. @@ -4115,7 +4136,7 @@ The setpoints are $D_y$ for the translation stage, $\theta_y$ for the tilt-stage In this section, the multi-body model of the micro-station is presented. Such model consists of several rigid bodies connected by springs and dampers. -The inertia of the solid bodies and the stiffness properties of the guiding mechanisms were first estimated based on the CAD model and data-sheets (Section\nbsp{}ref:ssec:ustation_model_simscape). +The inertia of the solid bodies and the stiffness properties of the guiding mechanisms were first estimated based on the 3D model and data-sheets (Section\nbsp{}ref:ssec:ustation_model_simscape). The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section\nbsp{}ref:ssec:ustation_model_comp_dynamics). @@ -4156,7 +4177,7 @@ The translation stage is connected to the granite by a 6-DoF joint, but the $D_y Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoF joint, with 1 imposed DoF each time. Finally, the positioning hexapod has 6-DoF. -The total number of "free" degrees of freedom is 27, so the model has 54 states. +The total number of "free" acrshortpl:dof is 27, so the model has 54 states. The springs and dampers values were first estimated from the joint/stage specifications and were later fined-tuned based on the measurements. The spring values are summarized in Table\nbsp{}ref:tab:ustation_6dof_stiffness_values. @@ -4176,9 +4197,9 @@ The spring values are summarized in Table\nbsp{}ref:tab:ustation_6dof_stiffness_ <> The dynamics of the micro-station was measured by placing accelerometers on each stage and by impacting the translation stage with an instrumented hammer in three directions. -The obtained FRFs were then projected at the CoM of each stage. +The obtained acrshortpl:frf were then projected at the CoM of each stage. -To gain a first insight into the accuracy of the obtained model, the FRFs from the hammer impacts to the acceleration of each stage were extracted from the multi-body model and compared with the measurements in Figure\nbsp{}ref:fig:ustation_comp_com_response. +To gain a first insight into the accuracy of the obtained model, the acrshortpl:frf from the hammer impacts to the acceleration of each stage were extracted from the multi-body model and compared with the measurements in Figure\nbsp{}ref:fig:ustation_comp_com_response. Even though there is some similarity between the model and the measurements (similar overall shapes and amplitudes), it is clear that the multi-body model does not accurately represent the complex micro-station dynamics. Tuning the numerous model parameters to better match the measurements is a highly non-linear optimization problem that is difficult to solve in practice. @@ -4272,8 +4293,8 @@ The equivalent forces and torques applied at center of $\{\mathcal{X}\}$ are the F_{\mathcal{X}} = \bm{J}_F^{\intercal} \cdot F_{\mathcal{L}} \end{equation} -Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the FRF from 6 forces/torques applied at the origin of frame $\{\mathcal{X}\}$ to the 6 linear/angular accelerations of the top platform expressed with respect to $\{\mathcal{X}\}$. -These FRFs were then used for comparison with the multi-body model. +Using the two Jacobian matrices, the acrshort:frf from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the acrshort:frf from 6 forces/torques applied at the origin of frame $\{\mathcal{X}\}$ to the 6 linear/angular accelerations of the top platform expressed with respect to $\{\mathcal{X}\}$. +These acrshortpl:frf were then used for comparison with the multi-body model. The compliance of the micro-station multi-body model was extracted by computing the transfer function from forces/torques applied on the hexapod's top platform to the "absolute" motion of the top platform. These results are compared with the measurements in Figure\nbsp{}ref:fig:ustation_frf_compliance_model. @@ -4325,7 +4346,7 @@ Therefore, from a control perspective, they are not important. ***** Ground Motion The ground motion was measured by using a sensitive 3-axis geophone shown in Figure\nbsp{}ref:fig:ustation_geophone_picture placed on the ground. -The generated voltages were recorded with a high resolution DAC, and converted to displacement using the Geophone sensitivity transfer function. +The generated voltages were recorded with a high resolution acrshort:adc, and converted to displacement using the Geophone sensitivity transfer function. The obtained ground motion displacement is shown in Figure\nbsp{}ref:fig:ustation_ground_disturbance. #+attr_latex: :options [b]{0.54\linewidth} @@ -4409,7 +4430,7 @@ The obtained results are shown in Figure\nbsp{}ref:fig:ustation_errors_spindle. A large fraction of the radial (Figure\nbsp{}ref:fig:ustation_errors_spindle_radial) and tilt (Figure\nbsp{}ref:fig:ustation_errors_spindle_tilt) errors is linked to the fact that the two spheres are not perfectly aligned with the rotation axis of the Spindle. 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 "point-of-interest" of the sample and the rotation axis will be considered because the alignment is not perfect in practice. +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). #+name: fig:ustation_errors_spindle @@ -4540,8 +4561,8 @@ 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 "point-of-interest" 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 "point-of-interest" with respect to the granite was recorded. +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. +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). @@ -4569,7 +4590,7 @@ 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 $\pm 4.5\, mm$. Both ground motion and translation stage vibrations were included in the simulation. -Similar to what was performed for the tomography simulation, the 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. +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 @@ -4603,8 +4624,8 @@ However, the complexity of its dynamic behavior poses significant challenges for Consequently, a multi-body modeling approach was adopted (Section\nbsp{}ref:sec:nhexa_model), facilitating seamless integration with the existing micro-station model. -The control of the Stewart platform introduces additional complexity due to its multi-input multi-output (MIMO) nature. -Section\nbsp{}ref:sec:nhexa_control explores how the High Authority Control/Low Authority Control (HAC-LAC) strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform. +The control of the Stewart platform introduces additional complexity due to its acrfull:mimo nature. +Section\nbsp{}ref:sec:nhexa_control explores how the acrfull:haclac strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform. This adaptation requires fundamental decisions regarding both the control architecture (centralized versus decentralized) and the control frame (Cartesian versus strut space). Through careful analysis of system interactions and plant characteristics in different frames, a control architecture combining decentralized Integral Force Feedback for active damping with a centralized high authority controller for positioning was developed, with both controllers implemented in the frame of the struts. @@ -4631,7 +4652,7 @@ The positioning of samples with respect to X-ray beam, that can be focused to si To overcome this limitation, external metrology systems have been implemented to measure sample positions with nanometer accuracy, enabling real-time feedback control for sample stabilization. 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 degrees of freedom. +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). @@ -4729,7 +4750,7 @@ The first key distinction of the NASS is in the continuous rotation of the activ 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. In addition, NASS implements a unique Long-Stroke/Short-Stroke architecture. In conventional systems, active platforms typically correct spindle positioning errors - for example, unwanted translations or tilts that occur during rotation, whereas the intended rotational motion ($R_z$) is performed by the spindle itself and is not corrected. -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. +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. @@ -4746,16 +4767,16 @@ 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 degrees of freedom ($D_x$, $D_y$, $D_z$, $R_x$, 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. +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 $2000\,N/\mu m$. -This limitation necessitates alternative control approaches, and the High Authority Control/Low Authority Control (HAC-LAC) strategy is proposed to address this challenge. +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. Two primary categories of positioning platform architectures are considered: serial and parallel mechanisms. -Serial robots, characterized by open-loop kinematic chains, typically dedicate one actuator per degree of freedom as shown in Figure\nbsp{}ref:fig:nhexa_serial_architecture_kenton. +Serial robots, characterized by open-loop kinematic chains, typically dedicate one actuator per acrshort:dof as shown in Figure\nbsp{}ref:fig:nhexa_serial_architecture_kenton. While offering large workspaces and high maneuverability, serial mechanisms suffer from several inherent limitations. These include low structural stiffness, cumulative positioning errors along the kinematic chain, high mass-to-payload ratios due to actuator placement, and limited payload capacity\nbsp{}[[cite:&taghirad13_paral]]. These limitations generally make serial architectures unsuitable for nano-positioning applications, except when handling very light samples, as was used in\nbsp{}[[cite:&nazaretski15_pushin_limit]] and shown in Figure\nbsp{}ref:fig:nhexa_stages_nazaretski. @@ -4763,7 +4784,7 @@ These limitations generally make serial architectures unsuitable for nano-positi In contrast, parallel mechanisms, which connect the mobile platform to the fixed base through multiple parallel struts, offer several advantages for precision positioning. Their closed-loop kinematic structure provides inherently higher structural stiffness, as the platform is simultaneously supported by multiple struts\nbsp{}[[cite:&taghirad13_paral]]. Although parallel mechanisms typically exhibit limited workspace compared to serial architectures, this limitation is not critical for NASS given its modest stroke requirements. -Numerous parallel kinematic architectures have been developed cite:dong07_desig_precis_compl_paral_posit to address various positioning requirements, with designs varying based on the desired degrees of freedom and specific application constraints. +Numerous parallel kinematic architectures have been developed\nbsp{}[[cite:&dong07_desig_precis_compl_paral_posit]] to address various positioning requirements, with designs varying based on the desired acrshortpl:dof and specific application constraints. Furthermore, hybrid architectures combining both serial and parallel elements have been proposed\nbsp{}[[cite:&shen19_dynam_analy_flexur_nanop_stage]], as illustrated in Figure\nbsp{}ref:fig:nhexa_serial_parallel_examples, offering potential compromises between the advantages of both approaches. #+name: fig:nhexa_serial_parallel_examples @@ -4785,7 +4806,7 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha #+end_figure After evaluating the different options, the Stewart platform architecture was selected for several reasons. -In addition to providing control over all required degrees of freedom, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints. +In addition to providing control over all required acrshortpl:dof, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints. Stewart platforms have been implemented in a wide variety of configurations, as illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_examples, which shows two distinct implementations: one implementing piezoelectric actuators for nano-positioning applications, and another based on voice coil actuators for vibration isolation. These examples demonstrate the architecture's versatility in terms of geometry, actuator selection, and scale, all of which can be optimized for specific applications. Furthermore, the successful implementation of Integral Force Feedback (IFF) control on Stewart platforms has been well documented\nbsp{}[[cite:&abu02_stiff_soft_stewar_platf_activ;&hanieh03_activ_stewar;&preumont07_six_axis_singl_stage_activ]], and the extensive body of research on this architecture enables thorough optimization specifically for the NASS. @@ -4847,8 +4868,8 @@ The typical configuration consists of a universal joint at one end and a spheric To facilitate the rigorous analysis of the Stewart platform, four reference frames were defined: - The fixed base frame $\{F\}$, which is located at the center of the base platform's bottom surface, serves as the mounting reference for the support structure. - The mobile frame $\{M\}$, which is located at the center of the top platform's upper platform, provides a reference for payload mounting. -- The point-of-interest frame $\{A\}$, fixed to the base but positioned at the workspace center. -- The moving point-of-interest frame $\{B\}$, attached to the mobile platform coincides with frame $\{A\}$ in the home position. +- The acrlong:poi frame $\{A\}$, fixed to the base but positioned at the workspace center. +- The moving acrlong:poi frame $\{B\}$, attached to the mobile platform coincides with frame $\{A\}$ in the home position. 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 $\{A\}$. @@ -5134,7 +5155,7 @@ 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 point of interest, denoted by frame $\{A\}$, is situated $150\,mm$ above the moving platform frame $\{M\}$. +The acrshort:poi, denoted by frame $\{A\}$, is situated $150\,mm$ above the moving platform frame $\{M\}$. The geometric parameters of the nano-hexapod are summarized in Table\nbsp{}ref:tab:nhexa_stewart_model_geometry. These parameters define the positions of all connection points in their respective coordinate frames. @@ -5183,7 +5204,7 @@ Both platforms were assigned a mass of $5\,kg$. The platform's joints play a crucial role in its dynamic behavior. At both the upper and lower connection points, various degrees of freedom can be modeled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components. -For each degree of freedom, stiffness characteristics can be incorporated into the model. +For each acrshort:dof, stiffness characteristics can be incorporated into the model. In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modeled as two-degree-of-freedom universal joints, while the top joints are represented as three-degree-of-freedom spherical joints. These joints are considered massless and exhibit no stiffness along their degrees of freedom. @@ -5261,7 +5282,7 @@ For the analytical model, the stiffness, damping, and mass matrices are defined The transfer functions from the actuator forces to the strut displacements are computed using these matrices according to equation\nbsp{}eqref:eq:nhexa_transfer_function_struts. These analytical transfer functions are then compared with those extracted from the multi-body model. -The developed multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform. +The developed multi-body model yields a state-space representation with 12 states, corresponding to the six acrshortpl:dof of the moving platform. Figure\nbsp{}ref:fig:nhexa_comp_multi_body_analytical presents a comparison between the analytical and multi-body transfer functions, specifically showing the response from the first actuator force to all six strut displacements. The close agreement between both approaches across the frequency spectrum validates the multi-body model's accuracy in capturing the system's dynamic behavior. @@ -5280,7 +5301,7 @@ The transfer functions from actuator forces $\bm{f}$ to both strut displacements The transfer functions relating actuator forces to strut displacements are presented in Figure\nbsp{}ref:fig:nhexa_multi_body_plant_dL. Due to the system's symmetrical design and identical strut configurations, all diagonal terms (transfer functions from force $f_i$ to displacement $l_i$ of the same strut) exhibit identical behavior. -While the system has six degrees of freedom, only four distinct resonance frequencies were observed in the frequency response. +While the system has six acrshortpl:dof, only four distinct resonance frequencies were observed in the acrshortpl:frf. This reduction from six to four observable modes is attributed to the system's symmetry, where two pairs of resonances occur at identical frequencies. The system's behavior can be characterized in three frequency regions. @@ -5325,7 +5346,7 @@ The validated multi-body model will serve as a valuable tool for predicting syst **** Introduction :ignore: The control of Stewart platforms presents distinct challenges compared to the uniaxial model due to their multi-input multi-output nature. -Although the uniaxial model demonstrated the effectiveness of the HAC-LAC strategy, its extension to Stewart platforms requires careful considerations discussed in this section. +Although the uniaxial model demonstrated the effectiveness of the acrshort:haclac strategy, its extension to Stewart platforms requires careful considerations discussed in this section. First, the distinction between centralized and decentralized control approaches is discussed in Section\nbsp{}ref:ssec:nhexa_control_centralized_decentralized. The impact of the control space selection - either Cartesian or strut space - is then analyzed in Section\nbsp{}ref:ssec:nhexa_control_space, highlighting the trade-offs between direction-specific tuning and implementation simplicity. @@ -5336,7 +5357,7 @@ This architecture, while simple, will be used to demonstrate the feasibility of **** Centralized and Decentralized Control <> -In the control of MIMO systems, and more specifically of Stewart platforms, a fundamental architectural decision lies in the choice between centralized and decentralized control strategies. +In the control of acrshort:mimo systems, and more specifically of Stewart platforms, a fundamental architectural decision lies in the choice between centralized and decentralized control strategies. In decentralized control, each actuator operates based on feedback from its associated sensor only, creating independent control loops, as illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_decentralized_control. While mechanical coupling between the struts exists, control decisions are made locally, with each controller processing information from a single sensor-actuator pair. @@ -5350,7 +5371,7 @@ For instance, when using external metrology systems that measure the platform's In the context of the nano-hexapod, 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) -- High-Authority Control (HAC), which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section\nbsp{}ref:ssec:nhexa_control_hac_lac) +- acrfull:hac, which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section\nbsp{}ref:ssec:nhexa_control_hac_lac) #+name: fig:nhexa_stewart_decentralized_control #+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity. @@ -5400,11 +5421,11 @@ Alternatively, control can be implemented directly in Cartesian space, as illust Here, the controller processes Cartesian errors $\bm{\epsilon}_{\mathcal{X}}$ to generate forces and torques $\bm{\mathcal{F}}$, which are then mapped to actuator forces using the transpose of the inverse Jacobian matrix\nbsp{}eqref:eq:nhexa_jacobian_forces. The plant behavior in Cartesian space, illustrated in Figure\nbsp{}ref:fig:nhexa_plant_frame_cartesian, reveals interesting characteristics. -Some degrees of freedom, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics. +Some acrshortpl:dof, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics. A key advantage of this approach is that the control performance can be tuned individually for each direction. -This is particularly valuable when performance requirements differ between degrees of freedom - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational degrees of freedom can tolerate larger errors than others. +This is particularly valuable when performance requirements differ between directions - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational acrshortpl:dof can tolerate larger errors than others. -However, significant coupling exists between certain degrees of freedom, particularly between rotations and translations (e.g., $\epsilon_{R_x}/\mathcal{F}_y$ or $\epsilon_{D_y}/\bm\mathcal{M}_x$). +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. More sophisticated control strategies will be explored during the detailed design phase. @@ -5526,9 +5547,9 @@ The loop gain of an individual control channel is shown in Figure\nbsp{}ref:fig: \end{bmatrix}, \quad K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}} \end{equation} -The stability of the MIMO feedback loop is analyzed through the /characteristic loci/ method. +The stability of the acrshort:mimo feedback loop is analyzed through the /characteristic loci/ method. Such characteristic loci represent the eigenvalues of the loop gain matrix $\bm{G}(j\omega)\bm{K}(j\omega)$ plotted in the complex plane as the frequency varies from $0$ to $\infty$. -For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point\nbsp{}[[cite:&skogestad07_multiv_feedb_contr]]. +For acrshort:mimo systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point\nbsp{}[[cite:&skogestad07_multiv_feedb_contr]]. As shown in Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_root_locus, all loci remain to the right of the $-1$ point, validating the stability of the closed-loop system. Additionally, the distance of the loci from the $-1$ point provides information about stability margins of the coupled system. @@ -5557,7 +5578,7 @@ The control architecture developed for the uniaxial and the rotating models was Two fundamental choices were first addressed: the selection between centralized and decentralized approaches and the choice of control space. While control in Cartesian space enables direction-specific performance tuning, implementation in strut space was selected for the conceptual design phase due to two key advantages: good decoupling at low frequencies and identical diagonal terms in the plant transfer functions, allowing a single controller design to be replicated across all struts. -The HAC-LAC strategy was then implemented. +The acrshort:haclac strategy was then implemented. The inner loop implements decentralized Integral Force Feedback for active damping. The collocated nature of the force sensors ensures stability despite strong coupling between struts at resonance frequencies, enabling effective damping of structural modes. The outer loop implements High Authority Control, enabling precise positioning of the mobile platform. @@ -5628,7 +5649,7 @@ As established in the previous section on Stewart platforms, the proposed contro For the Nano Active Stabilization System, computing the positioning errors in the frame of the struts involves three key steps. 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 point of interest needs to be positioned. +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. @@ -5682,9 +5703,9 @@ 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 degrees of freedom: three translations ($D_x$, $D_y$, $D_z$) and two rotations ($R_x$, $R_y$). +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 degree of freedom ($R_z$) is still required to compute the errors in the frame of the nano-hexapod struts (i.e. to compute the nano-hexapod inverse kinematics). +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 measured sample pose is represented by the homogeneous transformation matrix $\bm{T}_{\text{sample}}$, as shown in equation\nbsp{}eqref:eq:nass_sample_pose. @@ -5735,7 +5756,7 @@ 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 degrees of freedom, while the sixth degree of freedom (Rz) 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 nano-hexapod 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. @@ -5754,7 +5775,7 @@ Then, the high authority controller uses the computed errors in the frame of the <> **** Introduction :ignore: -Building on the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the HAC-LAC strategy. +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. @@ -5792,7 +5813,7 @@ The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_r 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. -However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining robust stability properties. +However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining good robustness. #+name: fig:nass_iff_plant_effect_rotation_payload #+caption: Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload}) @@ -5841,7 +5862,7 @@ The overall gain was then increased to obtain a large loop gain around the reson [[file:figs/nass_iff_loop_gain.png]] To verify stability, the root loci for the three payload configurations were computed, as shown in Figure\nbsp{}ref:fig:nass_iff_root_locus. -The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability of the applied decentralized IFF. +The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robustness of the applied decentralized IFF. #+name: fig:nass_iff_root_locus #+caption: Root Loci for Decentralized IFF for three payload masses. The closed-loop poles are shown by the black crosses. @@ -5973,7 +5994,7 @@ This coupling introduces complex behavior that is difficult to model and predict The soft nano-hexapod configuration was evaluated using a stiffness of $0.01\,N/\mu m$ 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 center of mass 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 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. #+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}) @@ -6032,9 +6053,9 @@ The Nano Active Stabilization System concept was validated through time-domain s 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). -The primary requirement stipulates that the point of interest must remain within beam dimensions throughout operation. +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 DAC and voltage amplifiers, were not included in this analysis, as these parameters will be optimized during the detailed design phase. +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). 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. @@ -6124,7 +6145,7 @@ The multi-body modeling approach proved essential for capturing the complex dyna 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. -The HAC-LAC control strategy was successfully adapted to address the unique challenges presented by the rotating NASS. +The acrshort:haclac control strategy was successfully adapted to address the unique challenges presented by the rotating NASS. 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. @@ -6147,7 +6168,7 @@ This chapter begins by determining the optimal geometric configuration for the n 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. -Section\nbsp{}ref:sec:detail_fem introduces a hybrid modeling methodology that combines finite element analysis 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 nano-hexapod 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. @@ -6257,7 +6278,7 @@ Stewart platforms incorporating force sensors are frequently used for vibration Inertial sensors (accelerometers and geophones) are commonly employed in vibration isolation applications\nbsp{}[[cite:&chen03_payload_point_activ_vibrat_isolat;&chi15_desig_exper_study_vcm_based]]. These sensors are predominantly aligned with the struts\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six;&li01_simul_fault_vibrat_isolat_point;&thayer02_six_axis_vibrat_isolat_system;&zhang11_six_dof;&jiao18_dynam_model_exper_analy_stewar;&tang18_decen_vibrat_contr_voice_coil]], although they may also be fixed to the top platform\nbsp{}[[cite:&wang16_inves_activ_vibrat_isolat_stewar]]. -For high-precision positioning applications, various displacement sensors are implemented, including LVDTs\nbsp{}[[cite:&thayer02_six_axis_vibrat_isolat_system;&kim00_robus_track_contr_desig_dof_paral_manip;&li01_simul_fault_vibrat_isolat_point;&thayer98_stewar]], capacitive sensors\nbsp{}[[cite:&ting07_measur_calib_stewar_microm_system;&ting13_compos_contr_desig_stewar_nanos_platf]], eddy current sensors\nbsp{}[[cite:&chen03_payload_point_activ_vibrat_isolat;&furutani04_nanom_cuttin_machin_using_stewar]], and strain gauges\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]]. +For high-precision positioning applications, various displacement sensors are implemented, including acrfullpl:lvdt \nbsp{}[[cite:&thayer02_six_axis_vibrat_isolat_system;&kim00_robus_track_contr_desig_dof_paral_manip;&li01_simul_fault_vibrat_isolat_point;&thayer98_stewar]], capacitive sensors\nbsp{}[[cite:&ting07_measur_calib_stewar_microm_system;&ting13_compos_contr_desig_stewar_nanos_platf]], eddy current sensors\nbsp{}[[cite:&chen03_payload_point_activ_vibrat_isolat;&furutani04_nanom_cuttin_machin_using_stewar]], and strain gauges\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]]. Notably, some designs incorporate external sensing methodologies rather than integrating sensors within the struts\nbsp{}[[cite:&li01_simul_fault_vibrat_isolat_point;&chen03_payload_point_activ_vibrat_isolat;&ting13_compos_contr_desig_stewar_nanos_platf]]. A recent design\nbsp{}[[cite:&naves21_desig_optim_large_strok_flexur_mechan]], although not strictly speaking a Stewart platform, has demonstrated the use of 3-phase rotary motors with rotary encoders for achieving long-stroke and highly repeatable positioning, as illustrated in Figure\nbsp{}ref:fig:detail_kinematics_naves. @@ -6505,7 +6526,7 @@ These conditions are studied in Section\nbsp{}ref:ssec:detail_kinematics_cubic_d \end{equation} In the frame of the struts, the equations of motion\nbsp{}eqref:eq:detail_kinematics_transfer_function_struts are well decoupled at low frequency. -This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is well decoupled and SISO control may be applied for each strut, independently of the payload being used. +This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is well decoupled and acrfull:siso control may be applied for each strut, independently of the payload being used. \begin{equation}\label{eq:detail_kinematics_transfer_function_struts} \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-\intercal} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1} @@ -6639,7 +6660,7 @@ In that case, the location of the top joints can be expressed by equation\nbsp{} The stiffness matrix is therefore diagonal when the considered $\{B\}$ frame is located at the center of the cube (shown by frame $\{C\}$). This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center). -This specific location where the stiffness matrix is diagonal is referred to as the "Center of Stiffness" (analogous to the "Center of Mass" where the mass matrix is diagonal). +This specific location where the stiffness matrix is diagonal is referred to as the acrfull:cok, analogous to the acrfull:com where the mass matrix is diagonal. ***** Effect of having frame $\{B\}$ off-centered @@ -6725,7 +6746,7 @@ At high frequency, the behavior is governed by the mass matrix (evaluated at fra \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1} \end{equation} -To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the $\{B\}$ frame, and the principal axes of inertia must align with the axes of the $\{B\}$ frame. +To achieve a diagonal mass matrix, the acrlong:com of the mobile components must coincide with the $\{B\}$ frame, and the principal axes of inertia must align with the axes of the $\{B\}$ frame. #+name: fig:detail_kinematics_cubic_payload #+caption: Cubic stewart platform with top cylindrical payload @@ -6734,8 +6755,8 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure\nbsp{}ref:fig:detail_kinematics_cubic_payload). Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames. -When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_com). -Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_cok). +When the $\{B\}$ frame was positioned at the acrlong:com, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_com). +Conversely, when positioned at the acrlong:cok, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_cok). #+name: fig:detail_kinematics_cubic_cart_coupling #+caption: Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the $\{B\}$ frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}). @@ -6757,8 +6778,8 @@ Conversely, when positioned at the center of stiffness, coupling occurred at hig ***** Payload's CoM at the cube's center -An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components\nbsp{}[[cite:&li01_simul_fault_vibrat_isolat_point]]. -This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload). +An effective strategy for improving dynamical performances involves aligning the cube's center (acrlong:cok) with the acrlong:com of the moving components\nbsp{}[[cite:&li01_simul_fault_vibrat_isolat_point]]. +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. @@ -6787,7 +6808,7 @@ If a design similar to Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_pay The analysis of dynamical properties of the cubic architecture yields several important conclusions. Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames $\{A\}$ and $\{B\}$ are positioned at the cube's center. Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements. -Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame $\{B\}$. +Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's acrlong:com with reference frame $\{B\}$. While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical. **** Decentralized Control @@ -6871,13 +6892,13 @@ Both the cubic and non-cubic configurations exhibited similar coupling character <> ***** Introduction :ignore: -As demonstrated in Section\nbsp{}ref:ssec:detail_kinematics_cubic_dynamic, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. +As demonstrated in Section\nbsp{}ref:ssec:detail_kinematics_cubic_dynamic, the cubic architecture can exhibit advantageous dynamical properties when the acrlong:com of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. As shown in Section\nbsp{}ref:ssec:detail_kinematics_cubic_static, the stiffness matrix is diagonal when the considered $\{B\}$ frame is located at the cube's center. However, the $\{B\}$ frame is typically positioned above the top platform where forces are applied and displacements are measured. This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration. -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 center of mass relative to the mobile platform (coincident with the cube's center). +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 $H_{CoM} = 20\,mm$. @@ -7013,19 +7034,19 @@ The analysis of decentralized control in the frame of the struts revealed more n While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described. Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar. -Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center. +Fully decoupled dynamics in the Cartesian frame can be achieved when the acrlong:com of the moving body coincides with the cube's center. However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications. 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 center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame. +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 <> **** 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 point of interest, which is positioned $150\,mm$ above the top platform. +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. 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 @@ -7081,8 +7102,8 @@ This geometry serves as the foundation for estimating required actuator stroke ( 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 center of mass coincides with the cube's center. -Given the impracticality of consistently aligning the center of mass with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. +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. **** Required Actuator stroke <> @@ -7126,7 +7147,7 @@ While cubic architectures are prevalent in the literature and attributed with be 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. -For the cubic configuration, complete dynamical decoupling in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center, but this arrangement is often impractical for real-world applications. +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. 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. @@ -7136,8 +7157,8 @@ This led to a practical design approach where struts were oriented more vertical <> *** Introduction :ignore: -During the nano-hexapod's detailed design phase, a hybrid modeling approach combining finite element analysis 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 FEM for real-time control scenarios. +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. 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). @@ -7148,20 +7169,20 @@ Through this approach, system-level dynamic behavior under closed-loop control c **** Introduction :ignore: Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models. -These components are traditionally analyzed using Finite Element Analysis (FEA) software. +These components are traditionally analyzed using acrshort:fea software. However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models\nbsp{}[[cite:&hatch00_vibrat_matlab_ansys]]. This combined multibody-FEA modeling approach presents significant advantages, as it enables the accurate FE modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system\nbsp{}[[cite:&rankers98_machin]]. The investigation of this hybrid modeling approach is structured in three sections. First, the fundamental principles and methodological approaches of this modeling framework are introduced (Section\nbsp{}ref:ssec:detail_fem_super_element_theory). -It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section\nbsp{}ref:ssec:detail_fem_super_element_example). +It is then illustrated through its practical application to the modelling of an acrfull:apa (Section\nbsp{}ref:ssec:detail_fem_super_element_example). Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section\nbsp{}ref:ssec:detail_fem_super_element_validation). **** Procedure <> -In this modeling approach, some components within the multi-body framework are represented as /reduced-order flexible bodies/, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models. -These matrices are generated via modal reduction techniques, specifically through the application of component mode synthesis (CMS), thus establishing this design approach as a combined multibody-FEA methodology. +In this modeling approach, some components within the multi-body framework are represented as /reduced-order flexible bodies/, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from acrshort:fea models. +These matrices are generated via modal reduction techniques, specifically through the application of component mode synthesis, thus establishing this design approach as a combined multibody-FEA methodology. Standard FEA implementations typically involve thousands or even hundreds of thousands of DoF, rendering direct integration into multi-body simulations computationally prohibitive. The objective of modal reduction is therefore to substantially decrease the number of DoF while preserving the essential dynamic characteristics of the component. @@ -7173,8 +7194,8 @@ These frames serve multiple functions, including connecting to other parts, appl Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method\nbsp{}[[cite:&craig68_coupl_subst_dynam_analy]] (also known as the "fixed-interface method"), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics. This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF. -The number of degrees of freedom in the reduced model is determined by\nbsp{}eqref:eq:detail_fem_model_order where $n$ represents the number of defined frames and $p$ denotes the number of additional modes to be modeled. -The outcome of this procedure is an $m \times m$ set of reduced mass and stiffness matrices, $m$ being the total retained number of degrees of freedom, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. +The number of acrshortpl:dof in the reduced model is determined by\nbsp{}eqref:eq:detail_fem_model_order where $n$ represents the number of defined frames and $p$ denotes the number of additional modes to be modeled. +The outcome of this procedure is an $m \times m$ set of reduced mass and stiffness matrices, $m$ being the total retained number of acrshortpl:dof, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. \begin{equation}\label{eq:detail_fem_model_order} m = 6 \times n + p @@ -7184,13 +7205,13 @@ m = 6 \times n + p <> ***** Introduction :ignore: -The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons. +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. 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 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]]. -The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework. -The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation. +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]]. +The selection of the acrshort:apa for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework. +The specific design of the acrshort:apa allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation. #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage @@ -7215,7 +7236,7 @@ The specific design of the APA allows for the simultaneous modeling of a mechani ***** Finite Element Model -The development of the finite element model for the APA95ML required the knowledge of the material properties, as summarized in Table\nbsp{}ref:tab:detail_fem_material_properties. +The development of the acrfull:fem for the APA95ML required the knowledge of the material properties, as summarized in Table\nbsp{}ref:tab:detail_fem_material_properties. The finite element mesh, shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_mesh, was then generated. #+name: tab:detail_fem_material_properties @@ -7314,7 +7335,7 @@ From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained. ***** Identification of the APA Characteristics -Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications. +Initial validation of the acrlong:fem and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications. 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. @@ -7334,7 +7355,7 @@ The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ rela 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 high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model. +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. **** Experimental Validation <> @@ -7345,8 +7366,8 @@ The goal was to measure the dynamics of the APA95ML and to compare it with predi 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 granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement $y$. -A digital-to-analog converter (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 analog-to-digital converter (ADC). +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. #+name: fig:detail_fem_apa95ml_bench_schematic #+caption: Test bench used to validate "reduced order solid bodies" using an APA95ML. @@ -7360,10 +7381,10 @@ The identification procedure required careful choice of the excitation signal\nb During all this experimental work, random noise excitation was predominantly employed. The designed excitation signal is then generated and both input and output signals are synchronously acquired. -From the obtained input and output data, the frequency response functions were derived. +From the obtained input and output data, the acrshortpl:frf were derived. To improve the quality of the obtained frequency domain data, averaging and windowing were used\nbsp{}[[cite:&pintelon12_system_ident, chap. 13]]. -The obtained frequency response functions 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 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 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. @@ -7401,7 +7422,7 @@ The IFF controller implementation, defined in equation\nbsp{}ref:eq:detail_fem_i The theoretical damped dynamics of the closed-loop system was estimated using the model by computed the root locus plot shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_iff_root_locus. For experimental validation, six gain values were tested: $g = [0,\,10,\,50,\,100,\,500,\,1000]$. -The measured frequency responses for each gain configuration were compared with model predictions, as presented in Figure\nbsp{}ref:fig:detail_fem_apa95ml_damped_plants. +The measured acrshortpl:frf for each gain configuration were compared with model predictions, as presented in Figure\nbsp{}ref:fig:detail_fem_apa95ml_damped_plants. The close agreement between experimental measurements and theoretical predictions across all gain configurations demonstrates the model's capability to accurately predict both open-loop and closed-loop system dynamics. @@ -7482,14 +7503,14 @@ Conventional piezoelectric stack actuators (shown in Figure\nbsp{}ref:fig:detail 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. -Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through a specific mechanical design. +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. Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic;&verma20_dynam_stabil_thin_apert_light]]. -Moreover, using APA for active damping has been successfully demonstrated in similar applications\nbsp{}[[cite:&hanieh03_activ_stewar]]. +Moreover, using acrshort:apa for active damping has been successfully demonstrated in similar applications\nbsp{}[[cite:&hanieh03_activ_stewar]]. -Several specific APA models were evaluated against the established specifications (Table\nbsp{}ref:tab:detail_fem_piezo_act_models). +Several specific acrshort:apa models were evaluated against the established specifications (Table\nbsp{}ref:tab:detail_fem_piezo_act_models). The APA300ML emerged as the optimal choice. -This selection was further reinforced by previous experience with APAs from the same manufacturer[fn:detail_fem_2], and particularly by the successful validation of the modeling methodology with a similar actuator (Section\nbsp{}ref:ssec:detail_fem_super_element_example). +This selection was further reinforced by previous experience with acrshortpl:apa from the same manufacturer[fn:detail_fem_2], and particularly by the successful validation of the modeling methodology with a similar actuator (Section\nbsp{}ref:ssec:detail_fem_super_element_example). 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 @@ -7553,19 +7574,19 @@ It considers only axial behavior, treating the actuator as infinitely rigid in o Several physical characteristics are not explicitly represented, including the mechanical amplification factor and the actual stress the piezoelectric stacks. Nevertheless, the model's primary advantage lies in its simplicity, adding only four states to the system model. -The model requires tuning of 8 parameters ($k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$, and $g_a$) to match the dynamics extracted from the finite element analysis. +The model requires tuning of 8 parameters ($k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$, and $g_a$) to match the dynamics extracted from the acrshort:fea. The shell parameters $k_1$ and $c_1$ were determined first through analysis of the zero in the $V_a$ to $V_s$ transfer function. The physical interpretation of this zero can be understood through Root Locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros. The open-loop zero therefore corresponds to the poles of the system with a theoretical infinite-gain controller that ensures zero force in the sensor stack. -This condition effectively represents the dynamics of an APA without the force sensor stack (i.e. an APA with only the shell). +This condition effectively represents the dynamics of an acrshort:apa without the force sensor stack (i.e. an acrshort:apa with only the shell). This physical interpretation enables straightforward parameter tuning: $k_1$ determines the frequency of the zero, while $c_1$ defines its damping characteristic. The stack parameters ($k_a$, $c_a$, $k_e$, $c_e$) were then derived from the first pole of the $V_a$ to $y$ response. Given that identical piezoelectric stacks are used for both sensing and actuation, the relationships $k_e = 2k_a$ and $c_e = 2c_a$ were enforced, reflecting the series configuration of the dual actuator stacks. Finally, the sensitivities $g_s$ and $g_a$ were adjusted to match the DC gains of the respective transfer functions. -The resulting parameters, listed in Table\nbsp{}ref:tab:detail_fem_apa300ml_2dof_parameters, yield dynamic behavior that closely matches the high-order finite element model, as demonstrated in Figure\nbsp{}ref:fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof. +The resulting parameters, listed in Table\nbsp{}ref:tab:detail_fem_apa300ml_2dof_parameters, yield dynamic behavior that closely matches the high-order acrshort:fem, as demonstrated in Figure\nbsp{}ref:fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof. While higher-order modes and non-axial flexibility are not captured, the model accurately represents the fundamental dynamics within the operational frequency range. #+name: tab:detail_fem_apa300ml_2dof_parameters @@ -7609,30 +7630,30 @@ The behavior of piezoelectric actuators is characterized by coupled constitutive To evaluate the impact of electrical boundary conditions on the system dynamics, experimental measurements were conducted using the APA95ML, comparing the transfer function from $V_a$ to $y$ under two distinct configurations. With the force sensor stack in open-circuit condition (analogous to voltage measurement with high input impedance) and in short-circuit condition (similar to charge measurement with low output impedance). As demonstrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_effect_electrical_boundaries, short-circuiting the force sensor stack results in a minor decrease in resonance frequency. -The developed models of the APA do not represent such behavior, but as this effect is quite small, this validates the simplifying assumption made in the models. +The developed models of the acrshort:apa do not represent such behavior, but as this effect is quite small, this validates the simplifying assumption made in the models. #+name: fig:detail_fem_apa95ml_effect_electrical_boundaries #+caption: Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency #+attr_latex: :scale 0.8 [[file:figs/detail_fem_apa95ml_effect_electrical_boundaries.png]] -However, the electrical characteristics of the APA remain crucial for instrumentation design. +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. **** Validation with the Nano-Hexapod <> -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 APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation. +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 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. Additionally, the model predicts no problematic high-frequency modes in the dynamics from $\bm{u}$ to $\bm{\epsilon}_{\mathcal{L}}$ (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant), maintaining consistency with earlier conceptual simulations. These findings suggest that the control performance targets established during the conceptual phase remain achievable with the selected actuator. -Comparative analysis between the high-order 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. +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 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 nano-hexapod. These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the nano-hexapod. @@ -7687,11 +7708,11 @@ For design simplicity and component standardization, identical joints are employ #+end_subfigure #+end_figure -While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other degrees of freedom, practical implementations exhibit parasitic stiffness that can impact control performance\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]]. +While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other acrshortpl:dof, practical implementations exhibit parasitic stiffness that can impact control performance\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]]. This section examines how these non-ideal characteristics affect system behavior, focusing particularly on bending/torsional stiffness (Section\nbsp{}ref:ssec:detail_fem_joint_bending) and axial compliance (Section\nbsp{}ref:ssec:detail_fem_joint_axial). 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 finite element analysis (Section\nbsp{}ref:ssec:detail_fem_joint_specs). +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. **** Bending and Torsional Stiffness @@ -7762,12 +7783,12 @@ Therefore, determining the minimum acceptable axial stiffness that maintains nan 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). -The resulting frequency responses (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_plants) reveal distinct effects on system dynamics. +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 frequency response data (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). +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). However, the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ demonstrates significant effects: internal strut modes appear at high frequencies, introducing substantial cross-coupling between axes. -This coupling is quantified through RGA analysis of the damped system (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_rga_hac_plant), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance. +This coupling is quantified through acrfull:rga analysis of the damped system (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_rga_hac_plant), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance. Above this resonance frequency, two critical limitations emerge. First, the system exhibits strong coupling between control channels, making decentralized control strategies ineffective. @@ -7834,11 +7855,11 @@ Based on the dynamic analysis presented in previous sections, quantitative speci 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. 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 degrees of freedom. +Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational acrshortpl:dof. -The joint geometry was optimized through parametric finite element analysis. +The joint geometry was optimized through parametric acrshort:fea. The optimization process revealed an inherent trade-off between maximizing axial stiffness and achieving sufficiently low bending/torsional stiffness, while maintaining material stresses within acceptable limits. -The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through finite element analysis and summarized in Table\nbsp{}ref:tab:detail_fem_joints_specs. +The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through acrshort:fea and summarized in Table\nbsp{}ref:tab:detail_fem_joints_specs. #+name: fig:detail_fem_joints_design #+caption: Designed flexible joints. @@ -7861,7 +7882,7 @@ The final design, featuring a neck dimension of 0.25mm, achieves mechanical prop **** Validation with the Nano-Hexapod <> -The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from finite element analysis. +The designed flexible joint was first validated through integration into the nano-hexapod 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. @@ -7875,7 +7896,7 @@ To improve computational efficiency, a low order representation was developed us 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 $k_a$). This simplification reduces the total model order to 48 states: 12 for the payload, 12 for the struts, and 24 for the joints (12 each for bottom and top joints). -While additional degrees of freedom could potentially capture more dynamic features, the selected configuration preserves essential system characteristics while minimizing computational complexity. +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}). @@ -7901,7 +7922,7 @@ While additional degrees of freedom could potentially capture more dynamic featu :END: <> -In this chapter, the methodology of combining finite element analysis 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 nano-hexapod 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. @@ -7920,15 +7941,15 @@ Such model reduction, guided by detailed understanding of component behavior, pr 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. During the conceptual design phase of the NASS, pragmatic approaches were implemented for each of these elements. -The High Authority Control-Low Authority Control (HAC-LAC) architecture was selected for combining sensors. -Control was implemented in the frame of the struts, leveraging the inherent low-frequency decoupling of the plant where all decoupled elements exhibited similar dynamics, thereby simplifying the Single-Input Single-Output (SISO) controller design process. +The acrfull:haclac architecture was selected for combining sensors. +Control was implemented in the frame of the struts, leveraging the inherent low-frequency decoupling of the plant where all decoupled elements exhibited similar dynamics, thereby simplifying the acrfull:siso controller design process. For these decoupled plants, open-loop shaping techniques were employed to tune the individual controllers. While these initial strategies proved effective in validating the NASS concept, this work explores alternative approaches with the potential to further enhance the performance. Section\nbsp{}ref:sec:detail_control_sensor examines different methods for combining multiple sensors, with particular emphasis on sensor fusion techniques that are based on complementary filters. A novel approach for designing these filters is proposed, which allows optimization of the sensor fusion effectiveness. -Section\nbsp{}ref:sec:detail_control_decoupling presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. +Section\nbsp{}ref:sec:detail_control_decoupling presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and acrfull:svd decoupling. Each method is evaluated in terms of its theoretical foundations, implementation requirements, and performance characteristics, providing insights into their respective advantages for different applications. Finally, Section\nbsp{}ref:sec:detail_control_cf addresses the challenge of controller design for decoupled plants. @@ -7943,13 +7964,13 @@ The literature review of Stewart platforms revealed a wide diversity of designs Control objectives (such as active damping, vibration isolation, or precise positioning) directly dictate sensor selection, whether inertial, force, or relative position sensors. In cases where multiple control objectives must be achieved simultaneously, as is the case for the Nano Active Stabilization System (NASS) where the Stewart platform must both position the sample and provide isolation from micro-station vibrations, combining multiple sensors within the control architecture has been demonstrated to yield significant performance benefits\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six]]. -From the literature, three principal approaches for combining sensors have been identified: High Authority Control-Low Authority Control (HAC-LAC), sensor fusion, and two-sensor control architectures. +From the literature, three principal approaches for combining sensors have been identified: acrlong:haclac, sensor fusion, and two-sensor control architectures. #+name: fig:detail_control_control_multiple_sensors #+caption: Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC LAC} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC-LAC} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 1 @@ -7971,7 +7992,7 @@ From the literature, three principal approaches for combining sensors have been #+end_subfigure #+end_figure -The HAC-LAC approach employs a dual-loop control strategy in which two control loops are using different sensors for distinct purposes (Figure\nbsp{}ref:fig:detail_control_sensor_arch_hac_lac). +The acrshort:haclac approach employs a dual-loop control strategy in which two control loops are using different sensors for distinct purposes (Figure\nbsp{}ref:fig:detail_control_sensor_arch_hac_lac). In\nbsp{}[[cite:&li01_simul_vibrat_isolat_point_contr]], vibration isolation is provided by accelerometers collocated with the voice coil actuators, while external rotational sensors are used to achieve pointing control. In\nbsp{}[[cite:&geng95_intel_contr_system_multip_degree]], force sensors collocated with the magnetostrictive actuators are used for active damping using decentralized IFF, and subsequently accelerometers are employed for adaptive vibration isolation. Similarly, in\nbsp{}[[cite:&wang16_inves_activ_vibrat_isolat_stewar]], piezoelectric actuators with collocated force sensors are used in a decentralized manner to provide active damping while accelerometers are implemented in an adaptive feedback loop to suppress periodic vibrations. @@ -7989,15 +8010,15 @@ In\nbsp{}[[cite:&thayer02_six_axis_vibrat_isolat_system]], the use of force sens Geophones are shown to provide better isolation performance than load cells but suffer from poor robustness. Conversely, the controller based on force sensors exhibited inferior performance (due to the presence of a pair of low frequency zeros), but demonstrated better robustness properties. A "two-sensor control" approach was proven to perform better than controllers based on individual sensors while maintaining better robustness. -A Linear Quadratic Regulator (LQG) was employed to optimize the two-input/one-output controller. +A acrfull:lqg was employed to optimize the two-input/one-output controller. Beyond these three main approaches, other control architectures have been proposed for different purposes. For instance, in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]], a first control loop based on force sensors and relative motion sensors is implemented to compensate for parasitic stiffness of the flexible joints. Subsequently, the system is decoupled in the modal space (facilitated by the removal of parasitic stiffness) and accelerometers are employed for vibration isolation. -The HAC-LAC architecture was previously investigated during the conceptual phase and successfully implemented to validate the NASS concept, demonstrating excellent performance. +The acrshort:haclac architecture was previously investigated during the conceptual phase and successfully implemented to validate the NASS concept, demonstrating excellent performance. At the other end of the spectrum, the two-sensor approach yields greater control design freedom but introduces increased complexity in tuning, and thus was not pursued in this study. -This work instead focuses on sensor fusion, which represents a promising middle ground between the proven HAC-LAC approach and the more complex two-sensor control strategy. +This work instead focuses on sensor fusion, which represents a promising middle ground between the proven acrshort:haclac approach and the more complex two-sensor control strategy. A review of sensor fusion is first presented in Section\nbsp{}ref:ssec:detail_control_sensor_review. Then, in Section\nbsp{}ref:ssec:detail_control_sensor_fusion_requirements, both the robustness of the fusion and the noise characteristics of the resulting "fused sensor" are derived and expressed as functions of the complementary filters' norms. @@ -8039,7 +8060,7 @@ Various design methods have been developed to optimize complementary filters. The most straightforward approach is based on analytical formulas, which depending on the application may be first order\nbsp{}[[cite:&corke04_inert_visual_sensin_system_small_auton_helic;&yeh05_model_contr_hydraul_actuat_two;&yong16_high_speed_vertic_posit_stage]], second order\nbsp{}[[cite:&baerveldt97_low_cost_low_weigh_attit;&stoten01_fusion_kinet_data_using_compos_filter;&jensen13_basic_uas]], or higher orders\nbsp{}[[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&stoten01_fusion_kinet_data_using_compos_filter;&collette15_sensor_fusion_method_high_perfor;&matichard15_seism_isolat_advan_ligo]]. Since the characteristics of the super sensor depend on proper complementary filter design\nbsp{}[[cite:&dehaeze19_compl_filter_shapin_using_synth]], several optimization techniques have emerged—ranging from optimizing parameters for analytical formulas\nbsp{}[[cite:&jensen13_basic_uas;&carreira15_compl_filter_desig_three_frequen_bands]] to employing convex optimization tools\nbsp{}[[cite:&hua04_polyp_fir_compl_filter_contr_system;&hua05_low_ligo]] such as linear matrix inequalities\nbsp{}[[cite:&pascoal99_navig_system_desig_using_time]]. As demonstrated in\nbsp{}[[cite:&plummer06_optim_compl_filter_their_applic_motion_measur]], complementary filter design can be linked to the standard mixed-sensitivity control problem, allowing powerful classical control theory tools to be applied. -For example, in\nbsp{}[[cite:&jensen13_basic_uas]], two gains of a Proportional Integral (PI) controller are optimized to minimize super sensor noise. +For example, in\nbsp{}[[cite:&jensen13_basic_uas]], two gains of a acrfull:pi controller are optimized to minimize super sensor noise. All these complementary filter design methods share the common objective of creating a super sensor with desired characteristics, typically in terms of noise and dynamics. As reported in\nbsp{}[[cite:&zimmermann92_high_bandw_orien_measur_contr;&plummer06_optim_compl_filter_their_applic_motion_measur]], phase shifts and magnitude bumps in the super sensor dynamics may occur if complementary filters are poorly designed or if sensors are improperly calibrated. @@ -8070,7 +8091,7 @@ The complementary property of filters $H_1(s)$ and $H_2(s)$ requires that the su ***** Sensor Models and Sensor Normalization To analyze sensor fusion architectures, appropriate sensor models are required. -The model shown in Figure\nbsp{}ref:fig:detail_control_sensor_model consists of a linear time invariant (LTI) system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing sensor noise. +The model shown in Figure\nbsp{}ref:fig:detail_control_sensor_model consists of a acrfull:lti system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing sensor noise. The model input $x$ is the measured physical quantity, and its output $\tilde{x}_i$ is the "raw" output of the sensor. Prior to filtering the sensor outputs $\tilde{x}_i$ with complementary filters, the sensors are typically normalized to simplify the fusion process. @@ -8134,7 +8155,7 @@ The estimation error $\epsilon_x$, defined as the difference between the sensor \epsilon_x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2 \end{equation} -As shown in\nbsp{}eqref:eq:detail_control_sensor_noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\epsilon_x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$. +As shown in\nbsp{}eqref:eq:detail_control_sensor_noise_filtering_psd, the acrfull:psd of the estimation error $\Phi_{\epsilon_x}$ depends both on the norm of the two complementary filters and on the acrshort:psd of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$. \begin{equation}\label{eq:detail_control_sensor_noise_filtering_psd} \Phi_{\epsilon_x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega) @@ -8351,7 +8372,7 @@ This straightforward example demonstrates that the proposed methodology for shap <> Certain applications necessitate the fusion of more than two sensors\nbsp{}[[cite:&stoten01_fusion_kinet_data_using_compos_filter;&carreira15_compl_filter_desig_three_frequen_bands]]. -At LIGO, for example, a super sensor is formed by merging three distinct sensors: an LVDT, a seismometer, and a geophone\nbsp{}[[cite:&matichard15_seism_isolat_advan_ligo]]. +At LIGO, for example, a super sensor is formed by merging three distinct sensors: a acrshort:lvdt, a seismometer, and a geophone\nbsp{}[[cite:&matichard15_seism_isolat_advan_ligo]]. For merging $n>2$ sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_fusion_three. Fusion can be implemented either "sequentially," using $n-1$ sets of two complementary filters (Figure\nbsp{}ref:fig:detail_control_sensor_fusion_three_sequential), or "in parallel," employing a single set of $n$ complementary filters (Figure\nbsp{}ref:fig:detail_control_sensor_fusion_three_parallel). @@ -8462,7 +8483,7 @@ This approach allows shaping of the filter magnitudes through the use of weighti This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude. Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters. -For the NASS, the HAC-LAC strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune. +For the NASS, the acrshort:haclac strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune. Looking forward, it would be interesting to investigate how sensor fusion (particularly between the force sensors and external metrology) compares to the HAC-IFF approach in terms of performance and robustness. *** Decoupling @@ -8470,20 +8491,20 @@ Looking forward, it would be interesting to investigate how sensor fusion (parti **** Introduction :ignore: -The control of parallel manipulators (and any MIMO system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of SISO control for the decoupled plant (discussed in section\nbsp{}ref:sec:detail_control_cf). +The control of parallel manipulators (and any acrshort:mimo system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of acrshort:siso control for the decoupled plant (discussed in section\nbsp{}ref:sec:detail_control_cf). When sensors are integrated within the struts, decentralized control may be applied, as the system is already well decoupled at low frequency. -For instance,\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar]] implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate PI controllers for each strut. +For instance,\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar]] implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate acrshort:pi controllers for each strut. A similar control architecture was proposed in\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]] using strain gauge sensors integrated in each strut. An alternative strategy involves decoupling the system in the Cartesian frame using Jacobian matrices. As demonstrated during the study of Stewart platform kinematics, Jacobian matrices can be used to map actuator forces to forces and torques applied on the top platform. This approach enables the implementation of controllers in a defined frame. It has been applied with various sensor types including force sensors\nbsp{}[[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], relative displacement sensors\nbsp{}[[cite:&kim00_robus_track_contr_desig_dof_paral_manip]], and inertial sensors\nbsp{}[[cite:&li01_simul_vibrat_isolat_point_contr;&abbas14_vibrat_stewar_platf]]. -The Cartesian frame in which the system is decoupled is typically chosen at the point of interest (i.e., where the motion is of interest) or at the center of mass. +The Cartesian frame in which the system is decoupled is typically chosen at the acrshort:poi (i.e., where the motion is of interest) or at the acrlong:com. Modal decoupling represents another noteworthy decoupling strategy, wherein the "local" plant inputs and outputs are mapped to the modal space. -In this approach, multiple SISO plants, each corresponding to a single mode, can be controlled independently. +In this approach, multiple acrshort:siso plants, each corresponding to a single mode, can be controlled independently. This decoupling strategy has been implemented for active damping applications\nbsp{}[[cite:&holterman05_activ_dampin_based_decoup_colloc_contr]], which is logical as it is often desirable to dampen specific modes. The strategy has also been employed in\nbsp{}[[cite:&pu11_six_degree_of_freed_activ]] for vibration isolation purposes using geophones, and in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]] using force sensors. @@ -8506,10 +8527,10 @@ Finally, a comparative analysis with concluding observations is provided in Sect Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis. The system illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test is used for this purpose. -It possesses three degrees of freedom (DoF) and incorporates three parallel struts. +It possesses three acrshortpl:dof and incorporates three parallel struts. Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform. -Two reference frames are defined within this model: frame $\{M\}$ with origin $O_M$ at the center of mass of the solid body, and frame $\{K\}$ with origin $O_K$ at the center of stiffness of the parallel manipulator. +Two reference frames are defined within this model: frame $\{M\}$ with origin $O_M$ at the acrlong:com of the solid body, and frame $\{K\}$ with origin $O_K$ at the acrlong:cok of the parallel manipulator. #+attr_latex: :options [b]{0.60\linewidth} #+begin_minipage @@ -8535,7 +8556,7 @@ Two reference frames are defined within this model: frame $\{M\}$ with origin $O #+latex: \captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters} #+end_minipage -The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass\nbsp{}eqref:eq:detail_control_decoupling_model_eom, where $\bm{\mathcal{X}}_{\{M\}}$ represents the two translations and one rotation with respect to the center of mass, and $\bm{\mathcal{F}}_{\{M\}}$ denotes the forces and torque applied at the center of mass. +The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass\nbsp{}eqref:eq:detail_control_decoupling_model_eom, where $\bm{\mathcal{X}}_{\{M\}}$ represents the two translations and one rotation with respect to the acrlong:com, and $\bm{\mathcal{F}}_{\{M\}}$ denotes the forces and torque applied at the acrlong:com. \begin{equation}\label{eq:detail_control_decoupling_model_eom} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t), \quad @@ -8643,11 +8664,11 @@ The transfer function from $\bm{\mathcal{F}}_{\{O\}$ to $\bm{\mathcal{X}}_{\{O\} \end{equation} The frame $\{O\}$ can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice. -Two natural reference frames are particularly relevant: the center of mass and the center of stiffness. +Two natural reference frames are particularly relevant: the acrlong:com and the acrlong:cok. ***** Center Of Mass -When the decoupling frame is located at the center of mass (frame $\{M\}$ in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test), the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoM_inverse. +When the decoupling frame is located at the acrlong:com (frame $\{M\}$ in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test), the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoM_inverse. \begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse} \bm{J}_{\{M\}} = \begin{bmatrix} @@ -8681,7 +8702,7 @@ Consequently, the plant exhibits effective decoupling at frequencies above the h This strategy is typically employed in systems with low-frequency suspension modes\nbsp{}[[cite:&butler11_posit_contr_lithog_equip]], where the plant approximates decoupled mass lines. The low-frequency coupling observed in this configuration has a clear physical interpretation. -When a static force is applied at the center of mass, the suspended mass rotates around the center of stiffness. +When a static force is applied at the acrlong:com, the suspended mass rotates around the acrlong:cok. This rotation is due to torque induced by the stiffness of the first actuator (i.e. the one on the left side), which is not aligned with the force application point. This phenomenon is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test_CoM. @@ -8705,7 +8726,7 @@ This phenomenon is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling ***** Center Of Stiffness -When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoK_inverse. +When the decoupling frame is located at the acrlong:cok, the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoK_inverse. \begin{equation}\label{eq:detail_control_decoupling_jacobian_CoK_inverse} \bm{J}_{\{K\}} = \begin{bmatrix} @@ -8721,7 +8742,7 @@ When the decoupling frame is located at the center of stiffness, the Jacobian ma The frame $\{K\}$ was selected based on physical reasoning, positioned in line with the side strut and equidistant between the two vertical struts. However, it could alternatively be determined through analytical methods to ensure that $\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}$ forms a diagonal matrix. -It should be noted that the existence of such a center of stiffness (i.e. a frame $\{K\}$ for which $\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}$ is diagonal) is not guaranteed for arbitrary systems. +It should be noted that the existence of such a acrlong:cok (i.e. a frame $\{K\}$ for which $\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}$ is diagonal) is not guaranteed for arbitrary systems. This property is typically achievable only in systems exhibiting specific symmetrical characteristics, as is the case in the present example. The analytical expression for the plant in this configuration was then computed\nbsp{}eqref:eq:detail_control_decoupling_plant_CoK. @@ -8730,7 +8751,7 @@ The analytical expression for the plant in this configuration was then computed\ \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1} \end{equation} -Figure\nbsp{}ref:fig:detail_control_decoupling_jacobian_plant_CoK_results presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the center of stiffness. +Figure\nbsp{}ref:fig:detail_control_decoupling_jacobian_plant_CoK_results presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the acrlong:cok. The plant is well decoupled below the suspension mode with the lowest frequency\nbsp{}eqref:eq:detail_control_decoupling_plant_CoK_low_freq, making it particularly suitable for systems with high stiffness. \begin{equation}\label{eq:detail_control_decoupling_plant_CoK_low_freq} @@ -8738,7 +8759,7 @@ The plant is well decoupled below the suspension mode with the lowest frequency\ \end{equation} The physical reason for high-frequency coupling is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test_CoK. -When a high-frequency force is applied at a point not aligned with the center of mass, it induces rotation around the center of mass. +When a high-frequency force is applied at a point not aligned with the acrlong:com, it induces rotation around the acrlong:com. #+name: fig:detail_control_decoupling_jacobian_plant_CoK_results #+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}). @@ -8859,7 +8880,7 @@ For real matrices $\bm{X}$, the resulting $\bm{U}$ and $\bm{V}$ matrices are als ***** Decoupling using the SVD -The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a Frequency Response Function (FRF), which yields a complex matrix $\bm{G}(\omega_i)$ for multiple frequency points $\omega_i$. +The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a acrfull:frf, which yields a complex matrix $\bm{G}(\omega_i)$ for multiple frequency points $\omega_i$. A specific frequency is then selected for optimal decoupling, with the targeted crossover frequency $\omega_c$ often serving as an appropriate choice. Since real matrices are required for the decoupling transformation, a real approximation of the complex measured response at the selected frequency must be computed. @@ -8877,7 +8898,7 @@ These singular input and output matrices are then applied to decouple the system #+caption: Decoupled plant $\bm{G}_{\text{SVD}}$ using the Singular Value Decomposition [[file:figs/detail_control_decoupling_svd.png]] -Implementation of SVD decoupling requires access to the system's FRF, at least in the vicinity of the desired decoupling frequency. +Implementation of SVD decoupling requires access to the system's acrshort:frf, at least in the vicinity of the desired decoupling frequency. This information can be obtained either experimentally or derived from a model. While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency. Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping. @@ -8953,7 +8974,7 @@ The phenomenon potentially relates to previous research on SVD controllers appli While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table\nbsp{}ref:tab:detail_control_decoupling_strategies_comp. -Each method employs a distinct conceptual framework: Jacobian decoupling is "topology-driven", relying on the geometric configuration of the system; modal decoupling is "physics-driven", based on the system's dynamical equations; and SVD decoupling is "data-driven", using measured frequency response functions. +Each method employs a distinct conceptual framework: Jacobian decoupling is "topology-driven", relying on the geometric configuration of the system; modal decoupling is "physics-driven", based on the system's dynamical equations; and SVD decoupling is "data-driven", using measured acrshortpl:frf. The physical interpretation of decoupled plant inputs and outputs varies considerably among these methods. With Jacobian decoupling, inputs and outputs retain clear physical meaning, corresponding to forces/torques and translations/rotations in a specified reference frame. @@ -8967,7 +8988,7 @@ Modal decoupling provides a natural framework when specific vibrational modes re SVD decoupling generally results in a loss of physical meaning for the "control space", potentially complicating the process of relating controller design to practical system requirements. The quality of decoupling achieved through these methods also exhibits distinct characteristics. -Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the center of stiffness, or at high frequencies when aligned with the center of mass. +Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the acrlong:cok, or at high frequencies when aligned with the acrlong:com. Systems designed with coincident centers of mass and stiffness may achieve excellent decoupling using this approach. Modal decoupling offers good decoupling across all frequencies, though its effectiveness relies on the model accuracy, with discrepancies potentially resulting in significant off-diagonal elements. SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point. @@ -9001,15 +9022,15 @@ SVD decoupling can be implemented using measured data without requiring a model, **** Introduction :ignore: -Once the system is properly decoupled using one of the approaches described in Section\nbsp{}ref:sec:detail_control_decoupling, SISO controllers can be individually tuned for each decoupled "directions". +Once the system is properly decoupled using one of the approaches described in Section\nbsp{}ref:sec:detail_control_decoupling, acrshort:siso controllers can be individually tuned for each decoupled "directions". Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented. -In some cases "fixed" controller structures are used, such as PI and PID controllers, whose parameters are manually tuned\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar;&du14_piezo_actuat_high_precis_flexib;&yang19_dynam_model_decoup_contr_flexib]]. +In some cases "fixed" controller structures are used, such as acrshort:pi and acrshort:pid controllers, whose parameters are manually tuned\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar;&du14_piezo_actuat_high_precis_flexib;&yang19_dynam_model_decoup_contr_flexib]]. Another popular method is Open-Loop shaping, which was used during the conceptual phase. Open-loop shaping involves tuning the controller through a series of "standard" filters (leads, lags, notches, low-pass filters, ...) to shape the open-loop transfer function $G(s)K(s)$ according to desired specifications, including bandwidth, gain and phase margins\nbsp{}[[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 4.4.7]]. Open-Loop shaping is very popular because the open-loop transfer function is a linear function of the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics. -Another key advantage is that controllers can be tuned directly from measured frequency response functions of the plant without requiring an explicit model. +Another key advantage is that controllers can be tuned directly from measured acrshortpl:frf of the plant without requiring an explicit model. However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions. Specifications can therefore be expressed in terms of the magnitude of closed-loop transfer functions, such as the sensitivity, plant sensitivity, and complementary sensitivity transfer functions\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 3]]. @@ -9022,7 +9043,7 @@ $\mathcal{H}_{\infty}\text{-synthesis}$ has been applied for the Stewart platfor In this section, an alternative controller synthesis scheme is proposed in which complementary filters are used for directly shaping the closed-loop transfer functions (i.e., directly addressing the closed-loop performances). In Section\nbsp{}ref:ssec:detail_control_cf_control_arch, the proposed control architecture is presented. In Section\nbsp{}ref:ssec:detail_control_cf_trans_perf, typical performance requirements are translated into the shape of the complementary filters. -The design of the complementary filters is briefly discussed in Section\nbsp{}ref:ssec:detail_control_cf_analytical_complementary_filters, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real time. +The design of the complementary filters is briefly discussed in Section\nbsp{}ref:ssec:detail_control_cf_analytical_complementary_filters, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real-time. Finally, in Section\nbsp{}ref:ssec:detail_control_cf_simulations, a numerical example is used to show how the proposed control architecture can be implemented in practice. **** Control Architecture @@ -9105,7 +9126,7 @@ Therefore, performance requirements must be translated into constraints on the s A closed-loop system is stable when all its elements (here $K$, $G^\prime$, and $H_L$) are stable and the sensitivity function $S = \frac{1}{1 + G^\prime K H_L}$ is stable. For the nominal system ($G^\prime = G$), the sensitivity transfer function equals the high-pass filter: $S(s) = H_H(s)$. -Nominal stability is therefore guaranteed when $H_L$, $H_H$, and $G$ are stable, and both $G$ and $H_H$ are minimum phase (ensuring $K$ is stable). +acrfull:ns is therefore guaranteed when $H_L$, $H_H$, and $G$ are stable, and both $G$ and $H_H$ are minimum phase (ensuring $K$ is stable). Consequently, stable and minimum phase complementary filters must be employed. ***** Nominal Performance (NP) @@ -9134,7 +9155,7 @@ Similarly, for noise attenuation, the magnitude of the complementary sensitivity Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function. Typically, maintaining $|S|_{\infty} \le 2$ ensures a gain margin of at least 2 and a phase margin of at least $\SI{29}{\degree}$. -Therefore, by carefully selecting the shape of the complementary filters, nominal performance specifications can be directly addressed in an intuitive manner. +Therefore, by carefully selecting the shape of the complementary filters, acrfull:np specifications can be directly addressed in an intuitive manner. ***** Robust Stability (RS) @@ -9166,7 +9187,7 @@ The set of possible plants $\Pi_i$ is described by\nbsp{}eqref:eq:detail_control #+end_subfigure #+end_figure -When considering input multiplicative uncertainty, robust stability can be derived graphically from the Nyquist plot (illustrated in Figure\nbsp{}ref:fig:detail_control_cf_nyquist_uncertainty), yielding to\nbsp{}eqref:eq:detail_control_cf_robust_stability_graphically, as demonstrated in\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 7.5.1]]. +When considering input multiplicative uncertainty, acrfull:rs can be derived graphically from the Nyquist plot (illustrated in Figure\nbsp{}ref:fig:detail_control_cf_nyquist_uncertainty), yielding to\nbsp{}eqref:eq:detail_control_cf_robust_stability_graphically, as demonstrated in\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 7.5.1]]. \begin{equation}\label{eq:detail_control_cf_robust_stability_graphically} \text{RS} \Longleftrightarrow \left|w_I(j\omega) L(j\omega) \right| \le \left| 1 + L(j\omega) \right| \quad \forall\omega @@ -9192,9 +9213,9 @@ Transforming this condition into constraints on the complementary filters yields \boxed{\text{RP} \Longleftrightarrow | w_H(j\omega) H_H(j\omega) | + | w_I(j\omega) H_L(j\omega) | \le 1, \ \forall\omega} \end{equation} -The robust performance condition effectively combines both nominal performance\nbsp{}eqref:eq:detail_control_cf_nominal_performance and robust stability conditions\nbsp{}eqref:eq:detail_control_cf_condition_robust_stability. +The acrfull:rp condition effectively combines both nominal performance\nbsp{}eqref:eq:detail_control_cf_nominal_performance and robust stability conditions\nbsp{}eqref:eq:detail_control_cf_condition_robust_stability. If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 7.6]]. -Therefore, for SISO systems, ensuring robust stability and nominal performance is typically sufficient. +Therefore, for acrshort:siso systems, ensuring robust stability and nominal performance is typically sufficient. **** Complementary filter design <> @@ -9225,7 +9246,7 @@ A significant advantage of using analytical formulas for complementary filters i This real-time tunability allows rapid testing of different control bandwidths to evaluate performance and robustness characteristics. #+name: fig:detail_control_cf_arch_tunable_params -#+caption: Implemented digital complementary filters with parameter $\omega_0$ that can be changed in real time +#+caption: Implemented digital complementary filters with parameter $\omega_0$ that can be changed in real-time [[file:figs/detail_control_cf_arch_tunable_params.png]] For many practical applications, first order complementary filters are not sufficient. @@ -9239,7 +9260,7 @@ For these cases, the complementary filters analytical formula in Equation\nbsp{} \end{align} \end{subequations} -The influence of parameters $\alpha$ and $\omega_0$ on the frequency response of these complementary filters is illustrated in Figure\nbsp{}ref:fig:detail_control_cf_analytical_effect. +The influence of parameters $\alpha$ and $\omega_0$ on the magnitude response of these complementary filters is illustrated in Figure\nbsp{}ref:fig:detail_control_cf_analytical_effect. The parameter $\alpha$ primarily affects the damping characteristics near the crossover frequency as well as high and low frequency magnitudes, while $\omega_0$ determines the frequency at which the transition between high-pass and low-pass behavior occurs. These filters can also be implemented in the digital domain with analytical formulas, preserving the ability to adjust $\alpha$ and $\omega_0$ in real-time. @@ -9364,7 +9385,7 @@ To ensure properness, low-pass filters with high corner frequencies are added as \end{equation} The Bode plot of the controller multiplied by the complementary low-pass filter, $K(s) \cdot H_L(s)$, is presented in Figure\nbsp{}ref:fig:detail_control_cf_bode_Kfb. -The frequency response reveals several important characteristics: +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 @@ -9401,13 +9422,13 @@ In this section, a control architecture in which complementary filters are used This approach differs from traditional open-loop shaping in that no controller is manually designed; rather, appropriate complementary filters are selected to achieve the desired closed-loop behavior. The method shares conceptual similarities with mixed-sensitivity $\mathcal{H}_{\infty}\text{-synthesis}$, as both approaches aim to shape closed-loop transfer functions, but with notable distinctions in implementation and complexity. -While $\mathcal{H}_{\infty}\text{-synthesis}$ offers greater flexibility and can be readily generalized to MIMO plants, the presented approach provides a simpler alternative that requires minimal design effort. +While $\mathcal{H}_{\infty}\text{-synthesis}$ offers greater flexibility and can be readily generalized to acrshort:mimo plants, the presented approach provides a simpler alternative that requires minimal design effort. Implementation only necessitates extracting a model of the plant and selecting appropriate analytical complementary filters, making it particularly interesting for applications where simplicity and intuitive parameter tuning are valued. Due to time constraints, an extensive literature review comparing this approach with similar existing architectures, such as Internal Model Control\nbsp{}[[cite:&saxena12_advan_inter_model_contr_techn]], was not conducted. Consequently, it remains unclear whether the proposed architecture offers significant advantages over existing methods in the literature. -The control architecture has been presented for SISO systems, but can be applied to MIMO systems when sufficient decoupling is achieved. +The control architecture has been presented for acrshort:siso systems, but can be applied to acrshort:mimo systems when sufficient decoupling is achieved. It will be experimentally validated with the NASS during the experimental phase. *** Conclusion @@ -9418,16 +9439,16 @@ It will be experimentally validated with the NASS during the experimental phase. In order to optimize the control of the Nano Active Stabilization System, several aspects of control theory were studied. Different approaches to combine sensors were compared in Section\nbsp{}ref:sec:detail_control_sensor. -While High Authority Control-Low Authority Control (HAC-LAC) was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters. +While acrfull:haclac was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters. It was demonstrated that the performance of such fusion depends significantly on the magnitude of the complementary filters. To address this challenge, a synthesis method based on $\mathcal{H}_\infty\text{-synthesis}$ was proposed, allowing for intuitive shaping of the complementary filters through weighting functions. -For the NASS, while HAC-LAC remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation. +For the NASS, while acrshort:haclac remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation. Various decoupling strategies for parallel manipulators were examined in Section\nbsp{}ref:sec:detail_control_decoupling, including decentralized control, Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. The main characteristics of each approach were highlighted, providing valuable insights into their respective strengths and limitations. Among the examined methods, Jacobian decoupling was determined to be most appropriate for the NASS, as it provides straightforward implementation while preserving the physical meaning of inputs and outputs. -With the system successfully decoupled, attention shifted to designing appropriate SISO controllers for each decoupled direction. +With the system successfully decoupled, attention shifted to designing appropriate acrshort:siso controllers for each decoupled direction. A control architecture for directly shaping closed-loop transfer functions was proposed. It is based on complementary filters that can be designed using either the proposed $\mathcal{H}_\infty\text{-synthesis}$ approach described earlier or through analytical formulas. Experimental validation of this method on the NASS will be conducted during the experimental tests on ID31. @@ -9441,8 +9462,8 @@ This chapter presents an approach to select and validate appropriate instrumenta Figure\nbsp{}ref:fig:detail_instrumentation_plant illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis. The selection process follows a three-stage methodology. -First, dynamic error budgeting is performed in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier). -This analysis is based on the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements. +First, dynamic error budgeting is performed in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting to establish maximum acceptable noise specifications for each instrumentation component (acrshort:adc, acrshort:dac, and voltage amplifier). +This analysis is based on the multi-body model with a 2DoF acrshort:apa model, focusing particularly on the vertical direction due to its more stringent requirements. From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived. Section\nbsp{}ref:sec:detail_instrumentation_choice then presents the selection of appropriate components based on these noise specifications and additional requirements. @@ -9452,7 +9473,7 @@ Each instrument is characterized individually, measuring actual noise levels and The measured noise characteristics are then incorporated into the multi-body model to confirm that the combined effect of all instrumentation noise sources remains within acceptable limits. #+name: fig:detail_instrumentation_plant -#+caption: Block diagram of the NASS with considered instrumentation. The real time controller is a Speedgoat machine. +#+caption: Block diagram of the NASS with considered instrumentation. The real-time controller is a Speedgoat machine. #+attr_latex: :width 0.9\linewidth [[file:figs/detail_instrumentation_plant.png]] @@ -9460,20 +9481,20 @@ The measured noise characteristics are then incorporated into the multi-body mod <> **** Introduction :ignore: -The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system. +The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (acrshort:adc, acrshort:dac, and voltage amplifier) that would result in acceptable vibration levels in the system. The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section\nbsp{}ref:ssec:detail_instrumentation_cl_sensitivity). This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs. Only the vertical direction is considered in this analysis as it presents the most stringent requirements; the horizontal directions are subject to less demanding constraints. -From these transfer functions, the maximum acceptable Amplitude Spectral Density (ASD) of the noise sources is derived (Section\nbsp{}ref:ssec:detail_instrumentation_max_noise_specs). -Since the voltage amplifier gain affects the amplification of DAC noise, an assumption of an amplifier gain of 20 was made. +From these transfer functions, the maximum acceptable acrfull:asd of the noise sources is derived (Section\nbsp{}ref:ssec:detail_instrumentation_max_noise_specs). +Since the voltage amplifier gain affects the amplification of acrshort:dac noise, an assumption of an amplifier gain of 20 was made. **** Closed-Loop Sensitivity to Instrumentation Disturbances <> Several key noise sources are considered in the analysis (Figure\nbsp{}ref:fig:detail_instrumentation_plant). -These include the output voltage noise of the DAC ($n_{da}$), the output voltage noise of the voltage amplifier ($n_{amp}$), and the voltage noise of the ADC measuring the force sensor stacks ($n_{ad}$). +These include the output voltage noise of the acrshort:dac ($n_{da}$), the output voltage noise of the voltage amplifier ($n_{amp}$), and the voltage noise of the acrshort:adc measuring the force sensor stacks ($n_{ad}$). Encoder noise, which is only used to estimate $R_z$, has been found to have minimal impact on the vertical sample error and is therefore omitted from this analysis for clarity. @@ -9490,9 +9511,9 @@ The transfer functions from these three noise sources (for one strut) to the ver The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of $100\,\text{nm}$, which corresponds to a maximum allowed vibration of $15\,\text{nm RMS}$. Several assumptions regarding the noise characteristics have been made. -The DAC, ADC, and amplifier noise are considered uncorrelated, which is a reasonable assumption. +The acrshort:dac, acrshort:adc, and amplifier noise are considered uncorrelated, which is a reasonable assumption. Similarly, the noise sources corresponding to each strut are also assumed to be uncorrelated. -This means that the power spectral densities (PSD) of the different noise sources are summed. +This means that the acrfullpl:psd of the different noise sources are summed. Since the effect of each strut on the vertical error is identical due to symmetry, only one strut is considered for this analysis, and the total effect of the six struts is calculated as six times the effect of one strut in terms of power, which translates to a factor of $\sqrt{6} \approx 2.5$ for RMS values. @@ -9501,10 +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: DAC maximum output noise ASD is established at $14\,\mu V/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise ASD at $280\,\mu V/\sqrt{\text{Hz}}$, and ADC maximum measurement noise ASD at $11\,\mu V/\sqrt{\text{Hz}}$. -In terms of RMS noise, these translate to less than $1\,\text{mV RMS}$ for the DAC, less than $20\,\text{mV RMS}$ for the voltage amplifier, and less than $0.8\,\text{mV RMS}$ for the ADC. +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 $11\,\mu V/\sqrt{\text{Hz}}$. +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 ADC, 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 15nm RMS. *** Choice of Instrumentation <> @@ -9513,7 +9534,7 @@ If the Amplitude Spectral Density of the noise of the ADC, DAC, and voltage ampl 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 DACs. +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. ***** Small signal Bandwidth and Output Impedance @@ -9567,7 +9588,7 @@ The specifications are summarized in Table\nbsp{}ref:tab:detail_instrumentation_ The most critical characteristics are the small signal bandwidth ($>5\,\text{kHz}$) and the output voltage noise ($<20\,\text{mV RMS}$). Several voltage amplifiers were considered, with their datasheet information presented in Table\nbsp{}ref:tab:detail_instrumentation_amp_choice. -One challenge encountered during the selection process was that manufacturers typically do not specify output noise as a function of frequency (i.e., the ASD of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies. +One challenge encountered during the selection process was that manufacturers typically do not specify output noise as a function of frequency (i.e., the acrshort:asd of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies. This approach does not account for the frequency dependency of the noise, which is crucial for accurate error budgeting. Additionally, the load conditions used to estimate bandwidth and noise specifications are often not explicitly stated. @@ -9603,8 +9624,8 @@ The proper selection of these components is critical for system performance. For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements\nbsp{}[[cite:&abramovitch22_pract_method_real_world_contr_system;&abramovitch23_tutor_real_time_comput_issues_contr_system]]. -Therefore, the ADC and DAC must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals. -Based on this requirement, priority was given to ADC and DAC components specifically marketed by Speedgoat to ensure optimal integration. +Therefore, the acrshort:adc and acrshort:dac must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals. +Based on this requirement, priority was given to acrshort:adc and acrshort:dac components specifically marketed by Speedgoat to ensure optimal integration. ***** Sampling Frequency, Bandwidth and delays @@ -9612,23 +9633,23 @@ Several requirements that may initially appear similar are actually distinct in 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. -Finally, /delay/ (or /latency/) refers to the time interval between the analog signal at the input of the ADC and the digital information transferred to the control system. +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 ADCs can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency. +Sigma-Delta acrshortpl:adc can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency. Typically, the latency can reach 20 times the sampling period\nbsp{}[[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 8.4]]. -Consequently, while Sigma-Delta ADCs are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor. +Consequently, while Sigma-Delta acrshortpl:adc are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor. -For real-time control applications, SAR-ADCs (Successive Approximation ADCs) remain the predominant choice due to their single-sample latency characteristics. +For real-time control applications, acrfull:sar remain the predominant choice due to their single-sample latency characteristics. ***** ADC Noise -Based on the dynamic error budget established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise ASD should not exceed $11\,\mu V/\sqrt{\text{Hz}}$. +Based on the dynamic error budget established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise acrshort:asd should not exceed $11\,\mu V/\sqrt{\text{Hz}}$. -ADCs are subject to various noise sources. +acrshortpl:adc are subject to various noise sources. Quantization noise, which results from the discrete nature of digital representation, is one of these sources. -To determine the minimum bit depth $n$ required to meet the noise specifications, an ideal ADC where quantization error is the only noise source is considered. +To determine the minimum bit depth $n$ required to meet the noise specifications, an ideal acrshort:adc where quantization error is the only noise source is considered. -The quantization step size, denoted as $q = \Delta V/2^n$, represents the voltage equivalent of the least significant bit, with $\Delta V$ the full range of the ADC in volts, and $F_s$ the sampling frequency in Hertz. +The quantization step size, denoted as $q = \Delta V/2^n$, represents the voltage equivalent of the acrfull:lsb, with $\Delta V$ the full range of the acrshort:adc in volts, and $F_s$ the sampling frequency in Hertz. The quantization noise ranges between $\pm q/2$, and its probability density function is constant across this range (uniform distribution). Since the integral of this probability density function $p(e)$ equals one, its value is $1/q$ for $-q/2 < e < q/2$, as illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_adc_quantization. @@ -9647,7 +9668,7 @@ To compute the power spectral density of the quantization noise, which is define Under this assumption, the autocorrelation function approximates a delta function in the time domain. Since the Fourier transform of a delta function equals one, the power spectral density becomes frequency-independent (white noise). -By Parseval's theorem, the power spectral density of the quantization noise $\Phi_q$ can be linked to the ADC sampling frequency and quantization step size\nbsp{}eqref:eq:detail_instrumentation_psd_quant_noise. +By Parseval's theorem, the power spectral density of the quantization noise $\Phi_q$ can be linked to the acrshort:adc sampling frequency and quantization step size\nbsp{}eqref:eq:detail_instrumentation_psd_quant_noise. \begin{equation}\label{eq:detail_instrumentation_psd_quant_noise} \int_{-F_s/2}^{F_s/2} \Phi_q(f) d f = \int_{-q/2}^{q/2} e^2 p(e) de \quad \Longrightarrow \quad \Phi_q = \frac{q^2}{12 F_s} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 F_s} \quad \text{in } \left[ \frac{V^2}{\text{Hz}} \right] @@ -9659,19 +9680,19 @@ 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 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 ADCs. +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. ***** DAC Output voltage noise -Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed $14\,\mu V/\sqrt{\text{Hz}}$. -This specification corresponds to a $\pm 10\,V$ 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 $14\,\mu V/\sqrt{\text{Hz}}$. +This specification corresponds to a $\pm 10\,V$ acrshort:dac with 13-bit resolution, which is easily attainable with current technology. ***** Choice of the ADC and DAC Board -Based on the preceding analysis, the selection of suitable ADC and DAC components is straightforward. +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, simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity. +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. Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements. @@ -9708,8 +9729,8 @@ These include optical encoders (Figure\nbsp{}ref:fig:detail_instrumentation_sens #+end_subfigure #+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 APA, as illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_capacitive_implementation. -In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure\nbsp{}ref:fig:detail_instrumentation_encoder_implementation. +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\nbsp{}ref:fig:detail_instrumentation_capacitive_implementation. +In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the acrshort:apa, as shown in Figure\nbsp{}ref:fig:detail_instrumentation_encoder_implementation. #+name: fig:detail_instrumentation_sensor_implementation #+caption: Implementation of relative displacement sensor to measure the motion of the APA @@ -9754,15 +9775,15 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic **** Analog to Digital Converters ***** Measured Noise -The measurement of 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 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 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. -All ADC channels demonstrated similar performance, so only one channel's noise profile is shown. +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. +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]]. To gain $w$ additional bits of resolution, the oversampling frequency $f_{os}$ should be set to $f_{os} = 4^w \cdot F_s$. -Given that the ADC can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4). -This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV)\nbsp{}[[cite:&hauser91_princ_overs_d_conver]]. +Given that the acrshort:adc can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4). +This approach is effective because the noise approximates white noise and its amplitude exceeds 1 acrshort:lsb (0.3 mV)\nbsp{}[[cite:&hauser91_princ_overs_d_conver]]. #+name: fig:detail_instrumentation_adc_noise_measured #+caption: Measured ADC noise (IO318) @@ -9779,18 +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 DAC, and the ADC signal was recorded. +Step signals with an amplitude of $1\,V$ 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. -These phenomena can be explained by examining the electrical schematic shown in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc, where the ADC has an input impedance $R_i$ and an input bias current $i_n$. +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 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 ADC. +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 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. +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. #+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}). @@ -9810,8 +9831,8 @@ With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu F$, #+end_subfigure #+end_figure -The constant voltage offset can be explained by the input bias current $i_n$ of the ADC, represented in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc. -At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the ADC, and therefore the input bias current $i_n$ passing through the internal resistance $R_i$ produces a constant voltage offset $V_{\text{off}} = R_i \cdot i_n$. +The constant voltage offset can be explained by the input bias current $i_n$ of the acrshort:adc, represented in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc. +At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the acrshort:adc, and therefore the input bias current $i_n$ passing through the internal resistance $R_i$ produces a constant voltage offset $V_{\text{off}} = R_i \cdot i_n$. The input bias current $i_n$ is estimated from $i_n = V_{\text{off}}/R_i = 1.5\mu A$. In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor $R_p$ can be added in parallel to the force sensor, as illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_R. @@ -9820,7 +9841,7 @@ 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 ($\tau = 0.45\,s$). -These results validate both the model of the ADC and the effectiveness of the added parallel resistor as a solution. +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 #+caption: Effect of an added resistor $R_p$ in parallel to the force sensor. The electrical schematic is shown in (\subref{fig:detail_instrumentation_force_sensor_adc_R}) and the measured signals in (\subref{fig:detail_instrumentation_step_response_force_sensor_R}). @@ -9842,16 +9863,16 @@ These results validate both the model of the ADC and the effectiveness of the ad **** Instrumentation Amplifier -Because the 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 ADC itself), a low noise instrumentation amplifier was employed. +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. 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 ADC (Figure\nbsp{}ref:fig:detail_instrumentation_femto_meas_setup). +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 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 ADC $q_{ad}$ is rendered negligible due to the high gain employed. -The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the (negligible) contribution of the ADC noise are presented in Figure\nbsp{}ref:fig:detail_instrumentation_femto_input_noise. +In this configuration, the noise contribution from the acrshort:adc $q_{ad}$ is rendered negligible due to the high gain employed. +The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the (negligible) contribution of the acrshort:adc noise are presented in Figure\nbsp{}ref:fig:detail_instrumentation_femto_input_noise. #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage @@ -9871,26 +9892,26 @@ The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the **** Digital to Analog Converters ***** Output Voltage Noise -To measure the output noise of the DAC, the setup schematically represented in Figure\nbsp{}ref:fig:detail_instrumentation_dac_setup was used. -The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the ADC. +To measure the output noise of the acrshort:dac, the setup schematically represented in Figure\nbsp{}ref:fig:detail_instrumentation_dac_setup was used. +The acrshort:dac was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the acrshort:adc. -The Amplitude Spectral Density $\Gamma_{n_{da}}(\omega)$ of the measured signal was computed, and verification was performed to confirm that the contributions of ADC noise and amplifier noise were negligible in the measurement. +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 ASD of $0.6\,\mu V/\sqrt{\text{Hz}}$. +The noise profile is predominantly white with an acrshort:asd of $0.6\,\mu V/\sqrt{\text{Hz}}$. 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 DAC channels demonstrated similar performance, so only one channel measurement is presented. +It should be noted that all acrshort:dac channels demonstrated similar performance, so only one channel measurement is presented. #+name: fig:detail_instrumentation_dac_setup #+caption: Measurement of the DAC output voltage noise. A pre-amplifier with a gain of 1000 is used before measuring the signal with the ADC. [[file:figs/detail_instrumentation_dac_setup.png]] ***** Delay from ADC to DAC -To measure the transfer function from DAC to ADC and verify that the bandwidth and latency of both instruments is sufficient, a direct connection was established between the DAC output and the ADC input. -A white noise signal was generated by the DAC, and the ADC response was recorded. +To measure the transfer function from acrshort:dac to acrshort:adc and verify that the bandwidth and latency of both instruments is sufficient, a direct connection was established between the acrshort:dac output and the acrshort:adc input. +A white noise signal was generated by the acrshort:dac, and the acrshort:adc response was recorded. -The resulting frequency response function from the digital DAC signal to the digital ADC signal is presented in Figure\nbsp{}ref:fig:detail_instrumentation_dac_adc_tf. -The observed frequency response function corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer. +The resulting acrshort:frf from the digital acrshort:dac signal to the digital acrshort:adc signal is presented in Figure\nbsp{}ref:fig:detail_instrumentation_dac_adc_tf. +The observed acrshort:frf corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer. #+name: fig:detail_instrumentation_dac #+caption: Measurement of the output voltage noise of the ADC (\subref{fig:detail_instrumentation_dac_output_noise}) and measured transfer function from DAC to ADC (\subref{fig:detail_instrumentation_dac_adc_tf}) which corresponds to a "1-sample" delay. @@ -9914,14 +9935,14 @@ The observed frequency response function corresponds to exactly one sample delay ***** Output Voltage Noise The measurement setup for evaluating the PD200 amplifier noise is illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_pd200_setup. The input of the PD200 amplifier was shunted with a $50\,\Ohm$ resistor to ensure that only the inherent noise of the amplifier itself was measured. -The pre-amplifier gain was increased to produce a signal substantially larger than the noise floor of the ADC. +The pre-amplifier gain was increased to produce a signal substantially larger than the noise floor of the acrshort:adc. Two piezoelectric stacks from the APA95ML were connected to the PD200 output to provide an appropriate load for the amplifier. #+name: fig:detail_instrumentation_pd200_setup #+caption: Setup used to measured the output voltage noise of the PD200 voltage amplifier. A gain $G_a = 1000$ was used for the instrumentation amplifier. [[file:figs/detail_instrumentation_pd200_setup.png]] -The Amplitude Spectral Density $\Gamma_{n}(\omega)$ of the signal measured by the ADC was computed. +The Amplitude Spectral Density $\Gamma_{n}(\omega)$ of the signal measured by the acrshort:adc was computed. From this, the Amplitude Spectral Density of the output voltage noise of the PD200 amplifier $n_p$ was derived, accounting for the gain of the pre-amplifier according to\nbsp{}eqref:eq:detail_instrumentation_amp_asd. \begin{equation}\label{eq:detail_instrumentation_amp_asd} @@ -9929,7 +9950,7 @@ From this, the Amplitude Spectral Density of the output voltage noise of the PD2 \end{equation} The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure\nbsp{}ref:fig:detail_instrumentation_pd200_noise. -Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or ADC noise. +Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or acrshort:adc noise. The measurements from all six amplifiers are displayed in this figure. The noise spectrum of the PD200 amplifiers exhibits several sharp peaks. @@ -9942,10 +9963,10 @@ While the exact cause of these peaks is not fully understood, their amplitudes r ***** Small Signal Bandwidth -The small signal dynamics of all six PD200 amplifiers were characterized through frequency response measurements. +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 DAC with an amplitude of $0.1\,V$, 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 DAC) was measured with a separate ADC channel of the Speedgoat system. +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 $5\,\text{kHz}$. +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. @@ -9987,7 +10008,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a **** Noise budgeting from measured instrumentation noise After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier. -The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure\nbsp{}ref:fig:detail_instrumentation_cl_noise_budget. +The vertical motion induced by the noise sources, specifically the acrshort:adc noise, acrshort:dac noise, and voltage amplifier noise, is presented in Figure\nbsp{}ref:fig:detail_instrumentation_cl_noise_budget. The total motion induced by all noise sources combined is approximately $1.5\,\text{nm RMS}$, which remains well within the specified limit of $15\,\text{nm RMS}$. This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application. @@ -10011,9 +10032,9 @@ The selection process revealed certain challenges, particularly with voltage amp Despite these challenges, suitable components were identified that theoretically met all requirements. 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 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. +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 ADC demonstrated noise levels of $5.6\,\mu V/\sqrt{\text{Hz}}$ (versus the $11\,\mu V/\sqrt{\text{Hz}}$ specification), the 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 $5.6\,\mu V/\sqrt{\text{Hz}}$ (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). 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. @@ -10043,7 +10064,7 @@ Finally, considerations for ease of mounting, alignment, and maintenance were in The strut design, illustrated in Figure\nbsp{}ref:fig:detail_design_strut, was driven by several factors. Stiff interfaces were required between the amplified piezoelectric actuator and the two flexible joints, as well as between the flexible joints and their respective mounting plates. Due to the limited angular stroke of the flexible joints, it was critical that the struts could be assembled such that the two cylindrical interfaces were coaxial while the flexible joints remained in their unstressed, nominal rest position. -To facilitate this alignment, cylindrical washers (Figure\nbsp{}ref:fig:detail_design_strut_without_enc) were integrated into the design to compensate for potential deviations from perfect flatness between the two APA interface planes (Figure\nbsp{}ref:fig:detail_design_apa). +To facilitate this alignment, cylindrical washers (Figure\nbsp{}ref:fig:detail_design_strut_without_enc) were integrated into the design to compensate for potential deviations from perfect flatness between the two acrshort:apa interface planes (Figure\nbsp{}ref:fig:detail_design_apa). Furthermore, a dedicated mounting bench was developed to enable precise alignment of each strut, even when accounting for typical machining inaccuracies. The mounting procedure is described in Section\nbsp{}ref:sec:test_struts_mounting. Lastly, the design needed to permit the fixation of an encoder parallel to the strut axis, as shown in Figure\nbsp{}ref:fig:detail_design_strut_with_enc. @@ -10066,14 +10087,14 @@ Lastly, the design needed to permit the fixation of an encoder parallel to the s #+end_subfigure #+end_figure -The flexible joints, shown in Figure\nbsp{}ref:fig:detail_design_flexible_joint, were manufactured using wire-cut electrical discharge machining (EDM). +The flexible joints, shown in Figure\nbsp{}ref:fig:detail_design_flexible_joint, were manufactured using wire-cut acrfull:edm. First, the part's inherent fragility, stemming from its $0.25\,\text{mm}$ neck dimension, makes it susceptible to damage from cutting forces typical in classical machining. -Furthermore, wire-cut EDM allows for the very tight machining tolerances critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure\nbsp{}ref:fig:detail_design_flexible_joint) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties. +Furthermore, wire-cut acrshort:edm allows for the very tight machining tolerances critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure\nbsp{}ref:fig:detail_design_flexible_joint) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties. The material chosen for the flexible joints is a stainless steel designated /X5CrNiCuNb16-4/ (alternatively known as F16Ph). This selection was based on its high specified yield strength (exceeding $1\,\text{GPa}$ after appropriate heat treatment) and its high fatigue resistance. -As shown in Figure\nbsp{}ref:fig:detail_design_flexible_joint, the interface designed to connect with the APA possesses a cylindrical shape, facilitating the use of the aforementioned cylindrical washers for alignment. -A slotted hole was incorporated to permit alignment of the flexible joint with the APA via a dowel pin. +As shown in Figure\nbsp{}ref:fig:detail_design_flexible_joint, the interface designed to connect with the acrshort:apa possesses a cylindrical shape, facilitating the use of the aforementioned cylindrical washers for alignment. +A slotted hole was incorporated to permit alignment of the flexible joint with the acrshort:apa via a dowel pin. Additionally, two threaded holes were included on the sides for mounting the encoder components. The interface connecting the flexible joint to the platform plates will be described subsequently. @@ -10151,13 +10172,13 @@ This characteristic is expected to permit repeated assembly and disassembly of t ***** Finite Element Analysis -A finite element analysis (FEA) of the complete active platform assembly was performed to identify modes that could potentially affect performance. +A acrfull:fea of the complete active platform 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. -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 APA actuation and their sensitivity to strut alignment. +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. #+name: fig:detail_design_fem_nano_hexapod @@ -10219,7 +10240,7 @@ This geometric discrepancy implies that if the relative motion control of the ac Prior to the procurement of mechanical components, the multi-body simulation model of the active platform was refined to incorporate the finalized design geometries. Two distinct configurations, corresponding to the two encoder mounting strategies discussed previously, were considered in the model, as displayed in Figure\nbsp{}ref:fig:detail_design_simscape: one with encoders fixed to the struts, and another with encoders fixed to the plates. -In these models, the top and bottom plates were represented as rigid bodies, with their inertial properties calculated directly from the 3D CAD geometry. +In these models, the top and bottom plates were represented as rigid bodies, with their inertial properties calculated directly from the 3D geometry. #+name: fig:detail_design_simscape #+caption: 3D representation of the multi-body model. There are two configurations: encoders fixed to the struts (\subref{fig:detail_design_simscape_encoder_struts}) and encoders fixed to the plates (\subref{fig:detail_design_simscape_encoder_plates}). @@ -10242,9 +10263,9 @@ In these models, the top and bottom plates were represented as rigid bodies, wit ***** Flexible Joints Several levels of detail were considered for modeling the flexible joints within the multi-body model. -Models with two degrees of freedom incorporating only bending stiffnesses, models with three degrees of freedom adding torsional stiffness, and models with four degrees of freedom further adding axial stiffness were evaluated. +Models with two acrshortpl:dof incorporating only bending stiffnesses, models with three acrshortpl:dof adding torsional stiffness, and models with four acrshortpl:dof further adding axial stiffness were evaluated. The multi-body representation corresponding to the 4DoF configuration is shown in Figure\nbsp{}ref:fig:detail_design_simscape_model_flexible_joint. -This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from finite element analysis of the joint component. +This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from acrshort:fea of the joint component. #+name: fig:detail_design_simscape_model_flexible_joint #+caption: 4DoF multi-body model of the flexible joints @@ -10253,8 +10274,8 @@ This model is composed of three distinct solid bodies interconnected by joints, ***** Amplified Piezoelectric Actuators -The amplified piezoelectric actuators (APAs) were incorporated into the multi-body model following the methodology detailed in Section\nbsp{}ref:sec:detail_fem_actuator. -Two distinct representations of the APA can be utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a finite element model. +The acrlongpl:apa were incorporated into the multi-body model following the methodology detailed in Section\nbsp{}ref:sec:detail_fem_actuator. +Two distinct representations of the acrshort:apa can be utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a acrshort:fem. ***** Encoders @@ -10309,7 +10330,7 @@ The geometry optimization began with a review of existing Stewart platform desig While cubic architectures are prevalent in the literature due to their purported advantages in decoupling and uniform stiffness, the analysis revealed that these benefits are more nuanced than commonly described. For the nano-hexapod application, struts were oriented more vertically than in a cubic configuration to address the stringent vertical performance requirements and to better match the micro-station's modal characteristics. -For component optimization, a hybrid modeling methodology was used that combined finite element analysis with multi-body dynamics. +For component optimization, a hybrid modeling methodology was used that combined acrshort:fea with multi-body dynamics. This approach, validated experimentally using an Amplified Piezoelectric Actuator, enabled both detailed component-level optimization and efficient system-level simulation. Through this methodology, the APA300ML was selected as the optimal actuator, offering the necessary combination of stroke, stiffness, and force sensing capabilities required for the application. Similarly, the flexible joints were designed with careful consideration of bending and axial stiffness requirements, resulting in a design that balances competing mechanical demands. @@ -10339,24 +10360,24 @@ Following the completion of this design phase and the subsequent procurement of The experimental validation follows a systematic approach, beginning with the characterization of individual components before advancing to evaluate the assembled system's performance (illustrated in Figure\nbsp{}ref:fig:chapter3_overview). Section\nbsp{}ref:sec:test_apa focuses on the Amplified Piezoelectric Actuator (APA300ML), examining its electrical properties, and dynamical behavior. -Two models are developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from finite element analysis. +Two models are developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from acrshort:fea. The implementation of Integral Force Feedback is also experimentally evaluated to assess its effectiveness in adding damping to the system. In Section\nbsp{}ref:sec:test_joints, the flexible joints are characterized to ensure they meet the required specifications for stiffness and stroke. A dedicated test bench is developed to measure the bending stiffness, with error analysis performed to validate the measurement accuracy. -Section\nbsp{}ref:sec:test_struts examines the assembly and testing of the struts, which integrate the APAs and flexible joints. +Section\nbsp{}ref:sec:test_struts examines the assembly and testing of the struts, which integrate the acrshortpl:apa and flexible joints. The mounting procedure is detailed, with particular attention to ensure consistent performance across multiple struts. Dynamical measurements are performed to verify whether the dynamics of the struts are corresponding to the multi-body model. The assembly and testing of the complete nano-hexapod is presented in Section\nbsp{}ref:sec:test_nhexa. A suspended table is developed to isolate the hexapod's dynamics from support dynamics, enabling accurate identification of its dynamical properties. -The experimental frequency response functions are compared with the multi-body model predictions to validate the modeling approach. +The experimental acrshortpl:frf are compared with the multi-body model predictions to validate the modeling approach. The effects of various payload masses are also investigated. Finally, Section\nbsp{}ref:sec:test_id31 presents the validation of the NASS on the ID31 beamline. A short-stroke metrology system is developed to measure the sample position relative to the granite base. -The HAC-LAC control architecture is implemented and tested under various experimental conditions, including payload masses up to $39\,\text{kg}$ and for typical experiments, including tomography scans, reflectivity measurements, and diffraction tomography. +The acrshort:haclac control architecture is implemented and tested under various experimental conditions, including payload masses up to $39\,\text{kg}$ and for typical experiments, including tomography scans, reflectivity measurements, and diffraction tomography. #+name: fig:chapter3_overview #+caption: Overview of the Experimental validation phase. The actuators and flexible joints and individual tested and then integrated into the struts. The Nano-hexapod is then mounted and the complete system is validated on the ID31 beamline. @@ -10372,17 +10393,17 @@ The HAC-LAC control architecture is implemented and tested under various experim In this chapter, the goal is to ensure that the received APA300ML (shown in Figure\nbsp{}ref:fig:test_apa_received) are complying with the requirements and that the dynamical models of the actuator accurately represent its dynamics. In section\nbsp{}ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks and the achievable stroke. -The flexible modes of the APA300ML, which were estimated using a finite element model, are compared with measurements. +The flexible modes of the APA300ML, which were estimated using a acrshort:fem, are compared with measurements. Using a dedicated test bench, dynamical measurements are performed (Section\nbsp{}ref:sec:test_apa_dynamics). -The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated. +The dynamics from the generated acrshort:dac voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated. Integral Force Feedback is experimentally applied, and the damped plants are estimated for several feedback gains. Two different models of the APA300ML are presented. First, in Section\nbsp{}ref:sec:test_apa_model_2dof, a two degrees-of-freedom model is presented, tuned, and compared with the measured dynamics. This model is proven to accurately represent the APA300ML's axial dynamics while having low complexity. -Then, in Section\nbsp{}ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a finite element model and imported into the multi-body model. +Then, in Section\nbsp{}ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a acrshort:fem and imported into the multi-body model. This more complex model also captures well capture the axial dynamics of the APA300ML. #+name: fig:test_apa_received @@ -10399,13 +10420,13 @@ Before measuring the dynamical characteristics of the APA300ML, simple measureme First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section\nbsp{}ref:ssec:test_apa_geometrical_measurements. Then, the capacitance of the piezoelectric stacks is measured in Section\nbsp{}ref:ssec:test_apa_electrical_measurements. The achievable stroke of the APA300ML is measured using a displacement probe in Section\nbsp{}ref:ssec:test_apa_stroke_measurements. -Finally, in Section\nbsp{}ref:ssec:test_apa_spurious_resonances, the flexible modes of the APA are measured and compared with a finite element model. +Finally, in Section\nbsp{}ref:ssec:test_apa_spurious_resonances, the flexible modes of the acrshort:apa are measured and compared with a acrshort:fem. **** 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\nbsp{}ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:test_apa_3] is used to measure the height of four points on each of the APA300ML interfaces. +As shown in Figure\nbsp{}ref:fig:test_apa_flatness_setup, the acrshort:apa is fixed to a clamp while a measuring probe[fn:test_apa_3] 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[fn:test_apa_4] a plane through all the points. The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatness_meas, are within the specifications. @@ -10476,7 +10497,7 @@ This may be because the manufacturer measures the capacitance with large signals **** Stroke and Hysteresis Measurement <> -To compare the stroke of the APA300ML with the datasheet specifications, one side of the 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. +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]. 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). @@ -10486,7 +10507,7 @@ Note that the voltage is slowly varied as the displacement probe has a very low #+attr_latex: :width 0.6\linewidth [[file:figs/test_apa_stroke_bench.jpg]] -The measured APA displacement is shown as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis. +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). @@ -10518,8 +10539,8 @@ From now on, only the six remaining amplified piezoelectric actuators that behav **** Flexible Mode Measurement <> -In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. -To experimentally estimate these modes, the APA is fixed at one end (see Figure\nbsp{}ref:fig:test_apa_meas_setup_modes). +In this section, the flexible modes of the APA300ML are investigated both experimentally and using a acrshort:fem. +To experimentally estimate these modes, the acrshort:apa is fixed at one end (see Figure\nbsp{}ref:fig:test_apa_meas_setup_modes). A Laser Doppler Vibrometer[fn:test_apa_6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:test_apa_7] is used to excite the flexible modes. Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated. @@ -10567,10 +10588,10 @@ The flexible modes for the same condition (i.e. one mechanical interface of the #+end_subfigure #+end_figure -The measured frequency response functions computed from the experimental setups of figures\nbsp{}ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure\nbsp{}ref:fig:test_apa_meas_freq_compare. +The measured acrshortpl:frf computed from the experimental setups of figures\nbsp{}ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure\nbsp{}ref:fig:test_apa_meas_freq_compare. The $y$ bending mode is observed at $280\,\text{Hz}$ and the $x$ bending mode is at $412\,\text{Hz}$. -These modes are measured at higher frequencies than the frequencies estimated from the Finite Element Model (see frequencies in Figure\nbsp{}ref:fig:test_apa_mode_shapes). -This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model). +These modes are measured at higher frequencies than the frequencies estimated from the acrshort:fem (see frequencies in Figure\nbsp{}ref:fig:test_apa_mode_shapes). +This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a acrshort:fem). This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used). Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades. @@ -10582,10 +10603,10 @@ Another explanation is the shape difference between the manufactured APA300ML an *** Dynamical measurements <> **** Introduction :ignore: -After the measurements on the 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. +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. 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 APA. +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. #+name: fig:test_bench_apa #+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML @@ -10607,7 +10628,7 @@ An encoder[fn:test_apa_8] is used to measure the relative movement between the t The bench is schematically shown in Figure\nbsp{}ref:fig:test_apa_schematic with the associated signals. It will be first used to estimate the hysteresis from the piezoelectric stack (Section\nbsp{}ref:ssec:test_apa_hysteresis) as well as the axial stiffness of the APA300ML (Section\nbsp{}ref:ssec:test_apa_stiffness). -The frequency response functions from the DAC voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section\nbsp{}ref:ssec:test_apa_meas_dynamics. +The acrshortpl:frf from the acrshort:dac voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section\nbsp{}ref:ssec:test_apa_meas_dynamics. The presence of a non-minimum phase zero found on the transfer function from $u$ to $V_s$ is investigated in Section\nbsp{}ref:ssec:test_apa_non_minimum_phase. To limit the low-frequency gain of the transfer function from $u$ to $V_s$, a resistor is added across the force sensor stack (Section\nbsp{}ref:ssec:test_apa_resistance_sensor_stack). Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section\nbsp{}ref:ssec:test_apa_iff_locus. @@ -10620,10 +10641,10 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a **** Hysteresis <> -Because the payload is vertically guided without friction, the hysteresis of the 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 DAC. +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. 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 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]]. +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]]. #+name: fig:test_apa_meas_hysteresis #+caption: Displacement as a function of applied voltage for multiple excitation amplitudes @@ -10633,8 +10654,8 @@ This is the typical behavior expected from a PZT stack actuator, where the hyste **** Axial stiffness <> -To estimate the stiffness of the 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 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. +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. \begin{equation} \label{eq:test_apa_stiffness} k_{\text{apa}} = \frac{m_a g}{\Delta d_e} @@ -10644,7 +10665,7 @@ The measured displacement $d_e$ as a function of time is shown in Figure\nbsp{}r It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep), and that the displacement does not return to the initial position after the mass is removed (probably due to piezoelectric hysteresis). These two effects induce some uncertainties in the measured stiffness. -The stiffnesses are computed for all APAs 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 $k_2$. +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 $k = 1.8\,N/\mu m$. #+attr_latex: :options [b]{0.57\textwidth} @@ -10682,7 +10703,7 @@ The obtained stiffness is $k \approx 2\,N/\mu m$ which is close to the values fo 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]]. To estimate this effect for the APA300ML, its stiffness is estimated using the "static deflection" method in two cases: -- $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance ADC) +- $k_{\text{os}}$: 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 $k_{\text{sc}} \approx 1.7\,N/\mu m$. @@ -10693,11 +10714,11 @@ The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ In this section, the dynamics from the excitation voltage $u$ to the encoder measured displacement $d_e$ and to the force sensor voltage $V_s$ is identified. 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 frequency response functions are similar to those of a (second order) mass-spring-damper system with: +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 $\approx -17\,\mu m/V$. - The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA + 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 APA support. +- 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. The flexible modes studied in section\nbsp{}ref:ssec:test_apa_spurious_resonances seem not to impact the measured axial motion of the actuator. The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure\nbsp{}ref:fig:test_apa_frf_force. @@ -10711,7 +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 finite element model). +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). 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. @@ -10769,7 +10790,7 @@ However, this is not so important here because the zero is lightly damped (i.e. 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 $\approx 5\,\mu F$). -As explained before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain. +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. The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor were measured and compared in Figure\nbsp{}ref:fig:test_apa_effect_resistance. It is confirmed that the added resistor has the effect of adding a high-pass filter with a cut-off frequency of $\approx 0.39\,\text{Hz}$. @@ -10782,7 +10803,7 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil **** Integral Force Feedback <> -To implement the Integral Force Feedback strategy, the measured frequency response function from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation\nbsp{}eqref:eq:test_apa_iff_manual_fit. +To implement the Integral Force Feedback strategy, the measured acrshort:frf from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation\nbsp{}eqref:eq:test_apa_iff_manual_fit. The parameters were manually tuned, and the obtained values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$. \begin{equation} \label{eq:test_apa_iff_manual_fit} @@ -10803,7 +10824,7 @@ It contains a high-pass filter (cut-off frequency of $2\,\text{Hz}$) to limit th K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000} \end{equation} -To estimate how the dynamics of the APA changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure\nbsp{}ref:fig:test_apa_iff_schematic is used. +To estimate how the dynamics of the acrshort:apa changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure\nbsp{}ref:fig:test_apa_iff_schematic is used. The transfer function from the "damped" plant input $u\prime$ to the encoder displacement $d_e$ is identified for several IFF controller gains $g$. #+name: fig:test_apa_iff_schematic @@ -10841,7 +10862,7 @@ The two obtained root loci are compared in Figure\nbsp{}ref:fig:test_apa_iff_roo <> ***** Introduction :ignore: -In this section, a multi-body model (Figure\nbsp{}ref:fig:test_apa_bench_model) of the measurement bench is used to tune the two degrees-of-freedom model of the APA using the measured frequency response functions. +In this section, a multi-body model (Figure\nbsp{}ref:fig:test_apa_bench_model) of the measurement bench is used to tune the two degrees-of-freedom model of the acrshort:apa using the measured acrshortpl:frf. This two degrees-of-freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and a low number of associated states. After the model is presented, the procedure for tuning the model is described, and the obtained model dynamics is compared with the measurements. @@ -10863,12 +10884,12 @@ It can be decomposed into three components: A sensor measures the stack strain $d_e$ which is then converted to a voltage $V_s$ using a sensitivity $g_s$ (in $V/m$) Such a simple model has some limitations: -- it only represents the axial characteristics of the APA as it is modeled as infinitely rigid in the other directions +- it only represents the axial characteristics of the acrshort:apa as it is modeled as infinitely rigid in the other directions - some physical insights are lost, such as the amplification factor and the real stress and strain in the piezoelectric stacks - the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear #+name: fig:test_apa_2dof_model -#+caption: Schematic of the two degrees-of-freedom model of the APA300ML, adapted from cite:souleille18_concep_activ_mount_space_applic +#+caption: Schematic of the two degrees-of-freedom model of the APA300ML, adapted from\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic]] [[file:figs/test_apa_2dof_model.png]] ***** Tuning of the APA model :ignore: @@ -10887,7 +10908,7 @@ Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resona 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. -In this case, the total stiffness of the APA model is described by\nbsp{}eqref:eq:test_apa_2dof_stiffness. +In this case, the total stiffness of the acrshort:apa model is described by\nbsp{}eqref:eq:test_apa_2dof_stiffness. \begin{equation}\label{eq:test_apa_2dof_stiffness} k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a @@ -10954,7 +10975,7 @@ This indicates that this model represents well the axial dynamics of the APA300M In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:test_apa_11]. It is then imported into multi-body (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in\nbsp{}ref:sec:test_apa_model_2dof. This procedure is illustrated in Figure\nbsp{}ref:fig:test_apa_super_element_simscape. -Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then made accessible in the multi-body software as shown at the right by the "frames" =F1= to =F5=. +Several /remote points/ are defined in the acrshort:fem (here illustrated by colorful planes and numbers from =1= to =5=) and are then made accessible in the multi-body software as shown at the right by the "frames" =F1= to =F5=. For the APA300ML /super element/, 5 /remote points/ are defined. Two /remote points/ (=1= and =2=) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used to connect the APA300ML with other mechanical elements. @@ -11005,9 +11026,9 @@ From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained, w ***** Comparison of the obtained dynamics -The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" are compared with the experimentally identified frequency response functions in Figure\nbsp{}ref:fig:test_apa_super_element_comp_frf. +The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" are compared with the experimentally identified acrshortpl:frf in Figure\nbsp{}ref:fig:test_apa_super_element_comp_frf. A good match between the model and the experimental results was observed. -It is however surprising that the model is "softer" than the measured system, as finite element models usually overestimate the stiffness (see Section\nbsp{}ref:ssec:test_apa_spurious_resonances for possible explanations). +It is however surprising that the model is "softer" than the measured system, as acrshortpl:fem usually overestimate the stiffness (see Section\nbsp{}ref:ssec:test_apa_spurious_resonances for possible explanations). Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever). @@ -11048,7 +11069,7 @@ In Section\nbsp{}ref:sec:test_apa_model_2dof, a simple two degrees-of-freedom ma After following a tuning procedure, the model dynamics was shown to match very well with the experiment. However, this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions. -In Section\nbsp{}ref:sec:test_apa_model_flexible, a /super element/ extracted from a finite element model was used to model the APA300ML. +In Section\nbsp{}ref:sec:test_apa_model_flexible, a /super element/ extracted from a acrshort:fem was used to model the APA300ML. Here, the /super element/ represents the dynamics of the APA300ML in all directions. However, only the axial dynamics could be compared with the experimental results, yielding a good match. The benefit of employing this model over the two degrees-of-freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved. @@ -11076,7 +11097,7 @@ During the detailed design phase, specifications in terms of stiffness and strok | Torsion Stiffness | $< 500\,Nm/\text{rad}$ | 260 | | Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | -After optimization using a finite element model, 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. +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. It serves several functions, as shown in Figure\nbsp{}ref:fig:test_joints_iso, such as: - Rigid interfacing with the nano-hexapod plates (yellow surfaces) @@ -11318,7 +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 finite element model 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 $k_s = 13\,N/\mu m$ from the acrshort:fem and an height $h=25\,mm$, the estimation errors of the bending stiffness due to shear is $\epsilon_s < 0.1\,\%$ ***** 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. @@ -11386,10 +11407,10 @@ To do so, an encoder[fn:test_joints_4] is used, which measures the motion of a r 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\nbsp{}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. -The obtained CAD design of the measurement bench is shown in Figure\nbsp{}ref:fig:test_joints_bench_overview while a zoom on the flexible joint with the associated important quantities is shown in Figure\nbsp{}ref:fig:test_joints_bench_side. +The obtained design of the measurement bench is shown in Figure\nbsp{}ref:fig:test_joints_bench_overview while a zoom on the flexible joint with the associated important quantities is shown in Figure\nbsp{}ref:fig:test_joints_bench_side. #+name: fig:test_joints_bench -#+caption: CAD view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side}) +#+caption: 3D view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_bench_overview} Schematic of the test bench to measure the bending stiffness of the flexible joints} @@ -11550,7 +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 finite element model ($k_{R_x} = k_{R_y} = 5\,Nm/\text{rad}$). +The measured bending stiffness values exhibited good agreement with the predictions from the acrshort:fem ($k_{R_x} = k_{R_y} = 5\,Nm/\text{rad}$). 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. @@ -11570,7 +11591,7 @@ The mounting procedure of the struts is explained in Section\nbsp{}ref:sec:test_ A mounting bench was used to ensure coaxiality between the two ends of the struts. In this way, no angular stroke is lost when mounted to the nano-hexapod. -The flexible modes of the struts were then experimentally measured and compared with a finite element model (Section\nbsp{}ref:sec:test_struts_flexible_modes). +The flexible modes of the struts were then experimentally measured and compared with a acrshort:fem (Section\nbsp{}ref:sec:test_struts_flexible_modes). Dynamic measurements of the strut are performed with the same test bench used to characterize the APA300ML dynamics (Section\nbsp{}ref:sec:test_struts_dynamical_meas). It was found that the dynamics from the acrshort:dac voltage to the displacement measured by the encoder is complex due to the flexible modes of the struts (Section\nbsp{}ref:sec:test_struts_flexible_modes). @@ -11587,10 +11608,10 @@ A mounting bench was developed to ensure: - Good coaxial alignment between the interfaces (cylinders) of the flexible joints. This is important not to loose to much angular stroke during their mounting into the nano-hexapod - Uniform length across all struts -- Precise alignment of the APA with the two flexible joints +- Precise alignment of the acrshort:apa with the two flexible joints - Reproducible and consistent assembly between all struts -A CAD view of the mounting bench is shown in Figure\nbsp{}ref:fig:test_struts_mounting_bench_first_concept. +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. 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. @@ -11600,7 +11621,7 @@ The strut length (defined by the distance between the rotation points of the two #+caption: Strut mounting bench #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_bench_first_concept}CAD view of the mounting bench} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_bench_first_concept}3D view of the mounting bench} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth @@ -11665,7 +11686,7 @@ The left sleeve has a thigh fit such that its orientation is fixed (it is roughl The cylindrical washers and the APA300ML are stacked on top of the flexible joints, as shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_2 and screwed together using a torque screwdriver. A dowel pin is used to laterally align the APA300ML with the flexible joints (see the dowel slot on the flexible joints in Figure\nbsp{}ref:fig:test_struts_mounting_joints). -Two cylindrical washers are used to allow proper mounting even when the two APA interfaces are not parallel. +Two cylindrical washers are used to allow proper mounting even when the two acrshort:apa interfaces are not parallel. The encoder and ruler are then fixed to the strut and properly aligned, as shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_3. @@ -11768,7 +11789,7 @@ These tests were performed with and without the encoder being fixed to the strut #+end_subfigure #+end_figure -The obtained frequency response functions for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_no_enc when the encoder is not fixed to the strut and in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_enc when the encoder is fixed to the strut. +The obtained acrshortpl:frf for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_no_enc when the encoder is not fixed to the strut and in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_enc when the encoder is fixed to the strut. #+name: fig:test_struts_spur_res_frf #+caption: Measured frequency response functions without the encoder\nbsp{}ref:fig:test_struts_spur_res_frf and with the encoder\nbsp{}ref:fig:test_struts_spur_res_frf_enc @@ -11858,7 +11879,7 @@ System identification was performed without the encoder being fixed to the strut #+end_subfigure #+end_figure -The obtained frequency response functions are compared in Figure\nbsp{}ref:fig:test_struts_effect_encoder. +The obtained acrshortpl:frf are compared in Figure\nbsp{}ref:fig:test_struts_effect_encoder. It was found that the encoder had very little effect on the transfer function from excitation voltage $u$ to the axial motion of the strut $d_a$ as measured by the interferometer (Figure\nbsp{}ref:fig:test_struts_effect_encoder_int). This means that the axial motion of the strut is unaffected by the presence of the encoder. Similarly, it has little effect on the transfer function from $u$ to the sensor stack voltage $V_s$ (Figure\nbsp{}ref:fig:test_struts_effect_encoder_iff). @@ -11941,9 +11962,9 @@ The reason for this variability will be studied in the next section thanks to th **** Introduction :ignore: The multi-body model of the strut was included in the multi-body model of the test bench (see Figure\nbsp{}ref:fig:test_struts_simscape_model). -The obtained model was first used to compare the measured FRF with the existing model (Section\nbsp{}ref:ssec:test_struts_comp_model). +The obtained model was first used to compare the measured acrshort:frf with the existing model (Section\nbsp{}ref:ssec:test_struts_comp_model). -Using a flexible APA model (extracted from a acrshort:fem), the effect of a misalignment of the APA with respect to flexible joints is studied (Section\nbsp{}ref:ssec:test_struts_effect_misalignment). +Using a flexible acrshort:apa model (extracted from a acrshort:fem), the effect of a misalignment of the acrshort:apa with respect to flexible joints is studied (Section\nbsp{}ref:ssec:test_struts_effect_misalignment). It was found that misalignment has a large impact on the dynamics from $u$ to $d_e$. This misalignment is estimated and measured in Section\nbsp{}ref:ssec:test_struts_meas_misalignment. The struts were then disassembled and reassemble a second time to optimize alignment (Section\nbsp{}ref:sec:test_struts_meas_all_aligned_struts). @@ -11957,9 +11978,9 @@ The struts were then disassembled and reassemble a second time to optimize align <> Two models of the APA300ML are used here: a simple two-degrees-of-freedom model and a model using a super-element extracted from a acrlong:fem. -These two models of the APA300ML were tuned to best match the measured frequency response functions of the APA alone. +These two models of the APA300ML were tuned to best match the measured acrshortpl:frf of the acrshort:apa alone. The flexible joints were modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness). -These two models are compared with the measured frequency responses in Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model. +These two models are compared using the measured acrshortpl:frf in Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model. The model dynamics from DAC voltage $u$ to the axial motion of the strut $d_a$ (Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model_int) and from DAC voltage $u$ to the force sensor voltage $V_s$ (Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model_iff) are well matching the experimental identification. @@ -11995,7 +12016,7 @@ 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 APA) can explain such dynamics. +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. @@ -12004,9 +12025,9 @@ In this case, the "x-bending" mode at 200Hz (see Figure\nbsp{}ref:fig:test_strut #+attr_latex: :width 0.8\linewidth [[file:figs/test_struts_misalign_schematic.png]] -To verify this assumption, the dynamics from the output DAC voltage $u$ to the measured displacement by the encoder $d_e$ is computed using the flexible APA model for several misalignments in the $y$ direction. +To verify this assumption, the dynamics from the output DAC voltage $u$ to the measured displacement by the encoder $d_e$ is computed using the flexible acrshort:apa model for several misalignments in the $y$ direction. The obtained dynamics are shown in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y. -The alignment of the APA with the flexible joints has a large influence on the dynamics from actuator voltage to the measured displacement by the encoder. +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 location of the complex conjugate zero between the first two resonances: @@ -12017,7 +12038,7 @@ The misalignment in the $y$ direction mostly influences: 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). -A comparison of the experimental frequency response functions 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. +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. #+name: fig:test_struts_effect_misalignment @@ -12041,17 +12062,17 @@ This similarity suggests that the identified differences in dynamics are caused **** Measured strut misalignment <> -During the initial mounting of the struts, as presented in Section\nbsp{}ref:sec:test_struts_mounting, the positioning pins that were used to position the APA with respect to the flexible joints in the $y$ directions were not used (not received at the time). +During the initial mounting of the struts, as presented in Section\nbsp{}ref:sec:test_struts_mounting, the positioning pins that were used to position the acrshort:apa with respect to the flexible joints in the $y$ directions were not used (not received at the time). Therefore, large $y$ misalignments are expected. -To estimate the misalignments between the two flexible joints and the APA: +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 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 APA shell is $20\,mm$, $0.5\,mm$ of height difference should be measured if the two are perfectly aligned +- 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 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 APA). +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. #+name: tab:test_struts_meas_y_misalignment @@ -12066,13 +12087,13 @@ Thickness differences for all the struts were found to be between $0.94\,mm$ and | 4 | -0.01 | 0.54 | | 5 | 0.15 | 0.02 | -By using the measured $y$ misalignment in the model with the flexible APA model, the model dynamics from $u$ to $d_e$ is closer to the measured dynamics, as shown in Figure\nbsp{}ref:fig:test_struts_comp_dy_tuned_model_frf_enc. +By using the measured $y$ misalignment in the model with the flexible acrshort:apa model, the model dynamics from $u$ to $d_e$ is closer to the measured dynamics, as shown in Figure\nbsp{}ref:fig:test_struts_comp_dy_tuned_model_frf_enc. A better match in the dynamics can be obtained by fine-tuning both the $x$ and $y$ misalignments (yellow curves in Figure\nbsp{}ref:fig:test_struts_comp_dy_tuned_model_frf_enc). -This confirms that misalignment between the APA and the strut axis (determined by the two flexible joints) is critical and inducing large variations in the dynamics from DAC voltage $u$ to encoder measured displacement $d_e$. -If encoders are fixed to the struts, the APA and flexible joints must be precisely aligned when mounting the struts. +This confirms that misalignment between the acrshort:apa and the strut axis (determined by the two flexible joints) is critical and inducing large variations in the dynamics from DAC voltage $u$ to encoder measured displacement $d_e$. +If encoders are fixed to the struts, the acrshort:apa and flexible joints must be precisely aligned when mounting the struts. -In the next section, the struts are re-assembled with a "positioning pin" to better align the APA with the flexible joints. +In the next section, the struts are re-assembled with a "positioning pin" to better align the acrshort:apa with the flexible joints. With a better alignment, the amplitude of the spurious resonances is expected to decrease, as shown in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y. #+name: fig:test_struts_comp_dy_tuned_model_frf_enc @@ -12084,7 +12105,7 @@ With a better alignment, the amplitude of the spurious resonances is expected to <> After receiving the positioning pins, the struts were mounted again with the positioning pins. -This should improve the alignment of the APA with the two flexible joints. +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). @@ -12271,7 +12292,7 @@ Then, four springs[fn:test_nhexa_6] were selected with low spring rate such that 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 -#+caption: CAD View of the vibration table. The purple cylinders are representing the soft springs. +#+caption: 3D View of the vibration table. The purple cylinders are representing the soft springs. #+attr_latex: :width 0.7\linewidth [[file:figs/test_nhexa_suspended_table_cad.jpg]] @@ -12360,7 +12381,7 @@ The obtained suspension modes of the multi-body model are compared with the meas **** Introduction :ignore: The Nano-Hexapod was then mounted on top of the suspended table, as shown in Figure\nbsp{}ref:fig:test_nhexa_hexa_suspended_table. -All instrumentation (Speedgoat with ADC, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) were configured and connected to the nano-hexapod using many cables. +All instrumentation (Speedgoat with acrshort:adc, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) were configured and connected to the nano-hexapod using many cables. #+name: fig:test_nhexa_hexa_suspended_table #+caption: Mounted Nano-Hexapod on top of the suspended table @@ -12371,7 +12392,7 @@ A modal analysis of the nano-hexapod is first performed in Section\nbsp{}ref:sse The results of the modal analysis will be useful to better understand the measured dynamics from actuators to sensors. A block diagram of the (open-loop) system is shown in Figure\nbsp{}ref:fig:test_nhexa_nano_hexapod_signals. -The frequency response functions from controlled signals $\bm{u}$ to the force sensors voltages $\bm{V}_s$ and to the encoders measured displacements $\bm{d}_e$ are experimentally identified in Section\nbsp{}ref:ssec:test_nhexa_identification. +The acrshortpl:frf from controlled signals $\bm{u}$ to the force sensors voltages $\bm{V}_s$ and to the encoders measured displacements $\bm{d}_e$ are experimentally identified in Section\nbsp{}ref:ssec:test_nhexa_identification. The effect of the payload mass on the dynamics is discussed in Section\nbsp{}ref:ssec:test_nhexa_added_mass. #+name: fig:test_nhexa_nano_hexapod_signals @@ -12432,7 +12453,7 @@ These modes are summarized in Table\nbsp{}ref:tab:test_nhexa_hexa_modal_modes_li The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) to the six measured displacement by the encoders ($d_{e1}$ to $d_{e6}$) and to the six force sensors ($V_{s1}$ to $V_{s6}$) were identified by generating low-pass filtered white noise for each command signal, one by one. -The $6 \times 6$ FRF matrix from $\bm{u}$ ot $\bm{d}_e$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de. +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. @@ -12453,7 +12474,7 @@ This would not have occurred if the encoders were fixed to the struts. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_identified_frf_de.png]] -Similarly, the $6 \times 6$ FRF matrix from $\bm{u}$ to $\bm{V}_s$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs. +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. 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. @@ -12477,7 +12498,7 @@ These three cylindrical masses on top of the nano-hexapod are shown in Figure\nb #+attr_latex: :width 0.8\linewidth [[file:figs/test_nhexa_table_mass_3.jpg]] -The obtained frequency response functions 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, 13kg, 26kg and 39kg) 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. @@ -12489,8 +12510,8 @@ 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 measured FRFs 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 FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust manner. +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. #+name: fig:test_nhexa_identified_frf_masses #+caption: Measured Frequency Response Functions from $u_i$ to $d_{ei}$ (\subref{fig:test_nhexa_identified_frf_de_masses}) and from $u_i$ to $V_{si}$ (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown. @@ -12532,15 +12553,15 @@ This is checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_masses. **** Nano-Hexapod model dynamics <> -The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF APA, and rigid top and bottom plates. -The stiffness values of the flexible joints were chosen based on the values estimated using the test bench and on the FEM. -The parameters of the APA model were determined from the test bench of the APA. +The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF acrshort:apa, and rigid top and bottom plates. +The stiffness values of the flexible joints were chosen based on the values estimated using the test bench and on the acrshort:fem. +The parameters of the acrshort:apa model were determined from the test bench of the acrshort:apa. The $6 \times 6$ transfer function matrices from $\bm{u}$ to $\bm{d}_e$ and from $\bm{u}$ to $\bm{V}_s$ are then extracted from the multi-body model. -First, is it evaluated how well the models matches the "direct" terms of the measured FRF matrix. -To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag. +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 APAs are here modeled as a simple uniaxial 2-DoF system. +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. 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 @@ -12564,8 +12585,8 @@ At higher frequencies, no resonances can be observed in the model, as the top pl **** Dynamical coupling <> -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 MIMO systems. -Instead of comparing the full 36 elements of the $6 \times 6$ 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 $d_{e6}$. +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. Similar results are observed for all other coupling terms and for the transfer function from $\bm{u}$ to $\bm{V}_s$. @@ -12575,7 +12596,7 @@ Similar results are observed for all other coupling terms and for the transfer f [[file:figs/test_nhexa_comp_simscape_de_all.png]] 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 FRF in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all_flex. +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. 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. @@ -12640,17 +12661,17 @@ Below the first suspension mode, good decoupling could be observed for the trans Many other modes were present above 700Hz, 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 frequency response functions 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). +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). This indicates that it is possible to implement decentralized Integral Force Feedback in a robust manner. The developed multi-body model of the nano-hexapod was found to accurately represents the suspension modes of the Nano-Hexapod (Section\nbsp{}ref:sec:test_nhexa_model). -Both FRF matrices from $\bm{u}$ to $\bm{V}_s$ and from $\bm{u}$ to $\bm{d}_e$ are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective. +Both acrshort:frf matrices from $\bm{u}$ to $\bm{V}_s$ and from $\bm{u}$ to $\bm{d}_e$ are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective. At frequencies above the suspension modes, the Nano-Hexapod model became inaccurate because the flexible modes were not modeled. It was found that modeling the APA300ML using a /super-element/ allows to model the internal resonances of the struts. The same can be done with the top platform and the encoder supports; however, the model order would be higher and may become unpractical for simulation. Obtaining a model that accurately represents the complex dynamics of the Nano-Hexapod was made possible by the modeling approach used in this study. -This approach involved tuning and validating models of individual components (such as the APA and flexible joints) using dedicated test benches. +This approach involved tuning and validating models of individual components (such as the acrshort:apa and flexible joints) using dedicated test benches. The different models could then be combined to form the Nano-Hexapod dynamical model. If a model of the nano-hexapod was developed in one time, it would be difficult to tune all the model parameters to match the measured dynamics, or even to know if the model "structure" would be adequate to represent the system dynamics. @@ -12669,7 +12690,7 @@ 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. These measurements were compared with predictions from the multi-body model to verify its accuracy and applicability to control design. -Sections\nbsp{}ref:sec:test_id31_iff and ref:sec:test_id31_hac focus on the implementation and validation of the HAC-LAC control architecture. +Sections\nbsp{}ref:sec:test_id31_iff and ref:sec:test_id31_hac focus on the implementation and validation of the acrshort:haclac control architecture. First, Section\nbsp{}ref:sec:test_id31_iff demonstrates the application of decentralized Integral Force Feedback for robust active damping of the nano-hexapod suspension modes. This is followed in Section\nbsp{}ref:sec:test_id31_hac by the implementation of the high authority controller, which addresses low-frequency disturbances and completes the control system design. @@ -12741,7 +12762,7 @@ 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 point of interest is indicated by the blue frame $\{B\}$, which is located $H = 150\,mm$ above the nano-hexapod's top platform. +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 $l_2 = 16.2\,mm$. 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. @@ -12922,7 +12943,7 @@ After amplification, the voltages across the piezoelectric stack actuators are $ From the setpoint of micro-station stages ($r_{D_y}$ for the translation stage, $r_{R_y}$ for the tilt stage and $r_{R_z}$ for the spindle), the reference pose of the sample $\bm{r}_{\mathcal{X}}$ is computed using the micro-station's kinematics. The sample's position $\bm{y}_\mathcal{X} = [D_x,\,D_y,\,D_z,\,R_x,\,R_y,\,R_z]$ is measured using multiple sensors. -First, the five interferometers $\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]$ are used to measure the $[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$ degrees of freedom of the sample. +First, the five interferometers $\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]$ are used to measure the $[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$ acrshortpl:dof of the sample. The $R_z$ position of the sample is computed from the spindle's setpoint $r_{R_z}$ and from the 6 encoders $\bm{d}_e$ integrated in the nano-hexapod. The sample's position $\bm{y}_{\mathcal{X}}$ is compared to the reference position $\bm{r}_{\mathcal{X}}$ to compute the position error in the frame of the (rotating) nano-hexapod $\bm{\epsilon\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]$. @@ -12945,7 +12966,7 @@ A comparison between the model and the measured dynamics is presented in Figure\ A good match can be observed for the diagonal dynamics (except the high frequency modes which are not modeled). However, the coupling of the transfer function from command signals $\bm{u}$ to the estimated strut motion from the external metrology $\bm{\epsilon\mathcal{L}}$ is larger than expected (Figure\nbsp{}ref:fig:test_id31_first_id_int). -The experimental time delay estimated from the FRF (Figure\nbsp{}ref:fig:test_id31_first_id_int) is larger than expected. +The experimental time delay estimated from the acrshort:frf (Figure\nbsp{}ref:fig:test_id31_first_id_int) is larger than expected. After investigation, it was found that the additional delay was due to a digital processing unit[fn:test_id31_3] that was used to get the interferometers' signals in the Speedgoat. This issue was later solved. @@ -12999,7 +13020,7 @@ Results shown in Figure\nbsp{}ref:fig:test_id31_Rz_align_correct are indeed indi The dynamics of the plant was identified again after fine alignment and compared with the model dynamics in Figure\nbsp{}ref:fig:test_id31_first_id_int_better_rz_align. Compared to the initial identification shown in Figure\nbsp{}ref:fig:test_id31_first_id_int, the obtained coupling was decreased and was close to the coupling obtained with the multi-body model. -At low frequency (below $10\,\text{Hz}$), all off-diagonal elements have an amplitude $\approx 100$ times lower than the diagonal elements, indicating that a low bandwidth feedback controller can be implemented in a decentralized manner (i.e. $6$ SISO controllers). +At low frequency (below $10\,\text{Hz}$), all off-diagonal elements have an amplitude $\approx 100$ times lower than the diagonal elements, indicating that a low bandwidth feedback controller can be implemented in a decentralized manner (i.e. $6$ acrshort:siso controllers). Between $650\,\text{Hz}$ and $1000\,\text{Hz}$, several modes can be observed, which are due to flexible modes of the top platform and the modes of the two spheres adjustment mechanism. The flexible modes of the top platform can be passively damped, whereas the modes of the two reference spheres should not be present in the final application. @@ -13075,7 +13096,7 @@ The obtained dynamics from command signal $u$ to estimated strut error $\epsilon Both direct terms (Figure\nbsp{}ref:fig:test_id31_effect_rotation_direct) and coupling terms (Figure\nbsp{}ref:fig:test_id31_effect_rotation_coupling) are unaffected by the rotation. The same can be observed for the dynamics from command signal to encoders and to force sensors. This confirms that spindle's rotation has no significant effect on plant dynamics. -This also indicates that the metrology kinematics is correct and is working in real time. +This also indicates that the metrology kinematics is correct and is working in real-time. #+name: fig:test_id31_effect_rotation #+caption: Effect of the spindle rotation on the plant dynamics from $u$ to $\epsilon\mathcal{L}$. Three rotational velocities are tested ($0\,\text{deg}/s$, $36\,\text{deg}/s$ and $180\,\text{deg}/s$). Both direct terms (\subref{fig:test_id31_effect_rotation_direct}) and coupling terms (\subref{fig:test_id31_effect_rotation_coupling}) are displayed. @@ -13100,7 +13121,7 @@ This also indicates that the metrology kinematics is correct and is working in r :UNNUMBERED: t :END: -The identified frequency response functions from command signals $\bm{u}$ to the force sensors $\bm{V}_s$ and to the estimated strut errors $\bm{\epsilon\mathcal{L}}$ are well matching the dynamics of the developed multi-body model. +The identified acrshortpl:frf from command signals $\bm{u}$ to the force sensors $\bm{V}_s$ and to the estimated strut errors $\bm{\epsilon\mathcal{L}}$ are well matching the dynamics of the developed multi-body model. The effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used. The spindle rotation had no visible effect on the measured dynamics, indicating that controllers should be robust against spindle rotation. @@ -13218,7 +13239,7 @@ The obtained damped plants are compared to the open-loop plants in Figure\nbsp{} The peak amplitudes corresponding to the suspension modes were approximately reduced by a factor $10$ for all considered payloads, indicating the effectiveness of the decentralized IFF control strategy. To experimentally validate the Decentralized IFF controller, it was implemented and the damped plants (i.e. the transfer function from $\bm{u}^\prime$ to $\bm{\epsilon\mathcal{L}}$) were identified for all payload conditions. -The obtained frequency response functions are compared with the model in Figure\nbsp{}ref:fig:test_id31_hac_plant_effect_mass verifying the good correlation between the predicted damped plant using the multi-body model and the experimental results. +The obtained acrshortpl:frf are compared with the model in Figure\nbsp{}ref:fig:test_id31_hac_plant_effect_mass verifying the good correlation between the predicted damped plant using the multi-body model and the experimental results. #+name: fig:test_id31_hac_plant_effect_mass_comp_model #+caption: Comparison of the open-loop plants and the damped plant with Decentralized IFF, estimated from the multi-body model (\subref{fig:test_id31_comp_ol_iff_plant_model}). Comparison of measured damped and modeled plants for all considered payloads (\subref{fig:test_id31_hac_plant_effect_mass}). Only "direct" terms ($\epsilon\mathcal{L}_i/u_i^\prime$) are displayed for simplificty @@ -13282,9 +13303,9 @@ Considering the complexity of the system's dynamics, the model can be considered [[file:figs/test_id31_comp_simscape_hac.png]] The challenge here is to tune a high authority controller such that it is robust to the change in dynamics due to different payloads being used. -Without using the HAC-LAC strategy, it would be necessary to design a controller that provides good performance for all undamped dynamics (blue curves in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants), which is a very complex control problem. -With the HAC-LAC strategy, the designed controller must be robust to all the damped dynamics (red curves in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants), which is easier from a control perspective. -This is one of the key benefits of using the HAC-LAC strategy. +Without using the acrshort:haclac strategy, it would be necessary to design a controller that provides good performance for all undamped dynamics (blue curves in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants), which is a very complex control problem. +With the acrshort:haclac strategy, the designed controller must be robust to all the damped dynamics (red curves in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants), which is easier from a control perspective. +This is one of the key benefits of using the acrshort:haclac strategy. #+name: fig:test_id31_comp_all_undamped_damped_plants #+caption: Comparison of the (six) direct terms for all (four) payload conditions in the undamped case (in blue) and the damped case (i.e. with the decentralized IFF being implemented, in red). @@ -13356,10 +13377,10 @@ However, small stability margins were observed for the highest mass, indicating **** Performance estimation with simulation of Tomography scans <> -To estimate the performances that can be expected with this HAC-LAC architecture and the designed controller, simulations of tomography experiments were performed[fn:test_id31_4]. +To estimate the performances that can be expected with this acrshort:haclac architecture and the designed controller, simulations of tomography experiments were performed[fn:test_id31_4]. The rotational velocity was set to $180\,\text{deg/s}$, and no payload was added on top of the nano-hexapod. An open-loop simulation and a closed-loop simulation were performed and compared in Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim. -The obtained closed-loop positioning accuracy was found to comply with the requirements as it succeeded to keep the point of interest on the beam (Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz). +The obtained closed-loop positioning accuracy was found to comply with the requirements as it succeeded to keep the acrshort:poi on the beam (Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz). #+name: fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim #+caption: Position error of the sample in the XY (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy}) and YZ (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}) planes during a simulation of a tomography experiment at $180\,\text{deg/s}$. No payload is placed on top of the nano-hexapod. @@ -13406,9 +13427,9 @@ This validation confirmed that the model can be reliably used to tune the feedba 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. -Finally, tomography experiments were simulated to validate the HAC-LAC architecture. +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). -With no payload at $180\,\text{deg/s}$, the NASS successfully maintained the sample point of interest in the beam, which fulfilled the specifications. +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. @@ -13431,7 +13452,7 @@ Unless explicitly stated, all closed-loop experiments were performed using the r 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 point of interest 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. +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. 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). @@ -13454,8 +13475,8 @@ First, tomography scans were performed with a rotational velocity of $6\,\text{d Each experimental sequence consisted of two complete spindle rotations: an initial open-loop rotation followed by a closed-loop rotation. The experimental results for the $26\,\text{kg}$ payload are presented in Figure\nbsp{}ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit. -Due to the static deformation of the micro-station stages under payload loading, a significant eccentricity was observed between the point of interest and the spindle rotation axis. -To establish a theoretical lower bound for open-loop errors, an ideal scenario was assumed, where the point of interest perfectly aligns with the spindle rotation axis. +Due to the static deformation of the micro-station stages under payload loading, a significant eccentricity was observed between the acrshort:poi and the spindle rotation axis. +To establish a theoretical lower bound for open-loop errors, an ideal scenario was assumed, where the acrshort:poi perfectly aligns with the spindle rotation axis. This idealized case was simulated by first calculating the eccentricity through circular fitting (represented by the dashed black circle in Figure\nbsp{}ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit), and then subtracting it from the measured data, as shown in Figure\nbsp{}ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed. 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. @@ -13478,7 +13499,7 @@ While this approach likely underestimates actual open-loop errors, as perfect al #+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\nbsp{}ref:fig:test_id31_tomo_Wz36_results. -Results are indicating the NASS succeeds in keeping the sample's point of interest on the beam, except for the highest mass of $39\,\text{kg}$ for which the lateral motion is a bit too high. +Results are indicating the NASS succeeds in keeping the sample's acrshort:poi on the beam, except for the highest mass of $39\,\text{kg}$ for which the lateral motion is a bit too high. These experimental findings are consistent with the predictions from the tomography simulations presented in Section\nbsp{}ref:ssec:test_id31_iff_hac_robustness. #+name: fig:test_id31_tomo_Wz36_results @@ -13489,7 +13510,7 @@ These experimental findings are consistent with the predictions from the tomogra ***** Fast Tomography scans A tomography experiment was then performed with the highest rotational velocity of the Spindle: $180\,\text{deg/s}$[fn:test_id31_7]. -The trajectory of the point of interest during the fast tomography scan is shown in Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp. +The trajectory of the acrshort:poi during the fast tomography scan is shown in Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp. Although the experimental results closely match the simulation results (Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim), the actual performance was slightly lower than predicted. Nevertheless, even with this robust (i.e. conservative) HAC implementation, the system performance was already close to the specified requirements. @@ -13513,8 +13534,8 @@ Nevertheless, even with this robust (i.e. conservative) HAC implementation, the ***** Cumulative Amplitude Spectra -A comparative analysis was conducted using three tomography scans at $180\,\text{deg/s}$ to evaluate the effectiveness of the HAC-LAC strategy in reducing positioning errors. -The scans were performed under three conditions: open-loop, with decentralized IFF control, and with the complete HAC-LAC strategy. +A comparative analysis was conducted using three tomography scans at $180\,\text{deg/s}$ to evaluate the effectiveness of the acrshort:haclac strategy in reducing positioning errors. +The scans were performed under three conditions: open-loop, with decentralized IFF control, and with the complete acrshort:haclac strategy. For this specific measurement, an enhanced high authority controller (discussed in Section\nbsp{}ref:ssec:test_id31_cf_control) was optimized for low payload masses to meet the performance requirements. Figure\nbsp{}ref:fig:test_id31_hac_cas_cl presents the cumulative amplitude spectra of the position errors for all three cases. @@ -13553,7 +13574,7 @@ 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). -The results confirmed that the NASS successfully maintained the point of interest within the specified beam parameters throughout the scanning process. +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 #+caption: Reflectivity scan ($R_y$) with a rotational velocity of $100\,\mu \text{rad}/s$. @@ -13821,7 +13842,7 @@ A schematic of the proposed control architecture is illustrated in Figure\nbsp{} # TODO - Add link to 2DoF model Implementation of this control architecture necessitates a plant model, which must subsequently be inverted. -This plant model was derived from the multi-body model incorporating the previously detailed 2-DoF APA model, such that the model order stays relatively low. +This plant model was derived from the multi-body model incorporating the previously detailed 2-DoF acrshort:apa model, such that the model order stays relatively low. Proposed analytical formulas for complementary filters having $40\,\text{dB/dec}$ were used during this experimental validation. # TODO - Add link to the analytical formulas @@ -13887,7 +13908,7 @@ A comprehensive series of experimental validations was conducted to evaluate the 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 $R_y$). 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 HAC-LAC 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 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 $100\,\mu m/s$. @@ -13935,7 +13956,7 @@ The identified limitations, primarily related to high-speed lateral scanning and <> This chapter presented a comprehensive experimental validation of the Nano Active Stabilization System (NASS) on the ID31 beamline, demonstrating its capability to maintain precise sample positioning during various experimental scenarios. -The implementation and testing followed a systematic approach, beginning with the development of a short-stroke metrology system to measure the sample position, followed by the successful implementation of a HAC-LAC control architecture, and concluding in extensive performance validation across diverse experimental conditions. +The implementation and testing followed a systematic approach, beginning with the development of a short-stroke metrology system to measure the sample position, followed by the successful implementation of a acrshort:haclac control architecture, and concluding in extensive performance validation across diverse experimental conditions. The short-stroke metrology system, while designed as a temporary solution, proved effective in providing high-bandwidth and low-noise 5-DoF position measurements. The careful alignment of the fibered interferometers targeting the two reference spheres ensured reliable measurements throughout the testing campaign. @@ -13963,7 +13984,7 @@ A methodical approach was employed—first characterizing individual components Initially, the Amplified Piezoelectric Actuators (APA300ML) were characterized, revealing consistent mechanical and electrical properties across multiple units. The implementation of Integral Force Feedback was shown to add significant damping to the system. -Two models of the APA300ML were developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from finite element analysis. +Two models of the APA300ML were developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from acrshort:fea. Both models accurately represented the axial dynamics of the actuators, with the super-element model additionally capturing flexible modes. The flexible joints were examined for geometric accuracy and bending stiffness, with measurements confirming compliance with design specifications. @@ -13975,10 +13996,10 @@ This finding led to the strategic decision to mount encoders on the nano-hexapod The nano-hexapod was then assembled and mounted on a suspended table to characterize its dynamic behavior. The measurement setup enabled isolation of the hexapod's dynamics from potential influence of complex support dynamics. -The experimental frequency response functions exhibited good correlation with the multi-body model, confirming that the model can be used for control system design. +The experimental acrshortpl:frf exhibited good correlation with the multi-body model, confirming that the model can be used for control system design. Finally, the complete NASS was validated on the ID31 beamline using a short-stroke metrology system. -The HAC-LAC control architecture successfully provided robust active damping of suspension modes and rejection of low-frequency disturbances across various payload conditions. +The acrshort:haclac control architecture successfully provided robust active damping of suspension modes and rejection of low-frequency disturbances across various payload conditions. Comprehensive testing under typical experimental scenarios—including tomography scans, reflectivity scans, and diffraction tomography—demonstrated the NASS ability to maintain the positioning errors within specifications ($30\,\text{nm RMS}$ in lateral direction, $15\,\text{nm RMS}$ in vertical direction, and $250\,\text{nrad RMS}$ in tilt direction). The system performed exceptionally well during vertical scans, though some limitations were identified during rapid lateral scanning and with heavier payloads. @@ -14001,15 +14022,15 @@ The conceptual design phase rigorously evaluated the feasibility of the NASS con Through progressive modeling, from simplified uniaxial representations to complex multi-body dynamic simulations, key design insights were obtained. It was determined that an active platform with moderate stiffness offered an optimal compromise, decoupling the system from micro-station dynamics while mitigating gyroscopic effects from continuous rotation. The multi-body modeling approach, informed by experimental modal analysis of the micro-station, was essential for capturing the system's complex dynamics. -The Stewart platform architecture was selected for the active stage, and its viability was confirmed through closed-loop simulations employing a High-Authority Control / Low-Authority Control (HAC-LAC) strategy. +The Stewart platform architecture was selected for the active stage, and its viability was confirmed through closed-loop simulations employing a acrfull:haclac strategy. This strategy used a modified form of Integral Force Feedback (IFF), adapted to provide robust active damping despite the platform rotation and varying payloads. These simulations demonstrated the NASS concept could meet the nanometer-level stability requirements under realistic operating conditions. Following the conceptual validation, the detailed design phase focused on translating the NASS concept into an optimized, physically realizable system. Geometric optimization studies refined the Stewart platform configuration. -A hybrid modeling technique combining Finite Element Analysis (FEA) with multi-body dynamics simulation was applied and experimentally validated. -This approach enabled detailed optimization of components, such as Amplified Piezoelectric Actuators (APA) and flexible joints, while efficiently simulating the complete system dynamics. -By dedicating one stack of the APA specifically to force sensing, excellent collocation with the actuator stacks was achieved, which is critical for implementing robust decentralized IFF. +A hybrid modeling technique combining acrfull:fea with multi-body dynamics simulation was applied and experimentally validated. +This approach enabled detailed optimization of components, such as acrfull:apa and flexible joints, while efficiently simulating the complete system dynamics. +By dedicating one stack of the acrshort:apa specifically to force sensing, excellent collocation with the actuator stacks was achieved, which is critical for implementing robust decentralized IFF. Work was also undertaken on the optimization of the control strategy for the active platform. Instrumentation selection (sensors, actuators, control hardware) was guided by dynamic error budgeting to ensure component noise levels met the overall nanometer-level performance target. @@ -14017,7 +14038,7 @@ The final phase of the project was dedicated to the experimental validation of t Component tests confirmed the performance of the selected actuators and flexible joints, validated their respective models. Dynamic testing of the assembled nano-hexapod on an isolated test bench provided essential experimental data that correlated well with the predictions of the multi-body model. The final validation was performed on the ID31 beamline, using a short-stroke metrology system to assess performance under realistic experimental conditions. -These tests demonstrated that the NASS, operating with the implemented HAC-LAC control architecture, successfully achieved the target positioning stability – maintaining residual errors below $30\,\text{nm RMS}$ laterally, $15\,\text{nm RMS}$ vertically, and $250\,\text{nrad RMS}$ in tilt – during various experiments, including tomography scans with significant payloads. +These tests demonstrated that the NASS, operating with the implemented acrshort:haclac control architecture, successfully achieved the target positioning stability – maintaining residual errors below $30\,\text{nm RMS}$ laterally, $15\,\text{nm RMS}$ vertically, and $250\,\text{nrad RMS}$ in tilt – during various experiments, including tomography scans with significant payloads. Crucially, the system's robustness to variations in payload mass and operational modes was confirmed. ** Perspectives @@ -14027,7 +14048,7 @@ Although this research successfully validated the NASS concept, it concurrently ***** Automatic tuning of a multi-body model from an experimental modal analysis The manual tuning process employed to match the multi-body model dynamics with experimental measurements was found to be laborious. -Systems like the micro-station can be conceptually modeled as interconnected solid bodies, springs, and dampers, with component inertia readily obtainable from CAD models. +Systems like the micro-station can be conceptually modeled as interconnected solid bodies, springs, and dampers, with component inertia readily obtainable from 3D models. An interesting perspective is the development of methods for the automatic tuning of the multi-body model's stiffness matrix (representing the interconnecting spring stiffnesses) directly from experimental modal analysis data. Such a capability would enable the rapid generation of accurate dynamic models for existing end-stations, which could subsequently be used for detailed system analysis and simulation studies. @@ -14045,15 +14066,15 @@ 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 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. +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. Nevertheless, a more rigorous analysis of this control architecture and its comparison with similar approaches documented in the literature is necessary to fully understand its capabilities and limitations. ***** Sensor Fusion -While the HAC-LAC approach demonstrated a simple and comprehensive methodology for controlling the NASS, sensor fusion represents an interesting alternative that is worth investigating. +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. One potential approach involves fusing external metrology (used at low frequencies) with force sensors (employed at high frequencies). @@ -14069,8 +14090,8 @@ This challenge is particularly complex when continuous rotation is combined with Yet, the development of such metrology systems is considered critical for future end-stations, especially for future tomography end stations where nano-meter accuracy is desired across larger strokes. Promising approaches have been presented in the literature. -A ball lens retroreflector is used in [[cite:&schropp20_ptynam]], providing a $\approx 1\,\text{mm}^3$ measuring volume, but does not fully accommodate complete rotation. -In [[cite:&geraldes23_sapot_carnaub_sirius_lnls]], an interesting metrology approach is presented, using interferometers for long stroke/non-rotated movements and capacitive sensors for short stroke/rotated positioning. +A ball lens retroreflector is used in\nbsp{}[[cite:&schropp20_ptynam]], providing a $\approx 1\,\text{mm}^3$ measuring volume, but does not fully accommodate complete rotation. +In\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]], an interesting metrology approach is presented, using interferometers for long stroke/non-rotated movements and capacitive sensors for short stroke/rotated positioning. ***** Alternative Architecture for the NASS @@ -14089,14 +14110,14 @@ One possible configuration, illustrated in Figure ref:fig:conclusion_nass_archit #+attr_latex: :options [h!tbp] [[file:figs/conclusion_nass_architecture.png]] -With this architecture, the online metrology could be divided into two systems, as proposed by [[cite:&geraldes23_sapot_carnaub_sirius_lnls]]: a long-stroke metrology system potentially using interferometers, and a short-stroke metrology system using capacitive sensors, as successfully demonstrated by [[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]]. +With this architecture, the online metrology could be divided into two systems, as proposed by\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]]: a long-stroke metrology system potentially using interferometers, and a short-stroke metrology system using capacitive sensors, as successfully demonstrated by\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]]. ***** 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 $3\,mm$ 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 [[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. +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. However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature. diff --git a/phd-thesis.tex b/phd-thesis.tex index 2f8378a..3e46316 100644 --- a/phd-thesis.tex +++ b/phd-thesis.tex @@ -1,28 +1,55 @@ -% Created 2025-04-22 Tue 16:24 +% Created 2025-04-22 Tue 18:56 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \input{config.tex} -\newacronym{haclac}{HAC-LAC}{High Authority Control - Low Authority Control} +\newacronym{adc}{ADC}{Analog to Digital Converter} +\newacronym{apa}{APA}{Amplified Piezoelectric Actuator} +\newacronym{asd}{ASD}{Amplitude Spectrum Density} +\newacronym{cas}{CAS}{Cumulative Amplitude Spectrum} +\newacronym{cl}{CL}{Closed Loop} +\newacronym{cmif}{CMIF}{Complex Modal Indication Function} +\newacronym{cok}{CoK}{Center of Stiffness} +\newacronym{com}{CoM}{Center of Mass} +\newacronym{dac}{DAC}{Digital to Analog Converter} +\newacronym{dvf}{DVF}{Direct Velocity Feedback} +\newacronym{dof}{DoF}{Degrees of Freedom} +\newacronym{ebs}{EBS}{Extremely Brilliant Source} +\newacronym{edm}{EDM}{Electrical Discharge Machining} +\newacronym{esrf}{ESRF}{European Synchrotron Radiation Facility} +\newacronym{fea}{FEA}{Finite Element Analysis} +\newacronym{fem}{FEM}{Finite Element Model} +\newacronym{frf}{FRF}{Frequency Response Function} \newacronym{hac}{HAC}{High Authority Control} \newacronym{lac}{LAC}{Low Authority Control} -\newacronym{nass}{NASS}{Nano Active Stabilization System} -\newacronym{asd}{ASD}{Amplitude Spectral Density} -\newacronym{psd}{PSD}{Power Spectral Density} -\newacronym{cps}{CPS}{Cumulative Power Spectrum} -\newacronym{cas}{CAS}{Cumulative Amplitude Spectrum} -\newacronym{frf}{FRF}{Frequency Response Function} +\newacronym{haclac}{HAC-LAC}{High Authority Control / Low Authority Control} +\newacronym{hpf}{HPF}{High Pass Filter} \newacronym{iff}{IFF}{Integral Force Feedback} +\newacronym{lpf}{LPF}{Low Pass Filter} +\newacronym{lqg}{LQG}{Linear Quadratic Gaussian} +\newacronym{lsb}{LSB}{Least Significant Bit} +\newacronym{lti}{LTI}{Linear Time Invariant} +\newacronym{lvdt}{LVDT}{Linear Variable Differential Transformer} +\newacronym{mif}{MIF}{Modal Indication Function} +\newacronym{mimo}{MIMO}{Multi Inputs Multi Outputs} +\newacronym{nass}{NASS}{Nano Active Stabilization System} +\newacronym{np}{NP}{Nominal Performance} +\newacronym{ns}{NS}{Nominal Stability} +\newacronym{ol}{OL}{Open Loop} +\newacronym{pi}{PI}{Proportional Integral} +\newacronym{pid}{PID}{Proportional Integral Derivative} +\newacronym{psd}{PSD}{Power Spectral Density} +\newacronym{pzt}{PZT}{Lead Zirconate Titanate} +\newacronym{poi}{PoI}{Point of Interest} \newacronym{rdc}{RDC}{Relative Damping Control} \newacronym{rga}{RGA}{Relative Gain Array} -\newacronym{hpf}{HPF}{high-pass filter} -\newacronym{lpf}{LPF}{low-pass filter} -\newacronym{dof}{DoF}{Degree of freedom} +\newacronym{rms}{RMS}{Root Mean Square} +\newacronym{rp}{RP}{Robust Performance} +\newacronym{rs}{RS}{Robust Stability} +\newacronym{siso}{SISO}{Single Input Single Output} +\newacronym{sps}{SPS}{Samples per Second} \newacronym{svd}{SVD}{Singular Value Decomposition} -\newacronym{mif}{MIF}{Mode Indicator Functions} -\newacronym{dac}{DAC}{Digital to Analog Converter} -\newacronym{fem}{FEM}{Finite Element Model} -\newacronym{apa}{APA}{Amplified Piezoelectric Actuator} +\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}}} @@ -174,7 +201,7 @@ The fundamental objective has been to ensure that anyone should be capable of re To achieve this goal of reproducibility, comprehensive sharing of all elements has been implemented. This includes the mathematical models developed, raw experimental data collected, and scripts used to generate the figures. -For those wishing to engage with the reproducible aspects of this work, all data and code are freely accessible \cite{dehaeze25_nano_activ_stabil_zenodo}. +For those wishing to engage with the reproducible aspects of this work, all data and code are freely accessible~\cite{dehaeze25_nano_activ_stabil_zenodo}. The organization of the code mirrors that of the manuscript, with corresponding chapters and sections. All materials have been made available under the MIT License, permitting free reuse. @@ -213,8 +240,8 @@ This global distribution of such facilities underscores the significant utility These facilities fundamentally comprise two main parts: the accelerator and storage ring, where electron acceleration and light generation occur, and the beamlines, where the intense X-ray beams are conditioned and directed for experimental use. -The European Synchrotron Radiation Facility (ESRF), shown in Figure~\ref{fig:introduction_esrf_picture}, is a joint research institution supported by 19 member countries. -The ESRF started user operations in 1994 as the world's first third-generation synchrotron. +The \acrfull{esrf}, shown in Figure~\ref{fig:introduction_esrf_picture}, is a joint research institution supported by 19 member countries. +The \acrshort{esrf} started user operations in 1994 as the world's first third-generation synchrotron. Its accelerator complex, schematically depicted in Figure~\ref{fig:introduction_esrf_schematic}, includes a linear accelerator where electrons are initially generated and accelerated, a booster synchrotron to further accelerate the electrons, and an 844-meter circumference storage ring where electrons are maintained in a stable orbit. Synchrotron light are emitted in more than 40 beamlines surrounding the storage ring, each having specialized experimental stations. @@ -236,11 +263,11 @@ These beamlines host diverse instrumentation that enables a wide spectrum of sci \caption{\label{fig:instroduction_esrf}Schematic (\subref{fig:introduction_esrf_schematic}) and picture (\subref{fig:introduction_esrf_picture}) of the European Synchrotron Radiation Facility, situated in Grenoble, France} \end{figure} -In August 2020, following an extensive 20-month upgrade period, the ESRF inaugurated its Extremely Brilliant Source (EBS), establishing it as the world's premier fourth-generation synchrotron~\cite{raimondi21_commis_hybrid_multib_achrom_lattic}. +In August 2020, following an extensive 20-month upgrade period, the \acrshort{esrf} inaugurated its \acrfull{ebs}, establishing it as the world's premier fourth-generation synchrotron~\cite{raimondi21_commis_hybrid_multib_achrom_lattic}. This upgrade implemented a novel storage ring concept that substantially increases the brilliance and coherence of the X-ray beams. Brilliance, a measure of the photon flux, is a key figure of merit for synchrotron facilities. -It experienced an approximate 100-fold increase with the implementation of EBS, as shown in the historical evolution depicted in Figure~\ref{fig:introduction_moore_law_brillance}. +It experienced an approximate 100-fold increase with the implementation of \acrshort{ebs}, as shown in the historical evolution depicted in Figure~\ref{fig:introduction_moore_law_brillance}. While this enhanced beam quality presents unprecedented scientific opportunities, it concurrently introduces considerable engineering challenges, particularly regarding experimental instrumentation and sample positioning systems. \begin{figure}[htbp] @@ -276,7 +303,7 @@ These components are housed in multiple Optical Hutches, as depicted in Figure~\ Following the optical hutches, the conditioned beam enters the Experimental Hutch (Figure~\ref{fig:introduction_id31_cad}), where, for experiments pertinent to this work, focusing optics are used. The sample is mounted on a positioning stage, referred to as the ``end-station'', that enables precise alignment relative to the X-ray beam. Detectors are used to capture the X-rays transmitted through or scattered by the sample. -Throughout this thesis, the standard 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. +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). @@ -296,7 +323,7 @@ Each stage serves distinct positioning functions; for example, the micro-hexapod \end{center} \subcaption{\label{fig:introduction_micro_station_dof} Micro-Station} \end{subfigure} -\caption{\label{fig:introduction_micro_station}CAD 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. CAD view of the micro-station with associated degrees of freedom (\subref{fig:introduction_micro_station_dof}).} +\caption{\label{fig:introduction_micro_station}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}).} \end{figure} The ``stacked-stages'' configuration of the micro-station provides high mobility, enabling diverse scientific experiments and imaging techniques. @@ -304,7 +331,7 @@ Two illustrative examples are provided. Tomography experiments, schematically represented in Figure~\ref{fig:introduction_tomography_schematic}, involve placing a sample in the X-ray beam path while controlling its vertical rotation angle using a dedicated stage. Detector images are captured at numerous rotation angles, allowing the reconstruction of three-dimensional sample structure (Figure~\ref{fig:introduction_tomography_results})~\cite{schoeppler17_shapin_highl_regul_glass_archit}. -This reconstruction depends critically on maintaining the sample's point of interest within the beam throughout the rotation process. +This reconstruction depends critically on maintaining the sample's \acrfull{poi} within the beam throughout the rotation process. 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. @@ -364,7 +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 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 \(10\,\mu m\) \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. @@ -415,7 +442,7 @@ To contextualize the system developed within this thesis, a brief overview of ex The aim is to identify the specific characteristics that distinguish the proposed system from current state-of-the-art implementations. Positioning systems can be broadly categorized based on their kinematic architecture, typically serial or parallel, as exemplified by the 3-Degree-of-Freedom (DoF) platforms in Figure~\ref{fig:introduction_kinematics}. -Serial kinematics (Figure~\ref{fig:introduction_serial_kinematics}) is composed of stacked stages where each degree of freedom is controlled by a dedicated actuator. +Serial kinematics (Figure~\ref{fig:introduction_serial_kinematics}) is composed of stacked stages where each \acrshort{dof} is controlled by a dedicated actuator. This configuration offers great mobility, but positioning errors (e.g., guiding inaccuracies, thermal expansion) accumulate through the stack, compromising overall accuracy. Similarly, the overall dynamic performance (stiffness, resonant frequencies) is limited by the softest component in the stack, often resulting in poor dynamic behavior when many stages are combined. @@ -436,7 +463,7 @@ Similarly, the overall dynamic performance (stiffness, resonant frequencies) is \end{figure} Conversely, parallel kinematic architectures (Figure~\ref{fig:introduction_parallel_kinematics}) involve the coordinated motion of multiple actuators to achieve the desired end-effector motion. -While theoretically offering the same controlled degrees of freedom as stacked stages, parallel systems generally provide limited stroke but significantly enhanced stiffness and superior dynamic performance. +While theoretically offering the same controlled \acrshortpl{dof} as stacked stages, parallel systems generally provide limited stroke but significantly enhanced stiffness and superior dynamic performance. Most end stations, particularly those requiring extensive mobility, employ stacked stages. Their positioning performance consequently depends entirely on the accuracy of individual components. @@ -461,7 +488,7 @@ However, when a large number of DoFs are required, the cumulative errors and lim \end{figure} The concept of using an external metrology to measure and potentially correct for positioning errors is increasing used for nano-positioning end-stations. -Ideally, the relative position between the sample's point of interest and the X-ray beam focus would be measured directly. +Ideally, the relative position between the sample's \acrfull{poi} and the X-ray beam focus would be measured directly. In practice, direct measurement is often impossible; instead, the sample position is typically measured relative to a reference frame associated with the focusing optics, providing an indirect measurement. This measured position can be used in several ways: for post-processing correction of acquired data; for calibration routines to compensate for repeatable errors; or, most relevantly here, for real-time feedback control. @@ -488,12 +515,12 @@ The PtiNAMi microscope at DESY P06 (Figure~\ref{fig:introduction_stages_schroer} \end{figure} 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 voice coil actuators~\cite{kelly22_delta_robot_long_travel_nano,geraldes23_sapot_carnaub_sirius_lnls}. +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}. Figure~\ref{fig:introduction_active_stations} showcases two end-stations incorporating online metrology and active feedback control. -The ID16A system at 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 hexapod 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}. @@ -519,8 +546,8 @@ 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. -End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few degrees of freedom with strokes around \(100\,\mu m\). -Recently, voice coil actuators were used to increase the stroke up to \(3\,\text{mm}\) \cite{kelly22_delta_robot_long_travel_nano,geraldes23_sapot_carnaub_sirius_lnls} +End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few \acrshortpl{dof} with strokes around \(100\,\mu m\). +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. The short-stroke stage actively compensates for errors based on metrology feedback, while the long-stroke stage performs the larger movements. @@ -544,8 +571,8 @@ This approach allows combining extended travel with high precision and good dyna \section{Challenge definition} The advent of fourth-generation light sources, coupled with advancements in focusing optics and detector technology, imposes stringent new requirements on sample positioning systems. -With ID31's anticipated minimum beam dimensions of approximately \(200\,\text{nm}\times 100\,\text{nm}\), the primary experimental objective is maintaining the sample's point of interest within this beam. -This necessitates peak-to-peak positioning errors below \(200\,\text{nm}\) in \(D_y\) and \(200\,\text{nm}\) in \(D_z\), corresponding to RMS errors of \(30\,\text{nm}\) and \(15\,\text{nm}\), respectively. +With ID31's anticipated minimum beam dimensions of approximately \(200\,\text{nm}\times 100\,\text{nm}\), the primary experimental objective is maintaining the sample's \acrshort{poi} within this beam. +This necessitates peak-to-peak positioning errors below \(200\,\text{nm}\) in \(D_y\) and \(200\,\text{nm}\) in \(D_z\), corresponding to \acrfull{rms} errors of \(30\,\text{nm}\) and \(15\,\text{nm}\), respectively. 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. @@ -556,8 +583,8 @@ The existing micro-station, despite being composed of high-performance stages, e 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} -To address these challenges, the concept of a Nano Active Stabilization System (NASS) is proposed. -As schematically illustrated in Figure~\ref{fig:introduction_nass_concept_schematic}, the NASS comprises four principal components integrated with the existing micro-station (yellow): a 5-DoF online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple). +To address these challenges, the concept of a \acrfull{nass} is proposed. +As schematically illustrated in Figure~\ref{fig:introduction_nass_concept_schematic}, the \acrshort{nass} comprises four principal components integrated with the existing micro-station (yellow): a 5-DoF online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple). This system essentially functions as a high-performance, multi-axis vibration isolation and error correction platform situated between the micro-station and the sample. It actively compensates for positioning errors measured by the external metrology system. @@ -568,7 +595,7 @@ It actively compensates for positioning errors measured by the external metrolog \end{figure} \paragraph{Online Metrology system} -The performance of the NASS is fundamentally reliant on the accuracy and bandwidth of its online metrology system, as the active control is based directly on these measurements. +The performance of the \acrshort{nass} is fundamentally reliant on the accuracy and bandwidth of its online metrology system, as the active control is based directly on these measurements. This metrology system must fulfill several criteria: measure the sample position in 5 DoF (excluding rotation about the vertical Z-axis); possess a measurement range compatible with the micro-station's extensive mobility and continuous spindle rotation; achieve an accuracy compatible with the sub-100 nm positioning target; and offer high bandwidth for real-time control. \begin{figure}[htbp] @@ -586,7 +613,7 @@ For the work presented herein, the metrology system is assumed to provide accura \paragraph{Active Stabilization Platform Design} The active stabilization platform, positioned between the micro-station top plate and the sample, must satisfy several demanding requirements. -It needs to provide active motion compensation in 5 degrees of freedom (\(D_x\), \(D_y\), \(D_z\), \(R_x\) and \(R_y\)). +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. @@ -615,7 +642,7 @@ A more detailed review of Stewart platform and its main components will be given \paragraph{Robust Control} The control system must compute the position measurements from the online metrology system and computes the reference positions derived from each micro-station desired movement. -It then commands the active platform in real time to stabilize the sample and compensate for all error sources, including stage imperfections, thermal drifts, and vibrations. +It then commands the active platform in real-time to stabilize the sample and compensate for all error sources, including stage imperfections, thermal drifts, and vibrations. Ensuring the stability and robustness of these feedback loops is crucial, especially within the demanding operational context of a synchrotron beamline, which requires reliable 24/7 operation with minimal intervention. Several factors complicate the design of robust feedback control for the NASS. @@ -627,7 +654,7 @@ Designing for robustness against large payload variations typically necessitates Consequently, high-performance positioning stages often work with well-characterized payload, as seen in systems like wafer-scanners or atomic force microscopes. Furthermore, unlike many systems where the active stage and sample are significantly lighter than the underlying coarse stages, the NASS payload mass can be substantially greater than the mass of the micro-station's top stage. -This leads to strong dynamic coupling between the active platform and the micro-station structure, resulting in a more complex multi-inputs multi-outputs (MIMO) system with significant cross-talk between axes. +This leads to strong dynamic coupling between the active platform and the micro-station structure, resulting in a more complex \acrfull{mimo} system with significant cross-talk between axes. These variations in operating conditions and payload translate into significant uncertainty or changes in the plant dynamics that the controller must handle. Therefore, the feedback controller must be designed to be robust against this plant uncertainty while still delivering the required nanometer-level performance. @@ -652,7 +679,7 @@ This thesis documents this process chronologically, illustrating how models of v While the resulting system is highly specific, the documented effectiveness of this design approach may contribute to the broader adoption of mechatronic methodologies in the design of future synchrotron instrumentation. \paragraph{Experimental validation of multi-body simulations with reduced order flexible bodies obtained by FEA} -A key tool employed extensively in this work was a combined multi-body simulation and Finite Element Analysis technique, specifically using Component Mode Synthesis to represent flexible bodies within the multi-body framework~\cite{brumund21_multib_simul_reduc_order_flexib_bodies_fea}. +A key tool employed extensively in this work was a combined multi-body simulation and \acrfull{fea} technique, specifically using Component Mode Synthesis to represent flexible bodies within the multi-body framework~\cite{brumund21_multib_simul_reduc_order_flexib_bodies_fea}. This hybrid approach, while established, was experimentally validated in this work for components critical to the NASS, namely amplified piezoelectric actuators and flexible joints. It proved invaluable for designing and optimizing components intended for integration into a larger, complex dynamic system. This methodology, detailed in Section~\ref{sec:detail_fem}, is presented as a potentially useful tool for future mechatronic instrument development. @@ -665,7 +692,7 @@ Furthermore, decoupling strategies for parallel manipulators were compared (Sect Consequently, the specified performance targets were met using controllers which, facilitated by this design approach, proved to be robust, readily tunable, and easily maintained. \paragraph{Active Damping of rotating mechanical systems using Integral Force Feedback} -During conceptual design, it was found that the guaranteed stability property of the established active damping technique known as Integral Force Feedback (IFF) is compromised when applied to rotating platforms like the NASS. +During conceptual design, it was found that the guaranteed stability property of the established active damping technique known as \acrfull{iff} is compromised when applied to rotating platforms like the NASS. To address this instability issue, two modifications to the classical IFF control scheme were proposed and analyzed. The first involves a minor adjustment to the control law itself, while the second incorporates physical springs in parallel with the force sensors. Stability conditions and optimal parameter tuning guidelines were derived for both modified schemes. @@ -709,7 +736,7 @@ The chapter concludes (Section~\ref{sec:detail_design}) by presenting the final, \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 (IFF) tested (Section~\ref{sec:test_apa}). +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}). @@ -772,7 +799,7 @@ 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 point of interest (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 nano-hexapod 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. @@ -780,9 +807,9 @@ In this section, a uniaxial model of the micro-station is tuned to match measure 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 \(\approx 1\,\text{Hz}\)}. -Three frequency response functions 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 \(x_{g}\). +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 frequency response functions 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 200Hz (solid lines in Figure~\ref{fig:uniaxial_comp_frf_meas_model}). \begin{figure}[htbp] \begin{subfigure}{0.69\textwidth} @@ -801,7 +828,7 @@ Due to the poor coherence at low frequencies, these frequency response functions \end{figure} \subsubsection{Uniaxial Model} The uniaxial model of the micro-station is shown in Figure~\ref{fig:uniaxial_model_micro_station}. -It consists of a mass spring damper system with three degrees of freedom. +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 \(k_h\). @@ -829,7 +856,7 @@ The hammer impacts \(F_{h}, F_{g}\) are shown in blue, whereas the measured iner \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}. -Because the uniaxial model has three degrees of freedom, only three modes with frequencies at \(70\,\text{Hz}\), \(140\,\text{Hz}\) and \(320\,\text{Hz}\) are modeled. +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}). However, the goal is not to have a perfect match with the measurement (this would require a much more complex model), but to have a first approximation. More accurate models will be used later on. @@ -845,7 +872,7 @@ A model of the nano-hexapod and sample is now added on top of the uniaxial model 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 point of interest and the nano-hexapod actuator will be considered in Section~\ref{sec:uniaxial_payload_dynamics}. +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}. \begin{figure}[htbp] \begin{subfigure}{0.39\textwidth} @@ -918,7 +945,7 @@ 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 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 \(2.4\,nV/\sqrt{\text{Hz}}\)} with a gain of 60dB 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. @@ -936,8 +963,8 @@ G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi \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.} \end{figure} -The amplitude spectral density of the floor motion \(\Gamma_{x_f}\) can be computed from the amplitude spectral density of measured voltage \(\Gamma_{\hat{V}_{x_f}}\) using~\eqref{eq:uniaxial_asd_floor_motion}. -The estimated amplitude spectral density \(\Gamma_{x_f}\) of the floor motion \(x_f\) is shown in Figure~\ref{fig:uniaxial_asd_floor_motion_id31}. +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}. +The estimated \acrshort{asd} \(\Gamma_{x_f}\) of the floor motion \(x_f\) is shown in Figure~\ref{fig:uniaxial_asd_floor_motion_id31}. \begin{equation}\label{eq:uniaxial_asd_floor_motion} \Gamma_{x_f}(\omega) = \frac{\Gamma_{\hat{V}_{x_f}}(\omega)}{|G_{geo}(j\omega)| \cdot g_0} \quad \left[ m/\sqrt{\text{Hz}} \right] @@ -1029,7 +1056,7 @@ The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses \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 Cumulative Amplitude Spectrum 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. +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 \(5\,\text{Hz}\). 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}. @@ -1066,7 +1093,7 @@ Therefore, while the \emph{open-loop} vibration is the lowest for the stiff nano 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}. \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 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}. 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}). @@ -1100,7 +1127,7 @@ The Integral Force Feedback strategy consists of using a force sensor in series \boxed{K_{\text{IFF}}(s) = \frac{g}{s}} \end{equation} -The mechanical equivalent of this IFF strategy is a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain \(k/g\) (see Figure~\ref{fig:uniaxial_active_damping_iff_equiv}). +The mechanical equivalent of this \acrshort{iff} strategy is a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain \(k/g\) (see Figure~\ref{fig:uniaxial_active_damping_iff_equiv}). \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -1172,10 +1199,10 @@ This is usually referred to as ``\emph{sky hook damper}''. The plant dynamics for all three active damping techniques are shown in Figure~\ref{fig:uniaxial_plant_active_damping_techniques}. 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 RDC cases because the sensors are collocated with the actuator~\cite[Chapter 7]{preumont18_vibrat_contr_activ_struc_fourt_edition}. -For the 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}. +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}. -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 DVF transfer functions. +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. @@ -1218,9 +1245,9 @@ There is even some damping authority on micro-station modes in the following cas \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 DVF in Figure~\ref{fig:uniaxial_active_damping_dvf_equiv}) is connected to the micro-station top platform through the stiff nano-hexapod. +\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 RDC (see mechanical equivalent in Figure~\ref{fig:uniaxial_active_damping_rdc_equiv}) which allows some damping of the micro-station. +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. \end{description} \begin{figure}[htbp] @@ -1288,7 +1315,7 @@ Several conclusions can be drawn by comparing the obtained sensitivity transfer 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}). 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 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 nano-hexapod (Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping_xf}). \end{itemize} \begin{figure}[htbp] @@ -1336,14 +1363,14 @@ All three active damping methods give similar results. \end{center} \subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,N/\mu m$} \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 (OL in black, IFF in blue, RDC in red and DVF in yellow).} +\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} \subsubsection{Conclusion} 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 DVF. +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}). Even some damping can be applied to some micro-station modes in specific cases. The obtained damped plants were found to be similar. @@ -1455,7 +1482,7 @@ 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~\ref{fig:uniaxial_nyquist_hac}). An arbitrary minimum modulus margin of \(0.25\) was chosen when designing the controllers. -These high authority controllers 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 low pass filter to increase the robustness to high frequency dynamics. +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}. \begin{subequations} \label{eq:uniaxial_hac_formulas} @@ -1490,7 +1517,7 @@ K_{\text{stiff}}(s) &= g \cdot \end{tabularx} \end{table} -The loop gains corresponding to the designed high authority controllers for the three nano-hexapod are shown in Figure~\ref{fig:uniaxial_loop_gain_hac}. +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. 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 \(\approx 100\,\text{Hz}\). @@ -1540,12 +1567,12 @@ The goal is to have a first estimation of the attainable performance. \end{center} \subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,N/\mu m$} \end{subfigure} -\caption{\label{fig:uniaxial_loop_gain_hac}Loop gain for the High Authority Controller} +\caption{\label{fig:uniaxial_loop_gain_hac}Loop gains for the High Authority Controllers} \end{figure} \subsubsection{Closed-Loop Noise Budgeting} \label{ssec:uniaxial_position_control_cl_noise_budget} -The high authority position feedback controllers are then implemented and the closed-loop sensitivities to disturbances are computed. +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). As expected, the sensitivity to disturbances decreased in the controller bandwidth and slightly increased outside this bandwidth. @@ -1568,10 +1595,10 @@ 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 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}. 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})} \end{figure} -The cumulative amplitude spectrum of the motion \(d\) is computed for all nano-hexapod configurations, all sample masses and in the open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases. +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 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 \(\approx 70\,nm\)). @@ -1811,7 +1838,7 @@ Even though the added sample's flexibility still shifts the high frequency mass \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 point-of-interest 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 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). 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 \(\omega_n = 200\,\text{Hz}\)). @@ -1825,7 +1852,7 @@ In this case, the measured (i.e., controlled) distance \(d\) is no longer equal 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. -The High Authority Controllers used in Section~\ref{sec:uniaxial_position_control} are then implemented on the damped plants. +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. Finally, closed-loop noise budgeting is computed for the obtained closed-loop system, and the cumulative amplitude spectrum of \(d\) and \(y\) are shown in Figure~\ref{fig:uniaxial_sample_flexibility_noise_budget_y}. @@ -1859,7 +1886,7 @@ Such additional dynamics can induce stability issues depending on their position 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. 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 point of interest (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}. +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. \subsection*{Conclusion} \label{sec:uniaxial_conclusion} @@ -1883,9 +1910,9 @@ This model is simple enough to be able to derive its dynamics analytically and t \acrfull{iff} is then applied to the rotating platform, and it is shown that the unconditional stability of \acrshort{iff} is lost due to the gyroscopic effects induced by the rotation (Section~\ref{sec:rotating_iff_pure_int}). Two modifications of the Integral Force Feedback are then proposed. -The first modification involves adding a high-pass filter to the \acrshort{iff} controller (Section~\ref{sec:rotating_iff_pseudo_int}). +The first modification involves adding a \acrfull{hpf} to the \acrshort{iff} controller (Section~\ref{sec:rotating_iff_pseudo_int}). It is shown that the \acrshort{iff} controller is stable for some gain values, and that damping can be added to the suspension modes. -The optimal high-pass filter cut-off frequency is computed. +The optimal \acrlong{hpf} cut-off frequency is computed. The second modification consists of adding a stiffness in parallel to the force sensors (Section~\ref{sec:rotating_iff_parallel_stiffness}). Under certain conditions, the unconditional stability of the IFF controller is regained. The optimal parallel stiffness is then computed. @@ -1901,7 +1928,7 @@ The goal is to determine whether the rotation imposes performance limitation on \subsection{System Description and Analysis} \label{sec:rotating_system_description} -The system used to study gyroscopic effects consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure~\ref{fig:rotating_3dof_model_schematic}). +The system used to study gyroscopic effects consists of a 2-\acrshortpl{dof} translation stage on top of a rotating stage (Figure~\ref{fig:rotating_3dof_model_schematic}). The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\s}\). The suspended platform consists of two orthogonal actuators, each represented by three elements in parallel: a spring with a stiffness \(k\) in \(\si{\newton\per\meter}\), a dashpot with a damping coefficient \(c\) in \(\si{\newton\per(\meter\per\second)}\) and an ideal force source \(F_u, F_v\). A payload with a mass \(m\) in \(\si{\kilo\gram}\), is mounted on the (rotating) suspended platform. @@ -1945,7 +1972,7 @@ The uniform rotation of the system induces two \emph{gyroscopic effects} as show \item \emph{Centrifugal forces}: that can be seen as an added \emph{negative stiffness} \(- m \Omega^2\) along \(\vec{i}_u\) and \(\vec{i}_v\) \item \emph{Coriolis forces}: that adds \emph{coupling} between the two orthogonal directions. \end{itemize} -One can verify that without rotation (\(\Omega = 0\)), the system becomes equivalent to two \emph{uncoupled} one degree of freedom mass-spring-damper systems. +One can verify that without rotation (\(\Omega = 0\)), the system becomes equivalent to two \emph{uncoupled} one \acrshort{dof} mass-spring-damper systems. To study the dynamics of the system, the two differential equations of motions~\eqref{eq:rotating_eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\bm{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation~\eqref{eq:rotating_Gd_mimo_tf} is obtained. The four transfer functions in \(\bm{G}_d\) are shown in equation~\eqref{eq:rotating_Gd_indiv_el}. @@ -2035,17 +2062,17 @@ For \(\Omega > \omega_0\), the low-frequency pair of complex conjugate poles \(p \label{sec:rotating_iff_pure_int} The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation. -As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat} and to make the plant easier to control for the high authority controller. +As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances~\cite{collette11_review_activ_vibrat_isolat_strat} and to make the plant easier to control for the high authority controller. -Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} 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}. +Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF)~\cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF)~\cite{preumont91_activ} 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~\cite{preumont91_activ}, the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system. -When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros, which guarantees the stability of the closed-loop system \cite{preumont02_force_feedb_versus_accel_feedb}. -It was later shown that this property holds for multiple collated actuator/sensor pairs \cite{preumont08_trans_zeros_struc_contr_with}. +When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros, which guarantees the stability of the closed-loop system~\cite{preumont02_force_feedb_versus_accel_feedb}. +It was later shown that this property holds for multiple collated actuator/sensor pairs~\cite{preumont08_trans_zeros_struc_contr_with}. -The main advantages of IFF over other active damping techniques are the guaranteed stability even in the presence of flexible dynamics, good performance, and robustness properties \cite{preumont02_force_feedb_versus_accel_feedb}. +The main advantages of IFF over other active damping techniques are the guaranteed stability even in the presence of flexible dynamics, good performance, and robustness properties~\cite{preumont02_force_feedb_versus_accel_feedb}. -Several improvements to the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping \cite{teo15_optim_integ_force_feedb_activ_vibrat_contr} or adding a high-pass filter to recover the loss of compliance at low-frequency \cite{chesne16_enhan_dampin_flexib_struc_using_force_feedb}. -Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to derive optimal gains for the IFF controller \cite{zhao19_optim_integ_force_feedb_contr}. \par +Several improvements to the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping~\cite{teo15_optim_integ_force_feedb_activ_vibrat_contr} or adding a \acrshort{hpf} to recover the loss of compliance at low-frequency~\cite{chesne16_enhan_dampin_flexib_struc_using_force_feedb}. +Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to derive optimal gains for the IFF controller~\cite{zhao19_optim_integ_force_feedb_contr}. However, none of these studies have been applied to rotating systems. In this section, the \acrshort{iff} strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alter the system dynamics and that IFF cannot be applied as is. @@ -2157,20 +2184,20 @@ The decentralized \acrshort{iff} controller \(\bm{K}_F\) corresponds to a diagon \end{equation} To determine how the \acrshort{iff} controller affects the poles of the closed-loop system, a Root Locus plot (Figure~\ref{fig:rotating_root_locus_iff_pure_int}) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain \(g\) varies from \(0\) to \(\infty\) for the two controllers \(K_{F}\) simultaneously. -As explained in \cite{preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr}, the closed-loop poles start at the open-loop poles (shown by crosses) for \(g = 0\) and coincide with the transmission zeros (shown by circles) as \(g \to \infty\). +As explained in~\cite{preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr}, the closed-loop poles start at the open-loop poles (shown by crosses) for \(g = 0\) and coincide with the transmission zeros (shown by circles) as \(g \to \infty\). -Whereas collocated IFF is usually associated with unconditional stability \cite{preumont91_activ}, this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null. +Whereas collocated IFF is usually associated with unconditional stability~\cite{preumont91_activ}, this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null. This can be seen in the Root Locus plot (Figure~\ref{fig:rotating_root_locus_iff_pure_int}) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in \(K_{F}\) and the finite gain of the plant (Figure~\ref{fig:rotating_iff_bode_plot_effect_rot}). The control system is thus cancels the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. -\subsection{Integral Force Feedback with a High-Pass Filter} +\subsection{Integral Force Feedback with a High-Pass Filter} \label{sec:rotating_iff_pseudo_int} As explained in the previous section, the instability of the IFF controller applied to the rotating system is due to the high gain of the integrator at low-frequency. To limit the low-frequency controller gain, a \acrfull{hpf} can be added to the controller, as shown in equation~\eqref{eq:rotating_iff_lhf}. This is equivalent to slightly shifting the controller pole to the left along the real axis. -This modification of the IFF controller is typically performed to avoid saturation associated with the pure integrator \cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}. -This is however not the reason why this high-pass filter is added here. +This modification of the IFF controller is typically performed to avoid saturation associated with the pure integrator~\cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}. +This is however not the reason why this \acrlong{hpf} is added here. \begin{equation}\label{eq:rotating_iff_lhf} \boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} @@ -2201,7 +2228,7 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the contro \end{center} \subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root Locus} \end{subfigure} -\caption{\label{fig:rotating_iff_modified_loop_gain_root_locus}Comparison of the IFF with pure integrator and modified IFF with added high-pass filter (\(\Omega = 0.1\omega_0\)). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with \(\omega_i = 0.1 \omega_0\) and \(g = 2\). The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large})} +\caption{\label{fig:rotating_iff_modified_loop_gain_root_locus}Comparison of the IFF with pure integrator and modified IFF with added high-pass filter (\(\Omega = 0.1\omega_0\)). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with \(\omega_i = 0.1 \omega_0\) and \(g = 2\). The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large})} \end{figure} \subsubsection{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)} Two parameters can be tuned for the modified controller in equation~\eqref{eq:rotating_iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\). @@ -2228,7 +2255,7 @@ For larger values of \(\omega_i\), the attainable damping ratio decreases as a f \end{center} \subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown} \end{subfigure} -\caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as \(\omega_i\) increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})} +\caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as \(\omega_i\) increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})} \end{figure} \subsubsection{Obtained Damped Plant} To study how the parameter \(\omega_i\) affects the damped plant, the obtained damped plants for several \(\omega_i\) are compared in Figure~\ref{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}. @@ -2326,7 +2353,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif \end{figure} \subsubsection{Effect of \(k_p\) on the attainable damping} Even though the parallel stiffness \(k_p\) has no impact on the open-loop poles (as the overall stiffness \(k\) is kept constant), it has a large impact on the transmission zeros. -Moreover, as the attainable damping is generally proportional to the distance between poles and zeros \cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the parallel stiffness \(k_p\) is expected to have some impact on the attainable damping. +Moreover, as the attainable damping is generally proportional to the distance between poles and zeros~\cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the parallel stiffness \(k_p\) is expected to have some impact on the attainable damping. To study this effect, Root Locus plots for several parallel stiffnesses \(k_p > m \Omega^2\) are shown in Figure~\ref{fig:rotating_iff_kp_root_locus_effect_kp}. The frequencies of the transmission zeros of the system increase with an increase in the parallel stiffness \(k_p\) (thus getting closer to the poles), and the associated attainable damping is reduced. Therefore, even though the parallel stiffness \(k_p\) should be larger than \(m \Omega^2\) for stability reasons, it should not be taken too large as this would limit the attainable damping. @@ -2353,7 +2380,7 @@ The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared Even though the two resonances are well damped, the IFF changes the low-frequency behavior of the plant, which is usually not desired. This is because ``pure'' integrators are used which are inducing large low-frequency loop gains. -To lower the low-frequency gain, a high-pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): +To lower the low-frequency gain, a \acrshort{hpf} is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): \begin{equation} K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} 1 & 0 \\ @@ -2361,11 +2388,11 @@ To lower the low-frequency gain, a high-pass filter is added to the IFF control \end{bmatrix} \end{equation} -To determine how the high-pass filter impacts the attainable damping, the controller gain \(g\) is kept constant while \(\omega_i\) is changed, and the minimum damping ratio of the damped plant is computed. +To determine how the \acrshort{hpf} impacts the attainable damping, the controller gain \(g\) is kept constant while \(\omega_i\) is changed, and the minimum damping ratio of the damped plant is computed. The obtained damping ratio as a function of \(\omega_i/\omega_0\) (where \(\omega_0\) is the resonance of the system without rotation) is shown in Figure~\ref{fig:rotating_iff_kp_added_hpf_effect_damping}. It is shown that the attainable damping ratio reduces as \(\omega_i\) is increased (same conclusion than in Section~\ref{sec:rotating_iff_pseudo_int}). Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compare the obtained damped plant again with the undamped and with the ``pure'' IFF in Figure~\ref{fig:rotating_iff_kp_added_hpf_damped_plant}. -The added high-pass filter gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior. +The added \acrshort{hpf} gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior. \begin{figure}[htbp] \begin{subfigure}{0.34\linewidth} @@ -2380,14 +2407,14 @@ The added high-pass filter gives almost the same damping properties to the suspe \end{center} \subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF} \end{subfigure} -\caption{\label{fig:rotating_iff_optimal_hpf}Effect of high-pass filter cut-off frequency on the obtained damping} +\caption{\label{fig:rotating_iff_optimal_hpf}Effect of high-pass filter cut-off frequency on the obtained damping} \end{figure} \subsection{Relative Damping Control} \label{sec:rotating_relative_damp_control} To apply a ``Relative Damping Control'' strategy, relative motion sensors are added in parallel with the actuators as shown in Figure~\ref{fig:rotating_3dof_model_schematic_rdc}. Two controllers \(K_d\) are used to feed back the relative motion to the actuator. -These controllers are in principle pure derivators (\(K_d = s\)), but to be implemented in practice they are usually replaced by a high-pass filter~\eqref{eq:rotating_rdc_controller}. +These controllers are in principle pure derivators (\(K_d = s\)), but to be implemented in practice they are usually replaced by a high-pass filter~\eqref{eq:rotating_rdc_controller}. \begin{equation}\label{eq:rotating_rdc_controller} K_d(s) = g \cdot \frac{s}{s + \omega_d} @@ -2518,7 +2545,7 @@ The parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \ 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 coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). +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] \begin{subfigure}{0.33\textwidth} @@ -2541,8 +2568,8 @@ The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually es \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} \end{figure} -\subsubsection{Optimal IFF with a High-Pass Filter} -Integral Force Feedback with an added high-pass filter is applied to the three nano-hexapods. +\subsubsection{Optimal IFF with a High-Pass Filter} +Integral Force Feedback with an added \acrlong{hpf} is applied to the three nano-hexapods. 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} @@ -2576,7 +2603,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar \end{figure} \begin{table}[htbp] -\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}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.} +\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}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.} \centering \begin{tabularx}{0.3\linewidth}{Xccc} \toprule @@ -2591,7 +2618,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar \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). -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 (\(\omega_i = \omega_0/10\)). +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. @@ -2822,7 +2849,7 @@ In this study, the gyroscopic effects induced by the spindle's rotation have bee Decentralized \acrlong{iff} with pure integrators was shown to be unstable when applied to rotating platforms. Two modifications of the classical \acrshort{iff} control have been proposed to overcome this issue. -The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators. +The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators. This is equivalent to moving the controller pole to the left along the real axis. This allows the closed-loop system to be stable up to some value of the controller gain. @@ -2850,7 +2877,7 @@ Although the inertia of each solid body can easily be estimated from its geometr Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station. The tuning approach for the multi-body model based on measurements is illustrated in Figure~\ref{fig:modal_vibration_analysis_procedure}. -First, a \emph{response model} is obtained, which corresponds to a set of frequency response functions computed from experimental measurements. +First, a \emph{response model} is obtained, which corresponds to a set of \acrshortpl{frf} computed from experimental measurements. From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes. This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies. @@ -2863,8 +2890,8 @@ This modal model can then be used to tune the spatial model (i.e. the multi-body The measurement setup used to obtain the response model is described in Section~\ref{sec:modal_meas_setup}. This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals. -In Section~\ref{sec:modal_frf_processing}, the obtained frequency response functions between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed. -These measurements are projected at the center of mass of each considered solid body to facilitate the further use of the results. +In Section~\ref{sec:modal_frf_processing}, the obtained \acrshortpl{frf} between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed. +These measurements are projected at the \acrfull{com} of each considered solid body to facilitate the further use of the results. The solid body assumption is then verified, validating the use of the multi-body model. Finally, the modal analysis is performed in Section~\ref{sec:modal_analysis}. @@ -2920,11 +2947,11 @@ If these local feedback controls were turned off, this would have resulted in ve The top part representing the active stabilization stage was disassembled as the active stabilization stage will be added in the multi-body model afterwards. -To perform the modal analysis from the measured responses, the \(n \times n\) frequency response function matrix \(\bm{H}\) needs to be measured, where \(n\) is the considered number of degrees of freedom. -The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the frequency response function from a force \(F_k\) applied at \acrfull{dof} \(k\) to the displacement of the structure \(X_j\) at \acrshort{dof} \(j\). +To perform the modal analysis from the measured responses, the \(n \times n\) \acrshort{frf} matrix \(\bm{H}\) needs to be measured, where \(n\) is the considered number of \acrshortpl{dof}. +The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the \acrshort{frf} from a force \(F_k\) applied at \acrfull{dof} \(k\) to the displacement of the structure \(X_j\) at \acrshort{dof} \(j\). Measuring this \acrshort{frf} matrix is time consuming as it requires to make \(n \times n\) measurements. However, due to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\bm{H}\) \cite[chapt. 5.2]{ewins00_modal}. -Therefore, a minimum set of \(n\) frequency response functions is required. +Therefore, a minimum set of \(n\) \acrshortpl{frf} is required. This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}. It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode). @@ -2934,7 +2961,7 @@ 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. -The positions of the accelerometers are visually shown on a CAD model in Figure~\ref{fig:modal_location_accelerometers} and their precise locations with respect to a frame located at the point of interest are summarized in Table~\ref{tab:modal_position_accelerometers}. +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}. 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. @@ -3036,7 +3063,7 @@ 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). -Similar results were obtained for all measured frequency response functions. +Similar results were obtained for all measured \acrshortpl{frf}. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -3054,7 +3081,7 @@ Similar results were obtained for all measured frequency response functions. \caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer 1 in the \(x\) direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the \(z\) direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})} \end{figure} -The frequency response function from the applied force to the measured acceleration is then computed and shown Figure~\ref{fig:modal_frf_acc_force}. +The \acrshort{frf} from the applied force to the measured acceleration is then computed and shown Figure~\ref{fig:modal_frf_acc_force}. The quality of the obtained data can be estimated using the \emph{coherence} function (Figure~\ref{fig:modal_coh_acc_force}). Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which corresponds to the frequency range of interest. @@ -3096,11 +3123,11 @@ For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that li \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. -Therefore, only \(6 \times 6 = 36\) degrees of freedom are of interest. +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. The coordinate transformation from accelerometers \acrshort{dof} to the solid body 6 \acrshortpl{dof} (three translations and three rotations) is performed in Section~\ref{ssec:modal_acc_to_solid_dof}. -The \(69 \times 3 \times 801\) frequency response matrix is then reduced to a \(36 \times 3 \times 801\) frequency response matrix where the motion of each solid body is expressed with respect to its center of mass. +The \(69 \times 3 \times 801\) frequency response matrix is then reduced to a \(36 \times 3 \times 801\) frequency response matrix where the motion of each solid body is expressed with respect to its \acrlong{com}. To validate this reduction of \acrshort{dof} and the solid body assumption, the frequency response function at the accelerometer location are ``reconstructed'' from the reduced frequency response matrix and are compared with the initial measurements in Section~\ref{ssec:modal_solid_body_assumption}. \subsubsection{From accelerometer DOFs to solid body DOFs} @@ -3155,8 +3182,8 @@ The motion of the solid body expressed in a chosen frame \(\{O\}\) can be determ Note that this matrix inversion is equivalent to resolving a mean square problem. Therefore, having more accelerometers permits better approximation of the motion of a solid body. -From the CAD model, the position of the center of mass of each solid body is computed (see Table~\ref{tab:modal_com_solid_bodies}). -The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined. +From the 3D model, the position of the \acrlong{com} of each solid body is computed (see Table~\ref{tab:modal_com_solid_bodies}). +The position of each accelerometer with respect to the \acrlong{com} of the corresponding solid body can easily be determined. \begin{table}[htbp] \caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''} @@ -3175,7 +3202,7 @@ Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\ \end{tabularx} \end{table} -Using~\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 center of mass 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 \(\bm{H}\). +Using~\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 \(\bm{H}\). \begin{equation}\label{eq:modal_frf_matrix_com} \bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix} @@ -3198,9 +3225,9 @@ In particular, the responses at the locations of the four accelerometers can be 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 original frequency response functions and those computed from the CoM responses match well in the frequency range of interest. +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 degrees of freedom from 69 (23 accelerometers with each 3 \acrshort{dof}) to 36 (6 solid bodies with 6 \acrshort{dof}). +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}). \begin{figure}[htbp] \centering @@ -3213,7 +3240,7 @@ The goal here is to extract the modal parameters describing the modes of the mic This is performed from the \acrshort{frf} matrix previously extracted from the measurements. In order to perform the modal parameter extraction, the order of the modal model has to be estimated (i.e. the number of modes in the frequency band of interest). -This is achived using the \acrfull{mif} in section~\ref{ssec:modal_number_of_modes}. +This is achieved using the \acrfull{mif} in section~\ref{ssec:modal_number_of_modes}. In section~\ref{ssec:modal_parameter_extraction}, the modal parameter extraction is performed. The graphical display of the mode shapes can be computed from the modal model, which is quite useful for physical interpretation of the modes. @@ -3223,7 +3250,7 @@ To validate the quality of the modal model, the full \acrshort{frf} matrix is co \label{ssec:modal_number_of_modes} The \acrshort{mif} is applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)). -The complex modal indication function is defined in equation~\eqref{eq:modal_cmif} where the diagonal matrix \(\Sigma\) is obtained from a \acrlong{svd} of the \acrshort{frf} matrix as shown in equation~\eqref{eq:modal_svd}. +The \acrfull{cmif} is defined in equation~\eqref{eq:modal_cmif} where the diagonal matrix \(\Sigma\) is obtained from a \acrfull{svd} of the \acrshort{frf} matrix as shown in equation~\eqref{eq:modal_svd}. \begin{equation} \label{eq:modal_cmif} [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^{\intercal} [\Sigma(\omega)]_{n\times p} \end{equation} @@ -3309,7 +3336,7 @@ From the obtained modal parameters, the mode shapes are computed and can be disp These animations are useful for visually obtaining a better understanding of the system's dynamic behavior. For instance, the mode shape of the first mode at \(11\,\text{Hz}\) (figure~\ref{fig:modal_mode1_animation}) indicates an issue with the lower granite. It turns out that four \emph{Airloc Levelers} are used to level the lower granite (figure~\ref{fig:modal_airloc}). -These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it has a low frequency ``tilt mode''. +These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it has a low frequency ``tilt mode''. The levelers were then better adjusted. \begin{figure}[htbp] @@ -3356,10 +3383,10 @@ With \(\bm{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the response \bm{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m} \end{equation} -A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure~\ref{fig:modal_comp_acc_frf_modal}. +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 frequency response function 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 micro-hexapod) in the \(x\) direction. \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -3386,11 +3413,11 @@ This can be seen in Figure~\ref{fig:modal_comp_acc_frf_modal_3} that shows the f \label{sec:modal_conclusion} In this study, a modal analysis of the micro-station was performed. -Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained. -The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture. +Thanks to an adequate choice of instrumentation and proper set of measurements, high quality \acrshortpl{frf} can be obtained. +The obtained \acrshortpl{frf} indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture. It shows a lot of coupling between stages and different directions, and many modes. -By measuring 12 degrees of freedom on each ``stage'', it could be verified that in the frequency range of interest, each stage behaved as a rigid body. +By measuring 12 \acrshortpl{dof} on each ``stage'', it could be verified that in the frequency range of interest, each stage behaved as a rigid body. This confirms that a multi-body model can be used to properly model the micro-station. Although a lot of effort was put into this experimental modal analysis of the micro-station, it was difficult to obtain an accurate modal model. @@ -3425,7 +3452,7 @@ Such a stacked architecture allows high mobility, but the overall stiffness is r \begin{figure}[htbp] \centering \includegraphics[scale=1,width=\linewidth]{figs/ustation_cad_view.png} -\caption{\label{fig:ustation_cad_view}CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple).} +\caption{\label{fig:ustation_cad_view}3D view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple).} \end{figure} There are different ways of modeling the stage dynamics in a multi-body model. @@ -3453,7 +3480,7 @@ Each linear guide is very stiff in radial directions such that the only DoF with This stage is mainly used in \emph{reflectivity} experiments where the sample \(R_y\) angle is scanned. This stage can also be used to tilt the rotation axis of the Spindle. -To precisely control the \(R_y\) angle, a stepper motor and two optical encoders are used in a PID feedback loop. +To precisely control the \(R_y\) angle, a stepper motor and two optical encoders are used in a \acrfull{pid} feedback loop. \begin{minipage}[b]{0.48\linewidth} \begin{center} @@ -3479,8 +3506,8 @@ Additional rotary unions and slip-rings are used to be able to pass electrical s 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. -This stage is used to position the point of interest of the sample with respect to the spindle rotation axis. -It can also be used to precisely position the PoI vertically with respect to the x-ray. +This stage is used to position the \acrshort{poi} of the sample with respect to the spindle rotation axis. +It can also be used to precisely position the \acrfull{poi} vertically with respect to the x-ray. \begin{minipage}[t]{0.49\linewidth} \begin{center} @@ -3664,7 +3691,7 @@ Another key advantage of homogeneous transformation is the easy inverse transfor \label{ssec:ustation_kinematics} Each stage is described by two frames; one is attached to the fixed platform \(\{A\}\) while the other is fixed to the mobile platform \(\{B\}\). -At ``rest'' position, the two have the same pose and coincide with the point of interest (\(O_A = O_B\)). +At ``rest'' position, the two have the same pose and coincide with the \acrshort{poi} (\(O_A = O_B\)). An example of the tilt stage is shown in Figure~\ref{fig:ustation_stage_motion}. The mobile frame of the translation stage is equal to the fixed frame of the tilt stage: \(\{B_{D_y}\} = \{A_{R_y}\}\). Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the spindle: \(\{B_{R_y}\} = \{A_{R_z}\}\). @@ -3735,7 +3762,7 @@ The setpoints are \(D_y\) for the translation stage, \(\theta_y\) for the tilt-s \label{sec:ustation_modeling} In this section, the multi-body model of the micro-station is presented. Such model consists of several rigid bodies connected by springs and dampers. -The inertia of the solid bodies and the stiffness properties of the guiding mechanisms were first estimated based on the CAD model and data-sheets (Section~\ref{ssec:ustation_model_simscape}). +The inertia of the solid bodies and the stiffness properties of the guiding mechanisms were first estimated based on the 3D model and data-sheets (Section~\ref{ssec:ustation_model_simscape}). The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section~\ref{ssec:ustation_model_comp_dynamics}). @@ -3776,7 +3803,7 @@ The translation stage is connected to the granite by a 6-DoF joint, but the \(D_ Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoF joint, with 1 imposed DoF each time. Finally, the positioning hexapod has 6-DoF. -The total number of ``free'' degrees of freedom is 27, so the model has 54 states. +The total number of ``free'' \acrshortpl{dof} is 27, so the model has 54 states. The springs and dampers values were first estimated from the joint/stage specifications and were later fined-tuned based on the measurements. The spring values are summarized in Table~\ref{tab:ustation_6dof_stiffness_values}. @@ -3799,9 +3826,9 @@ Hexapod & \(10\,N/\mu m\) & \(10\,N/\mu m\) & \(100\,N/\mu m\) & \(1.5\,Nm/rad\) \label{ssec:ustation_model_comp_dynamics} The dynamics of the micro-station was measured by placing accelerometers on each stage and by impacting the translation stage with an instrumented hammer in three directions. -The obtained FRFs were then projected at the CoM of each stage. +The obtained \acrshortpl{frf} were then projected at the CoM of each stage. -To gain a first insight into the accuracy of the obtained model, the FRFs from the hammer impacts to the acceleration of each stage were extracted from the multi-body model and compared with the measurements in Figure~\ref{fig:ustation_comp_com_response}. +To gain a first insight into the accuracy of the obtained model, the \acrshortpl{frf} from the hammer impacts to the acceleration of each stage were extracted from the multi-body model and compared with the measurements in Figure~\ref{fig:ustation_comp_com_response}. Even though there is some similarity between the model and the measurements (similar overall shapes and amplitudes), it is clear that the multi-body model does not accurately represent the complex micro-station dynamics. Tuning the numerous model parameters to better match the measurements is a highly non-linear optimization problem that is difficult to solve in practice. @@ -3894,8 +3921,8 @@ The equivalent forces and torques applied at center of \(\{\mathcal{X}\}\) are t F_{\mathcal{X}} = \bm{J}_F^{\intercal} \cdot F_{\mathcal{L}} \end{equation} -Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the FRF from 6 forces/torques applied at the origin of frame \(\{\mathcal{X}\}\) to the 6 linear/angular accelerations of the top platform expressed with respect to \(\{\mathcal{X}\}\). -These FRFs were then used for comparison with the multi-body model. +Using the two Jacobian matrices, the \acrshort{frf} from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the \acrshort{frf} from 6 forces/torques applied at the origin of frame \(\{\mathcal{X}\}\) to the 6 linear/angular accelerations of the top platform expressed with respect to \(\{\mathcal{X}\}\). +These \acrshortpl{frf} were then used for comparison with the multi-body model. The compliance of the micro-station multi-body model was extracted by computing the transfer function from forces/torques applied on the hexapod's top platform to the ``absolute'' motion of the top platform. These results are compared with the measurements in Figure~\ref{fig:ustation_frf_compliance_model}. @@ -3939,7 +3966,7 @@ Therefore, from a control perspective, they are not important. \paragraph{Ground Motion} The ground motion was measured by using a sensitive 3-axis geophone shown in Figure~\ref{fig:ustation_geophone_picture} placed on the ground. -The generated voltages were recorded with a high resolution DAC, and converted to displacement using the Geophone sensitivity transfer function. +The generated voltages were recorded with a high resolution \acrshort{adc}, and converted to displacement using the Geophone sensitivity transfer function. The obtained ground motion displacement is shown in Figure~\ref{fig:ustation_ground_disturbance}. \begin{minipage}[b]{0.54\linewidth} @@ -4017,7 +4044,7 @@ The obtained results are shown in Figure~\ref{fig:ustation_errors_spindle}. A large fraction of the radial (Figure~\ref{fig:ustation_errors_spindle_radial}) and tilt (Figure~\ref{fig:ustation_errors_spindle_tilt}) errors is linked to the fact that the two spheres are not perfectly aligned with the rotation axis of the Spindle. 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 ``point-of-interest'' of the sample and the rotation axis will be considered because the alignment is not perfect in practice. +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}). \begin{figure}[htbp] @@ -4134,8 +4161,8 @@ 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 ``point-of-interest'' 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 ``point-of-interest'' with respect to the granite was recorded. +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. +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}). @@ -4160,7 +4187,7 @@ 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 \(\pm 4.5\, mm\). Both ground motion and translation stage vibrations were included in the simulation. -Similar to what was performed for the tomography simulation, the PoI position with respect to the granite was recorded and compared with the experimental measurements in Figure~\ref{fig:ustation_errors_model_dy_vertical}. +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. \begin{figure}[htbp] @@ -4188,8 +4215,8 @@ However, the complexity of its dynamic behavior poses significant challenges for Consequently, a multi-body modeling approach was adopted (Section~\ref{sec:nhexa_model}), facilitating seamless integration with the existing micro-station model. -The control of the Stewart platform introduces additional complexity due to its multi-input multi-output (MIMO) nature. -Section~\ref{sec:nhexa_control} explores how the High Authority Control/Low Authority Control (HAC-LAC) strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform. +The control of the Stewart platform introduces additional complexity due to its \acrfull{mimo} nature. +Section~\ref{sec:nhexa_control} explores how the \acrfull{haclac} strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform. This adaptation requires fundamental decisions regarding both the control architecture (centralized versus decentralized) and the control frame (Cartesian versus strut space). Through careful analysis of system interactions and plant characteristics in different frames, a control architecture combining decentralized Integral Force Feedback for active damping with a centralized high authority controller for positioning was developed, with both controllers implemented in the frame of the struts. \subsection{Active Vibration Platforms} @@ -4212,7 +4239,7 @@ The positioning of samples with respect to X-ray beam, that can be focused to si To overcome this limitation, external metrology systems have been implemented to measure sample positions with nanometer accuracy, enabling real-time feedback control for sample stabilization. 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 degrees of freedom. +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}). @@ -4310,7 +4337,7 @@ The first key distinction of the NASS is in the continuous rotation of the activ 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. In addition, NASS implements a unique Long-Stroke/Short-Stroke architecture. In conventional systems, active platforms typically correct spindle positioning errors - for example, unwanted translations or tilts that occur during rotation, whereas the intended rotational motion (\(R_z\)) is performed by the spindle itself and is not corrected. -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. +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. @@ -4326,16 +4353,16 @@ 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 degrees of freedom (\(D_x\), \(D_y\), \(D_z\), \(R_x\), 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. +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 \(2000\,N/\mu m\). -This limitation necessitates alternative control approaches, and the High Authority Control/Low Authority Control (HAC-LAC) strategy is proposed to address this challenge. +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. Two primary categories of positioning platform architectures are considered: serial and parallel mechanisms. -Serial robots, characterized by open-loop kinematic chains, typically dedicate one actuator per degree of freedom as shown in Figure~\ref{fig:nhexa_serial_architecture_kenton}. +Serial robots, characterized by open-loop kinematic chains, typically dedicate one actuator per \acrshort{dof} as shown in Figure~\ref{fig:nhexa_serial_architecture_kenton}. While offering large workspaces and high maneuverability, serial mechanisms suffer from several inherent limitations. These include low structural stiffness, cumulative positioning errors along the kinematic chain, high mass-to-payload ratios due to actuator placement, and limited payload capacity~\cite{taghirad13_paral}. These limitations generally make serial architectures unsuitable for nano-positioning applications, except when handling very light samples, as was used in~\cite{nazaretski15_pushin_limit} and shown in Figure~\ref{fig:nhexa_stages_nazaretski}. @@ -4343,7 +4370,7 @@ These limitations generally make serial architectures unsuitable for nano-positi In contrast, parallel mechanisms, which connect the mobile platform to the fixed base through multiple parallel struts, offer several advantages for precision positioning. Their closed-loop kinematic structure provides inherently higher structural stiffness, as the platform is simultaneously supported by multiple struts~\cite{taghirad13_paral}. Although parallel mechanisms typically exhibit limited workspace compared to serial architectures, this limitation is not critical for NASS given its modest stroke requirements. -Numerous parallel kinematic architectures have been developed \cite{dong07_desig_precis_compl_paral_posit} to address various positioning requirements, with designs varying based on the desired degrees of freedom and specific application constraints. +Numerous parallel kinematic architectures have been developed~\cite{dong07_desig_precis_compl_paral_posit} to address various positioning requirements, with designs varying based on the desired \acrshortpl{dof} and specific application constraints. Furthermore, hybrid architectures combining both serial and parallel elements have been proposed~\cite{shen19_dynam_analy_flexur_nanop_stage}, as illustrated in Figure~\ref{fig:nhexa_serial_parallel_examples}, offering potential compromises between the advantages of both approaches. \begin{figure}[h!tbp] @@ -4363,7 +4390,7 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha \end{figure} After evaluating the different options, the Stewart platform architecture was selected for several reasons. -In addition to providing control over all required degrees of freedom, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints. +In addition to providing control over all required \acrshortpl{dof}, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints. Stewart platforms have been implemented in a wide variety of configurations, as illustrated in Figure~\ref{fig:nhexa_stewart_examples}, which shows two distinct implementations: one implementing piezoelectric actuators for nano-positioning applications, and another based on voice coil actuators for vibration isolation. These examples demonstrate the architecture's versatility in terms of geometry, actuator selection, and scale, all of which can be optimized for specific applications. Furthermore, the successful implementation of Integral Force Feedback (IFF) control on Stewart platforms has been well documented~\cite{abu02_stiff_soft_stewar_platf_activ,hanieh03_activ_stewar,preumont07_six_axis_singl_stage_activ}, and the extensive body of research on this architecture enables thorough optimization specifically for the NASS. @@ -4421,8 +4448,8 @@ To facilitate the rigorous analysis of the Stewart platform, four reference fram \begin{itemize} \item The fixed base frame \(\{F\}\), which is located at the center of the base platform's bottom surface, serves as the mounting reference for the support structure. \item The mobile frame \(\{M\}\), which is located at the center of the top platform's upper platform, provides a reference for payload mounting. -\item The point-of-interest frame \(\{A\}\), fixed to the base but positioned at the workspace center. -\item The moving point-of-interest frame \(\{B\}\), attached to the mobile platform coincides with frame \(\{A\}\) in the home position. +\item The \acrlong{poi} frame \(\{A\}\), fixed to the base but positioned at the workspace center. +\item The moving \acrlong{poi} frame \(\{B\}\), attached to the mobile platform coincides with frame \(\{A\}\) in the home position. \end{itemize} Frames \(\{F\}\) and \(\{M\}\) serve primarily to define the joint locations. @@ -4701,7 +4728,7 @@ 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~\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 point of interest, denoted by frame \(\{A\}\), is situated \(150\,mm\) above the moving platform frame \(\{M\}\). +The \acrshort{poi}, denoted by frame \(\{A\}\), is situated \(150\,mm\) above the moving platform frame \(\{M\}\). The geometric parameters of the nano-hexapod are summarized in Table~\ref{tab:nhexa_stewart_model_geometry}. These parameters define the positions of all connection points in their respective coordinate frames. @@ -4749,7 +4776,7 @@ Both platforms were assigned a mass of \(5\,kg\). The platform's joints play a crucial role in its dynamic behavior. At both the upper and lower connection points, various degrees of freedom can be modeled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components. -For each degree of freedom, stiffness characteristics can be incorporated into the model. +For each \acrshort{dof}, stiffness characteristics can be incorporated into the model. In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modeled as two-degree-of-freedom universal joints, while the top joints are represented as three-degree-of-freedom spherical joints. These joints are considered massless and exhibit no stiffness along their degrees of freedom. @@ -4824,7 +4851,7 @@ For the analytical model, the stiffness, damping, and mass matrices are defined The transfer functions from the actuator forces to the strut displacements are computed using these matrices according to equation~\eqref{eq:nhexa_transfer_function_struts}. These analytical transfer functions are then compared with those extracted from the multi-body model. -The developed multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform. +The developed multi-body model yields a state-space representation with 12 states, corresponding to the six \acrshortpl{dof} of the moving platform. Figure~\ref{fig:nhexa_comp_multi_body_analytical} presents a comparison between the analytical and multi-body transfer functions, specifically showing the response from the first actuator force to all six strut displacements. The close agreement between both approaches across the frequency spectrum validates the multi-body model's accuracy in capturing the system's dynamic behavior. @@ -4843,7 +4870,7 @@ The transfer functions from actuator forces \(\bm{f}\) to both strut displacemen The transfer functions relating actuator forces to strut displacements are presented in Figure~\ref{fig:nhexa_multi_body_plant_dL}. Due to the system's symmetrical design and identical strut configurations, all diagonal terms (transfer functions from force \(f_i\) to displacement \(l_i\) of the same strut) exhibit identical behavior. -While the system has six degrees of freedom, only four distinct resonance frequencies were observed in the frequency response. +While the system has six \acrshortpl{dof}, only four distinct resonance frequencies were observed in the \acrshortpl{frf}. This reduction from six to four observable modes is attributed to the system's symmetry, where two pairs of resonances occur at identical frequencies. The system's behavior can be characterized in three frequency regions. @@ -4882,7 +4909,7 @@ The validated multi-body model will serve as a valuable tool for predicting syst \subsection{Control of Stewart Platforms} \label{sec:nhexa_control} The control of Stewart platforms presents distinct challenges compared to the uniaxial model due to their multi-input multi-output nature. -Although the uniaxial model demonstrated the effectiveness of the HAC-LAC strategy, its extension to Stewart platforms requires careful considerations discussed in this section. +Although the uniaxial model demonstrated the effectiveness of the \acrshort{haclac} strategy, its extension to Stewart platforms requires careful considerations discussed in this section. First, the distinction between centralized and decentralized control approaches is discussed in Section~\ref{ssec:nhexa_control_centralized_decentralized}. The impact of the control space selection - either Cartesian or strut space - is then analyzed in Section~\ref{ssec:nhexa_control_space}, highlighting the trade-offs between direction-specific tuning and implementation simplicity. @@ -4892,7 +4919,7 @@ This architecture, while simple, will be used to demonstrate the feasibility of \subsubsection{Centralized and Decentralized Control} \label{ssec:nhexa_control_centralized_decentralized} -In the control of MIMO systems, and more specifically of Stewart platforms, a fundamental architectural decision lies in the choice between centralized and decentralized control strategies. +In the control of \acrshort{mimo} systems, and more specifically of Stewart platforms, a fundamental architectural decision lies in the choice between centralized and decentralized control strategies. In decentralized control, each actuator operates based on feedback from its associated sensor only, creating independent control loops, as illustrated in Figure~\ref{fig:nhexa_stewart_decentralized_control}. While mechanical coupling between the struts exists, control decisions are made locally, with each controller processing information from a single sensor-actuator pair. @@ -4907,7 +4934,7 @@ For instance, when using external metrology systems that measure the platform's In the context of the nano-hexapod, 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 High-Authority Control (HAC), which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section~\ref{ssec:nhexa_control_hac_lac}) +\item \acrfull{hac}, which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section~\ref{ssec:nhexa_control_hac_lac}) \end{itemize} \begin{figure}[htbp] @@ -4954,11 +4981,11 @@ Alternatively, control can be implemented directly in Cartesian space, as illust Here, the controller processes Cartesian errors \(\bm{\epsilon}_{\mathcal{X}}\) to generate forces and torques \(\bm{\mathcal{F}}\), which are then mapped to actuator forces using the transpose of the inverse Jacobian matrix~\eqref{eq:nhexa_jacobian_forces}. The plant behavior in Cartesian space, illustrated in Figure~\ref{fig:nhexa_plant_frame_cartesian}, reveals interesting characteristics. -Some degrees of freedom, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics. +Some \acrshortpl{dof}, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics. A key advantage of this approach is that the control performance can be tuned individually for each direction. -This is particularly valuable when performance requirements differ between degrees of freedom - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational degrees of freedom can tolerate larger errors than others. +This is particularly valuable when performance requirements differ between directions - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational \acrshortpl{dof} can tolerate larger errors than others. -However, significant coupling exists between certain degrees of freedom, particularly between rotations and translations (e.g., \(\epsilon_{R_x}/\mathcal{F}_y\) or \(\epsilon_{D_y}/\bm\mathcal{M}_x\)). +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. More sophisticated control strategies will be explored during the detailed design phase. @@ -5074,9 +5101,9 @@ The loop gain of an individual control channel is shown in Figure~\ref{fig:nhexa \end{bmatrix}, \quad K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}} \end{equation} -The stability of the MIMO feedback loop is analyzed through the \emph{characteristic loci} method. +The stability of the \acrshort{mimo} feedback loop is analyzed through the \emph{characteristic loci} method. Such characteristic loci represent the eigenvalues of the loop gain matrix \(\bm{G}(j\omega)\bm{K}(j\omega)\) plotted in the complex plane as the frequency varies from \(0\) to \(\infty\). -For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point~\cite{skogestad07_multiv_feedb_contr}. +For \acrshort{mimo} systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point~\cite{skogestad07_multiv_feedb_contr}. As shown in Figure~\ref{fig:nhexa_decentralized_hac_iff_root_locus}, all loci remain to the right of the \(-1\) point, validating the stability of the closed-loop system. Additionally, the distance of the loci from the \(-1\) point provides information about stability margins of the coupled system. @@ -5102,7 +5129,7 @@ The control architecture developed for the uniaxial and the rotating models was Two fundamental choices were first addressed: the selection between centralized and decentralized approaches and the choice of control space. While control in Cartesian space enables direction-specific performance tuning, implementation in strut space was selected for the conceptual design phase due to two key advantages: good decoupling at low frequencies and identical diagonal terms in the plant transfer functions, allowing a single controller design to be replicated across all struts. -The HAC-LAC strategy was then implemented. +The \acrshort{haclac} strategy was then implemented. The inner loop implements decentralized Integral Force Feedback for active damping. The collocated nature of the force sensors ensures stability despite strong coupling between struts at resonance frequencies, enabling effective damping of structural modes. The outer loop implements High Authority Control, enabling precise positioning of the mobile platform. @@ -5164,7 +5191,7 @@ As established in the previous section on Stewart platforms, the proposed contro For the Nano Active Stabilization System, computing the positioning errors in the frame of the struts involves three key steps. 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 point of interest needs to be positioned. +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}. @@ -5216,9 +5243,9 @@ Using these reference signals, the desired sample position relative to the fixed \label{ssec:nass_sample_pose_error} 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 degrees of freedom: three translations (\(D_x\), \(D_y\), \(D_z\)) and two rotations (\(R_x\), \(R_y\)). +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 degree of freedom (\(R_z\)) is still required to compute the errors in the frame of the nano-hexapod struts (i.e. to compute the nano-hexapod inverse kinematics). +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 measured sample pose is represented by the homogeneous transformation matrix \(\bm{T}_{\text{sample}}\), as shown in equation~\eqref{eq:nass_sample_pose}. @@ -5267,7 +5294,7 @@ 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 degrees of freedom, while the sixth degree of freedom (Rz) 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 nano-hexapod 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. @@ -5283,7 +5310,7 @@ Then, the high authority controller uses the computed errors in the frame of the \end{figure} \subsection{Decentralized Active Damping} \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 HAC-LAC strategy. +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. @@ -5318,7 +5345,7 @@ The effect of rotation, as shown in Figure~\ref{fig:nass_iff_plant_effect_rotati 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. -However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining robust stability properties. +However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining good robustness. \begin{figure}[h!tbp] \begin{subfigure}{0.48\textwidth} @@ -5364,7 +5391,7 @@ The overall gain was then increased to obtain a large loop gain around the reson \end{figure} To verify stability, the root loci for the three payload configurations were computed, as shown in Figure~\ref{fig:nass_iff_root_locus}. -The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability of the applied decentralized IFF. +The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robustness of the applied decentralized IFF. \begin{figure}[h!tbp] \begin{subfigure}{0.33\textwidth} @@ -5485,7 +5512,7 @@ This coupling introduces complex behavior that is difficult to model and predict The soft nano-hexapod configuration was evaluated using a stiffness of \(0.01\,N/\mu m\) 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 center of mass 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 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. \begin{figure}[h!tbp] \begin{subfigure}{0.48\textwidth} @@ -5538,9 +5565,9 @@ The Nano Active Stabilization System concept was validated through time-domain s 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). -The primary requirement stipulates that the point of interest must remain within beam dimensions throughout operation. +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 DAC and voltage amplifiers, were not included in this analysis, as these parameters will be optimized during the detailed design phase. +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). 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. @@ -5618,7 +5645,7 @@ The multi-body modeling approach proved essential for capturing the complex dyna 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. -The HAC-LAC control strategy was successfully adapted to address the unique challenges presented by the rotating NASS. +The \acrshort{haclac} control strategy was successfully adapted to address the unique challenges presented by the rotating NASS. 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. @@ -5636,7 +5663,7 @@ This chapter begins by determining the optimal geometric configuration for the n 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. -Section~\ref{sec:detail_fem} introduces a hybrid modeling methodology that combines finite element analysis 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 nano-hexapod 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. @@ -5735,7 +5762,7 @@ Stewart platforms incorporating force sensors are frequently used for vibration Inertial sensors (accelerometers and geophones) are commonly employed in vibration isolation applications~\cite{chen03_payload_point_activ_vibrat_isolat,chi15_desig_exper_study_vcm_based}. These sensors are predominantly aligned with the struts~\cite{hauge04_sensor_contr_space_based_six,li01_simul_fault_vibrat_isolat_point,thayer02_six_axis_vibrat_isolat_system,zhang11_six_dof,jiao18_dynam_model_exper_analy_stewar,tang18_decen_vibrat_contr_voice_coil}, although they may also be fixed to the top platform~\cite{wang16_inves_activ_vibrat_isolat_stewar}. -For high-precision positioning applications, various displacement sensors are implemented, including LVDTs~\cite{thayer02_six_axis_vibrat_isolat_system,kim00_robus_track_contr_desig_dof_paral_manip,li01_simul_fault_vibrat_isolat_point,thayer98_stewar}, capacitive sensors~\cite{ting07_measur_calib_stewar_microm_system,ting13_compos_contr_desig_stewar_nanos_platf}, eddy current sensors~\cite{chen03_payload_point_activ_vibrat_isolat,furutani04_nanom_cuttin_machin_using_stewar}, and strain gauges~\cite{du14_piezo_actuat_high_precis_flexib}. +For high-precision positioning applications, various displacement sensors are implemented, including \acrfullpl{lvdt} ~\cite{thayer02_six_axis_vibrat_isolat_system,kim00_robus_track_contr_desig_dof_paral_manip,li01_simul_fault_vibrat_isolat_point,thayer98_stewar}, capacitive sensors~\cite{ting07_measur_calib_stewar_microm_system,ting13_compos_contr_desig_stewar_nanos_platf}, eddy current sensors~\cite{chen03_payload_point_activ_vibrat_isolat,furutani04_nanom_cuttin_machin_using_stewar}, and strain gauges~\cite{du14_piezo_actuat_high_precis_flexib}. Notably, some designs incorporate external sensing methodologies rather than integrating sensors within the struts~\cite{li01_simul_fault_vibrat_isolat_point,chen03_payload_point_activ_vibrat_isolat,ting13_compos_contr_desig_stewar_nanos_platf}. A recent design~\cite{naves21_desig_optim_large_strok_flexur_mechan}, although not strictly speaking a Stewart platform, has demonstrated the use of 3-phase rotary motors with rotary encoders for achieving long-stroke and highly repeatable positioning, as illustrated in Figure~\ref{fig:detail_kinematics_naves}. @@ -5956,7 +5983,7 @@ These conditions are studied in Section~\ref{ssec:detail_kinematics_cubic_dynami \end{equation} In the frame of the struts, the equations of motion~\eqref{eq:detail_kinematics_transfer_function_struts} are well decoupled at low frequency. -This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is well decoupled and SISO control may be applied for each strut, independently of the payload being used. +This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is well decoupled and \acrfull{siso} control may be applied for each strut, independently of the payload being used. \begin{equation}\label{eq:detail_kinematics_transfer_function_struts} \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-\intercal} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1} @@ -6085,7 +6112,7 @@ In that case, the location of the top joints can be expressed by equation~\eqref The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)). This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center). -This specific location where the stiffness matrix is diagonal is referred to as the ``Center of Stiffness'' (analogous to the ``Center of Mass'' where the mass matrix is diagonal). +This specific location where the stiffness matrix is diagonal is referred to as the \acrfull{cok}, analogous to the \acrfull{com} where the mass matrix is diagonal. \paragraph{Effect of having frame \(\{B\}\) off-centered} When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix. @@ -6165,7 +6192,7 @@ At high frequency, the behavior is governed by the mass matrix (evaluated at fra \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1} \end{equation} -To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the \(\{B\}\) frame, and the principal axes of inertia must align with the axes of the \(\{B\}\) frame. +To achieve a diagonal mass matrix, the \acrlong{com} of the mobile components must coincide with the \(\{B\}\) frame, and the principal axes of inertia must align with the axes of the \(\{B\}\) frame. \begin{figure}[htbp] \centering @@ -6175,8 +6202,8 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure~\ref{fig:detail_kinematics_cubic_payload}). Transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) were computed for two specific locations of the \(\{B\}\) frames. -When the \(\{B\}\) frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_com}). -Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_cok}). +When the \(\{B\}\) frame was positioned at the \acrlong{com}, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_com}). +Conversely, when positioned at the \acrlong{cok}, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_cok}). \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -6195,8 +6222,8 @@ Conversely, when positioned at the center of stiffness, coupling occurred at hig \end{figure} \paragraph{Payload's CoM at the cube's center} -An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components~\cite{li01_simul_fault_vibrat_isolat_point}. -This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure~\ref{fig:detail_kinematics_cubic_centered_payload}). +An effective strategy for improving dynamical performances involves aligning the cube's center (\acrlong{cok}) with the \acrlong{com} of the moving components~\cite{li01_simul_fault_vibrat_isolat_point}. +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. @@ -6222,7 +6249,7 @@ If a design similar to Figure~\ref{fig:detail_kinematics_cubic_centered_payload} The analysis of dynamical properties of the cubic architecture yields several important conclusions. Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames \(\{A\}\) and \(\{B\}\) are positioned at the cube's center. Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements. -Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame \(\{B\}\). +Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's \acrlong{com} with reference frame \(\{B\}\). While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical. \subsubsection{Decentralized Control} \label{ssec:detail_kinematics_decentralized_control} @@ -6296,13 +6323,13 @@ The presented results do not demonstrate the pronounced decoupling advantages of Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement for decentralized control is less obvious than often reported in the literature. \subsubsection{Cubic architecture with Cube's center above the top platform} \label{ssec:detail_kinematics_cubic_design} -As demonstrated in Section~\ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. +As demonstrated in Section~\ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can exhibit advantageous dynamical properties when the \acrlong{com} of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. As shown in Section~\ref{ssec:detail_kinematics_cubic_static}, the stiffness matrix is diagonal when the considered \(\{B\}\) frame is located at the cube's center. However, the \(\{B\}\) frame is typically positioned above the top platform where forces are applied and displacements are measured. This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration. -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 center of mass relative to the mobile platform (coincident with the cube's center). +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 \(H_{CoM} = 20\,mm\). @@ -6426,16 +6453,16 @@ The analysis of decentralized control in the frame of the struts revealed more n While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described. Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar. -Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center. +Fully decoupled dynamics in the Cartesian frame can be achieved when the \acrlong{com} of the moving body coincides with the cube's center. However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications. 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 center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame. +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} \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 point of interest, which is positioned \(150\,mm\) above the top platform. +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. 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} @@ -6485,8 +6512,8 @@ While minor refinements may occur during detailed mechanical design to address m 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 center of mass coincides with the cube's center. -Given the impracticality of consistently aligning the center of mass with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. +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. \subsubsection{Required Actuator stroke} \label{ssec:detail_kinematics_nano_hexapod_actuator_stroke} @@ -6525,15 +6552,15 @@ While cubic architectures are prevalent in the literature and attributed with be 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. -For the cubic configuration, complete dynamical decoupling in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center, but this arrangement is often impractical for real-world applications. +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. 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. 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 finite element analysis 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 FEM for real-time control scenarios. +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. 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}). @@ -6541,19 +6568,19 @@ Through this approach, system-level dynamic behavior under closed-loop control c \subsection{Reduced order flexible bodies} \label{sec:detail_fem_super_element} Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models. -These components are traditionally analyzed using Finite Element Analysis (FEA) software. +These components are traditionally analyzed using \acrshort{fea} software. However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models~\cite{hatch00_vibrat_matlab_ansys}. This combined multibody-FEA modeling approach presents significant advantages, as it enables the accurate FE modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system~\cite{rankers98_machin}. The investigation of this hybrid modeling approach is structured in three sections. First, the fundamental principles and methodological approaches of this modeling framework are introduced (Section~\ref{ssec:detail_fem_super_element_theory}). -It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section~\ref{ssec:detail_fem_super_element_example}). +It is then illustrated through its practical application to the modelling of an \acrfull{apa} (Section~\ref{ssec:detail_fem_super_element_example}). Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section~\ref{ssec:detail_fem_super_element_validation}). \subsubsection{Procedure} \label{ssec:detail_fem_super_element_theory} -In this modeling approach, some components within the multi-body framework are represented as \emph{reduced-order flexible bodies}, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models. -These matrices are generated via modal reduction techniques, specifically through the application of component mode synthesis (CMS), thus establishing this design approach as a combined multibody-FEA methodology. +In this modeling approach, some components within the multi-body framework are represented as \emph{reduced-order flexible bodies}, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from \acrshort{fea} models. +These matrices are generated via modal reduction techniques, specifically through the application of component mode synthesis, thus establishing this design approach as a combined multibody-FEA methodology. Standard FEA implementations typically involve thousands or even hundreds of thousands of DoF, rendering direct integration into multi-body simulations computationally prohibitive. The objective of modal reduction is therefore to substantially decrease the number of DoF while preserving the essential dynamic characteristics of the component. @@ -6565,21 +6592,21 @@ These frames serve multiple functions, including connecting to other parts, appl Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method~\cite{craig68_coupl_subst_dynam_analy} (also known as the ``fixed-interface method''), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics. This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF. -The number of degrees of freedom in the reduced model is determined by~\eqref{eq:detail_fem_model_order} where \(n\) represents the number of defined frames and \(p\) denotes the number of additional modes to be modeled. -The outcome of this procedure is an \(m \times m\) set of reduced mass and stiffness matrices, \(m\) being the total retained number of degrees of freedom, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. +The number of \acrshortpl{dof} in the reduced model is determined by~\eqref{eq:detail_fem_model_order} where \(n\) represents the number of defined frames and \(p\) denotes the number of additional modes to be modeled. +The outcome of this procedure is an \(m \times m\) set of reduced mass and stiffness matrices, \(m\) being the total retained number of \acrshortpl{dof}, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. \begin{equation}\label{eq:detail_fem_model_order} m = 6 \times n + p \end{equation} \subsubsection{Example with an Amplified Piezoelectric Actuator} \label{ssec:detail_fem_super_element_example} -The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons. +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}. 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 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}. -The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework. -The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation. +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}. +The selection of the \acrshort{apa} for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework. +The specific design of the \acrshort{apa} allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation. \begin{minipage}[b]{0.48\linewidth} \begin{center} @@ -6604,7 +6631,7 @@ Stiffness & \(21\,N/\mu m\)\\ \end{minipage} \paragraph{Finite Element Model} -The development of the finite element model for the APA95ML required the knowledge of the material properties, as summarized in Table~\ref{tab:detail_fem_material_properties}. +The development of the \acrfull{fem} for the APA95ML required the knowledge of the material properties, as summarized in Table~\ref{tab:detail_fem_material_properties}. The finite element mesh, shown in Figure~\ref{fig:detail_fem_apa95ml_mesh}, was then generated. \begin{table}[htbp] @@ -6710,7 +6737,7 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine \end{table} \paragraph{Identification of the APA Characteristics} -Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications. +Initial validation of the \acrlong{fem} and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications. 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. @@ -6731,7 +6758,7 @@ The piezoelectric stacks, exhibiting a typical strain response of \(0.1\,\%\) re 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 high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model. +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} \label{ssec:detail_fem_super_element_validation} Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation. @@ -6739,8 +6766,8 @@ The goal was to measure the dynamics of the APA95ML and to compare it with predi 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 granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement \(y\). -A digital-to-analog converter (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 analog-to-digital converter (ADC). +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}. \begin{figure}[htbp] \centering @@ -6754,10 +6781,10 @@ The identification procedure required careful choice of the excitation signal~\c During all this experimental work, random noise excitation was predominantly employed. The designed excitation signal is then generated and both input and output signals are synchronously acquired. -From the obtained input and output data, the frequency response functions were derived. +From the obtained input and output data, the \acrshortpl{frf} were derived. To improve the quality of the obtained frequency domain data, averaging and windowing were used~\cite[, chap. 13]{pintelon12_system_ident}. -The obtained frequency response functions 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 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 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. @@ -6792,7 +6819,7 @@ The IFF controller implementation, defined in equation~\ref{eq:detail_fem_iff_co The theoretical damped dynamics of the closed-loop system was estimated using the model by computed the root locus plot shown in Figure~\ref{fig:detail_fem_apa95ml_iff_root_locus}. For experimental validation, six gain values were tested: \(g = [0,\,10,\,50,\,100,\,500,\,1000]\). -The measured frequency responses for each gain configuration were compared with model predictions, as presented in Figure~\ref{fig:detail_fem_apa95ml_damped_plants}. +The measured \acrshortpl{frf} for each gain configuration were compared with model predictions, as presented in Figure~\ref{fig:detail_fem_apa95ml_damped_plants}. The close agreement between experimental measurements and theoretical predictions across all gain configurations demonstrates the model's capability to accurately predict both open-loop and closed-loop system dynamics. @@ -6867,14 +6894,14 @@ Conventional piezoelectric stack actuators (shown in Figure~\ref{fig:detail_fem_ 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. -Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through a specific mechanical design. +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. Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF~\cite{souleille18_concep_activ_mount_space_applic,verma20_dynam_stabil_thin_apert_light}. -Moreover, using APA for active damping has been successfully demonstrated in similar applications~\cite{hanieh03_activ_stewar}. +Moreover, using \acrshort{apa} for active damping has been successfully demonstrated in similar applications~\cite{hanieh03_activ_stewar}. -Several specific APA models were evaluated against the established specifications (Table~\ref{tab:detail_fem_piezo_act_models}). +Several specific \acrshort{apa} models were evaluated against the established specifications (Table~\ref{tab:detail_fem_piezo_act_models}). The APA300ML emerged as the optimal choice. -This selection was further reinforced by previous experience with APAs from the same manufacturer\footnote{Cedrat technologies}, and particularly by the successful validation of the modeling methodology with a similar actuator (Section~\ref{ssec:detail_fem_super_element_example}). +This selection was further reinforced by previous experience with \acrshortpl{apa} from the same manufacturer\footnote{Cedrat technologies}, and particularly by the successful validation of the modeling methodology with a similar actuator (Section~\ref{ssec:detail_fem_super_element_example}). 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] @@ -6940,19 +6967,19 @@ It considers only axial behavior, treating the actuator as infinitely rigid in o Several physical characteristics are not explicitly represented, including the mechanical amplification factor and the actual stress the piezoelectric stacks. Nevertheless, the model's primary advantage lies in its simplicity, adding only four states to the system model. -The model requires tuning of 8 parameters (\(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\), and \(g_a\)) to match the dynamics extracted from the finite element analysis. +The model requires tuning of 8 parameters (\(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\), and \(g_a\)) to match the dynamics extracted from the \acrshort{fea}. The shell parameters \(k_1\) and \(c_1\) were determined first through analysis of the zero in the \(V_a\) to \(V_s\) transfer function. The physical interpretation of this zero can be understood through Root Locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros. The open-loop zero therefore corresponds to the poles of the system with a theoretical infinite-gain controller that ensures zero force in the sensor stack. -This condition effectively represents the dynamics of an APA without the force sensor stack (i.e. an APA with only the shell). +This condition effectively represents the dynamics of an \acrshort{apa} without the force sensor stack (i.e. an \acrshort{apa} with only the shell). This physical interpretation enables straightforward parameter tuning: \(k_1\) determines the frequency of the zero, while \(c_1\) defines its damping characteristic. The stack parameters (\(k_a\), \(c_a\), \(k_e\), \(c_e\)) were then derived from the first pole of the \(V_a\) to \(y\) response. Given that identical piezoelectric stacks are used for both sensing and actuation, the relationships \(k_e = 2k_a\) and \(c_e = 2c_a\) were enforced, reflecting the series configuration of the dual actuator stacks. Finally, the sensitivities \(g_s\) and \(g_a\) were adjusted to match the DC gains of the respective transfer functions. -The resulting parameters, listed in Table~\ref{tab:detail_fem_apa300ml_2dof_parameters}, yield dynamic behavior that closely matches the high-order finite element model, as demonstrated in Figure~\ref{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}. +The resulting parameters, listed in Table~\ref{tab:detail_fem_apa300ml_2dof_parameters}, yield dynamic behavior that closely matches the high-order \acrshort{fem}, as demonstrated in Figure~\ref{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}. While higher-order modes and non-axial flexibility are not captured, the model accurately represents the fundamental dynamics within the operational frequency range. \begin{table}[htbp] @@ -6997,7 +7024,7 @@ The behavior of piezoelectric actuators is characterized by coupled constitutive To evaluate the impact of electrical boundary conditions on the system dynamics, experimental measurements were conducted using the APA95ML, comparing the transfer function from \(V_a\) to \(y\) under two distinct configurations. With the force sensor stack in open-circuit condition (analogous to voltage measurement with high input impedance) and in short-circuit condition (similar to charge measurement with low output impedance). As demonstrated in Figure~\ref{fig:detail_fem_apa95ml_effect_electrical_boundaries}, short-circuiting the force sensor stack results in a minor decrease in resonance frequency. -The developed models of the APA do not represent such behavior, but as this effect is quite small, this validates the simplifying assumption made in the models. +The developed models of the \acrshort{apa} do not represent such behavior, but as this effect is quite small, this validates the simplifying assumption made in the models. \begin{figure}[htbp] \centering @@ -7005,22 +7032,22 @@ The developed models of the APA do not represent such behavior, but as this effe \caption{\label{fig:detail_fem_apa95ml_effect_electrical_boundaries}Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency} \end{figure} -However, the electrical characteristics of the APA remain crucial for instrumentation design. +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} \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 APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation. +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 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. Additionally, the model predicts no problematic high-frequency modes in the dynamics from \(\bm{u}\) to \(\bm{\epsilon}_{\mathcal{L}}\) (Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}), maintaining consistency with earlier conceptual simulations. These findings suggest that the control performance targets established during the conceptual phase remain achievable with the selected actuator. -Comparative analysis between the high-order 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. +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 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 nano-hexapod. These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the nano-hexapod. @@ -7068,11 +7095,11 @@ For design simplicity and component standardization, identical joints are employ \caption{\label{fig:detail_fem_joints_examples}Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical ``universal'' flexible joint used in~\cite{preumont07_six_axis_singl_stage_activ}. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in~\cite{yang19_dynam_model_decoup_contr_flexib}. (\subref{fig:detail_fem_joints_wire}) ``Thin'' flexible joints having ``notch curves'' are also used~\cite{du14_piezo_actuat_high_precis_flexib}.} \end{figure} -While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other degrees of freedom, practical implementations exhibit parasitic stiffness that can impact control performance~\cite{mcinroy02_model_desig_flexur_joint_stewar}. +While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other \acrshortpl{dof}, practical implementations exhibit parasitic stiffness that can impact control performance~\cite{mcinroy02_model_desig_flexur_joint_stewar}. This section examines how these non-ideal characteristics affect system behavior, focusing particularly on bending/torsional stiffness (Section~\ref{ssec:detail_fem_joint_bending}) and axial compliance (Section~\ref{ssec:detail_fem_joint_axial}). 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 finite element analysis (Section~\ref{ssec:detail_fem_joint_specs}). +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. \subsubsection{Bending and Torsional Stiffness} \label{ssec:detail_fem_joint_bending} @@ -7137,12 +7164,12 @@ Therefore, determining the minimum acceptable axial stiffness that maintains nan 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). -The resulting frequency responses (Figure~\ref{fig:detail_fem_joints_axial_stiffness_plants}) reveal distinct effects on system dynamics. +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 frequency response data (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_plant}) and root locus analysis (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_locus}). +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}). However, the transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) demonstrates significant effects: internal strut modes appear at high frequencies, introducing substantial cross-coupling between axes. -This coupling is quantified through RGA analysis of the damped system (Figure~\ref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance. +This coupling is quantified through \acrfull{rga} analysis of the damped system (Figure~\ref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance. Above this resonance frequency, two critical limitations emerge. First, the system exhibits strong coupling between control channels, making decentralized control strategies ineffective. @@ -7208,11 +7235,11 @@ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\ 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. 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 degrees of freedom. +Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational \acrshortpl{dof}. -The joint geometry was optimized through parametric finite element analysis. +The joint geometry was optimized through parametric \acrshort{fea}. The optimization process revealed an inherent trade-off between maximizing axial stiffness and achieving sufficiently low bending/torsional stiffness, while maintaining material stresses within acceptable limits. -The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through finite element analysis and summarized in Table~\ref{tab:detail_fem_joints_specs}. +The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through \acrshort{fea} and summarized in Table~\ref{tab:detail_fem_joints_specs}. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -7232,7 +7259,7 @@ The final design, featuring a neck dimension of 0.25mm, achieves mechanical prop \subsubsection{Validation with the Nano-Hexapod} \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 finite element analysis. +The designed flexible joint was first validated through integration into the nano-hexapod 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. @@ -7248,7 +7275,7 @@ To improve computational efficiency, a low order representation was developed us 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 \(k_a\)). This simplification reduces the total model order to 48 states: 12 for the payload, 12 for the struts, and 24 for the joints (12 each for bottom and top joints). -While additional degrees of freedom could potentially capture more dynamic features, the selected configuration preserves essential system characteristics while minimizing computational complexity. +While additional \acrshortpl{dof} could potentially capture more dynamic features, the selected configuration preserves essential system characteristics while minimizing computational complexity. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -7268,7 +7295,7 @@ While additional degrees of freedom could potentially capture more dynamic featu \subsection*{Conclusion} \label{sec:detail_fem_conclusion} -In this chapter, the methodology of combining finite element analysis 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 nano-hexapod 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. @@ -7284,15 +7311,15 @@ Such model reduction, guided by detailed understanding of component behavior, pr 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. During the conceptual design phase of the NASS, pragmatic approaches were implemented for each of these elements. -The High Authority Control-Low Authority Control (HAC-LAC) architecture was selected for combining sensors. -Control was implemented in the frame of the struts, leveraging the inherent low-frequency decoupling of the plant where all decoupled elements exhibited similar dynamics, thereby simplifying the Single-Input Single-Output (SISO) controller design process. +The \acrfull{haclac} architecture was selected for combining sensors. +Control was implemented in the frame of the struts, leveraging the inherent low-frequency decoupling of the plant where all decoupled elements exhibited similar dynamics, thereby simplifying the \acrfull{siso} controller design process. For these decoupled plants, open-loop shaping techniques were employed to tune the individual controllers. While these initial strategies proved effective in validating the NASS concept, this work explores alternative approaches with the potential to further enhance the performance. Section~\ref{sec:detail_control_sensor} examines different methods for combining multiple sensors, with particular emphasis on sensor fusion techniques that are based on complementary filters. A novel approach for designing these filters is proposed, which allows optimization of the sensor fusion effectiveness. -Section~\ref{sec:detail_control_decoupling} presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. +Section~\ref{sec:detail_control_decoupling} presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and \acrfull{svd} decoupling. Each method is evaluated in terms of its theoretical foundations, implementation requirements, and performance characteristics, providing insights into their respective advantages for different applications. Finally, Section~\ref{sec:detail_control_cf} addresses the challenge of controller design for decoupled plants. @@ -7304,14 +7331,14 @@ The literature review of Stewart platforms revealed a wide diversity of designs Control objectives (such as active damping, vibration isolation, or precise positioning) directly dictate sensor selection, whether inertial, force, or relative position sensors. In cases where multiple control objectives must be achieved simultaneously, as is the case for the Nano Active Stabilization System (NASS) where the Stewart platform must both position the sample and provide isolation from micro-station vibrations, combining multiple sensors within the control architecture has been demonstrated to yield significant performance benefits~\cite{hauge04_sensor_contr_space_based_six}. -From the literature, three principal approaches for combining sensors have been identified: High Authority Control-Low Authority Control (HAC-LAC), sensor fusion, and two-sensor control architectures. +From the literature, three principal approaches for combining sensors have been identified: \acrlong{haclac}, sensor fusion, and two-sensor control architectures. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} \includegraphics[scale=1,scale=1]{figs/detail_control_sensor_arch_hac_lac.png} \end{center} -\subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC LAC} +\subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC-LAC} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center} @@ -7330,7 +7357,7 @@ From the literature, three principal approaches for combining sensors have been \caption{\label{fig:detail_control_control_multiple_sensors}Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control})} \end{figure} -The HAC-LAC approach employs a dual-loop control strategy in which two control loops are using different sensors for distinct purposes (Figure~\ref{fig:detail_control_sensor_arch_hac_lac}). +The \acrshort{haclac} approach employs a dual-loop control strategy in which two control loops are using different sensors for distinct purposes (Figure~\ref{fig:detail_control_sensor_arch_hac_lac}). In~\cite{li01_simul_vibrat_isolat_point_contr}, vibration isolation is provided by accelerometers collocated with the voice coil actuators, while external rotational sensors are used to achieve pointing control. In~\cite{geng95_intel_contr_system_multip_degree}, force sensors collocated with the magnetostrictive actuators are used for active damping using decentralized IFF, and subsequently accelerometers are employed for adaptive vibration isolation. Similarly, in~\cite{wang16_inves_activ_vibrat_isolat_stewar}, piezoelectric actuators with collocated force sensors are used in a decentralized manner to provide active damping while accelerometers are implemented in an adaptive feedback loop to suppress periodic vibrations. @@ -7348,15 +7375,15 @@ In~\cite{thayer02_six_axis_vibrat_isolat_system}, the use of force sensors and g Geophones are shown to provide better isolation performance than load cells but suffer from poor robustness. Conversely, the controller based on force sensors exhibited inferior performance (due to the presence of a pair of low frequency zeros), but demonstrated better robustness properties. A ``two-sensor control'' approach was proven to perform better than controllers based on individual sensors while maintaining better robustness. -A Linear Quadratic Regulator (LQG) was employed to optimize the two-input/one-output controller. +A \acrfull{lqg} was employed to optimize the two-input/one-output controller. Beyond these three main approaches, other control architectures have been proposed for different purposes. For instance, in~\cite{yang19_dynam_model_decoup_contr_flexib}, a first control loop based on force sensors and relative motion sensors is implemented to compensate for parasitic stiffness of the flexible joints. Subsequently, the system is decoupled in the modal space (facilitated by the removal of parasitic stiffness) and accelerometers are employed for vibration isolation. -The HAC-LAC architecture was previously investigated during the conceptual phase and successfully implemented to validate the NASS concept, demonstrating excellent performance. +The \acrshort{haclac} architecture was previously investigated during the conceptual phase and successfully implemented to validate the NASS concept, demonstrating excellent performance. At the other end of the spectrum, the two-sensor approach yields greater control design freedom but introduces increased complexity in tuning, and thus was not pursued in this study. -This work instead focuses on sensor fusion, which represents a promising middle ground between the proven HAC-LAC approach and the more complex two-sensor control strategy. +This work instead focuses on sensor fusion, which represents a promising middle ground between the proven \acrshort{haclac} approach and the more complex two-sensor control strategy. A review of sensor fusion is first presented in Section~\ref{ssec:detail_control_sensor_review}. Then, in Section~\ref{ssec:detail_control_sensor_fusion_requirements}, both the robustness of the fusion and the noise characteristics of the resulting ``fused sensor'' are derived and expressed as functions of the complementary filters' norms. @@ -7397,7 +7424,7 @@ Various design methods have been developed to optimize complementary filters. The most straightforward approach is based on analytical formulas, which depending on the application may be first order~\cite{corke04_inert_visual_sensin_system_small_auton_helic,yeh05_model_contr_hydraul_actuat_two,yong16_high_speed_vertic_posit_stage}, second order~\cite{baerveldt97_low_cost_low_weigh_attit,stoten01_fusion_kinet_data_using_compos_filter,jensen13_basic_uas}, or higher orders~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,stoten01_fusion_kinet_data_using_compos_filter,collette15_sensor_fusion_method_high_perfor,matichard15_seism_isolat_advan_ligo}. Since the characteristics of the super sensor depend on proper complementary filter design~\cite{dehaeze19_compl_filter_shapin_using_synth}, several optimization techniques have emerged—ranging from optimizing parameters for analytical formulas~\cite{jensen13_basic_uas,carreira15_compl_filter_desig_three_frequen_bands} to employing convex optimization tools~\cite{hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo} such as linear matrix inequalities~\cite{pascoal99_navig_system_desig_using_time}. As demonstrated in~\cite{plummer06_optim_compl_filter_their_applic_motion_measur}, complementary filter design can be linked to the standard mixed-sensitivity control problem, allowing powerful classical control theory tools to be applied. -For example, in~\cite{jensen13_basic_uas}, two gains of a Proportional Integral (PI) controller are optimized to minimize super sensor noise. +For example, in~\cite{jensen13_basic_uas}, two gains of a \acrfull{pi} controller are optimized to minimize super sensor noise. All these complementary filter design methods share the common objective of creating a super sensor with desired characteristics, typically in terms of noise and dynamics. As reported in~\cite{zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur}, phase shifts and magnitude bumps in the super sensor dynamics may occur if complementary filters are poorly designed or if sensors are improperly calibrated. @@ -7426,7 +7453,7 @@ The complementary property of filters \(H_1(s)\) and \(H_2(s)\) requires that th \paragraph{Sensor Models and Sensor Normalization} To analyze sensor fusion architectures, appropriate sensor models are required. -The model shown in Figure~\ref{fig:detail_control_sensor_model} consists of a linear time invariant (LTI) system \(G_i(s)\) representing the sensor dynamics and an input \(n_i\) representing sensor noise. +The model shown in Figure~\ref{fig:detail_control_sensor_model} consists of a \acrfull{lti} system \(G_i(s)\) representing the sensor dynamics and an input \(n_i\) representing sensor noise. The model input \(x\) is the measured physical quantity, and its output \(\tilde{x}_i\) is the ``raw'' output of the sensor. Prior to filtering the sensor outputs \(\tilde{x}_i\) with complementary filters, the sensors are typically normalized to simplify the fusion process. @@ -7489,7 +7516,7 @@ The estimation error \(\epsilon_x\), defined as the difference between the senso \epsilon_x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2 \end{equation} -As shown in~\eqref{eq:detail_control_sensor_noise_filtering_psd}, the Power Spectral Density (PSD) of the estimation error \(\Phi_{\epsilon_x}\) depends both on the norm of the two complementary filters and on the PSD of the noise sources \(\Phi_{n_1}\) and \(\Phi_{n_2}\). +As shown in~\eqref{eq:detail_control_sensor_noise_filtering_psd}, the \acrfull{psd} of the estimation error \(\Phi_{\epsilon_x}\) depends both on the norm of the two complementary filters and on the \acrshort{psd} of the noise sources \(\Phi_{n_1}\) and \(\Phi_{n_2}\). \begin{equation}\label{eq:detail_control_sensor_noise_filtering_psd} \Phi_{\epsilon_x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega) @@ -7694,7 +7721,7 @@ This straightforward example demonstrates that the proposed methodology for shap \label{ssec:detail_control_sensor_hinf_three_comp_filters} Certain applications necessitate the fusion of more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,carreira15_compl_filter_desig_three_frequen_bands}. -At LIGO, for example, a super sensor is formed by merging three distinct sensors: an LVDT, a seismometer, and a geophone~\cite{matichard15_seism_isolat_advan_ligo}. +At LIGO, for example, a super sensor is formed by merging three distinct sensors: a \acrshort{lvdt}, a seismometer, and a geophone~\cite{matichard15_seism_isolat_advan_ligo}. For merging \(n>2\) sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure~\ref{fig:detail_control_sensor_fusion_three}. Fusion can be implemented either ``sequentially,'' using \(n-1\) sets of two complementary filters (Figure~\ref{fig:detail_control_sensor_fusion_three_sequential}), or ``in parallel,'' employing a single set of \(n\) complementary filters (Figure~\ref{fig:detail_control_sensor_fusion_three_parallel}). @@ -7800,25 +7827,25 @@ This approach allows shaping of the filter magnitudes through the use of weighti This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude. Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters. -For the NASS, the HAC-LAC strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune. +For the NASS, the \acrshort{haclac} strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune. Looking forward, it would be interesting to investigate how sensor fusion (particularly between the force sensors and external metrology) compares to the HAC-IFF approach in terms of performance and robustness. \subsection{Decoupling} \label{sec:detail_control_decoupling} -The control of parallel manipulators (and any MIMO system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of SISO control for the decoupled plant (discussed in section~\ref{sec:detail_control_cf}). +The control of parallel manipulators (and any \acrshort{mimo} system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of \acrshort{siso} control for the decoupled plant (discussed in section~\ref{sec:detail_control_cf}). When sensors are integrated within the struts, decentralized control may be applied, as the system is already well decoupled at low frequency. -For instance,~\cite{furutani04_nanom_cuttin_machin_using_stewar} implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate PI controllers for each strut. +For instance,~\cite{furutani04_nanom_cuttin_machin_using_stewar} implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate \acrshort{pi} controllers for each strut. A similar control architecture was proposed in~\cite{du14_piezo_actuat_high_precis_flexib} using strain gauge sensors integrated in each strut. An alternative strategy involves decoupling the system in the Cartesian frame using Jacobian matrices. As demonstrated during the study of Stewart platform kinematics, Jacobian matrices can be used to map actuator forces to forces and torques applied on the top platform. This approach enables the implementation of controllers in a defined frame. It has been applied with various sensor types including force sensors~\cite{mcinroy00_desig_contr_flexur_joint_hexap}, relative displacement sensors~\cite{kim00_robus_track_contr_desig_dof_paral_manip}, and inertial sensors~\cite{li01_simul_vibrat_isolat_point_contr,abbas14_vibrat_stewar_platf}. -The Cartesian frame in which the system is decoupled is typically chosen at the point of interest (i.e., where the motion is of interest) or at the center of mass. +The Cartesian frame in which the system is decoupled is typically chosen at the \acrshort{poi} (i.e., where the motion is of interest) or at the \acrlong{com}. Modal decoupling represents another noteworthy decoupling strategy, wherein the ``local'' plant inputs and outputs are mapped to the modal space. -In this approach, multiple SISO plants, each corresponding to a single mode, can be controlled independently. +In this approach, multiple \acrshort{siso} plants, each corresponding to a single mode, can be controlled independently. This decoupling strategy has been implemented for active damping applications~\cite{holterman05_activ_dampin_based_decoup_colloc_contr}, which is logical as it is often desirable to dampen specific modes. The strategy has also been employed in~\cite{pu11_six_degree_of_freed_activ} for vibration isolation purposes using geophones, and in~\cite{yang19_dynam_model_decoup_contr_flexib} using force sensors. @@ -7840,10 +7867,10 @@ Finally, a comparative analysis with concluding observations is provided in Sect Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis. The system illustrated in Figure~\ref{fig:detail_control_decoupling_model_test} is used for this purpose. -It possesses three degrees of freedom (DoF) and incorporates three parallel struts. +It possesses three \acrshortpl{dof} and incorporates three parallel struts. Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform. -Two reference frames are defined within this model: frame \(\{M\}\) with origin \(O_M\) at the center of mass of the solid body, and frame \(\{K\}\) with origin \(O_K\) at the center of stiffness of the parallel manipulator. +Two reference frames are defined within this model: frame \(\{M\}\) with origin \(O_M\) at the \acrlong{com} of the solid body, and frame \(\{K\}\) with origin \(O_K\) at the \acrlong{cok} of the parallel manipulator. \begin{minipage}[b]{0.60\linewidth} \begin{center} @@ -7870,7 +7897,7 @@ Two reference frames are defined within this model: frame \(\{M\}\) with origin \captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters} \end{minipage} -The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass~\eqref{eq:detail_control_decoupling_model_eom}, where \(\bm{\mathcal{X}}_{\{M\}}\) represents the two translations and one rotation with respect to the center of mass, and \(\bm{\mathcal{F}}_{\{M\}}\) denotes the forces and torque applied at the center of mass. +The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass~\eqref{eq:detail_control_decoupling_model_eom}, where \(\bm{\mathcal{X}}_{\{M\}}\) represents the two translations and one rotation with respect to the \acrlong{com}, and \(\bm{\mathcal{F}}_{\{M\}}\) denotes the forces and torque applied at the \acrlong{com}. \begin{equation}\label{eq:detail_control_decoupling_model_eom} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t), \quad @@ -7981,10 +8008,10 @@ The transfer function from \(\bm{\mathcal{F}}_{\{O\}\) to \(\bm{\mathcal{X}}_{\{ \end{equation} The frame \(\{O\}\) can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice. -Two natural reference frames are particularly relevant: the center of mass and the center of stiffness. +Two natural reference frames are particularly relevant: the \acrlong{com} and the \acrlong{cok}. \paragraph{Center Of Mass} -When the decoupling frame is located at the center of mass (frame \(\{M\}\) in Figure~\ref{fig:detail_control_decoupling_model_test}), the Jacobian matrix and its inverse are expressed as in~\eqref{eq:detail_control_decoupling_jacobian_CoM_inverse}. +When the decoupling frame is located at the \acrlong{com} (frame \(\{M\}\) in Figure~\ref{fig:detail_control_decoupling_model_test}), the Jacobian matrix and its inverse are expressed as in~\eqref{eq:detail_control_decoupling_jacobian_CoM_inverse}. \begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse} \bm{J}_{\{M\}} = \begin{bmatrix} @@ -8018,7 +8045,7 @@ Consequently, the plant exhibits effective decoupling at frequencies above the h This strategy is typically employed in systems with low-frequency suspension modes~\cite{butler11_posit_contr_lithog_equip}, where the plant approximates decoupled mass lines. The low-frequency coupling observed in this configuration has a clear physical interpretation. -When a static force is applied at the center of mass, the suspended mass rotates around the center of stiffness. +When a static force is applied at the \acrlong{com}, the suspended mass rotates around the \acrlong{cok}. This rotation is due to torque induced by the stiffness of the first actuator (i.e. the one on the left side), which is not aligned with the force application point. This phenomenon is illustrated in Figure~\ref{fig:detail_control_decoupling_model_test_CoM}. @@ -8039,7 +8066,7 @@ This phenomenon is illustrated in Figure~\ref{fig:detail_control_decoupling_mode \end{figure} \paragraph{Center Of Stiffness} -When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in~\eqref{eq:detail_control_decoupling_jacobian_CoK_inverse}. +When the decoupling frame is located at the \acrlong{cok}, the Jacobian matrix and its inverse are expressed as in~\eqref{eq:detail_control_decoupling_jacobian_CoK_inverse}. \begin{equation}\label{eq:detail_control_decoupling_jacobian_CoK_inverse} \bm{J}_{\{K\}} = \begin{bmatrix} @@ -8055,7 +8082,7 @@ When the decoupling frame is located at the center of stiffness, the Jacobian ma The frame \(\{K\}\) was selected based on physical reasoning, positioned in line with the side strut and equidistant between the two vertical struts. However, it could alternatively be determined through analytical methods to ensure that \(\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}\) forms a diagonal matrix. -It should be noted that the existence of such a center of stiffness (i.e. a frame \(\{K\}\) for which \(\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}\) is diagonal) is not guaranteed for arbitrary systems. +It should be noted that the existence of such a \acrlong{cok} (i.e. a frame \(\{K\}\) for which \(\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}\) is diagonal) is not guaranteed for arbitrary systems. This property is typically achievable only in systems exhibiting specific symmetrical characteristics, as is the case in the present example. The analytical expression for the plant in this configuration was then computed~\eqref{eq:detail_control_decoupling_plant_CoK}. @@ -8064,7 +8091,7 @@ The analytical expression for the plant in this configuration was then computed~ \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1} \end{equation} -Figure~\ref{fig:detail_control_decoupling_jacobian_plant_CoK_results} presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the center of stiffness. +Figure~\ref{fig:detail_control_decoupling_jacobian_plant_CoK_results} presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the \acrlong{cok}. The plant is well decoupled below the suspension mode with the lowest frequency~\eqref{eq:detail_control_decoupling_plant_CoK_low_freq}, making it particularly suitable for systems with high stiffness. \begin{equation}\label{eq:detail_control_decoupling_plant_CoK_low_freq} @@ -8072,7 +8099,7 @@ The plant is well decoupled below the suspension mode with the lowest frequency~ \end{equation} The physical reason for high-frequency coupling is illustrated in Figure~\ref{fig:detail_control_decoupling_model_test_CoK}. -When a high-frequency force is applied at a point not aligned with the center of mass, it induces rotation around the center of mass. +When a high-frequency force is applied at a point not aligned with the \acrlong{com}, it induces rotation around the \acrlong{com}. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -8184,7 +8211,7 @@ where \(\bm{U} \in \mathbb{C}^{n \times n}\) and \(\bm{V} \in \mathbb{C}^{m \tim For real matrices \(\bm{X}\), the resulting \(\bm{U}\) and \(\bm{V}\) matrices are also real, making them suitable for decoupling applications. \paragraph{Decoupling using the SVD} -The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a Frequency Response Function (FRF), which yields a complex matrix \(\bm{G}(\omega_i)\) for multiple frequency points \(\omega_i\). +The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a \acrfull{frf}, which yields a complex matrix \(\bm{G}(\omega_i)\) for multiple frequency points \(\omega_i\). A specific frequency is then selected for optimal decoupling, with the targeted crossover frequency \(\omega_c\) often serving as an appropriate choice. Since real matrices are required for the decoupling transformation, a real approximation of the complex measured response at the selected frequency must be computed. @@ -8204,7 +8231,7 @@ These singular input and output matrices are then applied to decouple the system \caption{\label{fig:detail_control_decoupling_svd}Decoupled plant \(\bm{G}_{\text{SVD}}\) using the Singular Value Decomposition} \end{figure} -Implementation of SVD decoupling requires access to the system's FRF, at least in the vicinity of the desired decoupling frequency. +Implementation of SVD decoupling requires access to the system's \acrshort{frf}, at least in the vicinity of the desired decoupling frequency. This information can be obtained either experimentally or derived from a model. While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency. Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping. @@ -8277,7 +8304,7 @@ The phenomenon potentially relates to previous research on SVD controllers appli While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table~\ref{tab:detail_control_decoupling_strategies_comp}. -Each method employs a distinct conceptual framework: Jacobian decoupling is ``topology-driven'', relying on the geometric configuration of the system; modal decoupling is ``physics-driven'', based on the system's dynamical equations; and SVD decoupling is ``data-driven'', using measured frequency response functions. +Each method employs a distinct conceptual framework: Jacobian decoupling is ``topology-driven'', relying on the geometric configuration of the system; modal decoupling is ``physics-driven'', based on the system's dynamical equations; and SVD decoupling is ``data-driven'', using measured \acrshortpl{frf}. The physical interpretation of decoupled plant inputs and outputs varies considerably among these methods. With Jacobian decoupling, inputs and outputs retain clear physical meaning, corresponding to forces/torques and translations/rotations in a specified reference frame. @@ -8291,7 +8318,7 @@ Modal decoupling provides a natural framework when specific vibrational modes re SVD decoupling generally results in a loss of physical meaning for the ``control space'', potentially complicating the process of relating controller design to practical system requirements. The quality of decoupling achieved through these methods also exhibits distinct characteristics. -Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the center of stiffness, or at high frequencies when aligned with the center of mass. +Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the \acrlong{cok}, or at high frequencies when aligned with the \acrlong{com}. Systems designed with coincident centers of mass and stiffness may achieve excellent decoupling using this approach. Modal decoupling offers good decoupling across all frequencies, though its effectiveness relies on the model accuracy, with discrepancies potentially resulting in significant off-diagonal elements. SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point. @@ -8327,15 +8354,15 @@ SVD decoupling can be implemented using measured data without requiring a model, \subsection{Closed-Loop Shaping using Complementary Filters} \label{sec:detail_control_cf} -Once the system is properly decoupled using one of the approaches described in Section~\ref{sec:detail_control_decoupling}, SISO controllers can be individually tuned for each decoupled ``directions''. +Once the system is properly decoupled using one of the approaches described in Section~\ref{sec:detail_control_decoupling}, \acrshort{siso} controllers can be individually tuned for each decoupled ``directions''. Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented. -In some cases ``fixed'' controller structures are used, such as PI and PID controllers, whose parameters are manually tuned~\cite{furutani04_nanom_cuttin_machin_using_stewar,du14_piezo_actuat_high_precis_flexib,yang19_dynam_model_decoup_contr_flexib}. +In some cases ``fixed'' controller structures are used, such as \acrshort{pi} and \acrshort{pid} controllers, whose parameters are manually tuned~\cite{furutani04_nanom_cuttin_machin_using_stewar,du14_piezo_actuat_high_precis_flexib,yang19_dynam_model_decoup_contr_flexib}. Another popular method is Open-Loop shaping, which was used during the conceptual phase. Open-loop shaping involves tuning the controller through a series of ``standard'' filters (leads, lags, notches, low-pass filters, \ldots{}) to shape the open-loop transfer function \(G(s)K(s)\) according to desired specifications, including bandwidth, gain and phase margins~\cite[, chapt. 4.4.7]{schmidt20_desig_high_perfor_mechat_third_revis_edition}. Open-Loop shaping is very popular because the open-loop transfer function is a linear function of the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics. -Another key advantage is that controllers can be tuned directly from measured frequency response functions of the plant without requiring an explicit model. +Another key advantage is that controllers can be tuned directly from measured \acrshortpl{frf} of the plant without requiring an explicit model. However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions. Specifications can therefore be expressed in terms of the magnitude of closed-loop transfer functions, such as the sensitivity, plant sensitivity, and complementary sensitivity transfer functions~\cite[, chapt. 3]{skogestad07_multiv_feedb_contr}. @@ -8348,7 +8375,7 @@ This approach requires a good model of the plant and expertise in selecting weig In this section, an alternative controller synthesis scheme is proposed in which complementary filters are used for directly shaping the closed-loop transfer functions (i.e., directly addressing the closed-loop performances). In Section~\ref{ssec:detail_control_cf_control_arch}, the proposed control architecture is presented. In Section~\ref{ssec:detail_control_cf_trans_perf}, typical performance requirements are translated into the shape of the complementary filters. -The design of the complementary filters is briefly discussed in Section~\ref{ssec:detail_control_cf_analytical_complementary_filters}, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real time. +The design of the complementary filters is briefly discussed in Section~\ref{ssec:detail_control_cf_analytical_complementary_filters}, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real-time. Finally, in Section~\ref{ssec:detail_control_cf_simulations}, a numerical example is used to show how the proposed control architecture can be implemented in practice. \subsubsection{Control Architecture} \label{ssec:detail_control_cf_control_arch} @@ -8426,7 +8453,7 @@ Therefore, performance requirements must be translated into constraints on the s A closed-loop system is stable when all its elements (here \(K\), \(G^\prime\), and \(H_L\)) are stable and the sensitivity function \(S = \frac{1}{1 + G^\prime K H_L}\) is stable. For the nominal system (\(G^\prime = G\)), the sensitivity transfer function equals the high-pass filter: \(S(s) = H_H(s)\). -Nominal stability is therefore guaranteed when \(H_L\), \(H_H\), and \(G\) are stable, and both \(G\) and \(H_H\) are minimum phase (ensuring \(K\) is stable). +\acrfull{ns} is therefore guaranteed when \(H_L\), \(H_H\), and \(G\) are stable, and both \(G\) and \(H_H\) are minimum phase (ensuring \(K\) is stable). Consequently, stable and minimum phase complementary filters must be employed. \paragraph{Nominal Performance (NP)} @@ -8454,7 +8481,7 @@ Similarly, for noise attenuation, the magnitude of the complementary sensitivity Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function. Typically, maintaining \(|S|_{\infty} \le 2\) ensures a gain margin of at least 2 and a phase margin of at least \(\SI{29}{\degree}\). -Therefore, by carefully selecting the shape of the complementary filters, nominal performance specifications can be directly addressed in an intuitive manner. +Therefore, by carefully selecting the shape of the complementary filters, \acrfull{np} specifications can be directly addressed in an intuitive manner. \paragraph{Robust Stability (RS)} Robust stability refers to a control system's ability to maintain stability despite discrepancies between the actual system \(G^\prime\) and the model \(G\) used for controller design. @@ -8483,7 +8510,7 @@ The set of possible plants \(\Pi_i\) is described by~\eqref{eq:detail_control_cf \caption{\label{fig:detail_control_cf_input_uncertainty_nyquist}Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty})} \end{figure} -When considering input multiplicative uncertainty, robust stability can be derived graphically from the Nyquist plot (illustrated in Figure~\ref{fig:detail_control_cf_nyquist_uncertainty}), yielding to~\eqref{eq:detail_control_cf_robust_stability_graphically}, as demonstrated in~\cite[, chapt. 7.5.1]{skogestad07_multiv_feedb_contr}. +When considering input multiplicative uncertainty, \acrfull{rs} can be derived graphically from the Nyquist plot (illustrated in Figure~\ref{fig:detail_control_cf_nyquist_uncertainty}), yielding to~\eqref{eq:detail_control_cf_robust_stability_graphically}, as demonstrated in~\cite[, chapt. 7.5.1]{skogestad07_multiv_feedb_contr}. \begin{equation}\label{eq:detail_control_cf_robust_stability_graphically} \text{RS} \Longleftrightarrow \left|w_I(j\omega) L(j\omega) \right| \le \left| 1 + L(j\omega) \right| \quad \forall\omega @@ -8508,9 +8535,9 @@ Transforming this condition into constraints on the complementary filters yields \boxed{\text{RP} \Longleftrightarrow | w_H(j\omega) H_H(j\omega) | + | w_I(j\omega) H_L(j\omega) | \le 1, \ \forall\omega} \end{equation} -The robust performance condition effectively combines both nominal performance~\eqref{eq:detail_control_cf_nominal_performance} and robust stability conditions~\eqref{eq:detail_control_cf_condition_robust_stability}. +The \acrfull{rp} condition effectively combines both nominal performance~\eqref{eq:detail_control_cf_nominal_performance} and robust stability conditions~\eqref{eq:detail_control_cf_condition_robust_stability}. If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2~\cite[, chapt. 7.6]{skogestad07_multiv_feedb_contr}. -Therefore, for SISO systems, ensuring robust stability and nominal performance is typically sufficient. +Therefore, for \acrshort{siso} systems, ensuring robust stability and nominal performance is typically sufficient. \subsubsection{Complementary filter design} \label{ssec:detail_control_cf_analytical_complementary_filters} @@ -8542,7 +8569,7 @@ This real-time tunability allows rapid testing of different control bandwidths t \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/detail_control_cf_arch_tunable_params.png} -\caption{\label{fig:detail_control_cf_arch_tunable_params}Implemented digital complementary filters with parameter \(\omega_0\) that can be changed in real time} +\caption{\label{fig:detail_control_cf_arch_tunable_params}Implemented digital complementary filters with parameter \(\omega_0\) that can be changed in real-time} \end{figure} For many practical applications, first order complementary filters are not sufficient. @@ -8556,7 +8583,7 @@ For these cases, the complementary filters analytical formula in Equation~\eqref \end{align} \end{subequations} -The influence of parameters \(\alpha\) and \(\omega_0\) on the frequency response of these complementary filters is illustrated in Figure~\ref{fig:detail_control_cf_analytical_effect}. +The influence of parameters \(\alpha\) and \(\omega_0\) on the magnitude response of these complementary filters is illustrated in Figure~\ref{fig:detail_control_cf_analytical_effect}. The parameter \(\alpha\) primarily affects the damping characteristics near the crossover frequency as well as high and low frequency magnitudes, while \(\omega_0\) determines the frequency at which the transition between high-pass and low-pass behavior occurs. These filters can also be implemented in the digital domain with analytical formulas, preserving the ability to adjust \(\alpha\) and \(\omega_0\) in real-time. @@ -8672,7 +8699,7 @@ To ensure properness, low-pass filters with high corner frequencies are added as \end{equation} The Bode plot of the controller multiplied by the complementary low-pass filter, \(K(s) \cdot H_L(s)\), is presented in Figure~\ref{fig:detail_control_cf_bode_Kfb}. -The frequency response reveals several important characteristics: +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) @@ -8707,29 +8734,29 @@ In this section, a control architecture in which complementary filters are used This approach differs from traditional open-loop shaping in that no controller is manually designed; rather, appropriate complementary filters are selected to achieve the desired closed-loop behavior. The method shares conceptual similarities with mixed-sensitivity \(\mathcal{H}_{\infty}\text{-synthesis}\), as both approaches aim to shape closed-loop transfer functions, but with notable distinctions in implementation and complexity. -While \(\mathcal{H}_{\infty}\text{-synthesis}\) offers greater flexibility and can be readily generalized to MIMO plants, the presented approach provides a simpler alternative that requires minimal design effort. +While \(\mathcal{H}_{\infty}\text{-synthesis}\) offers greater flexibility and can be readily generalized to \acrshort{mimo} plants, the presented approach provides a simpler alternative that requires minimal design effort. Implementation only necessitates extracting a model of the plant and selecting appropriate analytical complementary filters, making it particularly interesting for applications where simplicity and intuitive parameter tuning are valued. Due to time constraints, an extensive literature review comparing this approach with similar existing architectures, such as Internal Model Control~\cite{saxena12_advan_inter_model_contr_techn}, was not conducted. Consequently, it remains unclear whether the proposed architecture offers significant advantages over existing methods in the literature. -The control architecture has been presented for SISO systems, but can be applied to MIMO systems when sufficient decoupling is achieved. +The control architecture has been presented for \acrshort{siso} systems, but can be applied to \acrshort{mimo} systems when sufficient decoupling is achieved. It will be experimentally validated with the NASS during the experimental phase. \subsection*{Conclusion} \label{sec:detail_control_conclusion} In order to optimize the control of the Nano Active Stabilization System, several aspects of control theory were studied. Different approaches to combine sensors were compared in Section~\ref{sec:detail_control_sensor}. -While High Authority Control-Low Authority Control (HAC-LAC) was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters. +While \acrfull{haclac} was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters. It was demonstrated that the performance of such fusion depends significantly on the magnitude of the complementary filters. To address this challenge, a synthesis method based on \(\mathcal{H}_\infty\text{-synthesis}\) was proposed, allowing for intuitive shaping of the complementary filters through weighting functions. -For the NASS, while HAC-LAC remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation. +For the NASS, while \acrshort{haclac} remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation. Various decoupling strategies for parallel manipulators were examined in Section~\ref{sec:detail_control_decoupling}, including decentralized control, Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. The main characteristics of each approach were highlighted, providing valuable insights into their respective strengths and limitations. Among the examined methods, Jacobian decoupling was determined to be most appropriate for the NASS, as it provides straightforward implementation while preserving the physical meaning of inputs and outputs. -With the system successfully decoupled, attention shifted to designing appropriate SISO controllers for each decoupled direction. +With the system successfully decoupled, attention shifted to designing appropriate \acrshort{siso} controllers for each decoupled direction. A control architecture for directly shaping closed-loop transfer functions was proposed. It is based on complementary filters that can be designed using either the proposed \(\mathcal{H}_\infty\text{-synthesis}\) approach described earlier or through analytical formulas. Experimental validation of this method on the NASS will be conducted during the experimental tests on ID31. @@ -8739,8 +8766,8 @@ This chapter presents an approach to select and validate appropriate instrumenta Figure~\ref{fig:detail_instrumentation_plant} illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis. The selection process follows a three-stage methodology. -First, dynamic error budgeting is performed in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting} to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier). -This analysis is based on the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements. +First, dynamic error budgeting is performed in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting} to establish maximum acceptable noise specifications for each instrumentation component (\acrshort{adc}, \acrshort{dac}, and voltage amplifier). +This analysis is based on the multi-body model with a 2DoF \acrshort{apa} model, focusing particularly on the vertical direction due to its more stringent requirements. From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived. Section~\ref{sec:detail_instrumentation_choice} then presents the selection of appropriate components based on these noise specifications and additional requirements. @@ -8752,23 +8779,23 @@ The measured noise characteristics are then incorporated into the multi-body mod \begin{figure}[htbp] \centering \includegraphics[scale=1,width=0.9\linewidth]{figs/detail_instrumentation_plant.png} -\caption{\label{fig:detail_instrumentation_plant}Block diagram of the NASS with considered instrumentation. The real time controller is a Speedgoat machine.} +\caption{\label{fig:detail_instrumentation_plant}Block diagram of the NASS with considered instrumentation. The real-time controller is a Speedgoat machine.} \end{figure} \subsection{Dynamic Error Budgeting} \label{sec:detail_instrumentation_dynamic_error_budgeting} -The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system. +The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (\acrshort{adc}, \acrshort{dac}, and voltage amplifier) that would result in acceptable vibration levels in the system. The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section~\ref{ssec:detail_instrumentation_cl_sensitivity}). This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs. Only the vertical direction is considered in this analysis as it presents the most stringent requirements; the horizontal directions are subject to less demanding constraints. -From these transfer functions, the maximum acceptable Amplitude Spectral Density (ASD) of the noise sources is derived (Section~\ref{ssec:detail_instrumentation_max_noise_specs}). -Since the voltage amplifier gain affects the amplification of DAC noise, an assumption of an amplifier gain of 20 was made. +From these transfer functions, the maximum acceptable \acrfull{asd} of the noise sources is derived (Section~\ref{ssec:detail_instrumentation_max_noise_specs}). +Since the voltage amplifier gain affects the amplification of \acrshort{dac} noise, an assumption of an amplifier gain of 20 was made. \subsubsection{Closed-Loop Sensitivity to Instrumentation Disturbances} \label{ssec:detail_instrumentation_cl_sensitivity} Several key noise sources are considered in the analysis (Figure~\ref{fig:detail_instrumentation_plant}). -These include the output voltage noise of the DAC (\(n_{da}\)), the output voltage noise of the voltage amplifier (\(n_{amp}\)), and the voltage noise of the ADC measuring the force sensor stacks (\(n_{ad}\)). +These include the output voltage noise of the \acrshort{dac} (\(n_{da}\)), the output voltage noise of the voltage amplifier (\(n_{amp}\)), and the voltage noise of the \acrshort{adc} measuring the force sensor stacks (\(n_{ad}\)). Encoder noise, which is only used to estimate \(R_z\), has been found to have minimal impact on the vertical sample error and is therefore omitted from this analysis for clarity. @@ -8785,9 +8812,9 @@ The transfer functions from these three noise sources (for one strut) to the ver The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of \(100\,\text{nm}\), which corresponds to a maximum allowed vibration of \(15\,\text{nm RMS}\). Several assumptions regarding the noise characteristics have been made. -The DAC, ADC, and amplifier noise are considered uncorrelated, which is a reasonable assumption. +The \acrshort{dac}, \acrshort{adc}, and amplifier noise are considered uncorrelated, which is a reasonable assumption. Similarly, the noise sources corresponding to each strut are also assumed to be uncorrelated. -This means that the power spectral densities (PSD) of the different noise sources are summed. +This means that the \acrfullpl{psd} of the different noise sources are summed. Since the effect of each strut on the vertical error is identical due to symmetry, only one strut is considered for this analysis, and the total effect of the six struts is calculated as six times the effect of one strut in terms of power, which translates to a factor of \(\sqrt{6} \approx 2.5\) for RMS values. @@ -8796,16 +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: DAC maximum output noise ASD is established at \(14\,\mu V/\sqrt{\text{Hz}}\), voltage amplifier maximum output voltage noise ASD at \(280\,\mu V/\sqrt{\text{Hz}}\), and ADC maximum measurement noise ASD at \(11\,\mu V/\sqrt{\text{Hz}}\). -In terms of RMS noise, these translate to less than \(1\,\text{mV RMS}\) for the DAC, less than \(20\,\text{mV RMS}\) for the voltage amplifier, and less than \(0.8\,\text{mV RMS}\) for the ADC. +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 \(11\,\mu V/\sqrt{\text{Hz}}\). +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 ADC, 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 15nm 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 DACs. +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}. \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. @@ -8859,7 +8886,7 @@ The specifications are summarized in Table~\ref{tab:detail_instrumentation_amp_c The most critical characteristics are the small signal bandwidth (\(>5\,\text{kHz}\)) and the output voltage noise (\(<20\,\text{mV RMS}\)). Several voltage amplifiers were considered, with their datasheet information presented in Table~\ref{tab:detail_instrumentation_amp_choice}. -One challenge encountered during the selection process was that manufacturers typically do not specify output noise as a function of frequency (i.e., the ASD of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies. +One challenge encountered during the selection process was that manufacturers typically do not specify output noise as a function of frequency (i.e., the \acrshort{asd} of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies. This approach does not account for the frequency dependency of the noise, which is crucial for accurate error budgeting. Additionally, the load conditions used to estimate bandwidth and noise specifications are often not explicitly stated. @@ -8896,30 +8923,30 @@ The proper selection of these components is critical for system performance. For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements~\cite{abramovitch22_pract_method_real_world_contr_system,abramovitch23_tutor_real_time_comput_issues_contr_system}. -Therefore, the ADC and DAC must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals. -Based on this requirement, priority was given to ADC and DAC components specifically marketed by Speedgoat to ensure optimal integration. +Therefore, the \acrshort{adc} and \acrshort{dac} must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals. +Based on this requirement, priority was given to \acrshort{adc} and \acrshort{dac} components specifically marketed by Speedgoat to ensure optimal integration. \paragraph{Sampling Frequency, Bandwidth and delays} 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. -Finally, \emph{delay} (or \emph{latency}) refers to the time interval between the analog signal at the input of the ADC and the digital information transferred to the control system. +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 ADCs can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency. +Sigma-Delta \acrshortpl{adc} can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency. Typically, the latency can reach 20 times the sampling period~\cite[, chapt. 8.4]{schmidt20_desig_high_perfor_mechat_third_revis_edition}. -Consequently, while Sigma-Delta ADCs are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor. +Consequently, while Sigma-Delta \acrshortpl{adc} are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor. -For real-time control applications, SAR-ADCs (Successive Approximation ADCs) remain the predominant choice due to their single-sample latency characteristics. +For real-time control applications, \acrfull{sar} remain the predominant choice due to their single-sample latency characteristics. \paragraph{ADC Noise} -Based on the dynamic error budget established in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting}, the measurement noise ASD should not exceed \(11\,\mu V/\sqrt{\text{Hz}}\). +Based on the dynamic error budget established in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting}, the measurement noise \acrshort{asd} should not exceed \(11\,\mu V/\sqrt{\text{Hz}}\). -ADCs are subject to various noise sources. +\acrshortpl{adc} are subject to various noise sources. Quantization noise, which results from the discrete nature of digital representation, is one of these sources. -To determine the minimum bit depth \(n\) required to meet the noise specifications, an ideal ADC where quantization error is the only noise source is considered. +To determine the minimum bit depth \(n\) required to meet the noise specifications, an ideal \acrshort{adc} where quantization error is the only noise source is considered. -The quantization step size, denoted as \(q = \Delta V/2^n\), represents the voltage equivalent of the least significant bit, with \(\Delta V\) the full range of the ADC in volts, and \(F_s\) the sampling frequency in Hertz. +The quantization step size, denoted as \(q = \Delta V/2^n\), represents the voltage equivalent of the \acrfull{lsb}, with \(\Delta V\) the full range of the \acrshort{adc} in volts, and \(F_s\) the sampling frequency in Hertz. The quantization noise ranges between \(\pm q/2\), and its probability density function is constant across this range (uniform distribution). Since the integral of this probability density function \(p(e)\) equals one, its value is \(1/q\) for \(-q/2 < e < q/2\), as illustrated in Figure~\ref{fig:detail_instrumentation_adc_quantization}. @@ -8940,7 +8967,7 @@ To compute the power spectral density of the quantization noise, which is define Under this assumption, the autocorrelation function approximates a delta function in the time domain. Since the Fourier transform of a delta function equals one, the power spectral density becomes frequency-independent (white noise). -By Parseval's theorem, the power spectral density of the quantization noise \(\Phi_q\) can be linked to the ADC sampling frequency and quantization step size~\eqref{eq:detail_instrumentation_psd_quant_noise}. +By Parseval's theorem, the power spectral density of the quantization noise \(\Phi_q\) can be linked to the \acrshort{adc} sampling frequency and quantization step size~\eqref{eq:detail_instrumentation_psd_quant_noise}. \begin{equation}\label{eq:detail_instrumentation_psd_quant_noise} \int_{-F_s/2}^{F_s/2} \Phi_q(f) d f = \int_{-q/2}^{q/2} e^2 p(e) de \quad \Longrightarrow \quad \Phi_q = \frac{q^2}{12 F_s} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 F_s} \quad \text{in } \left[ \frac{V^2}{\text{Hz}} \right] @@ -8952,17 +8979,17 @@ 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 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 ADCs. +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}. \paragraph{DAC Output voltage noise} -Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed \(14\,\mu V/\sqrt{\text{Hz}}\). -This specification corresponds to a \(\pm 10\,V\) 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 \(14\,\mu V/\sqrt{\text{Hz}}\). +This specification corresponds to a \(\pm 10\,V\) \acrshort{dac} with 13-bit resolution, which is easily attainable with current technology. \paragraph{Choice of the ADC and DAC Board} -Based on the preceding analysis, the selection of suitable ADC and DAC components is straightforward. +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, simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity. +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. Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements. @@ -8996,8 +9023,8 @@ These include optical encoders (Figure~\ref{fig:detail_instrumentation_sensor_en \caption{\label{fig:detail_instrumentation_sensor_examples}Relative motion sensors considered for measuring the nano-hexapod 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 APA, as illustrated in Figure~\ref{fig:detail_instrumentation_capacitive_implementation}. -In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure~\ref{fig:detail_instrumentation_encoder_implementation}. +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}. +In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the \acrshort{apa}, as shown in Figure~\ref{fig:detail_instrumentation_encoder_implementation}. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -9043,15 +9070,15 @@ Digital Output & \(\times\) & & \\ \subsubsection{Analog to Digital Converters} \paragraph{Measured Noise} -The measurement of 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 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 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. -All ADC channels demonstrated similar performance, so only one channel's noise profile is shown. +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. +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}. To gain \(w\) additional bits of resolution, the oversampling frequency \(f_{os}\) should be set to \(f_{os} = 4^w \cdot F_s\). -Given that the ADC can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4). -This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV)~\cite{hauser91_princ_overs_d_conver}. +Given that the \acrshort{adc} can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4). +This approach is effective because the noise approximates white noise and its amplitude exceeds 1 \acrshort{lsb} (0.3 mV)~\cite{hauser91_princ_overs_d_conver}. \begin{figure}[htbp] \centering @@ -9070,18 +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 DAC, and the ADC signal was recorded. +Step signals with an amplitude of \(1\,V\) 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. -These phenomena can be explained by examining the electrical schematic shown in Figure~\ref{fig:detail_instrumentation_force_sensor_adc}, where the ADC has an input impedance \(R_i\) and an input bias current \(i_n\). +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 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 ADC. +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 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. +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. \begin{figure}[htbp] \begin{subfigure}{0.61\textwidth} @@ -9099,8 +9126,8 @@ With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\mu F\ \caption{\label{fig:detail_instrumentation_force_sensor}Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from~\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}).} \end{figure} -The constant voltage offset can be explained by the input bias current \(i_n\) of the ADC, represented in Figure~\ref{fig:detail_instrumentation_force_sensor_adc}. -At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the ADC, and therefore the input bias current \(i_n\) passing through the internal resistance \(R_i\) produces a constant voltage offset \(V_{\text{off}} = R_i \cdot i_n\). +The constant voltage offset can be explained by the input bias current \(i_n\) of the \acrshort{adc}, represented in Figure~\ref{fig:detail_instrumentation_force_sensor_adc}. +At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the \acrshort{adc}, and therefore the input bias current \(i_n\) passing through the internal resistance \(R_i\) produces a constant voltage offset \(V_{\text{off}} = R_i \cdot i_n\). The input bias current \(i_n\) is estimated from \(i_n = V_{\text{off}}/R_i = 1.5\mu A\). In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor \(R_p\) can be added in parallel to the force sensor, as illustrated in Figure~\ref{fig:detail_instrumentation_force_sensor_adc_R}. @@ -9109,7 +9136,7 @@ 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 (\(\tau = 0.45\,s\)). -These results validate both the model of the ADC and the effectiveness of the added parallel resistor as a solution. +These results validate both the model of the \acrshort{adc} and the effectiveness of the added parallel resistor as a solution. \begin{figure}[htbp] \begin{subfigure}{0.61\textwidth} @@ -9128,16 +9155,16 @@ These results validate both the model of the ADC and the effectiveness of the ad \end{figure} \subsubsection{Instrumentation Amplifier} -Because the 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 ADC itself), a low noise instrumentation amplifier was employed. +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. 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 ADC (Figure~\ref{fig:detail_instrumentation_femto_meas_setup}). +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 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 ADC \(q_{ad}\) is rendered negligible due to the high gain employed. -The resulting amplifier noise amplitude spectral density \(\Gamma_{n_a}\) and the (negligible) contribution of the ADC noise are presented in Figure~\ref{fig:detail_instrumentation_femto_input_noise}. +In this configuration, the noise contribution from the \acrshort{adc} \(q_{ad}\) is rendered negligible due to the high gain employed. +The resulting amplifier noise amplitude spectral density \(\Gamma_{n_a}\) and the (negligible) contribution of the \acrshort{adc} noise are presented in Figure~\ref{fig:detail_instrumentation_femto_input_noise}. \begin{minipage}[b]{0.48\linewidth} \begin{center} @@ -9154,15 +9181,15 @@ The resulting amplifier noise amplitude spectral density \(\Gamma_{n_a}\) and th \end{minipage} \subsubsection{Digital to Analog Converters} \paragraph{Output Voltage Noise} -To measure the output noise of the DAC, the setup schematically represented in Figure~\ref{fig:detail_instrumentation_dac_setup} was used. -The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the ADC. +To measure the output noise of the \acrshort{dac}, the setup schematically represented in Figure~\ref{fig:detail_instrumentation_dac_setup} was used. +The \acrshort{dac} was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the \acrshort{adc}. -The Amplitude Spectral Density \(\Gamma_{n_{da}}(\omega)\) of the measured signal was computed, and verification was performed to confirm that the contributions of ADC noise and amplifier noise were negligible in the measurement. +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 ASD of \(0.6\,\mu V/\sqrt{\text{Hz}}\). +The noise profile is predominantly white with an \acrshort{asd} of \(0.6\,\mu V/\sqrt{\text{Hz}}\). 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 DAC channels demonstrated similar performance, so only one channel measurement is presented. +It should be noted that all \acrshort{dac} channels demonstrated similar performance, so only one channel measurement is presented. \begin{figure}[htbp] \centering @@ -9170,11 +9197,11 @@ It should be noted that all DAC channels demonstrated similar performance, so on \caption{\label{fig:detail_instrumentation_dac_setup}Measurement of the DAC output voltage noise. A pre-amplifier with a gain of 1000 is used before measuring the signal with the ADC.} \end{figure} \paragraph{Delay from ADC to DAC} -To measure the transfer function from DAC to ADC and verify that the bandwidth and latency of both instruments is sufficient, a direct connection was established between the DAC output and the ADC input. -A white noise signal was generated by the DAC, and the ADC response was recorded. +To measure the transfer function from \acrshort{dac} to \acrshort{adc} and verify that the bandwidth and latency of both instruments is sufficient, a direct connection was established between the \acrshort{dac} output and the \acrshort{adc} input. +A white noise signal was generated by the \acrshort{dac}, and the \acrshort{adc} response was recorded. -The resulting frequency response function from the digital DAC signal to the digital ADC signal is presented in Figure~\ref{fig:detail_instrumentation_dac_adc_tf}. -The observed frequency response function corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer. +The resulting \acrshort{frf} from the digital \acrshort{dac} signal to the digital \acrshort{adc} signal is presented in Figure~\ref{fig:detail_instrumentation_dac_adc_tf}. +The observed \acrshort{frf} corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -9195,7 +9222,7 @@ The observed frequency response function corresponds to exactly one sample delay \paragraph{Output Voltage Noise} The measurement setup for evaluating the PD200 amplifier noise is illustrated in Figure~\ref{fig:detail_instrumentation_pd200_setup}. The input of the PD200 amplifier was shunted with a \(50\,\Ohm\) resistor to ensure that only the inherent noise of the amplifier itself was measured. -The pre-amplifier gain was increased to produce a signal substantially larger than the noise floor of the ADC. +The pre-amplifier gain was increased to produce a signal substantially larger than the noise floor of the \acrshort{adc}. Two piezoelectric stacks from the APA95ML were connected to the PD200 output to provide an appropriate load for the amplifier. \begin{figure}[htbp] @@ -9204,7 +9231,7 @@ Two piezoelectric stacks from the APA95ML were connected to the PD200 output to \caption{\label{fig:detail_instrumentation_pd200_setup}Setup used to measured the output voltage noise of the PD200 voltage amplifier. A gain \(G_a = 1000\) was used for the instrumentation amplifier.} \end{figure} -The Amplitude Spectral Density \(\Gamma_{n}(\omega)\) of the signal measured by the ADC was computed. +The Amplitude Spectral Density \(\Gamma_{n}(\omega)\) of the signal measured by the \acrshort{adc} was computed. From this, the Amplitude Spectral Density of the output voltage noise of the PD200 amplifier \(n_p\) was derived, accounting for the gain of the pre-amplifier according to~\eqref{eq:detail_instrumentation_amp_asd}. \begin{equation}\label{eq:detail_instrumentation_amp_asd} @@ -9212,7 +9239,7 @@ From this, the Amplitude Spectral Density of the output voltage noise of the PD2 \end{equation} The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure~\ref{fig:detail_instrumentation_pd200_noise}. -Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or ADC noise. +Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or \acrshort{adc} noise. The measurements from all six amplifiers are displayed in this figure. The noise spectrum of the PD200 amplifiers exhibits several sharp peaks. @@ -9225,10 +9252,10 @@ While the exact cause of these peaks is not fully understood, their amplitudes r \end{figure} \paragraph{Small Signal Bandwidth} -The small signal dynamics of all six PD200 amplifiers were characterized through frequency response measurements. +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 DAC with an amplitude of \(0.1\,V\), 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 DAC) was measured with a separate ADC channel of the Speedgoat system. +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 \(5\,\text{kHz}\). +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. @@ -9267,7 +9294,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a \subsubsection{Noise budgeting from measured instrumentation noise} After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier. -The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure~\ref{fig:detail_instrumentation_cl_noise_budget}. +The vertical motion induced by the noise sources, specifically the \acrshort{adc} noise, \acrshort{dac} noise, and voltage amplifier noise, is presented in Figure~\ref{fig:detail_instrumentation_cl_noise_budget}. The total motion induced by all noise sources combined is approximately \(1.5\,\text{nm RMS}\), which remains well within the specified limit of \(15\,\text{nm RMS}\). This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application. @@ -9288,9 +9315,9 @@ The selection process revealed certain challenges, particularly with voltage amp Despite these challenges, suitable components were identified that theoretically met all requirements. 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 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. +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 ADC demonstrated noise levels of \(5.6\,\mu V/\sqrt{\text{Hz}}\) (versus the \(11\,\mu V/\sqrt{\text{Hz}}\) specification), the 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 \(5.6\,\mu V/\sqrt{\text{Hz}}\) (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). 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. @@ -9317,7 +9344,7 @@ Finally, considerations for ease of mounting, alignment, and maintenance were in The strut design, illustrated in Figure~\ref{fig:detail_design_strut}, was driven by several factors. Stiff interfaces were required between the amplified piezoelectric actuator and the two flexible joints, as well as between the flexible joints and their respective mounting plates. Due to the limited angular stroke of the flexible joints, it was critical that the struts could be assembled such that the two cylindrical interfaces were coaxial while the flexible joints remained in their unstressed, nominal rest position. -To facilitate this alignment, cylindrical washers (Figure~\ref{fig:detail_design_strut_without_enc}) were integrated into the design to compensate for potential deviations from perfect flatness between the two APA interface planes (Figure~\ref{fig:detail_design_apa}). +To facilitate this alignment, cylindrical washers (Figure~\ref{fig:detail_design_strut_without_enc}) were integrated into the design to compensate for potential deviations from perfect flatness between the two \acrshort{apa} interface planes (Figure~\ref{fig:detail_design_apa}). Furthermore, a dedicated mounting bench was developed to enable precise alignment of each strut, even when accounting for typical machining inaccuracies. The mounting procedure is described in Section~\ref{sec:test_struts_mounting}. Lastly, the design needed to permit the fixation of an encoder parallel to the strut axis, as shown in Figure~\ref{fig:detail_design_strut_with_enc}. @@ -9338,14 +9365,14 @@ Lastly, the design needed to permit the fixation of an encoder parallel to the s \caption{\label{fig:detail_design_strut}Design of the Nano-Hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration.} \end{figure} -The flexible joints, shown in Figure~\ref{fig:detail_design_flexible_joint}, were manufactured using wire-cut electrical discharge machining (EDM). +The flexible joints, shown in Figure~\ref{fig:detail_design_flexible_joint}, were manufactured using wire-cut \acrfull{edm}. First, the part's inherent fragility, stemming from its \(0.25\,\text{mm}\) neck dimension, makes it susceptible to damage from cutting forces typical in classical machining. -Furthermore, wire-cut EDM allows for the very tight machining tolerances critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure~\ref{fig:detail_design_flexible_joint}) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties. +Furthermore, wire-cut \acrshort{edm} allows for the very tight machining tolerances critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure~\ref{fig:detail_design_flexible_joint}) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties. The material chosen for the flexible joints is a stainless steel designated \emph{X5CrNiCuNb16-4} (alternatively known as F16Ph). This selection was based on its high specified yield strength (exceeding \(1\,\text{GPa}\) after appropriate heat treatment) and its high fatigue resistance. -As shown in Figure~\ref{fig:detail_design_flexible_joint}, the interface designed to connect with the APA possesses a cylindrical shape, facilitating the use of the aforementioned cylindrical washers for alignment. -A slotted hole was incorporated to permit alignment of the flexible joint with the APA via a dowel pin. +As shown in Figure~\ref{fig:detail_design_flexible_joint}, the interface designed to connect with the \acrshort{apa} possesses a cylindrical shape, facilitating the use of the aforementioned cylindrical washers for alignment. +A slotted hole was incorporated to permit alignment of the flexible joint with the \acrshort{apa} via a dowel pin. Additionally, two threaded holes were included on the sides for mounting the encoder components. The interface connecting the flexible joint to the platform plates will be described subsequently. @@ -9419,13 +9446,13 @@ 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 finite element analysis (FEA) of the complete active platform assembly was performed to identify modes that could potentially affect performance. +A \acrfull{fea} of the complete active platform 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. -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 APA actuation and their sensitivity to strut alignment. +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}. \begin{figure}[htbp] @@ -9479,7 +9506,7 @@ This geometric discrepancy implies that if the relative motion control of the ac \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. Two distinct configurations, corresponding to the two encoder mounting strategies discussed previously, were considered in the model, as displayed in Figure~\ref{fig:detail_design_simscape}: one with encoders fixed to the struts, and another with encoders fixed to the plates. -In these models, the top and bottom plates were represented as rigid bodies, with their inertial properties calculated directly from the 3D CAD geometry. +In these models, the top and bottom plates were represented as rigid bodies, with their inertial properties calculated directly from the 3D geometry. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -9499,9 +9526,9 @@ In these models, the top and bottom plates were represented as rigid bodies, wit \paragraph{Flexible Joints} Several levels of detail were considered for modeling the flexible joints within the multi-body model. -Models with two degrees of freedom incorporating only bending stiffnesses, models with three degrees of freedom adding torsional stiffness, and models with four degrees of freedom further adding axial stiffness were evaluated. +Models with two \acrshortpl{dof} incorporating only bending stiffnesses, models with three \acrshortpl{dof} adding torsional stiffness, and models with four \acrshortpl{dof} further adding axial stiffness were evaluated. The multi-body representation corresponding to the 4DoF configuration is shown in Figure~\ref{fig:detail_design_simscape_model_flexible_joint}. -This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from finite element analysis of the joint component. +This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from \acrshort{fea} of the joint component. \begin{figure}[htbp] \centering @@ -9510,8 +9537,8 @@ This model is composed of three distinct solid bodies interconnected by joints, \end{figure} \paragraph{Amplified Piezoelectric Actuators} -The amplified piezoelectric actuators (APAs) were incorporated into the multi-body model following the methodology detailed in Section~\ref{sec:detail_fem_actuator}. -Two distinct representations of the APA can be utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex ``Reduced Order Flexible Body'' model derived from a finite element model. +The \acrlongpl{apa} were incorporated into the multi-body model following the methodology detailed in Section~\ref{sec:detail_fem_actuator}. +Two distinct representations of the \acrshort{apa} can be utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex ``Reduced Order Flexible Body'' model derived from a \acrshort{fem}. \paragraph{Encoders} In earlier modeling stages, the relative displacement sensors (encoders) were implemented as a direct measurement of the relative distance between the joint connection points \(\bm{a}_i\) and \(\bm{b}_i\). @@ -9557,7 +9584,7 @@ The geometry optimization began with a review of existing Stewart platform desig While cubic architectures are prevalent in the literature due to their purported advantages in decoupling and uniform stiffness, the analysis revealed that these benefits are more nuanced than commonly described. For the nano-hexapod application, struts were oriented more vertically than in a cubic configuration to address the stringent vertical performance requirements and to better match the micro-station's modal characteristics. -For component optimization, a hybrid modeling methodology was used that combined finite element analysis with multi-body dynamics. +For component optimization, a hybrid modeling methodology was used that combined \acrshort{fea} with multi-body dynamics. This approach, validated experimentally using an Amplified Piezoelectric Actuator, enabled both detailed component-level optimization and efficient system-level simulation. Through this methodology, the APA300ML was selected as the optimal actuator, offering the necessary combination of stroke, stiffness, and force sensing capabilities required for the application. Similarly, the flexible joints were designed with careful consideration of bending and axial stiffness requirements, resulting in a design that balances competing mechanical demands. @@ -9582,24 +9609,24 @@ Following the completion of this design phase and the subsequent procurement of \subsubsection*{Abstract} The experimental validation follows a systematic approach, beginning with the characterization of individual components before advancing to evaluate the assembled system's performance (illustrated in Figure~\ref{fig:chapter3_overview}). Section~\ref{sec:test_apa} focuses on the Amplified Piezoelectric Actuator (APA300ML), examining its electrical properties, and dynamical behavior. -Two models are developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from finite element analysis. +Two models are developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from \acrshort{fea}. The implementation of Integral Force Feedback is also experimentally evaluated to assess its effectiveness in adding damping to the system. In Section~\ref{sec:test_joints}, the flexible joints are characterized to ensure they meet the required specifications for stiffness and stroke. A dedicated test bench is developed to measure the bending stiffness, with error analysis performed to validate the measurement accuracy. -Section~\ref{sec:test_struts} examines the assembly and testing of the struts, which integrate the APAs and flexible joints. +Section~\ref{sec:test_struts} examines the assembly and testing of the struts, which integrate the \acrshortpl{apa} and flexible joints. The mounting procedure is detailed, with particular attention to ensure consistent performance across multiple struts. Dynamical measurements are performed to verify whether the dynamics of the struts are corresponding to the multi-body model. The assembly and testing of the complete nano-hexapod is presented in Section~\ref{sec:test_nhexa}. A suspended table is developed to isolate the hexapod's dynamics from support dynamics, enabling accurate identification of its dynamical properties. -The experimental frequency response functions are compared with the multi-body model predictions to validate the modeling approach. +The experimental \acrshortpl{frf} are compared with the multi-body model predictions to validate the modeling approach. The effects of various payload masses are also investigated. Finally, Section~\ref{sec:test_id31} presents the validation of the NASS on the ID31 beamline. A short-stroke metrology system is developed to measure the sample position relative to the granite base. -The HAC-LAC control architecture is implemented and tested under various experimental conditions, including payload masses up to \(39\,\text{kg}\) and for typical experiments, including tomography scans, reflectivity measurements, and diffraction tomography. +The \acrshort{haclac} control architecture is implemented and tested under various experimental conditions, including payload masses up to \(39\,\text{kg}\) and for typical experiments, including tomography scans, reflectivity measurements, and diffraction tomography. \begin{figure}[htbp] \centering @@ -9611,17 +9638,17 @@ The HAC-LAC control architecture is implemented and tested under various experim In this chapter, the goal is to ensure that the received APA300ML (shown in Figure~\ref{fig:test_apa_received}) are complying with the requirements and that the dynamical models of the actuator accurately represent its dynamics. In section~\ref{sec:test_apa_basic_meas}, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks and the achievable stroke. -The flexible modes of the APA300ML, which were estimated using a finite element model, are compared with measurements. +The flexible modes of the APA300ML, which were estimated using a \acrshort{fem}, are compared with measurements. Using a dedicated test bench, dynamical measurements are performed (Section~\ref{sec:test_apa_dynamics}). -The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated. +The dynamics from the generated \acrshort{dac} voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated. Integral Force Feedback is experimentally applied, and the damped plants are estimated for several feedback gains. Two different models of the APA300ML are presented. First, in Section~\ref{sec:test_apa_model_2dof}, a two degrees-of-freedom model is presented, tuned, and compared with the measured dynamics. This model is proven to accurately represent the APA300ML's axial dynamics while having low complexity. -Then, in Section~\ref{sec:test_apa_model_flexible}, a \emph{super element} of the APA300ML is extracted using a finite element model and imported into the multi-body model. +Then, in Section~\ref{sec:test_apa_model_flexible}, a \emph{super element} of the APA300ML is extracted using a \acrshort{fem} and imported into the multi-body model. This more complex model also captures well capture the axial dynamics of the APA300ML. \begin{figure}[htbp] @@ -9636,12 +9663,12 @@ Before measuring the dynamical characteristics of the APA300ML, simple measureme First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section~\ref{ssec:test_apa_geometrical_measurements}. Then, the capacitance of the piezoelectric stacks is measured in Section~\ref{ssec:test_apa_electrical_measurements}. The achievable stroke of the APA300ML is measured using a displacement probe in Section~\ref{ssec:test_apa_stroke_measurements}. -Finally, in Section~\ref{ssec:test_apa_spurious_resonances}, the flexible modes of the APA are measured and compared with a finite element model. +Finally, in Section~\ref{ssec:test_apa_spurious_resonances}, the flexible modes of the \acrshort{apa} are measured and compared with a \acrshort{fem}. \subsubsection{Geometrical Measurements} \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 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 \(\pm 0.5\,\mu m\)} 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. @@ -9712,7 +9739,7 @@ APA 7 & 4.85 & 9.85\\ \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 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 \(1\,\mu m\)} 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 \(20\,V/V\)}. 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}). @@ -9723,7 +9750,7 @@ Note that the voltage is slowly varied as the displacement probe has a very low \caption{\label{fig:test_apa_stroke_bench}Bench to measure the APA stroke} \end{figure} -The measured APA displacement is shown as a function of the applied voltage in Figure~\ref{fig:test_apa_stroke_hysteresis}. +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). @@ -9752,8 +9779,8 @@ From now on, only the six remaining amplified piezoelectric actuators that behav \subsubsection{Flexible Mode Measurement} \label{ssec:test_apa_spurious_resonances} -In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. -To experimentally estimate these modes, the APA is fixed at one end (see Figure~\ref{fig:test_apa_meas_setup_modes}). +In this section, the flexible modes of the APA300ML are investigated both experimentally and using a \acrshort{fem}. +To experimentally estimate these modes, the \acrshort{apa} is fixed at one end (see Figure~\ref{fig:test_apa_meas_setup_modes}). A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes. Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated. @@ -9797,10 +9824,10 @@ The flexible modes for the same condition (i.e. one mechanical interface of the \caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure the flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).} \end{figure} -The measured frequency response functions computed from the experimental setups of figures~\ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure~\ref{fig:test_apa_meas_freq_compare}. +The measured \acrshortpl{frf} computed from the experimental setups of figures~\ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure~\ref{fig:test_apa_meas_freq_compare}. The \(y\) bending mode is observed at \(280\,\text{Hz}\) and the \(x\) bending mode is at \(412\,\text{Hz}\). -These modes are measured at higher frequencies than the frequencies estimated from the Finite Element Model (see frequencies in Figure~\ref{fig:test_apa_mode_shapes}). -This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model). +These modes are measured at higher frequencies than the frequencies estimated from the \acrshort{fem} (see frequencies in Figure~\ref{fig:test_apa_mode_shapes}). +This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a \acrshort{fem}). This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used). Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades. @@ -9811,10 +9838,10 @@ Another explanation is the shape difference between the manufactured APA300ML an \end{figure} \subsection{Dynamical measurements} \label{sec:test_apa_dynamics} -After the measurements on the 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. +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. 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 APA. +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}. \begin{figure}[htbp] \begin{subfigure}{0.3\textwidth} @@ -9834,7 +9861,7 @@ An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is used to measu The bench is schematically shown in Figure~\ref{fig:test_apa_schematic} with the associated signals. It will be first used to estimate the hysteresis from the piezoelectric stack (Section~\ref{ssec:test_apa_hysteresis}) as well as the axial stiffness of the APA300ML (Section~\ref{ssec:test_apa_stiffness}). -The frequency response functions from the DAC voltage \(u\) to the displacement \(d_e\) and to the voltage \(V_s\) are measured in Section~\ref{ssec:test_apa_meas_dynamics}. +The \acrshortpl{frf} from the \acrshort{dac} voltage \(u\) to the displacement \(d_e\) and to the voltage \(V_s\) are measured in Section~\ref{ssec:test_apa_meas_dynamics}. The presence of a non-minimum phase zero found on the transfer function from \(u\) to \(V_s\) is investigated in Section~\ref{ssec:test_apa_non_minimum_phase}. To limit the low-frequency gain of the transfer function from \(u\) to \(V_s\), a resistor is added across the force sensor stack (Section~\ref{ssec:test_apa_resistance_sensor_stack}). Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section~\ref{ssec:test_apa_iff_locus}. @@ -9847,10 +9874,10 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a \subsubsection{Hysteresis} \label{ssec:test_apa_hysteresis} -Because the payload is vertically guided without friction, the hysteresis of the 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 DAC. +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}. 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 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}. +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}. \begin{figure}[htbp] \centering @@ -9860,8 +9887,8 @@ This is the typical behavior expected from a PZT stack actuator, where the hyste \subsubsection{Axial stiffness} \label{ssec:test_apa_stiffness} -To estimate the stiffness of the 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 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. +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. \begin{equation} \label{eq:test_apa_stiffness} k_{\text{apa}} = \frac{m_a g}{\Delta d_e} @@ -9871,7 +9898,7 @@ The measured displacement \(d_e\) as a function of time is shown in Figure~\ref{ It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep), and that the displacement does not return to the initial position after the mass is removed (probably due to piezoelectric hysteresis). These two effects induce some uncertainties in the measured stiffness. -The stiffnesses are computed for all APAs 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 \(k_2\). +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 \(k = 1.8\,N/\mu m\). \begin{minipage}[b]{0.57\textwidth} @@ -9911,7 +9938,7 @@ It is important to note that changes to the electrical impedance connected to th To estimate this effect for the APA300ML, its stiffness is estimated using the ``static deflection'' method in two cases: \begin{itemize} -\item \(k_{\text{os}}\): piezoelectric stacks left unconnected (or connect to the high impedance ADC) +\item \(k_{\text{os}}\): piezoelectric stacks left unconnected (or connect to the high impedance \acrshort{adc}) \item \(k_{\text{sc}}\): piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance) \end{itemize} @@ -9922,12 +9949,12 @@ The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\ In this section, the dynamics from the excitation voltage \(u\) to the encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified. First, the dynamics from \(u\) to \(d_e\) for the six APA300ML are compared in Figure~\ref{fig:test_apa_frf_encoder}. -The obtained frequency response functions are similar to those of a (second order) mass-spring-damper system with: +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 \(\approx -17\,\mu m/V\). -The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA +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 APA support. +\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. The flexible modes studied in section~\ref{ssec:test_apa_spurious_resonances} seem not to impact the measured axial motion of the actuator. \end{itemize} @@ -9942,7 +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 finite element model). +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}). 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. @@ -9994,7 +10021,7 @@ However, this is not so important here because the zero is lightly damped (i.e. 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 \(\approx 5\,\mu F\)). -As explained before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain. +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. The (low frequency) transfer function from \(u\) to \(V_s\) with and without this resistor were measured and compared in Figure~\ref{fig:test_apa_effect_resistance}. It is confirmed that the added resistor has the effect of adding a high-pass filter with a cut-off frequency of \(\approx 0.39\,\text{Hz}\). @@ -10007,7 +10034,7 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil \subsubsection{Integral Force Feedback} \label{ssec:test_apa_iff_locus} -To implement the Integral Force Feedback strategy, the measured frequency response function from \(u\) to \(V_s\) (Figure~\ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation~\eqref{eq:test_apa_iff_manual_fit}. +To implement the Integral Force Feedback strategy, the measured \acrshort{frf} from \(u\) to \(V_s\) (Figure~\ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation~\eqref{eq:test_apa_iff_manual_fit}. The parameters were manually tuned, and the obtained values are \(\omega_{\textsc{hpf}} = 0.4\, \text{Hz}\), \(\omega_{z} = 42.7\, \text{Hz}\), \(\xi_{z} = 0.4\,\%\), \(\omega_{p} = 95.2\, \text{Hz}\), \(\xi_{p} = 2\,\%\) and \(g_0 = 0.64\). \begin{equation} \label{eq:test_apa_iff_manual_fit} @@ -10029,7 +10056,7 @@ It contains a high-pass filter (cut-off frequency of \(2\,\text{Hz}\)) to limit K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000} \end{equation} -To estimate how the dynamics of the APA changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure~\ref{fig:test_apa_iff_schematic} is used. +To estimate how the dynamics of the \acrshort{apa} changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure~\ref{fig:test_apa_iff_schematic} is used. The transfer function from the ``damped'' plant input \(u\prime\) to the encoder displacement \(d_e\) is identified for several IFF controller gains \(g\). \begin{figure}[htbp] @@ -10065,7 +10092,7 @@ The two obtained root loci are compared in Figure~\ref{fig:test_apa_iff_root_loc \subsection{APA300ML - 2 degrees-of-freedom Model} \label{sec:test_apa_model_2dof} -In this section, a multi-body model (Figure~\ref{fig:test_apa_bench_model}) of the measurement bench is used to tune the two degrees-of-freedom model of the APA using the measured frequency response functions. +In this section, a multi-body model (Figure~\ref{fig:test_apa_bench_model}) of the measurement bench is used to tune the two degrees-of-freedom model of the \acrshort{apa} using the measured \acrshortpl{frf}. This two degrees-of-freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and a low number of associated states. After the model is presented, the procedure for tuning the model is described, and the obtained model dynamics is compared with the measurements. @@ -10090,7 +10117,7 @@ A sensor measures the stack strain \(d_e\) which is then converted to a voltage Such a simple model has some limitations: \begin{itemize} -\item it only represents the axial characteristics of the APA as it is modeled as infinitely rigid in the other directions +\item it only represents the axial characteristics of the \acrshort{apa} as it is modeled as infinitely rigid in the other directions \item some physical insights are lost, such as the amplification factor and the real stress and strain in the piezoelectric stacks \item the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear \end{itemize} @@ -10098,7 +10125,7 @@ Such a simple model has some limitations: \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/test_apa_2dof_model.png} -\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees-of-freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}} +\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees-of-freedom model of the APA300ML, adapted from~\cite{souleille18_concep_activ_mount_space_applic}} \end{figure} 9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure~\ref{fig:test_apa_2dof_model_simscape}) well represents the identified dynamics in Section~\ref{sec:test_apa_dynamics}. @@ -10116,7 +10143,7 @@ Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-reso 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. -In this case, the total stiffness of the APA model is described by~\eqref{eq:test_apa_2dof_stiffness}. +In this case, the total stiffness of the \acrshort{apa} model is described by~\eqref{eq:test_apa_2dof_stiffness}. \begin{equation}\label{eq:test_apa_2dof_stiffness} k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a @@ -10180,7 +10207,7 @@ This indicates that this model represents well the axial dynamics of the APA300M In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}. It is then imported into multi-body (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in~\ref{sec:test_apa_model_2dof}. This procedure is illustrated in Figure~\ref{fig:test_apa_super_element_simscape}. -Several \emph{remote points} are defined in the finite element model (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then made accessible in the multi-body software as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}. +Several \emph{remote points} are defined in the \acrshort{fem} (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then made accessible in the multi-body software as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}. For the APA300ML \emph{super element}, 5 \emph{remote points} are defined. Two \emph{remote points} (\texttt{1} and \texttt{2}) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used to connect the APA300ML with other mechanical elements. @@ -10234,9 +10261,9 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine \end{table} \paragraph{Comparison of the obtained dynamics} -The obtained dynamics using the \emph{super element} with the tuned ``sensor sensitivity'' and ``actuator sensitivity'' are compared with the experimentally identified frequency response functions in Figure~\ref{fig:test_apa_super_element_comp_frf}. +The obtained dynamics using the \emph{super element} with the tuned ``sensor sensitivity'' and ``actuator sensitivity'' are compared with the experimentally identified \acrshortpl{frf} in Figure~\ref{fig:test_apa_super_element_comp_frf}. A good match between the model and the experimental results was observed. -It is however surprising that the model is ``softer'' than the measured system, as finite element models usually overestimate the stiffness (see Section~\ref{ssec:test_apa_spurious_resonances} for possible explanations). +It is however surprising that the model is ``softer'' than the measured system, as \acrshortpl{fem} usually overestimate the stiffness (see Section~\ref{ssec:test_apa_spurious_resonances} for possible explanations). Using this simple test bench, it can be concluded that the \emph{super element} model of the APA300ML captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever). @@ -10271,7 +10298,7 @@ In Section~\ref{sec:test_apa_model_2dof}, a simple two degrees-of-freedom mass-s After following a tuning procedure, the model dynamics was shown to match very well with the experiment. However, this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions. -In Section~\ref{sec:test_apa_model_flexible}, a \emph{super element} extracted from a finite element model was used to model the APA300ML. +In Section~\ref{sec:test_apa_model_flexible}, a \emph{super element} extracted from a \acrshort{fem} was used to model the APA300ML. Here, the \emph{super element} represents the dynamics of the APA300ML in all directions. However, only the axial dynamics could be compared with the experimental results, yielding a good match. The benefit of employing this model over the two degrees-of-freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved. @@ -10300,7 +10327,7 @@ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\ \end{tabularx} \end{table} -After optimization using a finite element model, the geometry shown in Figure~\ref{fig:test_joints_schematic} has been obtained and the corresponding flexible joint characteristics are summarized in Table~\ref{tab:test_joints_specs}. +After optimization using a \acrshort{fem}, the geometry shown in Figure~\ref{fig:test_joints_schematic} has been obtained and the corresponding flexible joint characteristics are summarized in Table~\ref{tab:test_joints_specs}. This flexible joint is a monolithic piece of stainless steel\footnote{The alloy used is called \emph{F16PH}, also refereed as ``1.4542''} manufactured using wire electrical discharge machining. It serves several functions, as shown in Figure~\ref{fig:test_joints_iso}, such as: \begin{itemize} @@ -10529,7 +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 finite element model 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 \(k_s = 13\,N/\mu m\) from the \acrshort{fem} and an height \(h=25\,mm\), the estimation errors of the bending stiffness due to shear is \(\epsilon_s < 0.1\,\%\) \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. @@ -10596,7 +10623,7 @@ To do so, an encoder\footnote{Resolute\texttrademark{} encoder with \(1\,nm\) re 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. -The obtained CAD design of the measurement bench is shown in Figure~\ref{fig:test_joints_bench_overview} while a zoom on the flexible joint with the associated important quantities is shown in Figure~\ref{fig:test_joints_bench_side}. +The obtained design of the measurement bench is shown in Figure~\ref{fig:test_joints_bench_overview} while a zoom on the flexible joint with the associated important quantities is shown in Figure~\ref{fig:test_joints_bench_side}. \begin{figure}[htbp] \begin{subfigure}{0.78\textwidth} @@ -10611,7 +10638,7 @@ The obtained CAD design of the measurement bench is shown in Figure~\ref{fig:tes \end{center} \subcaption{\label{fig:test_joints_bench_side} Zoom} \end{subfigure} -\caption{\label{fig:test_joints_bench}CAD view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side})} +\caption{\label{fig:test_joints_bench}3D view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side})} \end{figure} \subsection{Bending Stiffness Measurement} \label{sec:test_joints_bending_stiffness_meas} @@ -10737,7 +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 finite element model (\(k_{R_x} = k_{R_y} = 5\,Nm/\text{rad}\)). +The measured bending stiffness values exhibited good agreement with the predictions from the \acrshort{fem} (\(k_{R_x} = k_{R_y} = 5\,Nm/\text{rad}\)). 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} @@ -10755,7 +10782,7 @@ The mounting procedure of the struts is explained in Section~\ref{sec:test_strut A mounting bench was used to ensure coaxiality between the two ends of the struts. In this way, no angular stroke is lost when mounted to the nano-hexapod. -The flexible modes of the struts were then experimentally measured and compared with a finite element model (Section~\ref{sec:test_struts_flexible_modes}). +The flexible modes of the struts were then experimentally measured and compared with a \acrshort{fem} (Section~\ref{sec:test_struts_flexible_modes}). Dynamic measurements of the strut are performed with the same test bench used to characterize the APA300ML dynamics (Section~\ref{sec:test_struts_dynamical_meas}). It was found that the dynamics from the \acrshort{dac} voltage to the displacement measured by the encoder is complex due to the flexible modes of the struts (Section~\ref{sec:test_struts_flexible_modes}). @@ -10772,11 +10799,11 @@ A mounting bench was developed to ensure: \item Good coaxial alignment between the interfaces (cylinders) of the flexible joints. This is important not to loose to much angular stroke during their mounting into the nano-hexapod \item Uniform length across all struts -\item Precise alignment of the APA with the two flexible joints +\item Precise alignment of the \acrshort{apa} with the two flexible joints \item Reproducible and consistent assembly between all struts \end{itemize} -A CAD view of the mounting bench is shown in Figure~\ref{fig:test_struts_mounting_bench_first_concept}. +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. @@ -10787,7 +10814,7 @@ The strut length (defined by the distance between the rotation points of the two \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_bench_first_concept.png} \end{center} -\subcaption{\label{fig:test_struts_mounting_bench_first_concept}CAD view of the mounting bench} +\subcaption{\label{fig:test_struts_mounting_bench_first_concept}3D view of the mounting bench} \end{subfigure} \begin{subfigure}{0.49\textwidth} \begin{center} @@ -10845,7 +10872,7 @@ The left sleeve has a thigh fit such that its orientation is fixed (it is roughl The cylindrical washers and the APA300ML are stacked on top of the flexible joints, as shown in Figure~\ref{fig:test_struts_mounting_step_2} and screwed together using a torque screwdriver. A dowel pin is used to laterally align the APA300ML with the flexible joints (see the dowel slot on the flexible joints in Figure~\ref{fig:test_struts_mounting_joints}). -Two cylindrical washers are used to allow proper mounting even when the two APA interfaces are not parallel. +Two cylindrical washers are used to allow proper mounting even when the two \acrshort{apa} interfaces are not parallel. The encoder and ruler are then fixed to the strut and properly aligned, as shown in Figure~\ref{fig:test_struts_mounting_step_3}. @@ -10941,7 +10968,7 @@ These tests were performed with and without the encoder being fixed to the strut \caption{\label{fig:test_struts_meas_modes}Measurement of strut flexible modes} \end{figure} -The obtained frequency response functions for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure~\ref{fig:test_struts_spur_res_frf_no_enc} when the encoder is not fixed to the strut and in Figure~\ref{fig:test_struts_spur_res_frf_enc} when the encoder is fixed to the strut. +The obtained \acrshortpl{frf} for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure~\ref{fig:test_struts_spur_res_frf_no_enc} when the encoder is not fixed to the strut and in Figure~\ref{fig:test_struts_spur_res_frf_enc} when the encoder is fixed to the strut. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -11025,7 +11052,7 @@ System identification was performed without the encoder being fixed to the strut \caption{\label{fig:test_struts_bench_leg_with_without_enc}Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front})} \end{figure} -The obtained frequency response functions are compared in Figure~\ref{fig:test_struts_effect_encoder}. +The obtained \acrshortpl{frf} are compared in Figure~\ref{fig:test_struts_effect_encoder}. It was found that the encoder had very little effect on the transfer function from excitation voltage \(u\) to the axial motion of the strut \(d_a\) as measured by the interferometer (Figure~\ref{fig:test_struts_effect_encoder_int}). This means that the axial motion of the strut is unaffected by the presence of the encoder. Similarly, it has little effect on the transfer function from \(u\) to the sensor stack voltage \(V_s\) (Figure~\ref{fig:test_struts_effect_encoder_iff}). @@ -11099,9 +11126,9 @@ The reason for this variability will be studied in the next section thanks to th \subsection{Strut Model} \label{sec:test_struts_simscape} The multi-body model of the strut was included in the multi-body model of the test bench (see Figure~\ref{fig:test_struts_simscape_model}). -The obtained model was first used to compare the measured FRF with the existing model (Section~\ref{ssec:test_struts_comp_model}). +The obtained model was first used to compare the measured \acrshort{frf} with the existing model (Section~\ref{ssec:test_struts_comp_model}). -Using a flexible APA model (extracted from a \acrshort{fem}), the effect of a misalignment of the APA with respect to flexible joints is studied (Section~\ref{ssec:test_struts_effect_misalignment}). +Using a flexible \acrshort{apa} model (extracted from a \acrshort{fem}), the effect of a misalignment of the \acrshort{apa} with respect to flexible joints is studied (Section~\ref{ssec:test_struts_effect_misalignment}). It was found that misalignment has a large impact on the dynamics from \(u\) to \(d_e\). This misalignment is estimated and measured in Section~\ref{ssec:test_struts_meas_misalignment}. The struts were then disassembled and reassemble a second time to optimize alignment (Section~\ref{sec:test_struts_meas_all_aligned_struts}). @@ -11115,9 +11142,9 @@ The struts were then disassembled and reassemble a second time to optimize align \label{ssec:test_struts_comp_model} Two models of the APA300ML are used here: a simple two-degrees-of-freedom model and a model using a super-element extracted from a \acrlong{fem}. -These two models of the APA300ML were tuned to best match the measured frequency response functions of the APA alone. +These two models of the APA300ML were tuned to best match the measured \acrshortpl{frf} of the \acrshort{apa} alone. The flexible joints were modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness). -These two models are compared with the measured frequency responses in Figure~\ref{fig:test_struts_comp_frf_flexible_model}. +These two models are compared using the measured \acrshortpl{frf} in Figure~\ref{fig:test_struts_comp_frf_flexible_model}. The model dynamics from DAC voltage \(u\) to the axial motion of the strut \(d_a\) (Figure~\ref{fig:test_struts_comp_frf_flexible_model_int}) and from DAC voltage \(u\) to the force sensor voltage \(V_s\) (Figure~\ref{fig:test_struts_comp_frf_flexible_model_iff}) are well matching the experimental identification. @@ -11150,7 +11177,7 @@ For the flexible model, it will be shown in the next section that by adding some \label{ssec:test_struts_effect_misalignment} 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 APA) can explain such dynamics. +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. @@ -11160,9 +11187,9 @@ In this case, the ``x-bending'' mode at 200Hz (see Figure~\ref{fig:test_struts_m \caption{\label{fig:test_struts_misalign_schematic}Mis-alignement between the joints and the APA} \end{figure} -To verify this assumption, the dynamics from the output DAC voltage \(u\) to the measured displacement by the encoder \(d_e\) is computed using the flexible APA model for several misalignments in the \(y\) direction. +To verify this assumption, the dynamics from the output DAC voltage \(u\) to the measured displacement by the encoder \(d_e\) is computed using the flexible \acrshort{apa} model for several misalignments in the \(y\) direction. The obtained dynamics are shown in Figure~\ref{fig:test_struts_effect_misalignment_y}. -The alignment of the APA with the flexible joints has a large influence on the dynamics from actuator voltage to the measured displacement by the encoder. +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}) @@ -11177,7 +11204,7 @@ The misalignment in the \(y\) direction mostly influences: 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}). -A comparison of the experimental frequency response functions 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. +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. \begin{figure}[htbp] @@ -11198,19 +11225,19 @@ This similarity suggests that the identified differences in dynamics are caused \subsubsection{Measured strut misalignment} \label{ssec:test_struts_meas_misalignment} -During the initial mounting of the struts, as presented in Section~\ref{sec:test_struts_mounting}, the positioning pins that were used to position the APA with respect to the flexible joints in the \(y\) directions were not used (not received at the time). +During the initial mounting of the struts, as presented in Section~\ref{sec:test_struts_mounting}, the positioning pins that were used to position the \acrshort{apa} with respect to the flexible joints in the \(y\) directions were not used (not received at the time). Therefore, large \(y\) misalignments are expected. -To estimate the misalignments between the two flexible joints and the APA: +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 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 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 \(\pm 0.5\,\mu m\)}, 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 \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 APA). +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}. \begin{table}[htbp] @@ -11229,13 +11256,13 @@ Thickness differences for all the struts were found to be between \(0.94\,mm\) a \end{tabularx} \end{table} -By using the measured \(y\) misalignment in the model with the flexible APA model, the model dynamics from \(u\) to \(d_e\) is closer to the measured dynamics, as shown in Figure~\ref{fig:test_struts_comp_dy_tuned_model_frf_enc}. +By using the measured \(y\) misalignment in the model with the flexible \acrshort{apa} model, the model dynamics from \(u\) to \(d_e\) is closer to the measured dynamics, as shown in Figure~\ref{fig:test_struts_comp_dy_tuned_model_frf_enc}. A better match in the dynamics can be obtained by fine-tuning both the \(x\) and \(y\) misalignments (yellow curves in Figure~\ref{fig:test_struts_comp_dy_tuned_model_frf_enc}). -This confirms that misalignment between the APA and the strut axis (determined by the two flexible joints) is critical and inducing large variations in the dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\). -If encoders are fixed to the struts, the APA and flexible joints must be precisely aligned when mounting the struts. +This confirms that misalignment between the \acrshort{apa} and the strut axis (determined by the two flexible joints) is critical and inducing large variations in the dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\). +If encoders are fixed to the struts, the \acrshort{apa} and flexible joints must be precisely aligned when mounting the struts. -In the next section, the struts are re-assembled with a ``positioning pin'' to better align the APA with the flexible joints. +In the next section, the struts are re-assembled with a ``positioning pin'' to better align the \acrshort{apa} with the flexible joints. With a better alignment, the amplitude of the spurious resonances is expected to decrease, as shown in Figure~\ref{fig:test_struts_effect_misalignment_y}. \begin{figure}[htbp] @@ -11247,7 +11274,7 @@ With a better alignment, the amplitude of the spurious resonances is expected to \label{sec:test_struts_meas_all_aligned_struts} After receiving the positioning pins, the struts were mounted again with the positioning pins. -This should improve the alignment of the APA with the two flexible joints. +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}). @@ -11423,7 +11450,7 @@ Finally, some interface elements were designed, and mechanical lateral mechanica \begin{figure}[htbp] \centering \includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_suspended_table_cad.jpg} -\caption{\label{fig:test_nhexa_suspended_table_cad}CAD View of the vibration table. The purple cylinders are representing the soft springs.} +\caption{\label{fig:test_nhexa_suspended_table_cad}3D View of the vibration table. The purple cylinders are representing the soft springs.} \end{figure} \subsubsection{Modal analysis of the suspended table} \label{ssec:test_nhexa_table_identification} @@ -11509,7 +11536,7 @@ Experimental & 1.3 Hz & 2.0 Hz & 6.9 Hz & 9.5 Hz\\ \label{sec:test_nhexa_dynamics} The Nano-Hexapod was then mounted on top of the suspended table, as shown in Figure~\ref{fig:test_nhexa_hexa_suspended_table}. -All instrumentation (Speedgoat with ADC, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) were configured and connected to the nano-hexapod using many cables. +All instrumentation (Speedgoat with \acrshort{adc}, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) were configured and connected to the nano-hexapod using many cables. \begin{figure}[htbp] \centering @@ -11521,7 +11548,7 @@ A modal analysis of the nano-hexapod is first performed in Section~\ref{ssec:tes The results of the modal analysis will be useful to better understand the measured dynamics from actuators to sensors. A block diagram of the (open-loop) system is shown in Figure~\ref{fig:test_nhexa_nano_hexapod_signals}. -The frequency response functions from controlled signals \(\bm{u}\) to the force sensors voltages \(\bm{V}_s\) and to the encoders measured displacements \(\bm{d}_e\) are experimentally identified in Section~\ref{ssec:test_nhexa_identification}. +The \acrshortpl{frf} from controlled signals \(\bm{u}\) to the force sensors voltages \(\bm{V}_s\) and to the encoders measured displacements \(\bm{d}_e\) are experimentally identified in Section~\ref{ssec:test_nhexa_identification}. The effect of the payload mass on the dynamics is discussed in Section~\ref{ssec:test_nhexa_added_mass}. \begin{figure}[htbp] @@ -11584,7 +11611,7 @@ These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}. The dynamics of the nano-hexapod from the six command signals (\(u_1\) to \(u_6\)) to the six measured displacement by the encoders (\(d_{e1}\) to \(d_{e6}\)) and to the six force sensors (\(V_{s1}\) to \(V_{s6}\)) were identified by generating low-pass filtered white noise for each command signal, one by one. -The \(6 \times 6\) FRF matrix from \(\bm{u}\) ot \(\bm{d}_e\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_de}. +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. @@ -11606,7 +11633,7 @@ This would not have occurred if the encoders were fixed to the struts. \caption{\label{fig:test_nhexa_identified_frf_de}Measured FRF for the transfer function from \(\bm{u}\) to \(\bm{d}_e\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.} \end{figure} -Similarly, the \(6 \times 6\) FRF matrix from \(\bm{u}\) to \(\bm{V}_s\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_Vs}. +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. 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. @@ -11630,7 +11657,7 @@ These three cylindrical masses on top of the nano-hexapod are shown in Figure~\r \caption{\label{fig:test_nhexa_table_mass_3}Picture of the nano-hexapod with the added three cylindrical masses for a total of \(\approx 40\,kg\)} \end{figure} -The obtained frequency response functions 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, 13kg, 26kg and 39kg) 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. @@ -11642,8 +11669,8 @@ 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 measured FRFs from \(u_i\) to \(V_{si}\) are shown in Figure~\ref{fig:test_nhexa_identified_frf_Vs_masses}. -For all tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust manner. +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. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -11680,15 +11707,15 @@ This is checked in Section~\ref{ssec:test_nhexa_comp_model_masses}. \subsubsection{Nano-Hexapod model dynamics} \label{ssec:test_nhexa_comp_model} -The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF APA, and rigid top and bottom plates. -The stiffness values of the flexible joints were chosen based on the values estimated using the test bench and on the FEM. -The parameters of the APA model were determined from the test bench of the APA. +The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF \acrshort{apa}, and rigid top and bottom plates. +The stiffness values of the flexible joints were chosen based on the values estimated using the test bench and on the \acrshort{fem}. +The parameters of the \acrshort{apa} model were determined from the test bench of the \acrshort{apa}. The \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(\bm{d}_e\) and from \(\bm{u}\) to \(\bm{V}_s\) are then extracted from the multi-body model. -First, is it evaluated how well the models matches the ``direct'' terms of the measured FRF matrix. -To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure~\ref{fig:test_nhexa_comp_simscape_diag}. +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 APAs are here modeled as a simple uniaxial 2-DoF system. +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. 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] @@ -11709,8 +11736,8 @@ At higher frequencies, no resonances can be observed in the model, as the top pl \subsubsection{Dynamical coupling} \label{ssec:test_nhexa_comp_model_coupling} -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 MIMO systems. -Instead of comparing the full 36 elements of the \(6 \times 6\) 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 \(d_{e6}\). +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. Similar results are observed for all other coupling terms and for the transfer function from \(\bm{u}\) to \(\bm{V}_s\). @@ -11721,7 +11748,7 @@ Similar results are observed for all other coupling terms and for the transfer f \end{figure} 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 FRF in Figure~\ref{fig:test_nhexa_comp_simscape_de_all_flex}. +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. 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. @@ -11781,17 +11808,17 @@ Below the first suspension mode, good decoupling could be observed for the trans Many other modes were present above 700Hz, 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 frequency response functions 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}). +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}). This indicates that it is possible to implement decentralized Integral Force Feedback in a robust manner. The developed multi-body model of the nano-hexapod was found to accurately represents the suspension modes of the Nano-Hexapod (Section~\ref{sec:test_nhexa_model}). -Both FRF matrices from \(\bm{u}\) to \(\bm{V}_s\) and from \(\bm{u}\) to \(\bm{d}_e\) are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective. +Both \acrshort{frf} matrices from \(\bm{u}\) to \(\bm{V}_s\) and from \(\bm{u}\) to \(\bm{d}_e\) are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective. At frequencies above the suspension modes, the Nano-Hexapod model became inaccurate because the flexible modes were not modeled. It was found that modeling the APA300ML using a \emph{super-element} allows to model the internal resonances of the struts. The same can be done with the top platform and the encoder supports; however, the model order would be higher and may become unpractical for simulation. Obtaining a model that accurately represents the complex dynamics of the Nano-Hexapod was made possible by the modeling approach used in this study. -This approach involved tuning and validating models of individual components (such as the APA and flexible joints) using dedicated test benches. +This approach involved tuning and validating models of individual components (such as the \acrshort{apa} and flexible joints) using dedicated test benches. The different models could then be combined to form the Nano-Hexapod dynamical model. If a model of the nano-hexapod was developed in one time, it would be difficult to tune all the model parameters to match the measured dynamics, or even to know if the model ``structure'' would be adequate to represent the system dynamics. \section{Nano Active Stabilization System} @@ -11807,7 +11834,7 @@ 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. These measurements were compared with predictions from the multi-body model to verify its accuracy and applicability to control design. -Sections~\ref{sec:test_id31_iff} and \ref{sec:test_id31_hac} focus on the implementation and validation of the HAC-LAC control architecture. +Sections~\ref{sec:test_id31_iff} and \ref{sec:test_id31_hac} focus on the implementation and validation of the \acrshort{haclac} control architecture. First, Section~\ref{sec:test_id31_iff} demonstrates the application of decentralized Integral Force Feedback for robust active damping of the nano-hexapod suspension modes. This is followed in Section~\ref{sec:test_id31_hac} by the implementation of the high authority controller, which addresses low-frequency disturbances and completes the control system design. @@ -11871,7 +11898,7 @@ 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 point of interest is indicated by the blue frame \(\{B\}\), which is located \(H = 150\,mm\) above the nano-hexapod's top platform. +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 \(l_2 = 16.2\,mm\). 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}. @@ -12043,7 +12070,7 @@ After amplification, the voltages across the piezoelectric stack actuators are \ From the setpoint of micro-station stages (\(r_{D_y}\) for the translation stage, \(r_{R_y}\) for the tilt stage and \(r_{R_z}\) for the spindle), the reference pose of the sample \(\bm{r}_{\mathcal{X}}\) is computed using the micro-station's kinematics. The sample's position \(\bm{y}_\mathcal{X} = [D_x,\,D_y,\,D_z,\,R_x,\,R_y,\,R_z]\) is measured using multiple sensors. -First, the five interferometers \(\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]\) are used to measure the \([D_x,\,D_y,\,D_z,\,R_x,\,R_y]\) degrees of freedom of the sample. +First, the five interferometers \(\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]\) are used to measure the \([D_x,\,D_y,\,D_z,\,R_x,\,R_y]\) \acrshortpl{dof} of the sample. The \(R_z\) position of the sample is computed from the spindle's setpoint \(r_{R_z}\) and from the 6 encoders \(\bm{d}_e\) integrated in the nano-hexapod. The sample's position \(\bm{y}_{\mathcal{X}}\) is compared to the reference position \(\bm{r}_{\mathcal{X}}\) to compute the position error in the frame of the (rotating) nano-hexapod \(\bm{\epsilon\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]\). @@ -12066,7 +12093,7 @@ A comparison between the model and the measured dynamics is presented in Figure~ A good match can be observed for the diagonal dynamics (except the high frequency modes which are not modeled). However, the coupling of the transfer function from command signals \(\bm{u}\) to the estimated strut motion from the external metrology \(\bm{\epsilon\mathcal{L}}\) is larger than expected (Figure~\ref{fig:test_id31_first_id_int}). -The experimental time delay estimated from the FRF (Figure~\ref{fig:test_id31_first_id_int}) is larger than expected. +The experimental time delay estimated from the \acrshort{frf} (Figure~\ref{fig:test_id31_first_id_int}) is larger than expected. After investigation, it was found that the additional delay was due to a digital processing unit\footnote{The ``PEPU''~\cite{hino18_posit_encod_proces_unit} was used for digital protocol conversion between the interferometers and the Speedgoat.} that was used to get the interferometers' signals in the Speedgoat. This issue was later solved. @@ -12115,7 +12142,7 @@ Results shown in Figure~\ref{fig:test_id31_Rz_align_correct} are indeed indicati The dynamics of the plant was identified again after fine alignment and compared with the model dynamics in Figure~\ref{fig:test_id31_first_id_int_better_rz_align}. Compared to the initial identification shown in Figure~\ref{fig:test_id31_first_id_int}, the obtained coupling was decreased and was close to the coupling obtained with the multi-body model. -At low frequency (below \(10\,\text{Hz}\)), all off-diagonal elements have an amplitude \(\approx 100\) times lower than the diagonal elements, indicating that a low bandwidth feedback controller can be implemented in a decentralized manner (i.e. \(6\) SISO controllers). +At low frequency (below \(10\,\text{Hz}\)), all off-diagonal elements have an amplitude \(\approx 100\) times lower than the diagonal elements, indicating that a low bandwidth feedback controller can be implemented in a decentralized manner (i.e. \(6\) \acrshort{siso} controllers). Between \(650\,\text{Hz}\) and \(1000\,\text{Hz}\), several modes can be observed, which are due to flexible modes of the top platform and the modes of the two spheres adjustment mechanism. The flexible modes of the top platform can be passively damped, whereas the modes of the two reference spheres should not be present in the final application. @@ -12186,7 +12213,7 @@ The obtained dynamics from command signal \(u\) to estimated strut error \(\epsi Both direct terms (Figure~\ref{fig:test_id31_effect_rotation_direct}) and coupling terms (Figure~\ref{fig:test_id31_effect_rotation_coupling}) are unaffected by the rotation. The same can be observed for the dynamics from command signal to encoders and to force sensors. This confirms that spindle's rotation has no significant effect on plant dynamics. -This also indicates that the metrology kinematics is correct and is working in real time. +This also indicates that the metrology kinematics is correct and is working in real-time. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -12204,7 +12231,7 @@ This also indicates that the metrology kinematics is correct and is working in r \caption{\label{fig:test_id31_effect_rotation}Effect of the spindle rotation on the plant dynamics from \(u\) to \(\epsilon\mathcal{L}\). Three rotational velocities are tested (\(0\,\text{deg}/s\), \(36\,\text{deg}/s\) and \(180\,\text{deg}/s\)). Both direct terms (\subref{fig:test_id31_effect_rotation_direct}) and coupling terms (\subref{fig:test_id31_effect_rotation_coupling}) are displayed.} \end{figure} \subsubsection*{Conclusion} -The identified frequency response functions from command signals \(\bm{u}\) to the force sensors \(\bm{V}_s\) and to the estimated strut errors \(\bm{\epsilon\mathcal{L}}\) are well matching the dynamics of the developed multi-body model. +The identified \acrshortpl{frf} from command signals \(\bm{u}\) to the force sensors \(\bm{V}_s\) and to the estimated strut errors \(\bm{\epsilon\mathcal{L}}\) are well matching the dynamics of the developed multi-body model. The effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used. The spindle rotation had no visible effect on the measured dynamics, indicating that controllers should be robust against spindle rotation. \subsection{Decentralized Integral Force Feedback} @@ -12315,7 +12342,7 @@ The obtained damped plants are compared to the open-loop plants in Figure~\ref{f The peak amplitudes corresponding to the suspension modes were approximately reduced by a factor \(10\) for all considered payloads, indicating the effectiveness of the decentralized IFF control strategy. To experimentally validate the Decentralized IFF controller, it was implemented and the damped plants (i.e. the transfer function from \(\bm{u}^\prime\) to \(\bm{\epsilon\mathcal{L}}\)) were identified for all payload conditions. -The obtained frequency response functions are compared with the model in Figure~\ref{fig:test_id31_hac_plant_effect_mass} verifying the good correlation between the predicted damped plant using the multi-body model and the experimental results. +The obtained \acrshortpl{frf} are compared with the model in Figure~\ref{fig:test_id31_hac_plant_effect_mass} verifying the good correlation between the predicted damped plant using the multi-body model and the experimental results. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -12371,9 +12398,9 @@ Considering the complexity of the system's dynamics, the model can be considered \end{figure} The challenge here is to tune a high authority controller such that it is robust to the change in dynamics due to different payloads being used. -Without using the HAC-LAC strategy, it would be necessary to design a controller that provides good performance for all undamped dynamics (blue curves in Figure~\ref{fig:test_id31_comp_all_undamped_damped_plants}), which is a very complex control problem. -With the HAC-LAC strategy, the designed controller must be robust to all the damped dynamics (red curves in Figure~\ref{fig:test_id31_comp_all_undamped_damped_plants}), which is easier from a control perspective. -This is one of the key benefits of using the HAC-LAC strategy. +Without using the \acrshort{haclac} strategy, it would be necessary to design a controller that provides good performance for all undamped dynamics (blue curves in Figure~\ref{fig:test_id31_comp_all_undamped_damped_plants}), which is a very complex control problem. +With the \acrshort{haclac} strategy, the designed controller must be robust to all the damped dynamics (red curves in Figure~\ref{fig:test_id31_comp_all_undamped_damped_plants}), which is easier from a control perspective. +This is one of the key benefits of using the \acrshort{haclac} strategy. \begin{figure}[htbp] \centering @@ -12442,10 +12469,10 @@ However, small stability margins were observed for the highest mass, indicating \subsubsection{Performance estimation with simulation of Tomography scans} \label{ssec:test_id31_iff_hac_perf} -To estimate the performances that can be expected with this HAC-LAC architecture and the designed controller, simulations of tomography experiments were performed\footnote{Note that the eccentricity of the ``point of interest'' with respect to the Spindle rotation axis has been tuned based on measurements.}. +To estimate the performances that can be expected with this \acrshort{haclac} architecture and the designed controller, simulations of tomography experiments were performed\footnote{Note that the eccentricity of the ``point of interest'' with respect to the Spindle rotation axis has been tuned based on measurements.}. The rotational velocity was set to \(180\,\text{deg/s}\), and no payload was added on top of the nano-hexapod. An open-loop simulation and a closed-loop simulation were performed and compared in Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim}. -The obtained closed-loop positioning accuracy was found to comply with the requirements as it succeeded to keep the point of interest on the beam (Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}). +The obtained closed-loop positioning accuracy was found to comply with the requirements as it succeeded to keep the \acrshort{poi} on the beam (Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}). \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -12485,9 +12512,9 @@ This validation confirmed that the model can be reliably used to tune the feedba 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. -Finally, tomography experiments were simulated to validate the HAC-LAC architecture. +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). -With no payload at \(180\,\text{deg/s}\), the NASS successfully maintained the sample point of interest in the beam, which fulfilled the specifications. +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. \subsection{Validation with Scientific experiments} @@ -12509,7 +12536,7 @@ Unless explicitly stated, all closed-loop experiments were performed using the r 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 point of interest 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. +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. 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}). @@ -12535,8 +12562,8 @@ First, tomography scans were performed with a rotational velocity of \(6\,\text{ Each experimental sequence consisted of two complete spindle rotations: an initial open-loop rotation followed by a closed-loop rotation. The experimental results for the \(26\,\text{kg}\) payload are presented in Figure~\ref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}. -Due to the static deformation of the micro-station stages under payload loading, a significant eccentricity was observed between the point of interest and the spindle rotation axis. -To establish a theoretical lower bound for open-loop errors, an ideal scenario was assumed, where the point of interest perfectly aligns with the spindle rotation axis. +Due to the static deformation of the micro-station stages under payload loading, a significant eccentricity was observed between the \acrshort{poi} and the spindle rotation axis. +To establish a theoretical lower bound for open-loop errors, an ideal scenario was assumed, where the \acrshort{poi} perfectly aligns with the spindle rotation axis. This idealized case was simulated by first calculating the eccentricity through circular fitting (represented by the dashed black circle in Figure~\ref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}), and then subtracting it from the measured data, as shown in Figure~\ref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}. 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. @@ -12557,7 +12584,7 @@ While this approach likely underestimates actual open-loop errors, as perfect al \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}. -Results are indicating the NASS succeeds in keeping the sample's point of interest on the beam, except for the highest mass of \(39\,\text{kg}\) for which the lateral motion is a bit too high. +Results are indicating the NASS succeeds in keeping the sample's \acrshort{poi} on the beam, except for the highest mass of \(39\,\text{kg}\) for which the lateral motion is a bit too high. These experimental findings are consistent with the predictions from the tomography simulations presented in Section~\ref{ssec:test_id31_iff_hac_robustness}. \begin{figure}[htbp] @@ -12568,7 +12595,7 @@ These experimental findings are consistent with the predictions from the tomogra \paragraph{Fast Tomography scans} A tomography experiment was then performed with the highest rotational velocity of the Spindle: \(180\,\text{deg/s}\)\footnote{The highest rotational velocity of \(360\,\text{deg/s}\) could not be tested due to an issue in the Spindle's controller.}. -The trajectory of the point of interest during the fast tomography scan is shown in Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}. +The trajectory of the \acrshort{poi} during the fast tomography scan is shown in Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}. Although the experimental results closely match the simulation results (Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim}), the actual performance was slightly lower than predicted. Nevertheless, even with this robust (i.e. conservative) HAC implementation, the system performance was already close to the specified requirements. @@ -12589,8 +12616,8 @@ Nevertheless, even with this robust (i.e. conservative) HAC implementation, the \end{figure} \paragraph{Cumulative Amplitude Spectra} -A comparative analysis was conducted using three tomography scans at \(180\,\text{deg/s}\) to evaluate the effectiveness of the HAC-LAC strategy in reducing positioning errors. -The scans were performed under three conditions: open-loop, with decentralized IFF control, and with the complete HAC-LAC strategy. +A comparative analysis was conducted using three tomography scans at \(180\,\text{deg/s}\) to evaluate the effectiveness of the \acrshort{haclac} strategy in reducing positioning errors. +The scans were performed under three conditions: open-loop, with decentralized IFF control, and with the complete \acrshort{haclac} strategy. For this specific measurement, an enhanced high authority controller (discussed in Section~\ref{ssec:test_id31_cf_control}) was optimized for low payload masses to meet the performance requirements. Figure~\ref{fig:test_id31_hac_cas_cl} presents the cumulative amplitude spectra of the position errors for all three cases. @@ -12624,7 +12651,7 @@ 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~\ref{fig:test_id31_reflectivity}). -The results confirmed that the NASS successfully maintained the point of interest within the specified beam parameters throughout the scanning process. +The results confirmed that the NASS successfully maintained the \acrshort{poi} within the specified beam parameters throughout the scanning process. \begin{figure}[htbp] \begin{subfigure}{0.33\textwidth} @@ -12867,7 +12894,7 @@ A schematic of the proposed control architecture is illustrated in Figure~\ref{f \end{figure} Implementation of this control architecture necessitates a plant model, which must subsequently be inverted. -This plant model was derived from the multi-body model incorporating the previously detailed 2-DoF APA model, such that the model order stays relatively low. +This plant model was derived from the multi-body model incorporating the previously detailed 2-DoF \acrshort{apa} model, such that the model order stays relatively low. Proposed analytical formulas for complementary filters having \(40\,\text{dB/dec}\) were used during this experimental validation. An initial experimental validation was conducted under no-payload conditions, with control applied solely to the \(D_y\), \(D_z\), and \(R_y\) directions. Increased control bandwidth was achieved for the \(D_z\) and \(R_y\) directions through appropriate tuning of the parameter \(\omega_0\). @@ -12923,7 +12950,7 @@ A comprehensive series of experimental validations was conducted to evaluate the 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 \(R_y\)). 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 HAC-LAC 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 \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 \(100\,\mu m/s\). @@ -12971,7 +12998,7 @@ Diffraction tomography (\(6\,\text{deg/s}\), \(1\,mm/s\)) & \(\bm{53}\) & \(10\) \label{ssec:test_id31_conclusion} This chapter presented a comprehensive experimental validation of the Nano Active Stabilization System (NASS) on the ID31 beamline, demonstrating its capability to maintain precise sample positioning during various experimental scenarios. -The implementation and testing followed a systematic approach, beginning with the development of a short-stroke metrology system to measure the sample position, followed by the successful implementation of a HAC-LAC control architecture, and concluding in extensive performance validation across diverse experimental conditions. +The implementation and testing followed a systematic approach, beginning with the development of a short-stroke metrology system to measure the sample position, followed by the successful implementation of a \acrshort{haclac} control architecture, and concluding in extensive performance validation across diverse experimental conditions. The short-stroke metrology system, while designed as a temporary solution, proved effective in providing high-bandwidth and low-noise 5-DoF position measurements. The careful alignment of the fibered interferometers targeting the two reference spheres ensured reliable measurements throughout the testing campaign. @@ -12995,7 +13022,7 @@ A methodical approach was employed—first characterizing individual components Initially, the Amplified Piezoelectric Actuators (APA300ML) were characterized, revealing consistent mechanical and electrical properties across multiple units. The implementation of Integral Force Feedback was shown to add significant damping to the system. -Two models of the APA300ML were developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from finite element analysis. +Two models of the APA300ML were developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from \acrshort{fea}. Both models accurately represented the axial dynamics of the actuators, with the super-element model additionally capturing flexible modes. The flexible joints were examined for geometric accuracy and bending stiffness, with measurements confirming compliance with design specifications. @@ -13007,10 +13034,10 @@ This finding led to the strategic decision to mount encoders on the nano-hexapod The nano-hexapod was then assembled and mounted on a suspended table to characterize its dynamic behavior. The measurement setup enabled isolation of the hexapod's dynamics from potential influence of complex support dynamics. -The experimental frequency response functions exhibited good correlation with the multi-body model, confirming that the model can be used for control system design. +The experimental \acrshortpl{frf} exhibited good correlation with the multi-body model, confirming that the model can be used for control system design. Finally, the complete NASS was validated on the ID31 beamline using a short-stroke metrology system. -The HAC-LAC control architecture successfully provided robust active damping of suspension modes and rejection of low-frequency disturbances across various payload conditions. +The \acrshort{haclac} control architecture successfully provided robust active damping of suspension modes and rejection of low-frequency disturbances across various payload conditions. Comprehensive testing under typical experimental scenarios—including tomography scans, reflectivity scans, and diffraction tomography—demonstrated the NASS ability to maintain the positioning errors within specifications (\(30\,\text{nm RMS}\) in lateral direction, \(15\,\text{nm RMS}\) in vertical direction, and \(250\,\text{nrad RMS}\) in tilt direction). The system performed exceptionally well during vertical scans, though some limitations were identified during rapid lateral scanning and with heavier payloads. @@ -13032,15 +13059,15 @@ The conceptual design phase rigorously evaluated the feasibility of the NASS con Through progressive modeling, from simplified uniaxial representations to complex multi-body dynamic simulations, key design insights were obtained. It was determined that an active platform with moderate stiffness offered an optimal compromise, decoupling the system from micro-station dynamics while mitigating gyroscopic effects from continuous rotation. The multi-body modeling approach, informed by experimental modal analysis of the micro-station, was essential for capturing the system's complex dynamics. -The Stewart platform architecture was selected for the active stage, and its viability was confirmed through closed-loop simulations employing a High-Authority Control / Low-Authority Control (HAC-LAC) strategy. +The Stewart platform architecture was selected for the active stage, and its viability was confirmed through closed-loop simulations employing a \acrfull{haclac} strategy. This strategy used a modified form of Integral Force Feedback (IFF), adapted to provide robust active damping despite the platform rotation and varying payloads. These simulations demonstrated the NASS concept could meet the nanometer-level stability requirements under realistic operating conditions. Following the conceptual validation, the detailed design phase focused on translating the NASS concept into an optimized, physically realizable system. Geometric optimization studies refined the Stewart platform configuration. -A hybrid modeling technique combining Finite Element Analysis (FEA) with multi-body dynamics simulation was applied and experimentally validated. -This approach enabled detailed optimization of components, such as Amplified Piezoelectric Actuators (APA) and flexible joints, while efficiently simulating the complete system dynamics. -By dedicating one stack of the APA specifically to force sensing, excellent collocation with the actuator stacks was achieved, which is critical for implementing robust decentralized IFF. +A hybrid modeling technique combining \acrfull{fea} with multi-body dynamics simulation was applied and experimentally validated. +This approach enabled detailed optimization of components, such as \acrfull{apa} and flexible joints, while efficiently simulating the complete system dynamics. +By dedicating one stack of the \acrshort{apa} specifically to force sensing, excellent collocation with the actuator stacks was achieved, which is critical for implementing robust decentralized IFF. Work was also undertaken on the optimization of the control strategy for the active platform. Instrumentation selection (sensors, actuators, control hardware) was guided by dynamic error budgeting to ensure component noise levels met the overall nanometer-level performance target. @@ -13048,7 +13075,7 @@ The final phase of the project was dedicated to the experimental validation of t Component tests confirmed the performance of the selected actuators and flexible joints, validated their respective models. Dynamic testing of the assembled nano-hexapod on an isolated test bench provided essential experimental data that correlated well with the predictions of the multi-body model. The final validation was performed on the ID31 beamline, using a short-stroke metrology system to assess performance under realistic experimental conditions. -These tests demonstrated that the NASS, operating with the implemented HAC-LAC control architecture, successfully achieved the target positioning stability – maintaining residual errors below \(30\,\text{nm RMS}\) laterally, \(15\,\text{nm RMS}\) vertically, and \(250\,\text{nrad RMS}\) in tilt – during various experiments, including tomography scans with significant payloads. +These tests demonstrated that the NASS, operating with the implemented \acrshort{haclac} control architecture, successfully achieved the target positioning stability – maintaining residual errors below \(30\,\text{nm RMS}\) laterally, \(15\,\text{nm RMS}\) vertically, and \(250\,\text{nrad RMS}\) in tilt – during various experiments, including tomography scans with significant payloads. Crucially, the system's robustness to variations in payload mass and operational modes was confirmed. \section{Perspectives} @@ -13056,7 +13083,7 @@ Although this research successfully validated the NASS concept, it concurrently \paragraph{Automatic tuning of a multi-body model from an experimental modal analysis} The manual tuning process employed to match the multi-body model dynamics with experimental measurements was found to be laborious. -Systems like the micro-station can be conceptually modeled as interconnected solid bodies, springs, and dampers, with component inertia readily obtainable from CAD models. +Systems like the micro-station can be conceptually modeled as interconnected solid bodies, springs, and dampers, with component inertia readily obtainable from 3D models. An interesting perspective is the development of methods for the automatic tuning of the multi-body model's stiffness matrix (representing the interconnecting spring stiffnesses) directly from experimental modal analysis data. Such a capability would enable the rapid generation of accurate dynamic models for existing end-stations, which could subsequently be used for detailed system analysis and simulation studies. \paragraph{Better addressing plant uncertainty from a change of payload} @@ -13072,14 +13099,14 @@ 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 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. +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. Nevertheless, a more rigorous analysis of this control architecture and its comparison with similar approaches documented in the literature is necessary to fully understand its capabilities and limitations. \paragraph{Sensor Fusion} -While the HAC-LAC approach demonstrated a simple and comprehensive methodology for controlling the NASS, sensor fusion represents an interesting alternative that is worth investigating. +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. One potential approach involves fusing external metrology (used at low frequencies) with force sensors (employed at high frequencies). @@ -13094,8 +13121,8 @@ This challenge is particularly complex when continuous rotation is combined with Yet, the development of such metrology systems is considered critical for future end-stations, especially for future tomography end stations where nano-meter accuracy is desired across larger strokes. Promising approaches have been presented in the literature. -A ball lens retroreflector is used in \cite{schropp20_ptynam}, providing a \(\approx 1\,\text{mm}^3\) measuring volume, but does not fully accommodate complete rotation. -In \cite{geraldes23_sapot_carnaub_sirius_lnls}, an interesting metrology approach is presented, using interferometers for long stroke/non-rotated movements and capacitive sensors for short stroke/rotated positioning. +A ball lens retroreflector is used in~\cite{schropp20_ptynam}, providing a \(\approx 1\,\text{mm}^3\) measuring volume, but does not fully accommodate complete rotation. +In~\cite{geraldes23_sapot_carnaub_sirius_lnls}, an interesting metrology approach is presented, using interferometers for long stroke/non-rotated movements and capacitive sensors for short stroke/rotated positioning. \paragraph{Alternative Architecture for the NASS} The original micro-station design was driven by optimizing positioning accuracy, using dedicated actuators for different DoFs (leading to simple kinematics and a stacked configuration), and maximizing stiffness. @@ -13114,13 +13141,13 @@ One possible configuration, illustrated in Figure \ref{fig:conclusion_nass_archi \caption{\label{fig:conclusion_nass_architecture}Proposed alternative configuration for an end-station including the Nano Active Stabilization System} \end{figure} -With this architecture, the online metrology could be divided into two systems, as proposed by \cite{geraldes23_sapot_carnaub_sirius_lnls}: a long-stroke metrology system potentially using interferometers, and a short-stroke metrology system using capacitive sensors, as successfully demonstrated by \cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}. +With this architecture, the online metrology could be divided into two systems, as proposed by~\cite{geraldes23_sapot_carnaub_sirius_lnls}: a long-stroke metrology system potentially using interferometers, and a short-stroke metrology system using capacitive sensors, as successfully demonstrated by~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}. \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 \(3\,mm\) 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. +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}. However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature.