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Cardona and M. G{\'e}radin}, + title = {A Superelement Formulation for Mechanism Analysis}, + journal = {Computer Methods in Applied Mechanics and Engineering}, + volume = 100, + number = 1, + pages = {1-29}, + year = 1992, + doi = {10.1016/0045-7825(92)90112-W}, + url = + {https://www.sciencedirect.com/science/article/pii/004578259290112W}, + issn = {0045-7825}, +} diff --git a/phd-thesis.org b/phd-thesis.org index 227dd97..3056e7c 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -574,13 +574,13 @@ 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 acrshortpl:dof as stacked stages, parallel systems generally provide limited stroke but significantly enhanced stiffness and superior dynamic performance. +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. Most end stations, particularly those requiring extensive mobility, employ stacked stages. Their positioning performance consequently depends entirely on the accuracy of individual components. Strategies include employing a limited number of high-performance stages, such as air-bearing spindles\nbsp{}[[cite:&riekel10_progr_micro_nano_diffr_at]], and maintaining extremely stable thermal environments within the experimental hutch, often requiring extended stabilization times\nbsp{}[[cite:&leake19_nanod_beaml_id01]]. Examples of such end-stations, including those at beamlines ID16B\nbsp{}[[cite:&martinez-criado16_id16b]] and ID11\nbsp{}[[cite:&wright20_new_oppor_at_mater_scien]], are shown in Figure\nbsp{}ref:fig:introduction_passive_stations. -However, when a large number of DoFs are required, the cumulative errors and limited dynamic stiffness of stacked configurations can make experiments with nano-focused beams extremely challenging or infeasible. +However, when a large number of degrees of freedom are required, the cumulative errors and limited dynamic stiffness of stacked configurations can make experiments with nano-focused beams extremely challenging or infeasible. #+name: fig:introduction_passive_stations #+caption: Example of two nano end-stations lacking online metrology for measuring the sample's position. @@ -663,15 +663,15 @@ 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 $50\,\text{kg}$) than previous stations with similar positioning requirements. -End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few acrshortpl:dof with strokes around $100\,\upmu\text{m}$. +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 $100\,\upmu\text{m}$. Recently, acrfull:vc actuators were used to increase the stroke up to $3\,\text{mm}$ [[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]] An alternative strategy involves a "long stroke-short stroke" architecture, illustrated conceptually in Figure\nbsp{}ref:fig:introduction_two_stage_schematic. In this configuration, a high-accuracy, high-bandwidth short-stroke stage is mounted on top of a less precise long-stroke stage. The short-stroke stage actively compensates for errors based on metrology feedback, while the long-stroke stage performs the larger movements. -This approach allows the combination of extended travel with high precision and good dynamical response, but is often implemented for only one or a few DoFs, as seen in Figures\nbsp{}ref:fig:introduction_two_stage_schematic and\nbsp{}ref:fig:introduction_two_stage_control_h_bridge. +This approach allows the combination of extended travel with high precision and good dynamical response, but is often implemented for only one or a few degrees of freedom, as seen in Figures\nbsp{}ref:fig:introduction_two_stage_schematic and\nbsp{}ref:fig:introduction_two_stage_control_h_bridge. #+name: fig:introduction_two_stage_example -#+caption: Schematic of a typical Long stroke-Short stroke control architecture (\subref{fig:introduction_two_stage_schematic}). A 3-DoFs long stroke-short stroke is shown in (\subref{fig:introduction_two_stage_control_h_bridge}) where $y_1$, $y_2$ and $x$ are 3-phase linear motors and short stroke actuators are voice coils. +#+caption: Schematic of a typical Long stroke-Short stroke control architecture (\subref{fig:introduction_two_stage_schematic}). A 3-DoF long stroke-short stroke is shown in (\subref{fig:introduction_two_stage_control_h_bridge}) where $y_1$, $y_2$ and $x$ are 3-phase linear motors and short stroke actuators are voice coils. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_two_stage_schematic} Typical Long Stroke-Short Stroke control architecture} @@ -707,7 +707,7 @@ 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 acrfull:nass is proposed. -As schematically illustrated in Figure\nbsp{}ref:fig:introduction_nass_concept_schematic, the acrshort:nass comprises three principal components integrated with the existing micro-station (yellow): a 5-DoFs online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple). +As schematically illustrated in Figure\nbsp{}ref:fig:introduction_nass_concept_schematic, the acrshort:nass comprises three 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. @@ -719,7 +719,7 @@ It actively compensates for positioning errors measured by the external metrolog ***** Online Metrology system 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-DoFs (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. +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 #+caption: 2D representation of the NASS metrology system. @@ -730,17 +730,17 @@ Fiber interferometers target both surfaces. A tracking system maintains perpendicularity between the interferometer beams and the mirror, such that Abbe errors are eliminated. Interferometers pointing at the spherical surface provides translation measurement, while the ones pointing at the flat bottom surface yield tilt angles. The development of this complex metrology system constitutes a significant mechatronic project in itself and is currently ongoing; it is not further detailed within this thesis. -For the work presented herein, the metrology system is assumed to provide accurate, high-bandwidth 5-DoFs position measurements. +For the work presented herein, the metrology system is assumed to provide accurate, high-bandwidth 5-DoF position measurements. ***** 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 acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$ and $R_y$). +It needs to provide active motion compensation in 5-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 $50\,\text{kg}$. -A suitable candidate architecture for this platform is the Stewart platform (also known as "hexapod"), a parallel kinematic mechanism capable of 6-DoFs motion. +A suitable candidate architecture for this platform is the Stewart platform (also known as "hexapod"), a parallel kinematic mechanism capable of 6-DoF motion. Stewart platforms are widely employed in positioning and vibration isolation applications due to their inherent stiffness and potential for high precision. Various designs exist, differing in geometry, actuation technology, sensing methods, and control strategies, as exemplified in Figure\nbsp{}ref:fig:introduction_stewart_platform_piezo. A central challenge addressed in this thesis is the optimal mechatronic design of such an active platform tailored to the specific requirements of the NASS. @@ -796,7 +796,7 @@ Key challenges within this approach include developing appropriate design method This thesis presents several original contributions aimed at addressing the challenges inherent in the design, control, and implementation of the Nano Active Stabilization System, primarily within the fields of Control Theory, Mechatronic Design, and Experimental Validation. -***** 6-DoFs vibration control of a rotating platform +***** 6-DoF vibration control of a rotating platform Traditional long-stroke/short-stroke architectures typically operate in one or two degrees of freedom. This work extends the concept to six degrees of freedom, with the active platform designed not only to correct rotational errors but to simultaneously compensate for errors originating from all underlying micro-station stages. @@ -846,7 +846,7 @@ The conclusion of this work involved the experimental implementation and validat Experimental results, presented in Section\nbsp{}ref:sec:test_id31, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native $\approx 10\,\upmu\text{m}$ level down to the target $\approx 100\,\text{nm}$ range during representative scientific experiments. Crucially, robustness to variations in sample mass and diverse experimental conditions was verified. The NASS thus provides a versatile end-station solution, uniquely combining high payload capacity with nanometer-level accuracy, enabling optimal use of the advanced capabilities of the ESRF-EBS beam and associated detectors. -To the author's knowledge, this represents the first demonstration of such a 5-DoFs active stabilization platform being used to enhance the accuracy of a complex positioning system to this level. +To the author's knowledge, this represents the first demonstration of such a 5-DoF active stabilization platform being used to enhance the accuracy of a complex positioning system to this level. ** Outline ***** Introduction :ignore: @@ -858,7 +858,7 @@ While the chapters follow this logical progression, care has been taken to struc The conceptual design phase, detailed in Chapter\nbsp{}ref:chap:concept, followed a methodical progression from simplified uniaxial models to more complex multi-body representations. Initial uniaxial analysis (Section\nbsp{}ref:sec:uniaxial) provided fundamental insights, particularly regarding the influence of active platform stiffness on performance. -The introduction of rotation in a 3-DoFs model (Section\nbsp{}ref:sec:rotating) allowed investigation of gyroscopic effects, revealing challenges for softer platform designs. +The introduction of rotation in a 3-DoF model (Section\nbsp{}ref:sec:rotating) allowed investigation of gyroscopic effects, revealing challenges for softer platform designs. Experimental modal analysis of the existing micro-station (Section\nbsp{}ref:sec:modal) confirmed its complex dynamics but supported a rigid-body assumption for the different stages, justifying the development of a detailed multi-body model. This model, tuned against experimental data and incorporating measured disturbances, was validated through simulation (Section\nbsp{}ref:sec:ustation). The Stewart platform architecture was selected for the active stage, and its kinematics, dynamics, and control were analyzed (Section\nbsp{}ref:sec:nhexa). @@ -915,7 +915,7 @@ To construct such a model, a comprehensive modal analysis was conducted, as deta This experimental modal analysis confirmed the complex nature of the micro-station dynamics while validating that each stage behaves predominantly as a rigid body within the frequency range of interest—thus supporting the subsequent development of a multi-body model. Section\nbsp{}ref:sec:ustation presents the development of this multi-body model for the micro-station. -Parameters were meticulously tuned to match measured compliance characteristics, and disturbance sources were carefully modeled based on experimental data. +Parameters were meticulously tuned to match measured compliance characteristics, and disturbance sources were carefully modelled based on experimental data. This refined model was then validated through simulations of scientific experiments, demonstrating its accuracy in representing the micro-station behavior under typical operating conditions. For the active stabilization stage, the Stewart platform architecture was selected after careful evaluation of various options. @@ -940,7 +940,7 @@ To have a relevant model, the micro-station dynamics is first identified and its Then, a model of the active platform is added on top of the micro-station. With the added sample and sensors, this gives a uniaxial dynamical model of the acrshort:nass that will be used for further analysis (Section\nbsp{}ref:sec:uniaxial_nano_station_model). -The disturbances affecting position stability are identified experimentally (Section\nbsp{}ref:sec:uniaxial_disturbances) and included in the model for dynamical noise budgeting (Section\nbsp{}ref:sec:uniaxial_noise_budgeting). +The disturbances affecting position stability are identified experimentally (Section\nbsp{}ref:sec:uniaxial_disturbances) and included in the model for dynamical error budgeting (Section\nbsp{}ref:sec:uniaxial_noise_budgeting). In all the following analysis, three active platform stiffnesses are considered to better understand the trade-offs and to find the most adequate active platform design. Three sample masses are also considered to verify the robustness of the applied control strategies with respect to a change of sample. @@ -950,7 +950,7 @@ 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 active platform and the sample's acrshort:poi (Section\nbsp{}ref:sec:uniaxial_payload_dynamics). +Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section\nbsp{}ref:sec:uniaxial_support_compliance) and the presence of dynamics between the active platform and the sample's acrlong:poi (Section\nbsp{}ref:sec:uniaxial_payload_dynamics). *** Micro Station Model <> @@ -986,12 +986,12 @@ Due to the poor coherence[fn:uniaxial_2] at low frequencies, these acrlongpl:frf ***** 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 acrshortpl:dof. +It consists of a mass spring damper system with three degrees of freedom. 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 positioning hexapod with mass $m_h$, damping $c_h$ and stiffness $k_h$. -The masses of the different stages are estimated from the 3D model, while the stiffnesses are from the data-sheet of the manufacturers. +The masses of the different stages are computed from the 3D model, while the stiffness values are taken from the manufacturers' datasheets for the various guiding elements used. The damping coefficients were tuned to match the damping identified from the measurements. The parameters obtained are summarized in Table\nbsp{}ref:tab:uniaxial_ustation_parameters. @@ -1011,7 +1011,7 @@ 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 positioning hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model. -Because the uniaxial model has three acrshortpl:dof, only three modes with frequencies at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$ are modeled. +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 modelled. 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. @@ -1137,7 +1137,7 @@ The estimated acrshort:asd $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in \end{equation} #+name: fig:uniaxial_asd_disturbance -#+caption: Estimated amplitude spectral density of the floor motion $x_f$ (\subref{fig:uniaxial_asd_floor_motion_id31}) and of the stage disturbances $f_t$ (\subref{fig:uniaxial_asd_disturbance_force}). +#+caption: Estimated amplitude spectral density of the floor motion $x_f$ (\subref{fig:uniaxial_asd_floor_motion_id31}) and of the stage disturbances $f_t$ (\subref{fig:uniaxial_asd_disturbance_force}). Data are shown between $1$ and $500\,\text{Hz}$, a frequency range for which the measurement quality is good. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_asd_floor_motion_id31}Estimated ASD of $x_f$} @@ -1175,12 +1175,12 @@ The amplitude spectral density $\Gamma_{f_{t}}$ of the disturbance force is them \Gamma_{f_{t}}(\omega) = \frac{\Gamma_{R_{z}}(\omega)}{|G_{f_t}(j\omega)|} \end{equation} -*** Open-Loop Dynamic Noise Budgeting +*** Open-Loop Dynamic Error Budgeting <> ***** Introduction :ignore: -Now that a model of the acrshort:nass has been obtained (see section\nbsp{}ref:sec:uniaxial_nano_station_model) and that the disturbances have been estimated (see section\nbsp{}ref:sec:uniaxial_disturbances), it is possible to perform an /open-loop dynamic noise budgeting/. +Now that a model of the acrshort:nass has been obtained (see section\nbsp{}ref:sec:uniaxial_nano_station_model) and that the disturbances have been estimated (see section\nbsp{}ref:sec:uniaxial_disturbances), it is possible to perform an /open-loop dynamic error budgeting/. -To perform such noise budgeting, the disturbances need to be modeled by their spectral densities (done in section\nbsp{}ref:sec:uniaxial_disturbances). +To perform such error budget, the disturbances need to be modelled by their spectral densities (done in section\nbsp{}ref:sec:uniaxial_disturbances). Then, the transfer functions from disturbances to the performance metric (here the distance $d$) are computed. Finally, these two types of information are combined to estimate the corresponding spectral density of the performance metric. This is very useful to identify what is limiting the performance of the system, and to compare the achievable performance with different system parameters. @@ -1222,7 +1222,7 @@ The obtained sensitivity to disturbances for the three active platform stiffness #+end_subfigure #+end_figure -***** Open-Loop Dynamic Noise Budgeting +***** Open-Loop Dynamic Error Budgeting Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion $d$ for each active platform and sample configuration. The acrfull:cas of the relative motion $d$ due to both floor motion $x_f$ and stage vibrations $f_t$ are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three active platform stiffnesses. It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft active platform below $5\,\text{Hz}$. @@ -1258,6 +1258,10 @@ However, what is more important is the /closed-loop/ residual vibration of $d$ ( The goal is to obtain a closed-loop residual vibration $\epsilon_d \approx 20\,\text{nm RMS}$ (represented by an horizontal dashed black line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses). The bandwidth of the feedback controller leading to a closed-loop residual vibration of $20\,\text{nm RMS}$ can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. +This is why, in this document, cumulative amplitude spectra are computed by integrating from high to low frequency. +Another important point is that cumulative \amplitude\ spectra are plotted instead of cumulative \power\ spectra, despite the warnings discussed in\nbsp{}[[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition Chapt. 8.1.5]]. +This choice comes at the cost of losing the straightforward interpretation of the relative importance of different frequencies, but it makes the plots easier to read and simplifies the estimation of the bandwidth required to achieve a desired residual RMS value. + A closed loop bandwidth of $\approx 10\,\text{Hz}$ is found for the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$), $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$), and $\approx 100\,\text{Hz}$ for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$). Therefore, while the /open-loop/ vibration is the lowest for the stiff active platform, it requires the largest feedback bandwidth to meet the specifications. @@ -1266,7 +1270,7 @@ The advantage of the soft active platform can be explained by its natural isolat *** Active Damping <> ***** Introduction :ignore: -In this section, three active damping techniques are applied to the active platform (see Figure\nbsp{}ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]], Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] and Direct Velocity Feedback (DVF)\nbsp{}[[cite:&karnopp74_vibrat_contr_using_semi_activ_force_gener;&serrand00_multic_feedb_contr_isolat_base_excit_vibrat;&preumont02_force_feedb_versus_accel_feedb]]. +In this section, three active damping techniques are applied to the active platform (see Figure\nbsp{}ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]], Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapt. 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 and are then compared in terms of achievable damping of the active platform mode, reduction of the effect of disturbances (i.e., $x_f$, $f_t$ and $f_s$) on the displacement $d$. @@ -1381,8 +1385,8 @@ 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 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 active platform 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 Chapt. 7]]. +For the acrshort:dvf controller, this property is not guaranteed, and may be lost if some flexibility between the active platform and the sample is considered\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapt. 8.4]]. When the active platform's suspension modes are at frequencies lower than the resonances of the micro-station (blue and red curves in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the IFF and acrshort:dvf transfer functions. For the stiff active platform (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques. @@ -1490,7 +1494,7 @@ All three active damping techniques yielded similar damped plants. #+end_subfigure #+end_figure -***** Sensitivity to Disturbances and Noise Budgeting +***** Sensitivity to Disturbances and Error Budget Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped. The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping. The comparison is done with the active platform having a stiffness $k_n = 1\,\text{N}/\upmu\text{m}$. @@ -1503,7 +1507,7 @@ Several conclusions can be drawn by comparing the obtained sensitivity transfer - both IFF and acrshort:dvf degrade the sensitivity to floor motion below the resonance of the active platform (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_xf). #+name: fig:uniaxial_sensitivity_dist_active_damping -#+caption: Change of sensitivity to disturbances for all three active damping strategies. Considered disturbances are $f_s$ 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_xf}). +#+caption: Change of sensitivity to disturbances for all three active damping strategies. Considered disturbances are $f_s$ 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_xf}). Sensitivity for IFF (displayed in blue) is superimposed with the sensitivity for DVF (yellow). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_fs}Direct forces} @@ -1591,7 +1595,7 @@ The acrfull:haclac architecture is shown in Figure\nbsp{}ref:fig:uniaxial_hac_la This corresponds to a /two step/ control strategy: - First, an active damping controller $\bm{K}_{\textsc{LAC}}$ is implemented (see Section\nbsp{}ref:sec:uniaxial_active_damping). It allows the vibration level to be reduced, and it also makes the damped plant (transfer function from $u^{\prime}$ to $y$) easier to control than the undamped plant (transfer function from $u$ to $y$). - This is called /low authority/ control as it only slightly affects the system poles\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 14.6]]. + This is called /low authority/ control as it only slightly affects the system poles\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapt. 14.6]]. - Then, a position controller $\bm{K}_{\textsc{HAC}}$ is implemented and is used to control the position $d$. This is called /high authority/ control as it usually relocates the system's poles. @@ -1620,7 +1624,7 @@ This control architecture applied to the uniaxial model is shown in Figure\nbsp{ The damped plants obtained for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses. For $k_n = 0.01\,\text{N}/\upmu\text{m}$ and $k_n = 1\,\text{N}/\upmu\text{m}$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). However, this is not the case for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). -This is due to the interaction between the micro-station (modeled modes at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$) and the active platform. +This is due to the interaction between the micro-station (modelled modes at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$) and the active platform. This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_compliance. #+name: fig:uniaxial_hac_iff_damped_plants_masses @@ -1655,7 +1659,7 @@ This controller must be robust to the change of sample's mass (from $1\,\text{kg The required feedback bandwidths were estimated in Section\nbsp{}ref:sec:uniaxial_noise_budgeting: - $f_b \approx 10\,\text{Hz}$ for the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$). Near this frequency, the plants (shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft) are equivalent to a mass line (i.e., slope of $-40\,\text{dB/dec}$ and a phase of -180 degrees). - The gain of this mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,\text{kg}$ up to $65\,\text{kg}$). + The gain of this mass line can vary up to a factor $\approx 5$ (suspended mass from $16\,\text{kg}$ up to $65\,\text{kg}$). This means that the designed controller will need to have /large gain margins/ to be robust to the change of sample's mass. - $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$). Similar to the soft active platform, the plants near the crossover frequency are equivalent to a mass line (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). @@ -1756,9 +1760,9 @@ The goal is to have a first estimation of the attainable performance. #+end_subfigure #+end_figure -***** Closed-Loop Noise Budgeting +***** Closed-Loop Error Budgeting -The acrlong:hac are then implemented and the closed-loop sensitivities to disturbances are computed. +The developed acrlongpl: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 active platform with $25\,\text{kg}$ sample's mass). As expected, the sensitivity to disturbances decreased in the controller bandwidth and slightly increased outside this bandwidth. @@ -1791,7 +1795,7 @@ 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 active platform ($\approx 25\,\text{nm}$ to $\approx 35\,\text{nm}$ depending on the sample's mass) than for the stiffer active platform (from $\approx 30\,\text{nm}$ to $\approx 70\,\text{nm}$). #+name: fig:uniaxial_cas_hac_lac -#+caption: Cumulative Amplitude Spectra for all three active platform stiffnesses in OL, with IFF and with acrshort:haclac. +#+caption: Cumulative Amplitude Spectra for all three active platform stiffnesses in OL, with IFF and with acrshort:haclac. The three lines of each color are corresponding to the considered three sample masses. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1816,7 +1820,7 @@ Obtained root mean square values of the distance $d$ are better for the soft act ***** Conclusion -On the basis of the open-loop noise budgeting made in Section\nbsp{}ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,\text{nm RMS}$ was estimated. +On the basis of the open-loop error budgeting made in Section\nbsp{}ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,\text{nm RMS}$ was estimated. To achieve such bandwidth, the acrshort:haclac strategy was followed, which consists of first using an active damping controller (studied in Section\nbsp{}ref:sec:uniaxial_active_damping) and then adding a high authority position feedback controller. In this section, feedback controllers were designed in such a way that the required closed-loop bandwidth was reached while being robust to changes in the payload mass. @@ -1974,12 +1978,12 @@ Note that the observations made in this section are also affected by the ratio b ***** Introduction :ignore: -Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_rigid_schematic). +Up to this section, the sample was modelled as a mass rigidly fixed to the active platform (as shown in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_rigid_schematic). However, such a sample may present internal dynamics, and its mounting on the active platform may have limited stiffness. To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_schematic are used. #+name: fig:uniaxial_payload_dynamics_models -#+caption: Models used to study the effect of payload dynamics. +#+caption: Models used to study the effect of payload dynamics. Active platform mass is $m_n = 15\,\text{kg}$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_rigid_schematic}Rigid payload} @@ -2002,13 +2006,13 @@ To study the impact of the flexibility between the active platform and the paylo Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_schematic are considered. The resonances of the payload are set at $\omega_s = 20\,\text{Hz}$ and at $\omega_s = 200\,\text{Hz}$ while its mass is either $m_s = 1\,\text{kg}$ or $m_s = 50\,\text{kg}$. -The transfer functions from the active platform force $F$ to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. +The transfer functions from the active platform force $F$ to the motion of the active platform top platform $L$ and $L^{\prime}$ are computed and are compared for a soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern). The frequency of the anti-resonance corresponds to the "free" resonance of the sample $\omega_s = \sqrt{k_s/m_s}$. The flexibility of the sample also changes the high-frequency gain (the mass line is shifted from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$). #+name: fig:uniaxial_payload_dynamics_soft_nano_hexapod -#+caption: Effect of the payload dynamics on the soft active platform with light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}). +#+caption: Effect of the payload dynamics on the soft active platform dynamics $L^{\prime}/F$ with light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,\text{N}/\upmu\text{m}$, $m_s = 1\,\text{kg}$} @@ -2050,7 +2054,7 @@ 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 active platform) and the acrshort:poi to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). +Having a flexibility between the measured position (i.e., the top platform of the active platform) and the acrlong:poi (i.e., the sample point to be position on 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 active platform and the sample (considered sample modes are at $\omega_s = 20\,\text{Hz}$ and $\omega_n = 200\,\text{Hz}$). @@ -2062,11 +2066,11 @@ In this case, the measured (i.e., controlled) distance $d$ is no longer equal to [[file:figs/uniaxial_sample_flexibility_control.png]] The system dynamics is computed and IFF is applied using the same gains as those used in Section\nbsp{}ref:sec:uniaxial_active_damping. -Due to the collocation between the active platform and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample. -The acrlong:hac used in Section\nbsp{}ref:sec:uniaxial_position_control are then implemented on the damped plants. +Due to the collocation of the active platform and the force sensor used for IFF, the damped plants remain stable, and damping values similar to those obtained with a rigid sample are observed. +The acrlong:hac used in Section\nbsp{}ref:sec:uniaxial_position_control is 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. +Finally, closed-loop error 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. The cumulative amplitude spectrum of the measured distance $d$ (Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_d) shows that the added flexibility at the sample location has very little effect on the control performance. However, the cumulative amplitude spectrum of the distance $y$ (Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_y) shows that the stability of $y$ is degraded when the sample flexibility is considered and is degraded as $\omega_s$ is lowered. @@ -2095,12 +2099,12 @@ What happens is that above $\omega_s$, even though the motion $d$ can be control Payload dynamics is usually a major concern when designing a positioning system. In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the active platform $\omega_n$. The larger the sample mass, the larger the effect (i.e., change of high-frequency gain, appearance of additional resonances and anti-resonances). -A zero/pole pattern is observed if $\omega_s > \omega_n$ and a pole/zero pattern if $\omega_s > \omega_n$. -Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in\nbsp{}[[cite:&rankers98_machin Section 4.2]]. +A zero/pole pattern is observed if $\omega_s > \omega_n$ and a pole/zero pattern if $\omega_s < \omega_n$. +Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in\nbsp{}[[cite:&rankers98_machin Chapt. 4.2]]. The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload. This is why high-bandwidth soft positioning stages are usually restricted to constant and calibrated payloads (CD-player, lithography machines, isolation system for gravitational wave detectors, ...), whereas stiff positioning systems are usually used when the control must be robust to a change of payload's mass (stiff piezo nano-positioning stages for instance). -Having some flexibility between the measurement point and the acrshort:poi (i.e., the sample point to be position on the x-ray) also degrades the position stability. +Having some flexibility between the measurement point and the acrshort:poi also degrades the position stability. Therefore, it is important to take special care when designing sampling environments, especially if a soft active platform is used. *** Conclusion @@ -2151,7 +2155,7 @@ 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-acrshortpl:dof 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-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. @@ -2160,13 +2164,13 @@ The position of the payload is represented by $(d_u, d_v, 0)$ expressed in the r After the dynamics of this system is studied, the objective will be to dampen the two suspension modes of the payload while the rotating stage performs a constant rotation. #+name: fig:rotating_3dof_model_schematic -#+caption: Schematic of the studied 2-DoFs translation stage on top of a rotation stage. +#+caption: Schematic of the studied 2-DoF translation stage on top of a rotation stage. #+attr_latex: :scale 0.8 [[file:figs/rotating_3dof_model_schematic.png]] ***** Equations of Motion and Transfer Functions To obtain the equations of motion for the system represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic, the Lagrangian equation\nbsp{}eqref:eq:rotating_lagrangian_equations is used. -$L = T - V$ is the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. +$L = T - V$ is the Lagrangian, $T$ the kinetic energy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. These terms are derived in\nbsp{}eqref:eq:rotating_energy_functions_lagrange. Note that the equation of motion corresponding to constant rotation around $\vec{i}_w$ is disregarded because this motion is imposed by the rotation stage. @@ -2988,12 +2992,12 @@ While quite simplistic, this allowed us to study the effects of rotation and the In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. ***** Nano Active Stabilization System Model -To have a more realistic dynamics model of the NASS, the 2-DoFs active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): +To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modelled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): - the floor whose motion is imposed -- a 2-DoFs granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\micro\m}$, $m_g = \SI{2500}{\kg}$) -- a 2-DoFs $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\micro\m}$, $m_t = \SI{600}{\kg}$) +- a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\micro\m}$, $m_g = \SI{2500}{\kg}$) +- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\micro\m}$, $m_t = \SI{600}{\kg}$) - a spindle (vertical rotation) stage whose rotation is imposed ($m_s = \SI{600}{\kg}$) -- a 2-DoFs positioning hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\micro\m}$, $m_h = \SI{15}{\kg}$) +- a 2-DoF positioning hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\micro\m}$, $m_h = \SI{15}{\kg}$) A payload is rigidly fixed to the active platform and the $x,y$ motion of the payload is measured with respect to the granite. @@ -3145,8 +3149,8 @@ As for the uniaxial model, three active platform stiffnesses values were conside The dynamics of the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). In addition, the attainable damping ratio of the soft active platform when using acrshort:iff is limited by gyroscopic effects. -To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been considered. -Results are similar to those of the uniaxial model except that some complexity is added for the soft active platform due to the spindle's rotation. +To better match the acrlong:nass dynamics, the limited compliance of the micro-station has been considered. +The results are similar to those of the uniaxial model, except for additional complexity introduced by the spindle’s rotation in the case of the soft active platform. For the moderately stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model. ** Micro Station - Modal Analysis @@ -3235,7 +3239,7 @@ 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 $n \times n$ acrshort:frf matrix $\bm{H}$ needs to be measured, where $n$ is the considered number of acrshortpl:dof. +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 degrees of freedom. 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 making $n \times n$ 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]]. @@ -3422,15 +3426,15 @@ 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 positioning hexapod. -Therefore, only $6 \times 6 = 36$ acrshortpl:dof are of interest. +Therefore, only $6 \times 6 = 36$ degrees of freedom 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 coordinate transformation from accelerometers acrshort:dof to the solid body 6-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 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. -**** From Accelerometer DOFs to Solid Body DOFs +**** From Accelerometer DoF to Solid Body DoF <> Let us consider the schematic shown in Figure\nbsp{}ref:fig:modal_local_to_global_coordinates where the motion of a solid body is measured at 4 distinct locations (in $x$, $y$ and $z$ directions). @@ -3442,7 +3446,7 @@ The goal here is to link these $4 \times 3 = 12$ measurements to the 6 acrshort: The motion of the rigid body of figure\nbsp{}ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$. -The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation\nbsp{}eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation\nbsp{}eqref:eq:modal_rotation_matrix\nbsp{}[[cite:&ewins00_modal chapt. 4.3.2]]. +The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation\nbsp{}eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation\nbsp{}eqref:eq:modal_rotation_matrix [[cite:&ewins00_modal chapt. 4.3.2]]. \begin{equation}\label{eq:modal_compute_point_response} \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\ @@ -3485,7 +3489,7 @@ The position of each accelerometer with respect to the acrlong:com of the corres #+name: tab:modal_com_solid_bodies #+caption: Center of mass of considered solid bodies with respect to the acrlong:poi. -#+attr_latex: :environment tabularx :width 0.45\linewidth :align Xccc +#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xccc #+attr_latex: :center t :booktabs t | | $X$ | $Y$ | $Z$ | |---------------------+-----------------+------------------+--------------------| @@ -3515,14 +3519,14 @@ Using\nbsp{}eqref:eq:modal_cart_to_acc, the frequency response matrix $\bm{H}_\t **** Verification of the Solid Body Assumption <> -From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\bm{H}_{\text{CoM}}$), and using equation\nbsp{}eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location. +From the response of one solid body expressed by its 6-DoF (i.e. from $\bm{H}_{\text{CoM}}$), and using equation\nbsp{}eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location. In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements $\bm{H}$. This is what is done here to check whether the solid body assumption is correct in the frequency band of interest. The comparison is made for the 4 accelerometers fixed on the positioning hexapod (Figure\nbsp{}ref:fig:modal_comp_acc_solid_body_frf). The original acrshortpl:frf and those computed from the CoM responses match well in the frequency range of interest. Similar results were obtained for the other solid bodies, indicating that the solid body assumption is valid and that a multi-body model can be used to represent the dynamics of the micro-station. -This also validates the reduction in the number of acrshortpl:dof from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof). +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). #+name: fig:modal_comp_acc_solid_body_frf #+caption: Comparison of the original accelerometer responses with responses reconstructed from the solid body response. Accelerometers 1 to 4, corresponding to the positioning hexapod, are shown. Input is a hammer force applied on the positioning hexapod in the $x$ direction. @@ -3546,7 +3550,7 @@ To validate the quality of the modal model, the full acrshort:frf matrix is comp **** Determination of the 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 acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement degrees of freedom (here $n=69$) and $p$ is the number of excitation degrees of freedom (here $p=3$). 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} @@ -3723,7 +3727,7 @@ Thanks to an adequate choice of instrumentation and proper set of measurements, 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 acrshortpl:dof on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body. +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. 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. @@ -3736,7 +3740,7 @@ However, the measurements are useful for tuning the parameters of the micro-stat From the start of this work, it became increasingly clear that an accurate micro-station model was necessary. First, during the uniaxial study, it became clear that the micro-station dynamics affects the active platform dynamics. -Then, using the 3-DoFs rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modeled. +Then, using the 3-DoF rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modelled. Finally, a modal analysis of the micro-station showed how complex the dynamics of the station is. The modal analysis also confirms that each stage behaves as a rigid body in the frequency range of interest. Therefore, a multi-body model is a good candidate to accurately represent the micro-station dynamics. @@ -3748,7 +3752,7 @@ The kinematics of the micro-station (i.e. how the motion of the stages are combi Then, the multi-body model is presented and tuned to match the measured dynamics of the micro-station (Section\nbsp{}ref:sec:ustation_modeling). -Disturbances affecting the positioning accuracy also need to be modeled properly. +Disturbances affecting the positioning accuracy also need to be modelled properly. To do so, the effects of these disturbances were first measured experimentally and then injected into the multi-body model (Section\nbsp{}ref:sec:ustation_disturbances). To validate the accuracy of the micro-station model, "real world" experiments are simulated and compared with measurements in Section\nbsp{}ref:sec:ustation_experiments. @@ -3767,11 +3771,10 @@ Such a stacked architecture allows high mobility, but the overall stiffness is r [[file:figs/ustation_cad_view.png]] There are different ways of modeling the stage dynamics in a multi-body model. -The one chosen in this work consists of modeling each stage by two solid bodies connected by one 6-DoFs joint. -The stiffness and damping properties of the joint -s can be tuned separately for each DoF. +The one chosen in this work consists of modeling each stage by two solid bodies connected by one 6-DoF joint. +The stiffness and damping properties of the joints can be tuned separately for each degree of freedom. -The "controlled" DoF of each stage (for instance the $D_y$ direction for the translation stage) is modeled as infinitely rigid (i.e. its motion is imposed by a "setpoint") while the other DoFs have limited stiffness to model the different micro-station modes. +The "controlled" degree of freedom of each stage (for instance the $D_y$ direction for the translation stage) is modelled as infinitely rigid (i.e. its motion is imposed by a "setpoint") while the other degrees of freedom have limited stiffness to model the different micro-station modes. **** Motion Stages <> @@ -3789,7 +3792,7 @@ Four cylindrical bearings[fn:ustation_4] are used to guide the motion (i.e. mini ***** Tilt Stage The tilt stage is guided by four linear motion guides[fn:ustation_1] which are placed such that the center of rotation coincide with the X-ray beam. -Each linear guide is very stiff in radial directions such that the only DoF with low stiffness is in $R_y$. +Each linear guide is very stiff in radial directions such that the only degree of freedom with low stiffness is in $R_y$. 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. @@ -3824,8 +3827,8 @@ 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 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. +This stage is used to position the acrfull:poi of the sample on the spindle rotation axis. +It can also be used to precisely position the acrshort:poi vertically with respect to the x-ray. #+attr_latex: :options [t]{0.49\linewidth} #+begin_minipage @@ -4025,7 +4028,7 @@ Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the [[file:figs/ustation_stage_motion.png]] The motion induced by a positioning stage can be described by a homogeneous transformation matrix from frame $\{A\}$ to frame $\{B\}$ as explain in Section\nbsp{}ref:ssec:ustation_kinematics. -As any motion stage induces parasitic motion in all 6-DoFs, the transformation matrix representing its induced motion can be written as in\nbsp{}eqref:eq:ustation_translation_stage_errors. +As any motion stage induces parasitic motion in all 6-DoF, the transformation matrix representing its induced motion can be written as in\nbsp{}eqref:eq:ustation_translation_stage_errors. \begin{equation}\label{eq:ustation_translation_stage_errors} {}^A\bm{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) = @@ -4091,8 +4094,8 @@ The inertia of the solid bodies and the stiffness properties of the guiding mech The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section\nbsp{}ref:ssec:ustation_model_comp_dynamics). -As the dynamics of the active platform is impacted by the micro-station compliance (see Section ref:sec:uniaxial_support_compliance), the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. -To do so, the 6-DoFs compliance of the micro-station is measured and then compared with the 6-DoFs compliance extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_model_compliance). +As the dynamics of the active platform is impacted by the micro-station compliance (see Section ref:sec:uniaxial_support_compliance), the most important dynamical characteristic that should be well modelled is the overall compliance of the micro-station. +To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_model_compliance). **** Multi-Body Model <> @@ -4106,13 +4109,13 @@ Joints are used to impose kinematic constraints between solid bodies and to spec External forces can be used to model disturbances, and "sensors" can be used to measure the relative pose between two defined frames. #+name: fig:ustation_simscape_stage_example -#+caption: Example of a stage (here the tilt-stage) represented in the multi-body model software (Simulink - Simscape). It is composed of two solid bodies connected by a 6-DoFs joint. One joint DoF (here the tilt angle) can be "controlled", the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included. +#+caption: Example of a stage (here the tilt-stage) represented in the multi-body model software (Simulink - Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint degree of freedom (here the tilt angle) can be "controlled", the other degrees of freedom are represented by springs and dampers. Additional disturbing forces for all degrees of freedom can be included. #+attr_latex: :scale 0.8 [[file:figs/ustation_simscape_stage_example.png]] -Therefore, the micro-station is modeled by several solid bodies connected by joints. -A typical stage (here the tilt-stage) is modeled as shown in Figure\nbsp{}ref:fig:ustation_simscape_stage_example where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoFs joint. -One DoF of the 6-DoFs joint is "imposed" by a setpoint (i.e. modeled as infinitely stiff), while the other 5 are each modeled by a spring and damper. +Therefore, the micro-station is modelled by several solid bodies connected by joints. +A typical stage (here the tilt-stage) is modelled as shown in Figure\nbsp{}ref:fig:ustation_simscape_stage_example where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoF joint. +One degree of freedom of the 6-DoF joint is "imposed" by a setpoint (i.e. modelled as infinitely stiff), while the other 5 are each modelled by a spring and damper. Additional forces can be used to model disturbances induced by the stage motion. The obtained 3D representation of the multi-body model is shown in Figure\nbsp{}ref:fig:ustation_simscape_model. @@ -4121,27 +4124,27 @@ The obtained 3D representation of the multi-body model is shown in Figure\nbsp{} #+attr_latex: :width 0.8\linewidth [[file:figs/ustation_simscape_model.jpg]] -The ground is modeled by a solid body connected to the "world frame" through a joint only allowing 3 translations. -The granite was then connected to the ground using a 6-DoFs joint. -The translation stage is connected to the granite by a 6-DoFs joint, but the $D_y$ motion is imposed. -Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoFs joint, with 1 imposed DoF each time. -Finally, the positioning hexapod has 6-DoFs. +The ground is modelled by a solid body connected to the "world frame" through a joint only allowing 3 translations. +The granite was then connected to the ground using a 6-DoF joint. +The translation stage is connected to the granite by a 6-DoF joint, but the $D_y$ motion is imposed. +Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoF joint, with 1 imposed degree of freedom each time. +Finally, the positioning hexapod has 6-DoF. -The total number of "free" acrshortpl:dof is 27, so the model has 54 states. +The total number of "free" degrees of freedom 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. #+name: tab:ustation_6dof_stiffness_values -#+caption: Summary of the stage stiffnesses. The constrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoFs joints are defined are indicated in figures found in Section ref:ssec:ustation_stages. +#+caption: Summary of the stage stiffnesses. The constrained degrees of freedom are indicated by "-". The frames in which the 6-DoF joints are defined are indicated in figures found in Section ref:ssec:ustation_stages. #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccccc #+attr_latex: :center t :booktabs t -| *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | -|-------------+-------------------------------+-------------------------------+-------------------------------+----------------------------------+---------------------------------+----------------------------------| -| Granite | $5\,\text{kN}/\upmu\text{m}$ | $5\,\text{kN}/\upmu\text{m}$ | $5\,\text{kN}/\upmu\text{m}$ | $25\,\text{Nm}/\upmu\text{rad}$ | $25\,\text{Nm}/\upmu\text{rad}$ | $10\,\text{Nm}/\upmu\text{rad}$ | -| Translation | $200\,\text{N}/\upmu\text{m}$ | - | $200\,\text{N}/\upmu\text{m}$ | $60\,\text{Nm}/\upmu\text{rad}$ | $90\,\text{Nm}/\upmu\text{rad}$ | $60\,\text{Nm}/\upmu\text{rad}$ | -| Tilt | $380\,\text{N}/\upmu\text{m}$ | $400\,\text{N}/\upmu\text{m}$ | $380\,\text{N}/\upmu\text{m}$ | $120\,\text{Nm}/\upmu\text{rad}$ | - | $120\,\text{Nm}/\upmu\text{rad}$ | -| Spindle | $700\,\text{N}/\upmu\text{m}$ | $700\,\text{N}/\upmu\text{m}$ | $2\,\text{kN}/\upmu\text{m}$ | $10\,\text{Nm}/\upmu\text{rad}$ | $10\,\text{Nm}/\upmu\text{rad}$ | - | -| Hexapod | $10\,\text{N}/\upmu\text{m}$ | $10\,\text{N}/\upmu\text{m}$ | $100\,\text{N}/\upmu\text{m}$ | $1.5\,\text{Nm/rad}$ | $1.5\,\text{Nm/rad}$ | $0.27\,\text{Nm/rad}$ | +| *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | +|-------------+-------------------------------+-------------------------------+-------------------------------+----------------------------------+----------------------------------+-----------------------------------| +| Granite | $5\,\text{kN}/\upmu\text{m}$ | $5\,\text{kN}/\upmu\text{m}$ | $5\,\text{kN}/\upmu\text{m}$ | $25\,\text{Nm}/\upmu\text{rad}$ | $25\,\text{Nm}/\upmu\text{rad}$ | $10\,\text{Nm}/\upmu\text{rad}$ | +| Translation | $200\,\text{N}/\upmu\text{m}$ | - | $200\,\text{N}/\upmu\text{m}$ | $60\,\text{Nm}/\upmu\text{rad}$ | $90\,\text{Nm}/\upmu\text{rad}$ | $60\,\text{Nm}/\upmu\text{rad}$ | +| Tilt | $380\,\text{N}/\upmu\text{m}$ | $400\,\text{N}/\upmu\text{m}$ | $380\,\text{N}/\upmu\text{m}$ | $120\,\text{Nm}/\upmu\text{rad}$ | - | $120\,\text{Nm}/\upmu\text{rad}$ | +| Spindle | $700\,\text{N}/\upmu\text{m}$ | $700\,\text{N}/\upmu\text{m}$ | $2\,\text{kN}/\upmu\text{m}$ | $10\,\text{Nm}/\upmu\text{rad}$ | $10\,\text{Nm}/\upmu\text{rad}$ | - | +| Hexapod | $10\,\text{N}/\upmu\text{m}$ | $10\,\text{N}/\upmu\text{m}$ | $100\,\text{N}/\upmu\text{m}$ | $1.5\,\text{Nm}/\upmu\text{rad}$ | $1.5\,\text{Nm}/\upmu\text{rad}$ | $0.27\,\text{Nm}/\upmu\text{rad}$ | **** Comparison with the Measured Dynamics <> @@ -4277,7 +4280,7 @@ These disturbance sources are then used during time domain simulations to accura The focus is on stochastic disturbances because, in principle, it is possible to calibrate the repeatable part of disturbances. Such disturbances include ground motions and vibrations induce by scanning the translation stage and the spindle. -In the multi-body model, stage vibrations are modeled as internal forces applied in the stage joint. +In the multi-body model, stage vibrations are modelled as internal forces applied in the stage joint. In practice, disturbance forces cannot be directly measured. Instead, the vibrations of the micro-station's top platform induced by the disturbances were measured (Section\nbsp{}ref:ssec:ustation_disturbances_meas). @@ -4290,7 +4293,7 @@ Finally, the obtained disturbance sources are compared in Section\nbsp{}ref:ssec In this section, ground motion is directly measured using geophones. Vibrations induced by scanning the translation stage and the spindle are also measured using dedicated setups. -The tilt stage and the positioning hexapod also have positioning errors; however, they are not modeled here because these two stages are only used for pre-positioning and not for scanning. +The tilt stage and the positioning hexapod also have positioning errors; however, they are not modelled here because these two stages are only used for pre-positioning and not for scanning. Therefore, from a control perspective, they are not important. ***** Ground Motion @@ -4361,7 +4364,7 @@ From the 5 measured displacements $[d_1,\,d_2,\,d_3,\,d_4,\,d_5]$, the translati #+caption: Experimental setup used to estimate the errors induced by the Spindle rotation (\subref{fig:ustation_rz_meas_lion}). The motion of the two reference spheres is measured using 5 capacitive sensors (\subref{fig:ustation_rz_meas_lion_zoom}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:ustation_rz_meas_lion}Micro-station and 5-DoFs metrology} +#+attr_latex: :caption \subcaption{\label{fig:ustation_rz_meas_lion}Micro-station and 5-DoF metrology} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :width 0.9\linewidth @@ -4377,7 +4380,7 @@ From the 5 measured displacements $[d_1,\,d_2,\,d_3,\,d_4,\,d_5]$, the translati A measurement was performed during a constant rotational velocity of the spindle of $60\,\text{rpm}$ and during 10 turns. 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. +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 are 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 acrshort:poi of the sample and the rotation axis will be considered because the alignment is not perfect in practice. @@ -4540,7 +4543,7 @@ 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 $\pm 4.5\,\text{mm}$. Both ground motion and translation stage vibrations were included in the simulation. -Similar to what was performed for the tomography simulation, the acrfull:poi position with respect to the granite was recorded and compared with the experimental measurements in Figure\nbsp{}ref:fig:ustation_errors_model_dy_vertical. +Similar to what was performed for the tomography simulation, the acrshort: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 @@ -4556,10 +4559,10 @@ A similar error amplitude was observed, thus indicating that the multi-body mode In this study, a multi-body model of the micro-station was developed. It was difficult to match the measured dynamics obtained from the modal analysis of the micro-station. -However, the most important dynamical characteristic to be modeled is the compliance, as it affects the dynamics of the NASS. +However, the most important dynamical characteristic to be modelled is the compliance, as it affects the dynamics of the NASS. After tuning the model parameters, a good match with the measured compliance was obtained (Figure\nbsp{}ref:fig:ustation_frf_compliance_model). -The disturbances affecting the sample position should also be well modeled. +The disturbances affecting the sample position should also be well modelled. After experimentally estimating the disturbances (Section\nbsp{}ref:sec:ustation_disturbances), the multi-body model was finally validated by performing a tomography simulation (Figure\nbsp{}ref:fig:ustation_errors_model_spindle) as well as a simulation in which the translation stage was scanned (Figure\nbsp{}ref:fig:ustation_errors_model_dy_vertical). ** Active Platform - Multi Body Model @@ -4602,7 +4605,7 @@ 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 acrshortpl:dof. +In many cases, sample position control is limited to translational degrees of freedom. At NSLS-II, for instance, a system capable of $100\,\upmu\text{m}$ stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure\nbsp{}ref:fig:nhexa_stages_nazaretski). Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately $100\,\text{Hz}$ (Figure\nbsp{}ref:fig:nhexa_stages_sapoti). @@ -4626,7 +4629,7 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac The integration of $R_z$ rotational capability, which is necessary for tomography experiments, introduces additional complexity. At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section\nbsp{}ref:sec:nhexa_stewart_platform) using piezoelectric actuators has been positioned below the spindle (Figure\nbsp{}ref:fig:nhexa_stages_villar). -While this configuration enables the correction of spindle motion errors through 5-DoFs control based on capacitive sensor measurements, the stroke is limited to $50\,\upmu\text{m}$ due to the inherent constraints of piezoelectric actuators. +While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to $50\,\upmu\text{m}$ due to the inherent constraints of piezoelectric actuators. In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\upmu\text{m}$ stroke (Figure\nbsp{}ref:fig:nhexa_stages_schroer). However, attempts to implement real-time feedback using YZ external metrology proved challenging, possibly due to the poor dynamical response of the serial stage configuration. @@ -4655,46 +4658,46 @@ Although direct performance comparisons between these systems are challenging du #+caption: End-Stations with integrated feedback loops based on online metrology. The stages used for feedback are indicated in bold font. Stages not used for scanning purposes are omitted or indicated between parentheses. The specifications for the NASS are indicated in the last row. #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccccc #+attr_latex: :placement [!ht] :center t :booktabs t -| *Stacked Stages* | *Specifications* | *Measured DoFs* | *Bandwidth* | *Reference* | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | 3 PID, n/a | APS | -| *XYZ stage (piezo)* | $D_{xyz}: 0.05\,\text{mm}$ | $D_{xyz}$ | | \nbsp{}[[cite:&nazaretski15_pushin_limit]] | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | light | Capacitive sensors | $\approx 10\,\text{Hz}$ | ESRF | -| Spindle | $R_z: \pm 90\,\text{deg}$ | $D_{xyz},\ R_{xy}$ | | ID16a | -| *Hexapod (piezo)* | $D_{xyz}: 0.05\,\text{mm}$ | | | \nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] | -| | $R_{xy}: 500\,\upmu\text{rad}$ | | | | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | n/a | PETRA III | -| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,\text{mm}$ | $D_{yz}$ | | P06 | -| Spindle | $R_z: 180\,\text{deg}$ | | | \nbsp{}[[cite:&schroer17_ptynam;&schropp20_ptynam]] | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | PID, n/a | PSI | -| Spindle | $R_z: \pm 182\,\text{deg}$ | $D_{yz},\ R_x$ | | OMNY | -| *Tripod (piezo)* | $D_{xyz}: 0.4\,\text{mm}$ | | | \nbsp{}[[cite:&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage]] | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | light | Interferometers | n/a | Soleil | -| (XY stage) | | $D_{xyz},\ R_{xy}$ | | Nanoprobe | -| Spindle | $R_z: 360\,\text{deg}$ | | | \nbsp{}[[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&engblom18_nanop_resul]] | -| *XYZ linear motors* | $D_{xyz}: 0.4\,\text{mm}$ | | | | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | up to $0.5\,\text{kg}$ | Interferometers | n/a | NSLS | -| Spindle | $R_z: 360\,\text{deg}$ | $D_{xyz}$ | | SRX | -| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,\text{mm}$ | | | \nbsp{}[[cite:&nazaretski22_new_kirkp_baez_based_scann]] | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | up to $0.35\,\text{kg}$ | Interferometers | $\approx 100\,\text{Hz}$ | Diamond, I14 | -| *Parallel XYZ VC* | $D_{xyz}: 3\,\text{mm}$ | $D_{xyz}$ | | \nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano]] | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | light | Capacitive sensors | $\approx 100\,\text{Hz}$ | LNLS | -| *Parallel XYZ VC* | $D_{xyz}: 3\,\text{mm}$ | and interferometers | | CARNAUBA | -| (Spindle) | $R_z: \pm 110 \,\text{deg}$ | $D_{xyz}$ | | \nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] | -|---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| -| Sample | up to $50\,\text{kg}$ | $D_{xyz},\ R_{xy}$ | | ESRF | -| *Active Platform* | | | | ID31 | -| (Hexapod) | | | | \nbsp{}[[cite:&dehaeze18_sampl_stabil_for_tomog_exper;&dehaeze21_mechat_approac_devel_nano_activ_stabil_system]] | -| Spindle | $R_z: 360\,\text{deg}$ | | | | -| Tilt-Stage | $R_y: \pm 3\,\text{deg}$ | | | | -| Translation Stage | $D_y: \pm 10\,\text{mm}$ | | | | +| *Stacked Stages* | *Specifications* | *Measured DoF* | *Bandwidth* | *Reference* | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | 3 PID, n/a | APS | +| *XYZ stage (piezo)* | $D_{xyz}: 0.05\,\text{mm}$ | $D_{xyz}$ | | \nbsp{}[[cite:&nazaretski15_pushin_limit]] | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | light | Capacitive sensors | $\approx 10\,\text{Hz}$ | ESRF | +| Spindle | $R_z: \pm 90\,\text{deg}$ | $D_{xyz},\ R_{xy}$ | | ID16a | +| *Hexapod (piezo)* | $D_{xyz}: 0.05\,\text{mm}$ | | | \nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] | +| | $R_{xy}: 500\,\upmu\text{rad}$ | | | | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | n/a | PETRA III | +| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,\text{mm}$ | $D_{yz}$ | | P06 | +| Spindle | $R_z: 180\,\text{deg}$ | | | \nbsp{}[[cite:&schroer17_ptynam;&schropp20_ptynam]] | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | PID, n/a | PSI | +| Spindle | $R_z: \pm 182\,\text{deg}$ | $D_{yz},\ R_x$ | | OMNY | +| *Tripod (piezo)* | $D_{xyz}: 0.4\,\text{mm}$ | | | \nbsp{}[[cite:&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage]] | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | light | Interferometers | n/a | Soleil | +| (XY stage) | | $D_{xyz},\ R_{xy}$ | | Nanoprobe | +| Spindle | $R_z: 360\,\text{deg}$ | | | \nbsp{}[[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&engblom18_nanop_resul]] | +| *XYZ linear motors* | $D_{xyz}: 0.4\,\text{mm}$ | | | | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | up to $0.5\,\text{kg}$ | Interferometers | n/a | NSLS | +| Spindle | $R_z: 360\,\text{deg}$ | $D_{xyz}$ | | SRX | +| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,\text{mm}$ | | | \nbsp{}[[cite:&nazaretski22_new_kirkp_baez_based_scann]] | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | up to $0.35\,\text{kg}$ | Interferometers | $\approx 100\,\text{Hz}$ | Diamond, I14 | +| *Parallel XYZ VC* | $D_{xyz}: 3\,\text{mm}$ | $D_{xyz}$ | | \nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano]] | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | light | Capacitive sensors | $\approx 100\,\text{Hz}$ | LNLS | +| *Parallel XYZ VC* | $D_{xyz}: 3\,\text{mm}$ | and interferometers | | CARNAUBA | +| (Spindle) | $R_z: \pm 110 \,\text{deg}$ | $D_{xyz}$ | | \nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] | +|-------------------+-----------------------------+---------------------+--------------------+--------------------------------------------------------------------------------------------------------| +| Sample | up to $50\,\text{kg}$ | $D_{xyz},\ R_{xy}$ | | ESRF | +| *Active Platform* | | | | ID31 | +| (Hexapod) | | | | \nbsp{}[[cite:&dehaeze18_sampl_stabil_for_tomog_exper;&dehaeze21_mechat_approac_devel_nano_activ_stabil_system]] | +| Spindle | $R_z: 360\,\text{deg}$ | | | | +| Tilt-Stage | $R_y: \pm 3\,\text{deg}$ | | | | +| Translation Stage | $D_y: \pm 10\,\text{mm}$ | | | | The first key distinction of the NASS is in the continuous rotation of the active vibration platform. This feature introduces significant complexity through gyroscopic effects and real-time changes in the platform orientation, which substantially impact both the system's kinematics and dynamics. @@ -4717,7 +4720,7 @@ The primary control requirements focus on $[D_y,\ D_z,\ R_y]$ motions; however, <> The choice of the active platform architecture for the NASS requires careful consideration of several critical specifications. -The platform must provide control over five acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\upmu\text{m}$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of $300\,\text{mm}$ diameter and $95\,\text{mm}$ height. +The platform must provide control over 5-DoF ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\upmu\text{m}$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of $300\,\text{mm}$ diameter and $95\,\text{mm}$ height. It must accommodate payloads up to $50\,\text{kg}$ while maintaining high dynamical performance. For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to $100\,\text{Hz}$. However, achieving such resonance frequencies with a $50\,\text{kg}$ payload would require unrealistic stiffness values of approximately $2000\,\text{N}/\upmu\text{m}$. @@ -4734,7 +4737,7 @@ 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\nbsp{}[[cite:&dong07_desig_precis_compl_paral_posit]] to address various positioning requirements, with designs varying based on the intended acrshortpl:dof 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 intended degrees of freedom 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 @@ -4747,7 +4750,7 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha #+attr_latex: :height 4.5cm [[file:figs/nhexa_serial_architecture_kenton.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoFs stage \cite{shen19_dynam_analy_flexur_nanop_stage}} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoF stage \cite{shen19_dynam_analy_flexur_nanop_stage}} #+attr_latex: :options {0.55\textwidth} #+begin_subfigure #+attr_latex: :height 4.5cm @@ -4756,7 +4759,7 @@ 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 allow 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. +In addition to allow 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. 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. @@ -4807,7 +4810,7 @@ These theoretical foundations form the basis for subsequent design decisions and <> The Stewart platform consists of two rigid platforms connected by six parallel struts (Figure\nbsp{}ref:fig:nhexa_stewart_architecture). -Each strut is modeled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. +Each strut is modelled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom[fn:nhexa_1]. #+name: fig:nhexa_stewart_architecture @@ -5143,25 +5146,25 @@ From these parameters, key kinematic properties can be derived: the strut orient ***** Inertia of Plates -The fixed base and moving platform were modeled as solid cylindrical bodies. +The fixed base and moving platform were modelled as solid cylindrical bodies. The base platform was characterized by a radius of $120\,\text{mm}$ and thickness of $15\,\text{mm}$, matching the dimensions of the positioning hexapod's top platform. -The moving platform was similarly modeled with a radius of $110\,\text{mm}$ and thickness of $15\,\text{mm}$. +The moving platform was similarly modelled with a radius of $110\,\text{mm}$ and thickness of $15\,\text{mm}$. Both platforms were assigned a mass of $5\,\text{kg}$. ***** Joints 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. +At both the upper and lower connection points, various degrees of freedom can be modelled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components. 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. +In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modelled 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. ***** Actuators The actuator model comprises several key elements (Figure\nbsp{}ref:fig:nhexa_actuator_model). -At its core, each actuator is modeled as a prismatic joint with internal stiffness $k_a$ and damping $c_a$, driven by a force source $f$. -Similarly to what was found using the rotating 3-DoFs model, a parallel stiffness $k_p$ is added in parallel with the force sensor to ensure stability when considering spindle rotation effects. +At its core, each actuator is modelled as a prismatic joint with internal stiffness $k_a$ and damping $c_a$, driven by a force source $f$. +Similarly to what was found using the rotating 3-DoF model, a parallel stiffness $k_p$ is added in parallel with the force sensor to ensure stability when considering spindle rotation effects. Each actuator is equipped with two sensors: a force sensor providing measurements $f_n$ and a relative motion sensor that measures the strut length $l_i$. The actuator parameters used in the conceptual phase are listed in Table\nbsp{}ref:tab:nhexa_actuator_parameters. @@ -5230,7 +5233,7 @@ 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 acrshortpl:dof of the moving platform. +The developed multi-body model yields a state-space representation with 12 states, corresponding to the 6-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. @@ -5249,7 +5252,7 @@ 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 acrshortpl:dof, only four distinct resonance frequencies were observed in the acrshortpl:frf. +While the system has six degrees of freedom, 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. @@ -5369,11 +5372,11 @@ 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 acrshortpl:dof, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics. +Some degrees of freedom, 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 directions - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational acrshortpl:dof 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 degrees of freedom can tolerate larger errors than others. -However, significant coupling exists between certain acrshortpl:dof, particularly between rotations and translations (e.g., $\epsilon_{R_x}/\mathcal{F}_y$ or $\epsilon_{D_y}/\bm\mathcal{M}_x$). +However, significant coupling exists between certain degrees of freedom, particularly between rotations and translations (e.g., $\epsilon_{R_x}/\mathcal{F}_y$ or $\epsilon_{D_y}/\bm\mathcal{M}_x$). For the conceptual validation of the acrshort:nass, control in the strut space was selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies. More sophisticated control strategies will be explored during the detailed design phase. @@ -5484,7 +5487,7 @@ This damping of structural resonances serves two purposes: it reduces vibrations #+end_figure Based upon the damped plant dynamics shown in Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_plant_damped, a high authority controller was designed with the structure given in\nbsp{}eqref:eq:nhexa_khac. -The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter for robustness against unmodeled high-frequency dynamics. +The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter for robustness against unmodelled high-frequency dynamics. The loop gain of an individual control channel is shown in Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_loop_gain. \begin{equation}\label{eq:nhexa_khac} @@ -5651,7 +5654,7 @@ Using these reference signals, the desired sample position relative to the fixed <> The external metrology system measures the sample position relative to the fixed granite. -Due to the system's symmetry, this metrology provides measurements for five acrshortpl:dof: three translations ($D_x$, $D_y$, $D_z$) and two rotations ($R_x$, $R_y$). +Due to the system's symmetry, this metrology provides measurements for 5-DoF: three translations ($D_x$, $D_y$, $D_z$) and two rotations ($R_x$, $R_y$). The sixth acrshort:dof ($R_z$) is still required to compute the errors in the frame of the active platform struts (i.e. to compute the active platform inverse kinematics). This $R_z$ rotation is estimated by combining measurements from the spindle encoder and the active platform's internal metrology. @@ -5706,7 +5709,7 @@ Finally, these errors are mapped to the strut space using the active platform Ja <> The complete control architecture is summarized in Figure\nbsp{}ref:fig:nass_control_architecture. -The sample pose is measured using external metrology for five acrshortpl:dof, while the sixth acrshort:dof ($R_z$) is estimated by combining measurements from the active platform encoders and spindle encoder. +The sample pose is measured using external metrology for 5-DoF, while the sixth acrshort:dof ($R_z$) is estimated by combining measurements from the active platform encoders and spindle encoder. The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and positioning hexapod. The position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the active platform frame. @@ -5726,7 +5729,7 @@ 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 acrshort:haclac strategy. -The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3-DoFs rotating model. +The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3-DoF rotating model. The objective here is to design a decentralized IFF controller that provides good damping of the active platform modes across payload masses ranging from $1$ to $50\,\text{kg}$ and rotational velocity up to $360\,\text{deg/s}$. The payloads used for validation have a cylindrical shape with $250\,\text{mm}$ height and with masses of $1\,\text{kg}$, $25\,\text{kg}$, and $50\,\text{kg}$. @@ -5759,7 +5762,7 @@ Although both cases show significant coupling around the resonances, stability i #+end_subfigure #+end_figure -The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,\text{N}/\upmu\text{m}$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3-DoFs rotating model). +The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,\text{N}/\upmu\text{m}$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3-DoF rotating model). Figure\nbsp{}ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics. The poles and zeros shift in frequency as the payload mass varies. @@ -5786,7 +5789,7 @@ However, their alternating pattern is preserved, which ensures the phase remains **** Controller Design <> -The previous analysis using the 3-DoFs rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation. +The previous analysis using the 3-DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation. This finding was also confirmed with the multi-body model of the NASS: the system was unstable when using pure integrators and without parallel stiffness. This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors. @@ -5933,7 +5936,7 @@ This result confirms effective dynamic decoupling between the active platform an **** Effect of Active Platform Stiffness on System Dynamics <> -The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3-DoFs) models. +The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degrees-of-freedom (3-DoF) models. These models suggest that a moderate stiffness of approximately $1\,\text{N}/\upmu\text{m}$ would provide better performance than either very stiff or very soft configurations. For the stiff active platform analysis, a system with an actuator stiffness of $100\,\text{N}/\upmu\text{m}$ was simulated with a $25\,\text{kg}$ payload. @@ -5968,7 +5971,7 @@ The current approach of controlling the position in the strut frame is inadequat <> A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure\nbsp{}ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at $10\,\text{Hz}$) for high performance operation. -The controller structure is defined in Equation\nbsp{}eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high-frequency modes. +The controller structure is defined in equation\nbsp{}eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high-frequency modes. \begin{equation}\label{eq:nass_robust_hac} 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}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right) @@ -6111,7 +6114,7 @@ As anticipated by the control analysis, some performance degradation was observe :END: Following the validation of the Nano Active Stabilization System concept in the previous chapter through simulated tomography experiments, this chapter addresses the refinement of the preliminary conceptual model into an optimized implementation. -The initial validation used an active platform with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise. +The initial validation used an active platform with arbitrary geometry, where components such as flexible joints and actuators were modelled as ideal elements, employing simplified control strategies without consideration for instrumentation noise. This detailed design phase aims to optimize each component while ensuring none will limit the system's overall performance. This chapter begins by determining the optimal geometric configuration for the active platform (Section\nbsp{}ref:sec:detail_kinematics). @@ -7095,7 +7098,7 @@ For the cubic configuration, complete dynamical decoupling in the Cartesian fram Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the active platform due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. For the active platform design, a key challenge was addressing the wide range of potential payloads (1 to $50\,\text{kg}$), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. -This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies. +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 (compared to the horizontal direction) by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies. ** Hybrid Modelling for Component Optimization <> @@ -7128,18 +7131,18 @@ Finally, the validity of this modeling approach is demonstrated through experime 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. +Standard FEA implementations typically involve thousands or even hundreds of thousands of degrees of freedom, rendering direct integration into multi-body simulations computationally prohibitive. +The objective of modal reduction is therefore to substantially decrease the number of degrees of freedom while preserving the essential dynamic characteristics of the component. The procedure for implementing this reduction involves several distinct stages. -Initially, the component is modeled in a finite element software with appropriate material properties and boundary conditions. +Initially, the component is modelled in a finite element software with appropriate material properties and boundary conditions. Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component. These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames. -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 still presenting the main dynamical characteristics. -This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100-DoFs. -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. +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 degrees of freedom while still presenting the main dynamical characteristics. +This transformation typically reduces the model complexity from hundreds of thousands to fewer than hundred degrees of freedom. +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 modelled. +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. \begin{equation}\label{eq:detail_fem_model_order} m = 6 \times n + p @@ -7498,19 +7501,19 @@ While this high order provides excellent accuracy for validation purposes, it pr The sensor and actuator "constants" ($g_s$ and $g_a$) derived in Section\nbsp{}ref:ssec:detail_fem_super_element_example for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks. -**** Simpler 2-DoFs Model of the APA300ML +**** Simpler 2-DoF Model of the APA300ML <> To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model, adapted from\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic]], was developed. This model, illustrated in Figure\nbsp{}ref:fig:detail_fem_apa_2dof_model, comprises three components. The mechanical shell is characterized by its axial stiffness $k_1$ and damping $c_1$. -The actuator is modeled with stiffness $k_a$ and damping $c_a$, incorporating a force source $f$. +The actuator is modelled with stiffness $k_a$ and damping $c_a$, incorporating a force source $f$. This force is related to the applied voltage $V_a$ through the actuator constant $g_a$. -The sensor stack is modeled with stiffness $k_e$ and damping $c_e$, with its deformation $d_L$ being converted to the output voltage $V_s$ through the sensor sensitivity $g_s$. +The sensor stack is modelled with stiffness $k_e$ and damping $c_e$, with its deformation $d_L$ being converted to the output voltage $V_s$ through the sensor sensitivity $g_s$. #+name: fig:detail_fem_apa_2dof_model -#+caption: Schematic of the 2-DoFs model of the Amplified Piezoelectric Actuator. +#+caption: Schematic of the 2-DoF model of the Amplified Piezoelectric Actuator. [[file:figs/detail_fem_apa_2dof_model.png]] While providing computational efficiency, this simplified model has inherent limitations. @@ -7534,7 +7537,7 @@ The resulting parameters, listed in Table\nbsp{}ref:tab:detail_fem_apa300ml_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 -#+caption: Summary of the obtained parameters for the 2-DoFs APA300ML model. +#+caption: Summary of the obtained parameters for the 2-DoF APA300ML model. #+attr_latex: :environment tabularx :width 0.25\linewidth :align cc #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | @@ -7549,7 +7552,7 @@ While higher-order modes and non-axial flexibility are not captured, the model a | $g_s$ | $0.53\,\text{V}/\upmu\text{m}$ | #+name: fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof -#+caption: Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2-DoFs model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}). +#+caption: Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2-DoF model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}from $V_a$ to $d_i$} @@ -7588,21 +7591,21 @@ These aspects will be addressed in the instrumentation chapter. **** Validation with the Active Platform <> -The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with acrshort:apa modeled as flexible bodies, and to validate the simplified 2-DoFs model through comparative analysis with the full acrshort:fem implementation. +The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with acrshort:apa modelled as flexible bodies, and to validate the simplified 2-DoF 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 acrshort:fem implementation and the simplified 2-DoFs 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 2-DoF model (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_plants) demonstrates remarkable agreement in the frequency range of interest. This validates the use of the simplified model for time-domain simulations. -The reduction in model order is substantial: while the acrshort:fem implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2-DoFs model requires only 24 states for the complete active platform. +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 2-DoF model requires only 24 states for the complete active platform. These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the active platform. #+name: fig:detail_fem_actuator_fem_vs_perfect_plants -#+caption: Comparison of the dynamics obtained between an active platform having the actuators modeled with FEM and an active platform having actuators modeled as 2-DoFs system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). +#+caption: Comparison of the dynamics obtained between an active platform having the actuators modelled with FEM and an active platform having actuators modelled as 2-DoF system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7652,7 +7655,7 @@ 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 acrshortpl:dof, 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 degrees of freedom, 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. @@ -7675,7 +7678,7 @@ This condition is more readily satisfied with the relatively stiff actuators sel For the force sensor plant, bending stiffness introduces complex conjugate zeros at low frequency (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_plant). This behavior resembles having parallel stiffness to the force sensor as was the case with the APA300ML (see Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant). -However, this time the parallel stiffness does not comes from the considered strut, but from the bending stiffness of the flexible joints of the other five struts. +However, this time the parallel stiffness does not come from the considered strut, but from the bending stiffness of the flexible joints of the other five struts. This characteristic impacts the achievable damping using decentralized Integral Force Feedback\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. This is confirmed by the root locus plot in Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_locus_1dof. This effect becomes less significant when using the selected APA300ML actuators (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml), which already incorporate parallel stiffness by design which is higher than the one induced by flexible joint stiffness. @@ -7701,7 +7704,7 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes #+end_figure #+name: fig:detail_fem_joints_bending_stiffness_iff_locus -#+caption: Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. For 1-DoF actuators (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2-DoFs model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}). +#+caption: Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. For 1-DoF actuators (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2-DoF model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1-DoF actuators} @@ -7799,7 +7802,7 @@ 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 active platform Jacobian matrix. Second, the design allows easy tuning of different directional stiffnesses through a limited number of geometric parameters. -Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational acrshortpl:dof. +Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational degrees of freedom. 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. @@ -7835,15 +7838,15 @@ The computed transfer functions from actuator forces to both force sensor measur [[file:figs/detail_fem_joints_frames.png]] While this detailed modeling approach provides high accuracy, it results in a significant increase in system model order. -The complete active platform model incorporates 240 states: 12 for the payload (6 DOF), 12 for the 2DOF struts, and 216 for the flexible joints (18 states for each of the 12 joints). -To improve computational efficiency, a low order representation was developed using simplified joint elements with selective compliance DoF. +The complete active platform model incorporates 240 states: 12 for the payload (6-DoF), 12 for the 2-DoF struts, and 216 for the flexible joints (18 states for each of the 12 joints). +To improve computational efficiency, a low order representation was developed using simplified joint elements with compliance only along the wanted degrees of freedom. 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 acrshortpl:dof could potentially capture more dynamic features, the selected configuration preserves essential system characteristics while minimizing computational complexity. +While additional degrees of freedom 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 an active platform including joints modeled with FEM and an active platform having 2-DoFs bottom joints and 3-DoFs top joints. 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_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}). +#+caption: Comparison of the dynamics obtained between an active platform including joints modelled with FEM and an active platform having 2-DoF bottom joints and 3-DoF top joints. 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_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7885,7 +7888,7 @@ Such model reduction, guided by detailed understanding of component behavior, pr Three critical elements for the control of parallel manipulators such as the active platform were identified: effective use and combination of multiple sensors, appropriate plant decoupling strategies, and robust controller design for the decoupled system. During the conceptual design phase of the NASS, pragmatic approaches were implemented for each of these elements. -The acrfull:haclac architecture was selected for combining sensors. +The acrlong: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. @@ -7907,11 +7910,11 @@ A method for directly shaping closed-loop transfer functions using complementary The literature review of Stewart platforms revealed a wide diversity of designs with various sensor and actuator configurations. 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]]. +In cases where multiple control objectives must be achieved simultaneously, as is the case for the acrshort: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: acrlong:haclac, sensor fusion, and two-sensor control architectures. #+name: fig:detail_control_control_multiple_sensors -#+caption: Different control architectures combining 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}) and Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}). +#+caption: Different control architectures combining multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}), Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}) and Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC-LAC} @@ -8466,12 +8469,12 @@ The decentralized plant (transfer functions from actuators to sensors integrated Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_jacobian), modal decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_modal), and Singular Value Decomposition (SVD) decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_svd). Finally, a comparative analysis with concluding observations is provided in Section\nbsp{}ref:ssec:detail_control_decoupling_comp. -**** 3-DoFs Test Model +**** 3-DoF Test Model <> Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate the analysis. The system illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test is used for this purpose. -It has three acrshortpl:dof and incorporates three parallel struts. +It has three degrees of freedom 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 acrlong:com of the solid body, and frame $\{K\}$ with origin $O_K$ at the acrlong:cok of the parallel manipulator. @@ -8728,7 +8731,7 @@ When a high-frequency force is applied at a point not aligned with the acrlong:c ***** Theory :ignore: Modal decoupling represents an approach based on the principle that a mechanical system's behavior can be understood as a combination of contributions from various modes\nbsp{}[[cite:&rankers98_machin]]. -To convert the dynamics in the modal space, the equation of motion are first written with respect to the center of mass\nbsp{}eqref:eq:detail_control_decoupling_equation_motion_CoM. +To convert the dynamics in the modal space, the equations of motion are first written with respect to the center of mass\nbsp{}eqref:eq:detail_control_decoupling_equation_motion_CoM. \begin{equation}\label{eq:detail_control_decoupling_equation_motion_CoM} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t) @@ -8847,7 +8850,7 @@ 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. -***** Test on the 3-DoFs model +***** Test on the 3-DoF model Plant decoupling using the Singular Value Decomposition was then applied on the test model. A decoupling frequency of $\SI{100}{Hz}$ was used. @@ -8959,7 +8962,7 @@ SVD decoupling can be implemented using measured data without requiring a model, |-----------------------+-----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------| | *Pros* | Retain physical meaning of inputs / outputs. Controller acts on a meaningfully "frame" | Ability to target specific modes. Simple $2^{nd}$ order diagonal plants | Good Decoupling near the crossover. Very General and requires no model | |-----------------------+-----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------| -| *Cons* | Good decoupling at all frequency can only be obtained for specific mechanical architecture | Relies on the accuracy of equation of motions. Robustness to unmodeled dynamics may be poor | Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor | +| *Cons* | Good decoupling at all frequency can only be obtained for specific mechanical architecture | Relies on the accuracy of equation of motions. Robustness to unmodelled dynamics may be poor | Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor | *** Closed-Loop Shaping using Complementary Filters <> @@ -9104,7 +9107,7 @@ Therefore, by carefully selecting the shape of the complementary filters, acrful ***** 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. -These discrepancies may arise from unmodeled dynamics or nonlinearities. +These discrepancies may arise from unmodelled dynamics or nonlinearities. To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure\nbsp{}ref:fig:detail_control_cf_input_uncertainty is employed. The set of possible plants $\Pi_i$ is described by\nbsp{}eqref:eq:detail_control_cf_multiplicative_uncertainty, with the weighting function $w_I$ selected such that all possible plants $G^\prime$ are contained within the set $\Pi_i$. @@ -9247,7 +9250,7 @@ The model of the plant $G(s)$ from actuator force $F$ to displacement $y$ is des G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = \SI{1}{\N/\micro\m},\ c = \SI{100}{\N\per(\m\per\s)} \end{equation} -The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics and payload dynamics. +The plant dynamics include uncertainties related to limited support compliance, unmodelled flexible dynamics and payload dynamics. These uncertainties are represented using a multiplicative input uncertainty weight\nbsp{}eqref:eq:detail_control_cf_test_plant_uncertainty, which specifies the magnitude of uncertainty as a function of frequency. \begin{equation}\label{eq:detail_control_cf_test_plant_uncertainty} @@ -9282,7 +9285,7 @@ For this example, the requirements are: - filtering of measurement noise above $\SI{300}{Hz}$, where sensor noise is significant (requiring a filtering factor of approximately 100 above this frequency) - maximizing disturbance rejection -Additionally, robust stability must be ensured, requiring the closed-loop system to remain stable despite the dynamic uncertainties modeled by $w_I$. +Additionally, robust stability must be ensured, requiring the closed-loop system to remain stable despite the dynamic uncertainties modelled by $w_I$. This condition is satisfied when the magnitude of the low-pass complementary filter $|H_L(j\omega)|$ remains below the inverse of the uncertainty weight magnitude $|w_I(j\omega)|$, as expressed in Equation\nbsp{}eqref:eq:detail_control_cf_condition_robust_stability. Robust performance is achieved when both nominal performance and robust stability conditions are simultaneously satisfied. @@ -9398,7 +9401,7 @@ Figure\nbsp{}ref:fig:detail_instrumentation_plant illustrates the control diagra 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 (acrshort:adc, acrshort:dac, and voltage amplifier). -This analysis is based on the multi-body model with a 2-DoFs acrshort:apa model, focusing particularly on the vertical direction due to its more stringent requirements. +This analysis is based on the multi-body model with a 2-DoF 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. @@ -9419,7 +9422,7 @@ The measured noise characteristics are then incorporated into the multi-body mod 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-DoFs Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs. +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 acrfull:asd of the noise sources is derived (Section\nbsp{}ref:ssec:detail_instrumentation_max_noise_specs). @@ -9457,8 +9460,8 @@ In order to derive specifications in terms of noise spectral density for each in The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met. While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications. -Based on this analysis, the obtained maximum noise levels are as follows: acrshort:dac maximum output noise acrshort:asd is established at $14\,\upmu\text{V}/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise acrshort:asd at $280\,\upmu\text{V}/\sqrt{\text{Hz}}$, and acrshort:adc maximum measurement noise acrshort:asd at $11\,\upmu\text{V}/\sqrt{\text{Hz}}$. -In terms of RMS noise, these translate to less than $1\,\text{mV RMS}$ for the acrshort:dac, less than $20\,\text{mV RMS}$ for the voltage amplifier, and less than $0.8\,\text{mV RMS}$ for the acrshort:adc. +Based on this analysis, the obtained maximum noise levels are as follows: acrshort:dac maximum output noise acrshort:asd is established at $32\,\upmu\text{V}/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise acrshort:asd at $650\,\upmu\text{V}/\sqrt{\text{Hz}}$, and acrshort:adc maximum measurement noise acrshort:asd at $35\,\upmu\text{V}/\sqrt{\text{Hz}}$. +In terms of RMS noise, these translate to $2.3\,\text{mV RMS}$ for the acrshort:dac, less than $46\,\text{mV RMS}$ for the voltage amplifier, and $2.5\,\text{mV RMS}$ for the acrshort:adc. If the Amplitude Spectral Density of the noise of the acrshort:adc, acrshort:dac, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below $15\,\text{nm RMS}$. @@ -9510,17 +9513,17 @@ Therefore, ideally, a voltage amplifier capable of providing $0.3\,\text{A}$ of ***** Output voltage noise -As established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the output noise of the voltage amplifier should be below $20\,\text{mV RMS}$. +As established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the output noise of the voltage amplifier should be below $46\,\text{mV RMS}$. It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure\nbsp{}ref:fig:detail_instrumentation_amp_output_impedance. Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement\nbsp{}[[cite:&spengen20_high_voltag_amplif]]. -For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\upmu\text{F}$ and a bandwidth exceeding $5\,\text{kHz}$. +For this application, the output noise must remain below $46\,\text{mV RMS}$ with a load of $8.8\,\upmu\text{F}$ and a bandwidth exceeding $5\,\text{kHz}$. ***** Choice of voltage amplifier The specifications are summarized in Table\nbsp{}ref:tab:detail_instrumentation_amp_choice. -The most critical characteristics are the small signal bandwidth ($>5\,\text{kHz}$) and the output voltage noise ($<20\,\text{mV RMS}$). +The most critical characteristics are the small signal bandwidth ($>5\,\text{kHz}$) and the output voltage noise ($<46\,\text{mV RMS}$). 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 acrshort:asd of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies. @@ -9536,19 +9539,19 @@ The PD200 amplifier from PiezoDrive was ultimately selected because it meets all #+caption: Specifications for the voltage amplifier and considered commercial products. #+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcccc #+attr_latex: :center t :booktabs t :float t -| *Specifications* | PD200 | WMA-200 | LA75B | E-505 | +| *Specifications* | PD200 | WMA-200 | LA75B | E-505 | | | PiezoDrive | Falco | Cedrat | PI | |--------------------------------------------+-------------------------------+------------------------------+---------------------+---------------------| -| Input Voltage Range: $\pm 10\,\text{V}$ | $\pm 10\,\text{V}$ | $\pm8.75\,\text{V}$ | $-1/7.5\,\text{V}$ | $-2/12\,\text{V}$ | -| Output Voltage Range: $-20/150\,\text{V}$ | $-50/150\,\text{V}$ | $\pm 175\,\text{V}$ | $-20/150\,\text{V}$ | $-30/130\,\text{V}$ | +| Input Voltage Range: $\pm 10\,\text{V}$ | $\pm 10\,\text{V}$ | $\pm8.75\,\text{V}$ | $-1/7.5\,\text{V}$ | $-2/12\,\text{V}$ | +| Output Voltage Range: $-20/150\,\text{V}$ | $-50/150\,\text{V}$ | $\pm 175\,\text{V}$ | $-20/150\,\text{V}$ | $-30/130\,\text{V}$ | | Gain $>15$ | 20 | 20 | 20 | 10 | | Output Current $> 300\,\text{mA}$ | $900\,\text{mA}$ | $150\,\text{mA}$ | $360\,\text{mA}$ | $215\,\text{mA}$ | | Slew Rate $> 34\,\text{V/ms}$ | $150\,\text{V}/\upmu\text{s}$ | $80\,\text{V}/\upmu\text{s}$ | n/a | n/a | -| Output noise $< 20\,\text{mV RMS}$ | $0.7\,\text{mV}$ | $0.05\,\text{mV}$ | $3.4\,\text{mV}$ | $0.6\,\text{mV}$ | +| Output noise $< 46\,\text{mV RMS}$ | $0.7\,\text{mV}$ | $0.05\,\text{mV}$ | $3.4\,\text{mV}$ | $0.6\,\text{mV}$ | | (10uF load) | ($10\,\upmu\text{F}$ load) | ($10\,\upmu\text{F}$ load) | (n/a) | (n/a) | | Small Signal Bandwidth $> 5\,\text{kHz}$ | $6.4\,\text{kHz}$ | $300\,\text{Hz}$ | $30\,\text{kHz}$ | n/a | | ($10\,\upmu\text{F}$ load) | ($10\,\upmu\text{F}$ load) | ($10\,\upmu\text{F}$ load) | (unloaded) | (n/a) | -| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a | +| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a | **** ADC and DAC ***** Introduction :ignore: @@ -9578,7 +9581,7 @@ For real-time control applications, successive-approximation ADC remain the pred ***** ADC Noise -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 $11\,\upmu V/\sqrt{\text{Hz}}$. +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 $35\,\upmu V/\sqrt{\text{Hz}}$. acrshortpl:adc are subject to various noise sources. Quantization noise, which results from the discrete nature of digital representation, is one of these sources. @@ -9615,12 +9618,12 @@ 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 $\Delta V = 20\,\text{V}$ and a maximum allowed acrshort:asd $\Gamma_{\text{max}} = 11\,\upmu\text{V}/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 12.4$, which is readily achievable with commercial acrshortpl:adc. +With a sampling frequency $F_s = 10\,\text{kHz}$, an input range $\Delta V = 20\,\text{V}$ and a maximum allowed acrshort:asd $\Gamma_{\text{max}} = 35\,\upmu\text{V}/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 10.7$, which is readily achievable with commercial acrshortpl:adc. ***** DAC Output voltage noise -Similar to the acrshort:adc requirements, the acrshort:dac output voltage noise acrshort:asd should not exceed $14\,\upmu\text{V}/\sqrt{\text{Hz}}$. -This specification corresponds to a $\pm 10\,\text{V}$ acrshort:dac with 13-bit resolution, which is easily attainable with current technology. +Similar to the acrshort:adc requirements, the acrshort:dac output voltage noise acrshort:asd should not exceed $32\,\upmu\text{V}/\sqrt{\text{Hz}}$. +This specification corresponds to a $\pm 10\,\text{V}$ acrshort:dac with 11-bit resolution, which is easily attainable with current technology. ***** Choice of the ADC and DAC Board @@ -9717,8 +9720,8 @@ All acrshort:adc channels demonstrated similar performance, so only one channel' 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 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]]. +Given that the acrshort:adc can operate at $200\,\text{kSPS}$ while the real-time controller runs at $10\,\text{kSPS}$, 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 ($0.3\,\text{mV}$)\nbsp{}[[cite:&hauser91_princ_overs_d_conver]]. #+name: fig:detail_instrumentation_adc_noise_measured #+caption: Measured ADC noise (IO318). @@ -9729,7 +9732,7 @@ This approach is effective because the noise approximates white noise and its am To further validate the ADC's capability to effectively measure voltage generated by a piezoelectric stack, a test was conducted using the APA95ML. The setup is illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_setup, where two stacks are used as actuators (connected in parallel) and one stack serves as a sensor. -The voltage amplifier employed in this setup has a gain of 20. +The voltage amplifier employed in this setup has a gain of $20$. #+name: fig:detail_instrumentation_force_sensor_adc_setup #+caption: Schematic of the setup to validate the use of the ADC for reading the force sensor voltage. @@ -9906,7 +9909,7 @@ This measurement approach eliminates the influence of ADC-DAC-related time delay All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to $500\,\text{Hz}$, well within the specified requirements. -The identified dynamics shown in Figure\nbsp{}ref:fig:detail_instrumentation_pd200_tf can be accurately modeled as either a first-order low-pass filter or as a simple constant gain. +The identified dynamics shown in Figure\nbsp{}ref:fig:detail_instrumentation_pd200_tf can be accurately modelled as either a first-order low-pass filter or as a simple constant gain. #+name: fig:detail_instrumentation_pd200_tf #+caption: Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier. @@ -9940,15 +9943,15 @@ The noise profile exhibits characteristics of white noise with an amplitude of a [[file:figs/detail_instrumentation_vionic_asd.png]] #+end_minipage -**** Noise Budgeting from Measured Instrumentation Noise +**** Error 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. 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 $15\,\text{nm RMS}$. +The total motion induced by all noise sources combined is approximately $0.7\,\text{nm RMS}$, 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. #+name: fig:detail_instrumentation_cl_noise_budget -#+caption: Closed-loop noise budgeting using measured noise of instrumentation. +#+caption: Closed-loop error budgeting using measured noise of instrumentation. #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_cl_noise_budget.png]] @@ -9963,16 +9966,17 @@ The multi-body model created earlier served as a key tool for embedding instrume From the most stringent requirement (i.e. the specification on vertical sample motion limited to $15\,\text{nm RMS}$), detailed specifications for each noise source were methodically derived through dynamic error budgeting. Based on these specifications, appropriate instrumentation components were selected for the system. -The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate noise budgeting, such as amplitude spectral densities under specific load conditions. +The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate error budgeting, such as amplitude spectral densities under specific load conditions. Despite these challenges, suitable components were identified that theoretically met all requirements. The selected instrumentation was procured and thoroughly characterized. Initial measurements of the acrshort:adc system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit. -All components were found to meet or exceed their respective specifications. The acrshort:adc demonstrated noise levels of $5.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus the $11\,\upmu\text{V}/\sqrt{\text{Hz}}$ specification), the acrshort:dac showed $0.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus $14\,\upmu\text{V}/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $280\,\upmu\text{V}/\sqrt{\text{Hz}}$ 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 $5.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus the $35\,\upmu\text{V}/\sqrt{\text{Hz}}$ specification), the acrshort:dac showed $0.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus $32\,\upmu\text{V}/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $650\,\upmu\text{V}/\sqrt{\text{Hz}}$ limit, and the encoders achieved $1\,\text{nm RMS}$ noise (versus the $6\,\text{nm RMS}$ specification). Finally, the measured noise characteristics of all instrumentation components were included into the multi-body model to predict the actual system performance. -The combined effect of all noise sources was estimated to induce vertical sample vibrations of only $1.5\,\text{nm RMS}$, which is substantially below the $15\,\text{nm RMS}$ requirement. +The combined effect of all noise sources was estimated to induce vertical sample vibrations of only $0.7\,\text{nm RMS}$, which is substantially below the $15\,\text{nm RMS}$ requirement. This rigorous methodology spanning requirement formulation, component selection, and experimental characterization validates the instrumentation's ability to fulfill the nano active stabilization system's demanding performance specifications. ** Obtained Design: the "Nano-Hexapod" @@ -9984,7 +9988,7 @@ Several primary objectives guided the mechanical design. First, to ensure a well-defined Jacobian matrix used in the control architecture, accurate positioning of the top flexible joint rotation points and correct orientation of the struts were required. Secondly, space constraints necessitated that the entire platform fit within a cylinder with a radius of $120\,\text{mm}$ and a height of $95\,\text{mm}$. Thirdly, because performance predicted by the multi-body model was fulfilling the requirements, the final design was intended to approximate the behavior of this "idealized" active platform as closely as possible. -This objective implies that the frequencies of (un-modeled) flexible modes potentially detrimental to control performance needed to be maximized. +This objective implies that the frequencies of (un-modelled) flexible modes potentially detrimental to control performance needed to be maximized. Finally, considerations for ease of mounting, alignment, and maintenance were incorporated, specifically ensuring that struts could be easily replaced in the event of failure. #+name: fig:detail_design_nano_hexapod_elements @@ -10197,20 +10201,19 @@ 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 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 4-DoFs configuration is shown in Figure\nbsp{}ref:fig:detail_design_simscape_model_flexible_joint. +Several levels of detail were considered for modeling the flexible joints within the multi-body model: 2-DoF models incorporating only bending stiffness, 3-DoF models including additional torsional stiffness, and 4-DoF models further incorporating axial stiffness were evaluated. +The multi-body representation corresponding to the 4-DoF 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 acrshort:fea of the joint component. #+name: fig:detail_design_simscape_model_flexible_joint -#+caption: 4-DoFs multi-body model of the flexible joints. Axial, bending and torsional stiffnesses are modeled. +#+caption: 4-DoF multi-body model of the flexible joints. Axial, bending and torsional stiffnesses are modelled. #+attr_latex: :scale 1 [[file:figs/detail_design_simscape_model_flexible_joint.png]] ***** Amplified Piezoelectric Actuators 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 used within the simulation: a simplified 2-DoFs model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a acrshort:fem. +Two distinct representations of the acrshort:apa can be used within the simulation: a simplified 2-DoF model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a acrshort:fem. ***** Encoders @@ -10295,7 +10298,7 @@ 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 acrshort:fea. +Two models are developed and validated: a simplified two-degree-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. @@ -10335,7 +10338,7 @@ The dynamics from the generated acrshort:dac voltage (going through the voltage 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. +First, in Section\nbsp{}ref:sec:test_apa_model_2dof, a two-degree-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 acrshort:fem and imported into the multi-body model. @@ -10353,8 +10356,8 @@ This more complex model also captures well capture the axial dynamics of the APA Before measuring the dynamical characteristics of the APA300ML, simple measurements are performed. 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. +Then, the capacitances of the piezoelectric stacks are measured in Section\nbsp{}ref:ssec:test_apa_electrical_measurements. +The achievable strokes of the APA300ML are 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 acrshort:apa are measured and compared with a acrshort:fem. **** Geometrical Measurements @@ -10670,7 +10673,7 @@ A longer measurement was performed using different excitation signals (noise, sl The coherence (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase_coherence) is good even in the vicinity of the lightly damped zero, and the phase (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase_zoom) clearly indicates non-minimum phase behavior. Such non-minimum phase zero when using load cells has also been observed on other mechanical systems\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. -It could be induced to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear. +It could be due to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear. However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the root locus plot in Figure\nbsp{}ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice. @@ -10763,13 +10766,13 @@ The two obtained root loci are compared in Figure\nbsp{}ref:fig:test_apa_iff_roo #+end_subfigure #+end_figure -*** Two degrees-of-freedom Model +*** Two-degree-of-freedom Model <> ***** 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 acrshort:apa using the measured acrshortpl:frf. +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-degree-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. +This two-degree-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. #+name: fig:test_apa_bench_model @@ -10777,7 +10780,7 @@ After the model is presented, the procedure for tuning the model is described, a #+attr_latex: :width 0.7\linewidth [[file:figs/test_apa_bench_model.png]] -***** Two degrees-of-freedom APA Model +***** Two-degree-of-freedom APA Model The model of the amplified piezoelectric actuator is shown in Figure\nbsp{}ref:fig:test_apa_2dof_model. It can be decomposed into three components: @@ -10789,12 +10792,12 @@ 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 $\text{V/m}$) Such a simple model has some limitations: -- it only represents the axial characteristics of the acrshort:apa as it is modeled as infinitely rigid in the other directions +- it only represents the axial characteristics of the acrshort:apa as it is modelled 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 +- the creep and hysteresis of the piezoelectric stacks are not modelled 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-degree-of-freedom model of the APA300ML, adapted from [[cite:&souleille18_concep_activ_mount_space_applic]]. [[file:figs/test_apa_2dof_model.png]] ***** Tuning of the APA model :ignore: @@ -10802,7 +10805,7 @@ Such a simple model has some limitations: 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\nbsp{}ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section\nbsp{}ref:sec:test_apa_dynamics. #+name: fig:test_apa_2dof_model_simscape -#+caption: Schematic of the two degrees-of-freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$. +#+caption: Schematic of the two-degree-of-freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$. [[file:figs/test_apa_2dof_model_simscape.png]] First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. @@ -10833,7 +10836,7 @@ In the last step, $g_s$ and $g_a$ can be tuned to match the gain of the identifi The obtained parameters of the model shown in Figure\nbsp{}ref:fig:test_apa_2dof_model_simscape are summarized in Table\nbsp{}ref:tab:test_apa_2dof_parameters. #+name: tab:test_apa_2dof_parameters -#+caption: Summary of the obtained parameters for the 2-DoFs APA300ML model. +#+caption: Summary of the obtained parameters for the 2-DoF APA300ML model. #+attr_latex: :environment tabularx :width 0.25\linewidth :align cc #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | @@ -10850,13 +10853,13 @@ The obtained parameters of the model shown in Figure\nbsp{}ref:fig:test_apa_2dof ***** Obtained Dynamics :ignore: -The dynamics of the two degrees-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table\nbsp{}ref:tab:test_apa_2dof_parameters) from the multi-body model. +The dynamics of the two-degree-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table\nbsp{}ref:tab:test_apa_2dof_parameters) from the multi-body model. This is compared with the experimental data in Figure\nbsp{}ref:fig:test_apa_2dof_comp_frf. A good match can be observed between the model and the experimental data, both for the encoder (Figure\nbsp{}ref:fig:test_apa_2dof_comp_frf_enc) and for the force sensor (Figure\nbsp{}ref:fig:test_apa_2dof_comp_frf_force). This indicates that this model represents well the axial dynamics of the APA300ML. #+name: fig:test_apa_2dof_comp_frf -#+caption: Comparison of the measured frequency response functions and the identified dynamics from the 2-DoFs model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_2dof_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_2dof_comp_frf_force}). +#+caption: Comparison of the measured frequency response functions and the identified dynamics from the 2-DoF model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_2dof_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_2dof_comp_frf_force}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_2dof_comp_frf_enc}from $u$ to $d_e$} @@ -10970,14 +10973,14 @@ This consistency indicates good manufacturing tolerances, facilitating the model Although a non-minimum zero was identified in the transfer function from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase), it was found not to be problematic because a large amount of damping could be added using the integral force feedback strategy (Figure\nbsp{}ref:fig:test_apa_iff). Then, two different models were used to represent the APA300ML dynamics. -In Section\nbsp{}ref:sec:test_apa_model_2dof, a simple two degrees-of-freedom mass-spring-damper model was presented and tuned based on the measured dynamics. +In Section\nbsp{}ref:sec:test_apa_model_2dof, a simple two-degree-of-freedom mass-spring-damper model was presented and tuned based on the measured dynamics. 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 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. +The benefit of employing this model over the two-degree-of-freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved. Nonetheless, the /super element/ model's value will become clear in subsequent sections, when its capacity to accurately model the APA300ML's flexibility across various directions will be important. ** Flexible Joints @@ -11054,7 +11057,7 @@ Sixteen flexible joints have been ordered (shown in Figure\nbsp{}ref:fig:test_jo #+end_subfigure #+end_figure -In this document, the received flexible joints are characterized to ensure that they fulfill the requirements and such that they can well be modeled. +In this document, the received flexible joints are characterized to ensure that they fulfill the requirements and such that they can well be modelled. First, the flexible joints are visually inspected, and the minimum gaps (responsible for most of the joint compliance) are measured (Section\nbsp{}ref:sec:test_joints_flex_dim_meas). Then, a test bench was developed to measure the bending stiffness of the flexible joints. @@ -11881,19 +11884,19 @@ The struts were then disassembled and reassemble a second time to optimize align **** Model Dynamics <> -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. +Two models of the APA300ML are used here: a simple two-degree-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 acrshortpl:frf of the acrshort:apa alone. -The flexible joints were modeled with the 4-DoFs model (axial stiffness, two bending stiffnesses and one torsion stiffness). +The flexible joints were modelled with the 4-DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness). 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. However, the transfer function from $u$ to encoder displacement $d_e$ are not well matching for both models. -For the 2-DoFs model, this is normal because the resonances affecting the dynamics are not modeled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis). +For the 2-DoF model, this is normal because the resonances affecting the dynamics are not modelled at all (the APA300ML is modelled as infinitely rigid in all directions except the translation along it's actuation axis). For the flexible model, it will be shown in the next section that by adding some misalignment between the flexible joints and the APA300ML, this model can better represent the observed dynamics. #+name: fig:test_struts_comp_frf_flexible_model -#+caption: Comparison of the measured dynamics of the struts (black) with dynamics extracted from the multi-body model using the 2-DoFs APA model (blue), and using the reduced order flexible model of the APA300ML model (red). +#+caption: Comparison of the measured dynamics of the struts (black) with dynamics extracted from the multi-body model using the 2-DoF APA model (blue), and using the reduced order flexible model of the APA300ML model (red). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$ (interferometer)} @@ -12063,7 +12066,7 @@ Prior to the nano-hexapod assembly, all the struts were mounted and individually In Section\nbsp{}ref:sec:test_nhexa_mounting, the assembly procedure of the nano-hexapod is presented. To identify the dynamics of the nano-hexapod, a special suspended table was developed, which consisted of a stiff "optical breadboard" suspended on top of four soft springs. -The Nano-Hexapod was then mounted on top of the suspended table such that its dynamics is not affected by complex dynamics except from the suspension modes of the table that can be well characterized and modeled (Section\nbsp{}ref:sec:test_nhexa_table). +The Nano-Hexapod was then mounted on top of the suspended table such that its dynamics is not affected by complex dynamics except from the suspension modes of the table that can be well characterized and modelled (Section\nbsp{}ref:sec:test_nhexa_table). The obtained nano-hexapod dynamics is analyzed in Section\nbsp{}ref:sec:test_nhexa_dynamics, and compared with the multi-body model in Section\nbsp{}ref:sec:test_nhexa_model. @@ -12262,7 +12265,7 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{ <> The multi-body model of the suspended table consists simply of two solid bodies connected by 4 springs. -The 4 springs are here modeled with "bushing joints" that have stiffness and damping properties in x, y, and z directions. +The 4 springs are here modelled with "bushing joints" that have stiffness and damping properties in x, y, and z directions. The model order is 12, which corresponds to the 6 suspension modes. The inertia properties of the parts were determined from the geometry and material densities. @@ -12457,16 +12460,16 @@ 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-DoFs flexible joints, 2-DoFs acrshort:apa, and rigid top and bottom plates. +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 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 $122\,\text{Hz}$, $143\,\text{Hz}$, $165\,\text{Hz}$ and $191\,\text{Hz}$) are well modeled. -The three resonances that were attributed to "internal" flexible modes of the struts (at $237\,\text{Hz}$, $349\,\text{Hz}$ and $395\,\text{Hz}$) cannot be seen in the model, which is reasonable because the acrshortpl:apa are here modeled as a simple uniaxial 2-DoFs 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. +It can be seen that the 4 suspension modes of the nano-hexapod (at $122\,\text{Hz}$, $143\,\text{Hz}$, $165\,\text{Hz}$ and $191\,\text{Hz}$) are well modelled. +The three resonances that were attributed to "internal" flexible modes of the struts (at $237\,\text{Hz}$, $349\,\text{Hz}$ and $395\,\text{Hz}$) cannot be seen in the model, which is reasonable because the acrshortpl:apa are here modelled 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 modelled as rigid bodies. #+name: fig:test_nhexa_comp_simscape_diag #+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag}). @@ -12491,22 +12494,22 @@ At higher frequencies, no resonances can be observed in the model, as the top pl Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of acrshort:mimo systems. Instead of comparing the full 36 elements of the $6 \times 6$ acrshort:frf matrix from $\bm{u}$ to $\bm{d}_e$, only the first "column" is compared (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$. -It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at $237\,\text{Hz}$. +It can be seen that the coupling in the model matches the measurements well up to the first un-modelled flexible mode at $237\,\text{Hz}$. Similar results are observed for all other coupling terms and for the transfer function from $\bm{u}$ to $\bm{V}_s$. #+name: fig:test_nhexa_comp_simscape_de_all -#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a 2-DoFs mass-spring-damper system. No payload us used. +#+caption: Comparison of the measured (in blue) and modelled (in red) FRFs from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modelled with a 2-DoF mass-spring-damper system. No payload us used. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all.png]] -The APA300ML was then modeled with a /super-element/ extracted from a FE-software. +The APA300ML was then modelled with a /super-element/ extracted from a FE-software. The obtained transfer functions from $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$ are compared with the measured acrshort:frf in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all_flex. -While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at $237\,\text{Hz}$ and $349\,\text{Hz}$ are well modeled. +While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at $237\,\text{Hz}$ and $349\,\text{Hz}$ are well modelled. Even the mode $395\,\text{Hz}$ can be observed in the model. -Therefore, if the modes of the struts are to be modeled, the /super-element/ of the APA300ML can be used at the cost of obtaining a much higher order model. +Therefore, if the modes of the struts are to be modelled, the /super-element/ of the APA300ML can be used at the cost of obtaining a much higher order model. #+name: fig:test_nhexa_comp_simscape_de_all_flex -#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a "super-element". No payload us used. +#+caption: Comparison of the measured (in blue) and modelled (in red) FRFs from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modelled with a "super-element". No payload us used. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all_flex.png]] @@ -12545,7 +12548,7 @@ Excellent match between experimental and model coupling is observed. Therefore, the model effectively represents the system coupling for different payloads. #+name: fig:test_nhexa_comp_simscape_de_all_high_mass -#+caption: Comparison of the measured (in blue) and modeled (in red) FRF from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. $39\,\text{kg}$ payload is used. +#+caption: Comparison of the measured (in blue) and modelled (in red) FRF from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. $39\,\text{kg}$ payload is used. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all_high_mass.png]] @@ -12557,7 +12560,7 @@ Therefore, the model effectively represents the system coupling for different pa The goal of this test bench was to obtain an accurate model of the nano-hexapod that could then be included on top of the micro-station model. The adopted strategy was to identify the nano-hexapod dynamics under conditions in which all factors that could have affected the nano-hexapod dynamics were considered. -This was achieved by developing a suspended table with low frequency suspension modes that can be accurately modeled (Section\nbsp{}ref:sec:test_nhexa_table). +This was achieved by developing a suspended table with low frequency suspension modes that can be accurately modelled (Section\nbsp{}ref:sec:test_nhexa_table). Although the dynamics of the nano-hexapod was indeed impacted by the dynamics of the suspended platform, this impact was also considered in the multi-body model. The dynamics of the nano-hexapod was then identified in Section\nbsp{}ref:sec:test_nhexa_dynamics. @@ -12570,7 +12573,7 @@ This indicates that it is possible to implement decentralized Integral Force Fee 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 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. +At frequencies above the suspension modes, the Nano-Hexapod model became inaccurate because the flexible modes were not modelled. 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. @@ -12586,7 +12589,7 @@ If a model of the nano-hexapod was developed in one time, it would be difficult To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure\nbsp{}ref:fig:test_id31_micro_station_nano_hexapod. This section presents a comprehensive experimental evaluation of the complete system's performance on the ID31 beamline, focusing on its ability to maintain precise sample positioning under various experimental conditions. -Initially, the project planned to develop a long-stroke ($\approx 1 \, \text{cm}^3$) 5-DoFs metrology system to measure the sample position relative to the granite base. +Initially, the project planned to develop a long-stroke ($\approx 1 \, \text{cm}^3$) 5-DoF metrology system to measure the sample position relative to the granite base. However, the complexity of this development prevented its completion before the experimental testing phase on ID31. To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\upmu\text{m}^3$) metrology system was developed instead, which is presented in Section\nbsp{}ref:sec:test_id31_metrology. @@ -12631,7 +12634,7 @@ This system comprises 5 capacitive sensors facing two reference spheres. However, as the gap between the capacitive sensors and the spheres is very small[fn:test_id31_1], the risk of damaging the spheres and the capacitive sensors is too high. #+name: fig:test_id31_short_stroke_metrology -#+caption: Short stroke metrology system used to measure the sample position with respect to the granite in 5-DoFs. The system is based on a "Spindle error analyzer" (\subref{fig:test_id31_lion}), but the capacitive sensors are replaced with fibered interferometers (\subref{fig:test_id31_interf}). One interferometer head is shown in (\subref{fig:test_id31_interf_head}). +#+caption: Short stroke metrology system used to measure the sample position with respect to the granite in 5-DoF. The system is based on a "Spindle error analyzer" (\subref{fig:test_id31_lion}), but the capacitive sensors are replaced with fibered interferometers (\subref{fig:test_id31_interf}). One interferometer head is shown in (\subref{fig:test_id31_interf_head}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_lion}Capacitive Sensors} @@ -12847,8 +12850,8 @@ 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 $[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$ 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. +First, the five interferometers $\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]$ are used to measure five degrees of freedom $[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$ of the sample. +The sixth DoF of the sample ($R_z$) 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 $\bm{\epsilon\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]$. Finally, the Jacobian matrix $\bm{J}$ of the nano-hexapod is used to map $\bm{\epsilon\mathcal{X}}$ in the frame of the nano-hexapod struts $\bm{\epsilon\mathcal{L}} = [\epsilon_{\mathcal{L}_1},\,\epsilon_{\mathcal{L}_2},\,\epsilon_{\mathcal{L}_3},\,\epsilon_{\mathcal{L}_4},\,\epsilon_{\mathcal{L}_5},\,\epsilon_{\mathcal{L}_6}]$. @@ -12867,7 +12870,7 @@ The dynamics of the plant is first identified for a fixed spindle angle (at $0\, The model dynamics is also identified under the same conditions. A comparison between the model and the measured dynamics is presented in Figure\nbsp{}ref:fig:test_id31_first_id. -A good match can be observed for the diagonal dynamics (except the high-frequency modes which are not modeled). +A good match can be observed for the diagonal dynamics (except the high-frequency modes which are not modelled). 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 acrshort:frf (Figure\nbsp{}ref:fig:test_id31_first_id_int) is larger than expected. @@ -13062,7 +13065,7 @@ Similar results were obtained for all other 30 elements and for the different pa This confirms that the multi-body model can be used to tune the IFF controller. #+name: fig:test_id31_comp_simscape_Vs -#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal $u_1$ to the six force sensor voltages $V_{s1}$ to $V_{s6}$. +#+caption: Comparison of the measured (in blue) and modelled (in red) FRFs from the first control signal $u_1$ to the six force sensor voltages $V_{s1}$ to $V_{s6}$. #+attr_latex: :scale 0.8 [[file:figs/test_id31_comp_simscape_Vs.png]] @@ -13146,7 +13149,7 @@ To experimentally validate the Decentralized IFF controller, it was implemented 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 plant 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 simplicity. +#+caption: Comparison of the open-loop plant 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 modelled 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 simplicity. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_ol_iff_plant_model}Effect of IFF on the plant} @@ -13172,7 +13175,7 @@ The implementation of a decentralized Integral Force Feedback controller was suc Using the multi-body model, the controller was designed and optimized to ensure stability across all payload conditions while providing significant damping of suspension modes. The experimental results validated the model predictions, showing a reduction in peak amplitudes by approximately a factor of 10 across the full payload range (0 to $39\,\text{kg}$). Although higher gains could achieve better damping performance for lighter payloads, the chosen fixed-gain configuration represents a robust compromise that maintains stability and performance under all operating conditions. -The good correlation between the modeled and measured damped plants confirms the effectiveness of using the multi-body model for both controller design and performance prediction. +The good correlation between the modelled and measured damped plants confirms the effectiveness of using the multi-body model for both controller design and performance prediction. *** High Authority Control in the Frame of the Struts <> @@ -13202,7 +13205,7 @@ To verify whether the multi-body model accurately represents the measured damped Considering the complexity of the system's dynamics, the model can be considered to represent the system's dynamics with good accuracy, and can therefore be used to tune the feedback controller and evaluate its performance. #+name: fig:test_id31_comp_simscape_hac -#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal ($u_1^\prime$) of the damped plant to the estimated errors ($\epsilon_{\mathcal{L}_i}$) in the frame of the six struts by the external metrology. +#+caption: Comparison of the measured (in blue) and modelled (in red) FRFs from the first control signal ($u_1^\prime$) of the damped plant to the estimated errors ($\epsilon_{\mathcal{L}_i}$) in the frame of the six struts by the external metrology. #+attr_latex: :scale 0.8 [[file:figs/test_id31_comp_simscape_hac.png]] @@ -13743,7 +13746,7 @@ A schematic of the proposed control architecture is illustrated in Figure\nbsp{} [[file:figs/test_id31_cf_control.png]] 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-DoFs acrshort:apa (Section\nbsp{}ref:sec:test_apa_model_2dof) model and 4-DoFs flexible joints, 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 (Section\nbsp{}ref:sec:test_apa_model_2dof) model and 4-DoF flexible joints, such that the model order stays relatively low. Analytical formulas for complementary filters having $40\,\text{dB/dec}$ slopes, proposed in Section\nbsp{}ref:ssec:detail_control_cf_analytical_complementary_filters, 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. @@ -13858,7 +13861,7 @@ 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 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-DoFs position measurements. +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. The implementation of the control architecture validated the theoretical framework developed earlier in this project. @@ -13884,7 +13887,7 @@ 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 acrshort:fea. +Two models of the APA300ML were developed and validated: a simplified two-degree-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. @@ -13948,7 +13951,7 @@ 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 3D models. +Systems like the micro-station can be conceptually modelled 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. @@ -13995,7 +13998,7 @@ In\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]], an interesting metrolog ***** 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. +The original micro-station design was driven by optimizing positioning accuracy, using dedicated actuators for different degrees of freedom (leading to simple kinematics and a stacked configuration), and maximizing stiffness. This design philosophy ensured that the micro-station would remain functional for micro-focusing applications even if the NASS project did not meet expectations. Analyzing the NASS as an complete system reveals that the positioning accuracy is primarily determined by the metrology system and the feedback control. @@ -14019,11 +14022,11 @@ Stages based on voice coils, offering nano-positioning capabilities with $3\,\te 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\nbsp{}ref:fig:conclusion_maglev. -Specifically, a compact 6-DoFs stage known as LevCube, providing a mobility of approximately $1\,\text{cm}^3$, is depicted in Figure\nbsp{}ref:fig:conclusion_maglev_heyman23, while a 6-DoFs stage featuring infinite rotation, is shown in Figure\nbsp{}ref:fig:conclusion_maglev_dyck15. +Specifically, a compact 6-DoF stage known as LevCube, providing a mobility of approximately $1\,\text{cm}^3$, is depicted in Figure\nbsp{}ref:fig:conclusion_maglev_heyman23, while a 6-DoF stage featuring infinite rotation, is shown in Figure\nbsp{}ref:fig:conclusion_maglev_dyck15. However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature. #+name: fig:conclusion_maglev -#+caption: Example of magnetic levitation stages. LevCube allowing for 6-DoFs control of the position with $\approx 1\,\text{cm}^3$ mobility (\subref{fig:conclusion_maglev_heyman23}). Magnetic levitation stage with infinite $R_z$ rotation mobility (\subref{fig:conclusion_maglev_dyck15}). +#+caption: Example of magnetic levitation stages. LevCube allowing for 6-DoF control of the position with $\approx 1\,\text{cm}^3$ mobility (\subref{fig:conclusion_maglev_heyman23}). Magnetic levitation stage with infinite $R_z$ rotation mobility (\subref{fig:conclusion_maglev_dyck15}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:conclusion_maglev_heyman23}LevCube with $\approx 1\,\text{cm}^3$ mobility \cite{heyman23_levcub}}