From 67e1d2f4198c32003efac484d3338e83c8113f38 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 23 Apr 2025 14:15:45 +0200 Subject: [PATCH] Christophe's review on control --- phd-thesis.org | 644 ++++++++++++++++++++++++------------------------- 1 file changed, 318 insertions(+), 326 deletions(-) diff --git a/phd-thesis.org b/phd-thesis.org index 5440bb1..26beb39 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -359,7 +359,7 @@ Synchrotron light are emitted in more than 40 beamlines surrounding the storage These beamlines host diverse instrumentation that enables a wide spectrum of scientific investigations, including structural biology, materials science, and study of matter under extreme conditions. #+name: fig:instroduction_esrf -#+caption: Schematic (\subref{fig:introduction_esrf_schematic}) and picture (\subref{fig:introduction_esrf_picture}) of the European Synchrotron Radiation Facility, situated in Grenoble, France +#+caption: Schematic (\subref{fig:introduction_esrf_schematic}) and picture (\subref{fig:introduction_esrf_picture}) of the European Synchrotron Radiation Facility, situated in Grenoble, France. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_esrf_schematic} Schematic of the ESRF. The linear accelerator is shown in blue, the booster synchrotron in purple and the storage ring in green. There are over 40 beamlines, the ID31 beamline is highlighted in red} @@ -400,7 +400,7 @@ The goal of the beamline is therefore to filter and shape the X-rays to the desi These components are housed in multiple Optical Hutches, as depicted in Figure\nbsp{}ref:fig:introduction_id31_oh. #+name: fig:introduction_id31_oh -#+caption: Schematic of the two ID31 optical hutches: OH1 (\subref{fig:introduction_id31_oh1}) and OH2 (\subref{fig:introduction_id31_oh2}). Distance from the source (the insertion device) is indicated in meters. +#+caption: Schematic of the two ID31 optical hutches: OH1 (\subref{fig:introduction_id31_oh1}) and OH2 (\subref{fig:introduction_id31_oh2}). Distance from the source (i.e. from the insertion device) is indicated in meters. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_id31_oh1}OH1} @@ -462,10 +462,10 @@ Positional instabilities, such as vibrations and thermal drifts, inevitably lead Other advanced imaging modalities practiced on ID31 include reflectivity, diffraction tomography, and small/wide-angle X-ray scattering (SAXS/WAXS). #+name: fig:introduction_tomography -#+caption: Exemple of a tomography experiment. The sample is rotated and images are taken at several angles (\subref{fig:introduction_tomography_schematic}). Example of one 3D image obtained after tomography (\subref{fig:introduction_tomography_results}). +#+caption: Exemple of a tomography experiment. The sample is rotated and images are taken at several angles (\subref{fig:introduction_tomography_schematic}). Example of one 3D image obtained using tomography (\subref{fig:introduction_tomography_results}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:introduction_tomography_schematic} Experimental setup} +#+attr_latex: :caption \subcaption{\label{fig:introduction_tomography_schematic} Typical tomography experimental setup} #+attr_latex: :options {0.65\textwidth} #+begin_subfigure #+attr_latex: :scale 0.9 @@ -483,7 +483,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra #+caption: Exemple of a scanning experiment. The sample is scanned in the Y-Z plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of $20\,\text{nm}$ (\subref{fig:introduction_scanning_results}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:introduction_scanning_schematic} Experimental setup} +#+attr_latex: :caption \subcaption{\label{fig:introduction_scanning_schematic} Typical experimental setup for scanning experiment} #+attr_latex: :options {0.65\textwidth} #+begin_subfigure #+attr_latex: :scale 0.9 @@ -529,7 +529,7 @@ The historical reduction in achievable spot sizes is represented in Figure\nbsp{ Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines. #+name: fig:introduction_moore_law_focus -#+caption: Evolution of the measured spot size for different hard X-ray focusing elements. Adapated from\nbsp{}[[cite:&barrett24_x_optic_accel_based_light_sourc]] +#+caption: Evolution of the measured spot size for different hard X-ray focusing elements. Adapated from\nbsp{}[[cite:&barrett24_x_optic_accel_based_light_sourc]]. #+attr_latex: :scale 0.9 #+attr_latex: :options [h!tbp] [[file:figs/introduction_moore_law_focus.png]] @@ -541,7 +541,7 @@ In this mode, the sample is moved to the desired position, the detector acquisit While effective for mitigating radiation damage, this sequential process can be time-consuming, especially for high-resolution maps requiring numerous points. #+name: fig:introduction_scan_mode -#+caption: Two acquisition modes. In step-by-step mode (\subref{fig:introduction_scan_step}), the motor moves at the wanted imaged position, the detector acquisition is started, the shutter is openned briefly to have the wanted exposition, the detector acquisition is stopped, and the motor can move to a new position. In /fly-scan/ mode (\subref{fig:introduction_scan_fly}), the shutter is openned while the sample is in motion, and the detector is acquired only at the wanted positions, on the /fly/. +#+caption: Two acquisition modes. In step-by-step mode (\subref{fig:introduction_scan_step}), the motor moves at the wanted imaged position, the detector acquisition is started, the shutter is openned briefly to have the wanted exposure, the detector acquisition is stopped, and the motor can move to a new position. In /fly-scan/ mode (\subref{fig:introduction_scan_fly}), the shutter is openned while the sample is in motion, and the detector is acquired only at the wanted positions, on the /fly/. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_scan_step} Step by step scan} @@ -586,7 +586,7 @@ This configuration offers great mobility, but positioning errors (e.g., guiding Similarly, the overall dynamic performance (stiffness, resonant frequencies) is limited by the softest component in the stack, often resulting in poor dynamic behavior when many stages are combined. #+name: fig:introduction_kinematics -#+caption: Two positioning platforms with $D_x/D_y/R_z$ degrees of freedom. One is using serial kinematics (\subref{fig:introduction_serial_kinematics}), while the other uses parallel kinematics (\subref{fig:introduction_parallel_kinematics}) +#+caption: Two positioning platforms with $D_x/D_y/R_z$ degrees of freedom. One is using serial kinematics (\subref{fig:introduction_serial_kinematics}), while the other uses parallel kinematics (\subref{fig:introduction_parallel_kinematics}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_serial_kinematics} Serial Kinematics} @@ -613,7 +613,7 @@ Examples of such end-stations, including those at beamlines ID16B\nbsp{}[[cite:& 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. #+name: fig:introduction_passive_stations -#+caption: Example of two nano end-stations without online metrology +#+caption: Example of two nano end-stations without online metrology measuring the sample's position. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_endstation_id16b}ID16b end-station \cite{martinez-criado16_id16b}} @@ -642,7 +642,7 @@ The system at NSLS X8C (Figure\nbsp{}ref:fig:introduction_stages_wang) used capa The PtiNAMi microscope at DESY P06 (Figure\nbsp{}ref:fig:introduction_stages_schroer) employs interferometers directed at a spherical target below the sample for position monitoring during tomography, with plans for future feedback loop implementation\nbsp{}[[cite:&schropp20_ptynam]]. #+name: fig:introduction_metrology_stations -#+caption: Two examples of end-station with integrated online metrology +#+caption: Two examples of end-station with integrated online metrology. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_stages_wang} NSLS X8C - TXM \cite{wang12_autom_marker_full_field_hard}} @@ -670,7 +670,7 @@ Another example, shown in Figure\nbsp{}ref:fig:introduction_stages_nazaretski, e A more comprehensive review of actively controlled end-stations is provided in Section\nbsp{}ref:sec:nhexa_platform_review. #+name: fig:introduction_active_stations -#+caption: Example of two end-stations with real-time position feedback based on an online metrology +#+caption: Example of two end-stations with real-time position feedback based on an online metrology. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_stages_villar} ESRF ID16a - HPZ. \textsc{KB} is the focusing optics, \textsc{S} the sample, \textsc{C} the capacitive sensors and \textsc{LM} is the light microscope \cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}} @@ -742,7 +742,7 @@ This system essentially functions as a high-performance, multi-axis vibration is It actively compensates for positioning errors measured by the external metrology system. #+name: fig:introduction_nass_concept_schematic -#+caption: The Nano Active Stabilization System concept +#+caption: The Nano Active Stabilization System concept. #+attr_latex: :options [h!tbp] [[file:figs/introduction_nass_concept_schematic.png]] @@ -777,7 +777,7 @@ A central challenge addressed in this thesis is the optimal mechatronic design o A more detailed review of Stewart platform and its main components will be given in Section\nbsp{}ref:sec:detail_kinematics_stewart_review. #+name: fig:introduction_stewart_platform_piezo -#+caption: Two examples of very different Stewart platforms geometries and strut configurations. +#+caption: Two examples of very different Stewart platforms geometries and struts' configuration. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_stewart_du14}Piezo based, for positioning purposes \cite{du14_piezo_actuat_high_precis_flexib}} @@ -1026,7 +1026,7 @@ The damping coefficients were tuned to match the damping identified from the mea The parameters obtained are summarized in Table\nbsp{}ref:tab:uniaxial_ustation_parameters. #+name: tab:uniaxial_ustation_parameters -#+caption: Physical parameters used for the micro-station uniaxial model +#+caption: Physical parameters used for the micro-station uniaxial model. #+attr_latex: :environment tabularx :width 0.6\linewidth :align Xccc #+attr_latex: :center t :booktabs t | *Stage* | *Mass* | *Stiffness* | *Damping* | @@ -1047,7 +1047,7 @@ However, the goal is not to have a perfect match with the measurement (this woul More accurate models will be used later on. #+name: fig:uniaxial_comp_frf_meas_model -#+caption: Comparison of the measured FRF and identified ones from the uniaxial model +#+caption: Comparison of the measured FRF and the uniaxial model dynamics. #+attr_latex: :scale 0.8 [[file:figs/uniaxial_comp_frf_meas_model.png]] @@ -1062,16 +1062,16 @@ The sample is here considered as a rigid body and rigidly fixed to the active pl The effect of resonances between the sample's acrshort:poi and the active platform actuator will be considered in Section\nbsp{}ref:sec:uniaxial_payload_dynamics. #+name: fig:uniaxial_model_micro_station_nass_with_tf -#+caption: Uniaxial model of the NASS (\subref{fig:uniaxial_model_micro_station_nass}) with the micro-station shown in black, the active platform represented in blue and the sample represented in green. Disturbances are shown in red. Extracted transfer function from $f$ to $d$ (\subref{fig:uniaxial_plant_first_params}). +#+caption: Uniaxial model of the NASS (\subref{fig:uniaxial_model_micro_station_nass}) with the micro-station shown in black, the active platform in blue, the sample in green and disturbances in red. Transfer function from $f$ to $d$ (\subref{fig:uniaxial_plant_first_params}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_model_micro_station_nass}Uniaxial mass-spring-damper model of the NASS} #+attr_latex: :options {0.39\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.8 [[file:figs/uniaxial_model_micro_station_nass.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_first_params}Bode Plot of the transfer function from actuator forces $f$ to measured displacement $d$ by the metrology} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_first_params}Plant Dynamics} #+attr_latex: :options {0.59\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1093,7 +1093,7 @@ The /plant/ (i.e., the transfer function from actuator force $f$ to measured dis For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass will be considered: three active platform stiffnesses ($k_n = 0.01\,\text{N}/\upmu\text{m}$, $k_n = 1\,\text{N}/\upmu\text{m}$ and $k_n = 100\,\text{N}/\upmu\text{m}$) combined with three sample's masses ($m_s = 1\,\text{kg}$, $m_s = 25\,\text{kg}$ and $m_s = 50\,\text{kg}$). #+name: fig:uniaxial_sensitivity_dist_first_params -#+caption: Sensitivity of the relative motion $d$ to disturbances: $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_first_params_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}) +#+caption: Sensitivity of the relative motion $d$ to the following disturbances: $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_first_params_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_first_params_xf}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_fs}Direct forces} @@ -1124,7 +1124,7 @@ One is located on the floor, another one on the granite, and the last one on the The geophone located on the floor was used to measure the floor motion $x_f$ while the other two geophones were used to measure vibrations introduced by scanning of the $T_y$ stage and $R_z$ stage (see Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup). #+name: fig:uniaxial_ustation_meas_disturbances_setup -#+caption: Identification of the disturbances coming from the micro-station. The measurement schematic is shown in (\subref{fig:uniaxial_ustation_meas_disturbances}). A picture of the setup is shown in (\subref{fig:uniaxial_ustation_dynamical_id_setup}) +#+caption: Identification of the disturbances coming from the micro-station. The measurement schematic is shown in (\subref{fig:uniaxial_ustation_meas_disturbances}). A picture of the setup is shown in (\subref{fig:uniaxial_ustation_dynamical_id_setup}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_ustation_meas_disturbances}Disturbance measurement setup - Schematic} @@ -1156,7 +1156,7 @@ G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi \end{equation} #+name: fig:uniaxial_geophone_meas_chain -#+caption: Measurement setup for one geophone. The inertial displacement $x$ is converted to a voltage $V$ by the geophone. This voltage is amplified by a factor $g_0 = 60\,\text{dB}$ using a low-noise voltage amplifier. It is then converted to a digital value $\hat{V}_x$ using a 16bit ADC. +#+caption: Measurement setup for one geophone. The inertial displacement $x_f$ is converted to a voltage $V_{x_f}$ by the geophone. This voltage is amplified by a factor $g_0 = 60\,\text{dB}$ using a low-noise voltage amplifier. It is then converted to a digital value $\hat{V}_{x_f}$ using a 16bit ADC. [[file:figs/uniaxial_geophone_meas_chain.png]] The acrfull:asd of the floor motion $\Gamma_{x_f}$ can be computed from the acrlong:asd of measured voltage $\Gamma_{\hat{V}_{x_f}}$ using\nbsp{}eqref:eq:uniaxial_asd_floor_motion. @@ -1167,7 +1167,7 @@ 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}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_asd_floor_motion_id31}Estimated ASD of $x_f$} @@ -1194,7 +1194,7 @@ It is shown that the spindle rotation increases the vibrations above $20\,\text{ The sharp peak observed at $24\,\text{Hz}$ is believed to be induced by electromagnetic interference between the currents in the spindle motor phases and the geophone cable because this peak is not observed when rotating the spindle "by hand". #+name: fig:uniaxial_asd_vibration_spindle_rotation -#+caption: Amplitude Spectral Density $\Gamma_{R_z}$ of the relative motion measured between the granite and the positioning hexapod's top platform during Spindle rotating +#+caption: Amplitude Spectral Density $\Gamma_{R_z}$ of the relative motion measured between the granite and the positioning hexapod's top platform during continuous Spindle rotation. #+attr_latex: :scale 0.8 [[file:figs/uniaxial_asd_vibration_spindle_rotation.png]] @@ -1229,7 +1229,7 @@ The obtained sensitivity to disturbances for the three active platform stiffness - Above the suspension mode of the active platform, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to $1$ (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf) #+name: fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses -#+caption: Sensitivity of $d$ to disturbances for three different nano-hexpod stiffnesses. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}) +#+caption: Sensitivity of $d$ to disturbances for three different active platform stiffnesses. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}Direct forces} @@ -1281,7 +1281,7 @@ The conclusion is that the sample mass has little effect on the cumulative ampli ***** Conclusion The open-loop residual vibrations of $d$ can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. -This residual vibration of $d$ is found to be in the order of $100\,\text{nm RMS}$ for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$), $200\,\text{nm RMS}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$) and $1\,\upmu\text{m}\,\text{RMS}$ for the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$). +This residual vibration of $d$ is found to be in the order of $100\,\text{nm RMS}$ for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$), $200\,\text{nm RMS}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$) and $1\,\upmu\text{m}\,\text{RMS}$ for the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$). From this analysis, it may be concluded that the stiffer the active platform the better. However, what is more important is the /closed-loop/ residual vibration of $d$ (i.e., while the feedback controller is used). @@ -1301,25 +1301,25 @@ In this section, three active damping techniques are applied to the active platf 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 $d$. #+name: fig:uniaxial_active_damping_strategies -#+caption: Three active damping strategies. Integral Force Feedback (\subref{fig:uniaxial_active_damping_strategies_iff}) using a force sensor, Relative Damping Control (\subref{fig:uniaxial_active_damping_strategies_rdc}) using a relative displacement sensor, and Direct Velocity Feedback (\subref{fig:uniaxial_active_damping_strategies_dvf}) using a geophone +#+caption: Three active damping strategies. Integral Force Feedback (\subref{fig:uniaxial_active_damping_strategies_iff}) using a force sensor, Relative Damping Control (\subref{fig:uniaxial_active_damping_strategies_rdc}) using a relative displacement sensor, and Direct Velocity Feedback (\subref{fig:uniaxial_active_damping_strategies_dvf}) using a geophone. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_strategies_iff}IFF} #+attr_latex: :options {0.37\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_strategies_iff.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_strategies_rdc}RDC} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_strategies_rdc.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_strategies_dvf}DVF} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_strategies_dvf.png]] #+end_subfigure #+end_figure @@ -1334,19 +1334,19 @@ The Integral Force Feedback strategy consists of using a force sensor in series The mechanical equivalent of this acrshort:iff strategy is a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain $k/g$ (see Figure\nbsp{}ref:fig:uniaxial_active_damping_iff_equiv). #+name: fig:uniaxial_active_damping_iff -#+caption: Integral Force Feedback (\subref{fig:uniaxial_active_damping_iff_schematic}) is equivalent to a damper in series with the actuator stiffness (\subref{fig:uniaxial_active_damping_iff_equiv}) +#+caption: Integral Force Feedback (\subref{fig:uniaxial_active_damping_iff_schematic}) is equivalent to a damper in series with the actuator stiffness (\subref{fig:uniaxial_active_damping_iff_equiv}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_iff_schematic}Integral Force Feedback} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_iff_schematic.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_iff_equiv}Equivalent mechanical representation} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_iff_equiv.png]] #+end_subfigure #+end_figure @@ -1361,19 +1361,19 @@ For the Relative Damping Control strategy, a relative motion sensor that measure The mechanical equivalent of acrshort:rdc is a dashpot in parallel with the actuator with a damping coefficient equal to the controller gain $g$ (see Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv). #+name: fig:uniaxial_active_damping_rdc -#+caption: Relative Damping Control (\subref{fig:uniaxial_active_damping_rdc_schematic}) is equivalent to a damper in parallel with the actuator (\subref{fig:uniaxial_active_damping_rdc_equiv}) +#+caption: Relative Damping Control (\subref{fig:uniaxial_active_damping_rdc_schematic}) is equivalent to a damper in parallel with the actuator (\subref{fig:uniaxial_active_damping_rdc_equiv}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_rdc_schematic}Relative motion control} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_rdc_schematic.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_rdc_equiv}Equivalent mechanical representation} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_rdc_equiv.png]] #+end_subfigure #+end_figure @@ -1390,19 +1390,19 @@ This is equivalent to a dashpot (with a damping coefficient equal to the control This is usually referred to as "/sky hook damper/". #+name: fig:uniaxial_active_damping_dvf -#+caption: Direct velocity Feedback (\subref{fig:uniaxial_active_damping_dvf_schematic}) is equivalent to a "sky hook damper" (\subref{fig:uniaxial_active_damping_dvf_equiv}) +#+caption: Direct velocity Feedback (\subref{fig:uniaxial_active_damping_dvf_schematic}) is equivalent to a "sky hook damper" (\subref{fig:uniaxial_active_damping_dvf_equiv}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_dvf_schematic}Direct velocity feedback} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_dvf_schematic.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_active_damping_dvf_equiv}Equivalent mechanical representation} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.9 [[file:figs/uniaxial_active_damping_dvf_equiv.png]] #+end_subfigure #+end_figure @@ -1419,7 +1419,7 @@ For the stiff active platform (yellow curves), the micro-station dynamics can be Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff active platform is used. #+name: fig:uniaxial_plant_active_damping_techniques -#+caption: Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), for three active platform stiffnesses ($k_n = 0.01\,\text{N}/\upmu\text{m}$ in blue, $k_n = 1\,\text{N}/\upmu\text{m}$ in red and $k_n = 100\,\text{N}/\upmu\text{m}$ in yellow) and three sample's masses ($m_s = 1\,\text{kg}$: solid curves, $m_s = 25\,\text{kg}$: dot-dashed curves, and $m_s = 50\,\text{kg}$: dashed curves). +#+caption: Plant dynamics for the three active damping techniques: IFF (\subref{fig:uniaxial_plant_active_damping_techniques_iff}), RDC (\subref{fig:uniaxial_plant_active_damping_techniques_rdc}) and DVF (\subref{fig:uniaxial_plant_active_damping_techniques_dvf})). Three active platform stiffnesses ($k_n = 0.01\,\text{N}/\upmu\text{m}$ in blue, $k_n = 1\,\text{N}/\upmu\text{m}$ in red and $k_n = 100\,\text{N}/\upmu\text{m}$ in yellow) and three sample's masses ($m_s = 1\,\text{kg}$: solid curves, $m_s = 25\,\text{kg}$: dot-dashed curves, and $m_s = 50\,\text{kg}$: dashed curves) are considered in each case. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_active_damping_techniques_iff}IFF} @@ -1465,7 +1465,7 @@ There is even some damping authority on micro-station modes in the following cas The micro-station and the active platform masses are connected through a large damper induced by acrshort:rdc (see mechanical equivalent in Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv) which allows some damping of the micro-station. #+name: fig:uniaxial_root_locus_damping_techniques -#+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The Root Loci are zoomed in the suspension mode of the active platform. +#+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The Root Loci are zoomed on the suspension mode of the active platform. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1497,7 +1497,7 @@ The transfer functions from the plant input $f$ to the relative displacement $d$ All three active damping techniques yielded similar damped plants. #+name: fig:uniaxial_damped_plant_three_active_damping_techniques -#+caption: Obtained damped transfer function from $f$ to $d$ for the three damping techniques. +#+caption: Obtained damped transfer functions from $f$ to $d$ for the three damping techniques. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1533,7 +1533,7 @@ 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 disturbance with all three active damping strategies. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_active_damping_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_active_damping_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_active_damping_fs}) +#+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}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_fs}Direct forces} @@ -1561,7 +1561,7 @@ The cumulative amplitude spectrum of the distance $d$ with all three active damp All three active damping methods give similar results. #+name: fig:uniaxial_cas_active_damping -#+caption: Comparison of the cumulative amplitude spectrum (CAS) of the distance $d$ for all three active damping techniques (acrshort:ol in black, IFF in blue, RDC in red and DVF in yellow). +#+caption: Comparison of the acrlong:cas of the distance $d$ for all three active damping techniques. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1599,7 +1599,7 @@ It is difficult to conclude on the best active damping strategy for the acrfull: Which one will be used will be determined by the use of more accurate models and will depend on which is the easiest to implement in practice #+name: tab:comp_active_damping -#+caption: Comparison of active damping strategies +#+caption: Comparison of active damping strategies for the NASS. #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccc #+attr_latex: :center t :booktabs t | | *IFF* | *RDC* | *DVF* | @@ -1629,19 +1629,19 @@ In this section, Integral Force Feedback is used as the Low Authority Controller This control architecture applied to the uniaxial model is shown in Figure\nbsp{}ref:fig:uniaxial_hac_lac_model. #+name: fig:uniaxial_hac_lac -#+caption: acrfull:haclac +#+caption: acrfull:haclac. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_lac_architecture}Typical HAC-LAC Architecture} #+attr_latex: :options {0.54\textwidth} #+begin_subfigure -#+attr_latex: :width \linewidth +#+attr_latex: :scale 0.8 [[file:figs/uniaxial_hac_lac_architecture.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-IFF strategy} #+attr_latex: :options {0.45\textwidth} #+begin_subfigure -#+attr_latex: :scale 1 +#+attr_latex: :scale 0.8 [[file:figs/uniaxial_hac_lac_model.png]] #+end_subfigure #+end_figure @@ -1654,7 +1654,7 @@ This is due to the interaction between the micro-station (modeled modes at $70\, This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_compliance. #+name: fig:uniaxial_hac_iff_damped_plants_masses -#+caption: Obtained damped plant using Integral Force Feedback for three sample masses +#+caption: Obtained damped plant using Integral Force Feedback for three sample masses. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1719,7 +1719,7 @@ K_{\text{stiff}}(s) &= g \cdot \end{subequations} #+name: tab:uniaxial_feedback_controller_parameters -#+caption: Parameters used for the position feedback controllers +#+caption: Parameters used for the position feedback controllers. #+attr_latex: :environment tabularx :width 0.75\linewidth :align Xccc #+attr_latex: :center t :booktabs t | | *Soft* | *Moderately stiff* | *Stiff* | @@ -1739,7 +1739,7 @@ Note that these controllers were not designed using any optimization methods. The goal is to have a first estimation of the attainable performance. #+name: fig:uniaxial_nyquist_hac -#+caption: Nyquist Plot for the high authority controller. The minimum modulus margin is illustrated by a black circle. +#+caption: Nyquist Plot for the High Authority Controllers. The modulus margin is illustrated by the black circles. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1763,7 +1763,7 @@ The goal is to have a first estimation of the attainable performance. #+end_figure #+name: fig:uniaxial_loop_gain_hac -#+caption: Loop gains for the High Authority Controllers +#+caption: Loop gains for the High Authority Controllers. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1793,7 +1793,7 @@ These are compared with the open-loop and damped plants cases in Figure\nbsp{}re As expected, the sensitivity to disturbances decreased in the controller bandwidth and slightly increased outside this bandwidth. #+name: fig:uniaxial_sensitivity_dist_hac_lac -#+caption: Change of sensitivity to disturbances with acrshort:lac and with acrshort:haclac. An active platform with $k_n = 1\,\text{N}/\upmu\text{m}$ and a sample mass of $25\,\text{kg}$ is used. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}) +#+caption: Change of sensitivity to disturbances with acrshort:lac and with acrshort:haclac. An active platform with $k_n = 1\,\text{N}/\upmu\text{m}$ and a sample mass of $25\,\text{kg}$ are used. Disturbances are: $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), $f_t$ the disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_xf}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_fs}Direct forces} @@ -1821,7 +1821,7 @@ 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 Spectrum for all three active platform stiffnesses - Comparison of OL, IFF and acrshort:haclac cases +#+caption: Cumulative Amplitude Spectra for all three active platform stiffnesses in OL, with IFF and with acrshort:haclac. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1862,13 +1862,13 @@ In this section, the impact of the compliance of the support (i.e., the micro-st This is a critical point because the dynamics of the micro-station is complex, depends on the considered direction (see measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model) and may vary with position and time. It would be much better to have a plant dynamics that is not impacted by the micro-station. -Therefore, the objective of this section is to obtain some guidance for the design of a active platform that will not be impacted by the complex micro-station dynamics. +Therefore, the objective of this section is to obtain some guidance for the design of an active platform that will not be impacted by the complex micro-station dynamics. To study this, two models are used (Figure\nbsp{}ref:fig:uniaxial_support_compliance_models). The first one consists of the active platform directly fixed on top of the granite, thus neglecting any support compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_nano_hexapod_only). The second one consists of the active platform fixed on top of the micro-station having some limited compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) #+name: fig:uniaxial_support_compliance_models -#+caption: Models used to study the effect of limited support compliance +#+caption: Models used to study the effect of limited support compliance. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_support_compliance_nano_hexapod_only}Active platform fixed directly on the Granite} @@ -1893,7 +1893,7 @@ Obtained transfer functions from $F$ to $L^\prime$ (shown in Figure\nbsp{}ref:fi When neglecting the support compliance, a large feedback bandwidth can be achieved for all three active platforms. #+name: fig:uniaxial_effect_support_compliance_neglected -#+caption: Obtained transfer functions from $F$ to $L^{\prime}$ when neglecting support compliance +#+caption: Obtained transfer functions from $F$ to $L^{\prime}$ when neglecting support compliance. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_soft}$\omega_{n} \ll \omega_{\mu}$} @@ -1929,7 +1929,7 @@ In such a case, the control of the piezoelectric stage using its integrated metr If a soft active platform is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_soft) which will impact the control robustness and performance. #+name: fig:uniaxial_effect_support_compliance_dynamics -#+caption: Effect of the support compliance on the transfer functions from $F$ to $L$ +#+caption: Effect of the support compliance on the transfer functions from $F$ to $L$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_soft}$\omega_{n} \ll \omega_{\mu}$} @@ -1959,7 +1959,7 @@ Indeed, using a "soft" active platform (i.e., with a suspension mode at lower fr Conversely, if a "stiff" active platform is used, the support dynamics appears in the plant dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_stiff). #+name: fig:uniaxial_effect_support_compliance_dynamics_d -#+caption: Effect of the support compliance on the transfer functions from $F$ to $d$ +#+caption: Effect of the support compliance on the transfer functions from $F$ to $d$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_soft}$\omega_{n} \ll \omega_{\mu}$} @@ -1991,7 +1991,7 @@ For the acrfull:nass, having the suspension mode of the active platform at lower Note that the observations made in this section are also affected by the ratio between the support mass $m_{\mu}$ and the active platform mass $m_n$ (the effect is more pronounced when the ratio $m_n/m_{\mu}$ increases). #+name: tab:uniaxial_effect_compliance -#+caption: Impact of the support dynamics on the plant dynamics +#+caption: Impact of the support dynamics on the plant dynamics. #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :center t :booktabs t | | $\omega_{\nu} \ll \omega_{\mu}$ | $\omega_{\nu} \approx \omega_{\mu}$ | $\omega_{\nu} \gg \omega_{\mu}$ | @@ -2009,7 +2009,7 @@ However, such a sample may present internal dynamics, and its fixation to the ac To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are used. #+name: fig:uniaxial_payload_dynamics_models -#+caption: Models used to study the effect of payload dynamics +#+caption: Models used to study the effect of payload dynamics. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_rigid_schematic}Rigid payload} @@ -2018,7 +2018,7 @@ To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref #+attr_latex: :scale 1 [[file:figs/uniaxial_paylaod_dynamics_rigid_schematic.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Payload with some flexibility} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Payload having some flexibility} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 1 @@ -2038,7 +2038,7 @@ The frequency of the anti-resonance corresponds to the "free" resonance of the 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. 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 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}$k_n = 0.01\,\text{N}/\upmu\text{m}$, $m_s = 1\,\text{kg}$} @@ -2061,7 +2061,7 @@ This changes the zero/pole pattern to a pole/zero pattern (the frequency of the Even though the added sample's flexibility still shifts the high frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy). #+name: fig:uniaxial_payload_dynamics_stiff_nano_hexapod -#+caption: Effect of the payload dynamics on the stiff active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}) +#+caption: Effect of the payload dynamics on the stiff active platform with light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,\text{N}/\upmu\text{m}$, $m_s = 1\,\text{kg}$} @@ -2087,7 +2087,8 @@ This is the same model that was used in Section\nbsp{}ref:sec:uniaxial_position_ In this case, the measured (i.e., controlled) distance $d$ is no longer equal to the real performance index (the distance $y$). #+name: fig:uniaxial_sample_flexibility_control -#+caption: Uniaxial model considering some flexibility between the active platform top platform and the sample. In this case, the measured and controlled distance $d$ is different from the distance $y$ which is the real performance index +#+caption: Uniaxial model considering some flexibility between the active platform top platform and the sample. In this case, the measured and controlled distance $d$ is different from the distance $y$ which is the real performance index. +#+attr_latex: :scale 0.8 [[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. @@ -2102,7 +2103,7 @@ However, the cumulative amplitude spectrum of the distance $y$ (Figure\nbsp{}ref What happens is that above $\omega_s$, even though the motion $d$ can be controlled perfectly, the sample's mass is "isolated" from the motion of the active platform and the control on $y$ is not effective. #+name: fig:uniaxial_sample_flexibility_noise_budget -#+caption: Cumulative Amplitude Spectrum of the distances $d$ and $y$. The effect of the sample's flexibility does not affect much $d$ but is detrimental to the stability of $y$. A sample mass $m_s = 1\,\text{kg}$ and a active platform stiffness of $100\,\text{N}/\upmu\text{m}$ are used for the simulations. +#+caption: Cumulative Amplitude Spectrum of the distances $d$ and $y$. The effect of the sample's flexibility does not affect much $d$ but is detrimental to the stability of $y$. A sample mass $m_s = 1\,\text{kg}$ and an active platform stiffness of $100\,\text{N}/\upmu\text{m}$ are used for the simulations. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_d}Cumulative Amplitude Spectrum of $d$} @@ -2191,7 +2192,7 @@ 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 system +#+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]] @@ -2275,7 +2276,7 @@ The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forces exceeds the spring stiffness $k$. #+name: fig:rotating_campbell_diagram -#+caption: Campbell diagram - Real (\subref{fig:rotating_campbell_diagram_real}) and Imaginary (\subref{fig:rotating_campbell_diagram_imag}) parts of the poles as a function of the rotating velocity $\Omega$. +#+caption: Campbell diagram: Real (\subref{fig:rotating_campbell_diagram_real}) and Imaginary (\subref{fig:rotating_campbell_diagram_imag}) parts of the poles as a function of the rotating velocity $\Omega$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_campbell_diagram_real}Real part} @@ -2300,7 +2301,7 @@ These plots confirm the expected behavior: the frequencies of the two pairs of c For $\Omega > \omega_0$, the low-frequency pair of complex conjugate poles $p_{-}$ becomes unstable (shown be the 180 degrees phase lead instead of phase lag). #+name: fig:rotating_bode_plot -#+caption: Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_direct}) terms for several rotating velocities +#+caption: Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_direct}) terms for several rotating velocities. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_bode_plot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$} @@ -2352,7 +2353,7 @@ Two identical controllers $K_F$ described by\nbsp{}eqref:eq:rotating_iff_control #+attr_latex: :caption \subcaption{\label{fig:rotating_3dof_model_schematic_iff}System with added Force Sensor in series with the actuators} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :width 0.95\linewidth +#+attr_latex: :scale 0.8 [[file:figs/rotating_3dof_model_schematic_iff.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_diagram}Control diagram} @@ -2420,7 +2421,7 @@ As expected from the derived equations of motion: - when $\omega_0 < \Omega$: the low-frequency pole becomes unstable. #+name: fig:rotating_iff_bode_plot_effect_rot -#+caption: Effect of the rotation velocity on the bode plot of the direct terms (\subref{fig:rotating_iff_bode_plot_effect_rot_direct}) and on the IFF root locus (\subref{fig:rotating_root_locus_iff_pure_int}) +#+caption: Effect of the rotation velocity on the bode plot of the direct terms (\subref{fig:rotating_iff_bode_plot_effect_rot_direct}) and on the IFF root locus (\subref{fig:rotating_root_locus_iff_pure_int}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_bode_plot_effect_rot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$} @@ -2484,7 +2485,7 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the controll \end{equation} #+name: fig:rotating_iff_modified_loop_gain_root_locus -#+caption: Comparison of the IFF with pure integrator and modified IFF with added high-pass filter ($\Omega = 0.1\omega_0$). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with $\omega_i = 0.1 \omega_0$ and $g = 2$. The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large}) +#+caption: Comparison of the IFF with pure integrator and modified IFF with added high-pass filter ($\Omega = 0.1\omega_0$). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with $\omega_i = 0.1 \omega_0$ and $g = 2$. The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_modified_loop_gain}Loop gain} @@ -2514,7 +2515,7 @@ For small values of $\omega_i$, the added damping is limited by the maximum allo For larger values of $\omega_i$, the attainable damping ratio decreases as a function of $\omega_i$ as was predicted from the root locus plot of Figure\nbsp{}ref:fig:rotating_iff_root_locus_hpf_large. #+name: fig:rotating_iff_modified_effect_wi -#+caption: Root Locus for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as $\omega_i$ increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain}) +#+caption: Root loci for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as $\omega_i$ increases (\subref{fig:rotating_iff_hpf_optimal_gain}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus} @@ -2538,7 +2539,7 @@ Therefore, there is a trade-off between achievable damping and added coupling wh The same trade-off can be seen between achievable damping and loss of compliance at low-frequency (see Figure\nbsp{}ref:fig:rotating_iff_hpf_effect_wi_compliance). #+name: fig:rotating_iff_hpf_damped_plant_effect_wi -#+caption: Effect of $\omega_i$ on the damped plant coupling +#+caption: Effect of $\omega_i$ on the damped plant coupling (\subref{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}) and on the compliance (\subref{fig:rotating_iff_hpf_effect_wi_compliance}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}Obtained plants} @@ -2563,7 +2564,7 @@ In this section it is proposed to add springs in parallel with the force sensors Such springs are schematically shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_iff_parallel_springs where $k_a$ is the stiffness of the actuator and $k_p$ the added stiffness in parallel with the actuator and force sensor. #+name: fig:rotating_3dof_model_schematic_iff_parallel_springs -#+caption: Studied system with additional springs in parallel with the actuators and force sensors (shown in red) +#+caption: Studied system with additional springs in parallel with the actuators and force sensors (shown in red). #+attr_latex: :scale 0.8 [[file:figs/rotating_3dof_model_schematic_iff_parallel_springs.png]] @@ -2616,16 +2617,16 @@ Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus shows the Root Locus plots for $ It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the unconditional stability of acrshort:iff is recovered. #+name: fig:rotating_iff_plant_effect_kp -#+caption: Effect of parallel stiffness on the IFF plant +#+caption: Effect of parallel stiffness on the IFF plant (\subref{fig:rotating_iff_effect_kp}) and on the control stability (\subref{fig:rotating_iff_kp_root_locus}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_effect_kp}Bode plots of $f_u/F_u$ without parallel spring (blue), with parallel spring $k_p < m \Omega^2$ (red) and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$ (yellow)} #+attr_latex: :options {0.55\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_effect_kp.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with soft parallel spring and with stiff parallel spring} #+attr_latex: :options {0.44\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2642,16 +2643,16 @@ Therefore, even though the parallel stiffness $k_p$ should be larger than $m \Om This is confirmed by the Figure\nbsp{}ref:fig:rotating_iff_kp_optimal_gain where the attainable closed-loop damping ratio $\xi_{\text{cl}}$ and the associated optimal control gain $g_\text{opt}$ are computed as a function of the parallel stiffness. #+name: fig:rotating_iff_optimal_kp -#+caption: Effect of parallel stiffness on the IFF plant +#+caption: Effect of the parallel stiffness on the achievable damping with IFF. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness, $\Omega = 0.1 \omega_0$} #+attr_latex: :options {0.49\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_kp_root_locus_effect_kp.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown because the system is unstable.} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown} #+attr_latex: :options {0.49\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2680,17 +2681,17 @@ Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compare the obtained damped pla The added acrshort:hpf gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior. #+name: fig:rotating_iff_optimal_hpf -#+caption:Effect of high-pass filter cut-off frequency on the obtained damping +#+caption:Effect of high-pass filter cut-off frequency on the obtained damping (\subref{fig:rotating_iff_kp_added_hpf_effect_damping}) and on the dampled plant (\subref{fig:rotating_iff_kp_added_hpf_damped_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$} -#+attr_latex: :options {0.34\linewidth} +#+attr_latex: :options {0.44\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_kp_added_hpf_effect_damping.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF} -#+attr_latex: :options {0.65\linewidth} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Effect of the added HPF on the damped plant} +#+attr_latex: :options {0.54\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_kp_added_hpf_damped_plant.png]] @@ -2701,16 +2702,16 @@ The added acrshort:hpf gives almost the same damping properties to the suspensio <> ***** Introduction :ignore: -To apply a "Relative Damping Control" strategy, relative motion sensors are added in parallel with the actuators as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_rdc. +To apply a acrfull:rdc strategy, relative motion sensors are added in parallel with the actuators as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_rdc. Two controllers $K_d$ are used to feed back the relative motion to the actuator. -These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter\nbsp{}eqref:eq:rotating_rdc_controller. +These controllers have in principle pure derivative action ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter\nbsp{}eqref:eq:rotating_rdc_controller. \begin{equation}\label{eq:rotating_rdc_controller} K_d(s) = g \cdot \frac{s}{s + \omega_d} \end{equation} #+name: fig:rotating_3dof_model_schematic_rdc -#+caption: System with relative motion sensor and decentralized "relative damping control" applied. +#+caption: System with relative motion sensors and decentralized acrfull:rdc applied. #+attr_latex: :scale 0.8 [[file:figs/rotating_3dof_model_schematic_rdc.png]] @@ -2748,7 +2749,7 @@ The rotating aspect does not add any complexity to the use of Relative Damping C It does not increase the low-frequency coupling as compared to the Integral Force Feedback. #+name: fig:rotating_rdc_result -#+caption: Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant}) +#+caption: Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control} @@ -2783,7 +2784,7 @@ This is not the case for the system in which the controller is augmented with an It is interesting to note that the maximum added damping is very similar for both modified IFF techniques. #+name: fig:rotating_comp_techniques -#+caption: Comparison of active damping techniques for rotating platform +#+caption: Comparison of active damping techniques for rotating platform. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_root_locus}Root Locus} @@ -2818,7 +2819,7 @@ Using IFF degrades the compliance at low frequencies, whereas using relative dam This is very well known characteristics of these common active damping techniques that hold when applied to rotating platforms. #+name: fig:rotating_comp_techniques_trans_compliance -#+caption: Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques +#+caption: Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_transmissibility}Transmissibility} @@ -2852,7 +2853,7 @@ This can be seen by the large shift of the resonance frequencies, and by the ind The coupling (or interaction) in a acrshort:mimo $2 \times 2$ system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). #+name: fig:rotating_nano_hexapod_dynamics -#+caption: Effect of rotation on the active platform dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity +#+caption: Effect of rotation on the active platform dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -2910,7 +2911,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar #+end_figure #+name: tab:rotating_iff_hpf_opt_iff_hpf_params_nass -#+caption: Obtained optimal parameters ($\omega_i$ and $g$) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping of the two modes $\xi$ is also shown. +#+caption: Obtained optimal parameters ($\omega_i$ and $g$) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping $\xi_{\text{opt}}$ of the two modes is also shown. #+attr_latex: :environment tabularx :width 0.3\linewidth :align Xccc #+attr_latex: :center t :booktabs t | $k_n$ | $\omega_i$ | $g$ | $\xi_\text{opt}$ | @@ -2937,7 +2938,7 @@ The corresponding optimal controller gains and achievable damping are summarized #+begin_minipage #+name: fig:rotating_iff_kp_nass_optimal_gain #+attr_latex: :scale 0.8 :float nil -#+caption: Maximum damping $\xi$ as a function of the parallel stiffness $k_p$ +#+caption: Maximum damping $\xi$ as a function of the parallel stiffness $k_p$. [[file:figs/rotating_iff_kp_nass_optimal_gain.png]] #+end_minipage \hfill @@ -2946,11 +2947,11 @@ The corresponding optimal controller gains and achievable damping are summarized #+latex: \centering #+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align cccc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| $k_n$ | $k_p$ | $g$ | $\xi_{\text{opt}}$ | -|-----------------------+-----------------------+---------+--------------------| -| $0.01\,\text{N}/\upmu\text{m}$ | $1\,\text{N/mm}$ | 47.9 | 0.44 | -| $1\,\text{N}/\upmu\text{m}$ | $0.01\,\text{N}/\upmu\text{m}$ | 465.57 | 0.97 | -| $100\,\text{N}/\upmu\text{m}$ | $1\,\text{N}/\upmu\text{m}$ | 4624.25 | 0.99 | +| $k_n$ | $k_p$ | $g$ | $\xi_{\text{opt}}$ | +|--------------------------------+--------------------------------+------+--------------------| +| $0.01\,\text{N}/\upmu\text{m}$ | $1\,\text{N/mm}$ | 48 | 0.44 | +| $1\,\text{N}/\upmu\text{m}$ | $0.01\,\text{N}/\upmu\text{m}$ | 465 | 0.97 | +| $100\,\text{N}/\upmu\text{m}$ | $1\,\text{N}/\upmu\text{m}$ | 4624 | 0.99 | #+latex: \captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses} #+end_minipage @@ -2962,7 +2963,7 @@ The gain is chosen such that 99% of modal damping is obtained (obtained gains ar #+begin_minipage #+name: fig:rotating_rdc_optimal_gain #+attr_latex: :scale 0.8 :float nil -#+caption: Maximum damping $\xi$ as a function of the RDC gain $g$ +#+caption: Maximum damping $\xi$ as a function of the RDC gain $g$. [[file:figs/rotating_rdc_optimal_gain.png]] #+end_minipage \hfill @@ -2976,7 +2977,7 @@ The gain is chosen such that 99% of modal damping is obtained (obtained gains ar | $0.01\,\text{N}/\upmu\text{m}$ | 1600 | 0.99 | | $1\,\text{N}/\upmu\text{m}$ | 8200 | 0.99 | | $100\,\text{N}/\upmu\text{m}$ | 80000 | 0.99 | -#+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC} +#+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the acrlong:rdc} #+end_minipage ***** Comparison of the Obtained Damped Plants @@ -2988,7 +2989,7 @@ Similar to what was concluded in the previous analysis: - Coupling is smaller for stiff active platforms #+name: fig:rotating_nass_damped_plant_comp -#+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three active platform stiffnesses are considered. For this analysis the rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$. +#+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three active platform stiffnesses are considered. Rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -3044,7 +3045,7 @@ It can be observed that: - The two proposed IFF modifications yield similar results #+name: fig:rotating_nass_plant_comp_stiffness -#+caption: Bode plot of the transfer function from active platform actuator to measured motion by the external metrology +#+caption: Bode plot of the transfer function from active platform actuator to measured motion by the external metrology. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -3104,7 +3105,7 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: #+end_figure #+name: fig:rotating_nass_effect_stage_vibration -#+caption: Effect of micro-station vibrations $f_{t,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft active platform suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc}) +#+caption: Effect of micro-station vibrations $f_{t,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft active platform suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -3227,7 +3228,7 @@ Here, 3-axis accelerometers[fn:modal_1] shown in figure\nbsp{}ref:fig:modal_acce These accelerometers were glued to the micro-station using a thin layer of wax for best results\nbsp{}[[cite:&ewins00_modal chapt. 3.5.7]]. #+name: fig:modal_analysis_instrumentation -#+caption: Instrumentation used for the modal analysis +#+caption: Instrumentation used for the modal analysis. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:modal_accelero_M393B05}3-axis accelerometer} @@ -3290,7 +3291,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrsh #+attr_latex: :options [b]{0.63\linewidth} #+begin_minipage #+name: fig:modal_location_accelerometers -#+caption: Position of the accelerometers +#+caption: Position of the accelerometers. #+attr_latex: :width 0.95\linewidth :float nil [[file:figs/modal_location_accelerometers.png]] #+end_minipage @@ -3329,7 +3330,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrsh #+end_minipage #+name: fig:modal_accelerometer_pictures -#+caption: Accelerometers fixed on the micro-station stages +#+caption: Accelerometers fixed on the micro-station stages. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_ty} $T_y$ stage} @@ -3355,7 +3356,7 @@ It was chosen to match the location of one accelerometer, because a /point measu The impacts were performed in three directions, as shown in figures\nbsp{}ref:fig:modal_impact_x, ref:fig:modal_impact_y and ref:fig:modal_impact_z. #+name: fig:modal_hammer_impacts -#+caption: The three hammer impacts used for the modal analysis +#+caption: The three hammer impacts used for the modal analysis. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:modal_impact_x} $X$ impact} @@ -3391,7 +3392,7 @@ These data are corresponding to a hammer impact in the vertical direction and to Similar results were obtained for all measured acrshortpl:frf. #+name: fig:modal_raw_meas_asd -#+caption: Raw measurement of the accelerometer 1 in the $x$ direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the $z$ direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force}) +#+caption: Raw measurement of the accelerometer 1 in the $x$ direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the $z$ direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals} @@ -3413,7 +3414,7 @@ The quality of the obtained data can be estimated using the /coherence/ function Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest. #+name: fig:modal_frf_coh_acc_force -#+caption: Computed frequency response function from the applied force $F_{z}$ to the measured response $X_{1,x}$ (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force}) +#+caption: Computed acrshort:frf from the applied force $F_{z}$ to the measured response $X_{1,x}$ (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function} @@ -3468,7 +3469,7 @@ Let us consider the schematic shown in Figure\nbsp{}ref:fig:modal_local_to_globa The goal here is to link these $4 \times 3 = 12$ measurements to the 6 acrshort:dof of the solid body expressed in the frame $\{O\}$. #+name: fig:modal_local_to_global_coordinates -#+caption: Schematic of the measured motions of a solid body +#+caption: Schematic of the measured motion of a solid body at 4 distinc locations. [[file:figs/modal_local_to_global_coordinates.png]] 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 $\{O\}$. @@ -3515,7 +3516,7 @@ From the 3D model, the position of the acrlong:com of each solid body is compute The position of each accelerometer with respect to the acrlong:com of the corresponding solid body can easily be determined. #+name: tab:modal_com_solid_bodies -#+caption: Center of mass of considered solid bodies with respect to the "point of interest" +#+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: :center t :booktabs t | | $X$ | $Y$ | $Z$ | @@ -3556,7 +3557,7 @@ Similar results were obtained for the other solid bodies, indicating that the so This also validates the reduction in the number of acrshortpl:dof from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof). #+name: fig:modal_comp_acc_solid_body_frf -#+caption: Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the positioning hexapod are shown. Input is a hammer force applied on the positioning hexapod in the $x$ direction +#+caption: Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4, which are corresponding to the positioning hexapod, are shown. Input is a hammer force applied on the positioning hexapod in the $x$ direction. #+attr_latex: :scale 0.8 [[file:figs/modal_comp_acc_solid_body_frf.png]] @@ -3598,7 +3599,7 @@ The obtained natural frequencies and associated modal damping are summarized in #+attr_latex: :options [b]{0.65\linewidth} #+begin_minipage #+name: fig:modal_indication_function -#+caption: Modal Indication Function +#+caption: Modal Indication Function. #+attr_latex: :float nil :scale 0.8 [[file:figs/modal_indication_function.png]] #+end_minipage @@ -3641,7 +3642,7 @@ It takes into account the fact that the properties of all individual curves are From the obtained modal parameters, the mode shapes are computed and can be displayed in the form of animations (three mode shapes are shown in Figure\nbsp{}ref:fig:modal_mode_animations). #+name: fig:modal_mode_animations -#+caption: Three obtained mode shape animations +#+caption: Three obtained mode shape animations. #+attr_latex: :options [hbtp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:modal_mode1_animation}$1^{st}$ mode at $11.9\,\text{Hz}$: tilt suspension mode of the granite} @@ -3671,7 +3672,7 @@ These are difficult to adjust and can lead to a situation in which the granite i The levelers were then better adjusted. #+name: fig:modal_airloc -#+caption: AirLoc used for the granite (2120-KSKC) +#+caption: AirLoc used for the granite (2120-KSKC). #+attr_latex: :width 0.6\linewidth [[file:figs/modal_airlock_picture.jpg]] @@ -3830,7 +3831,7 @@ To precisely control the $R_y$ angle, a stepper motor and two optical encoders a #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:ustation_ty_stage -#+caption: Translation Stage +#+caption: Translation Stage. #+attr_latex: :scale 1 :float nil [[file:figs/ustation_ty_stage.png]] #+end_minipage @@ -3838,7 +3839,7 @@ To precisely control the $R_y$ angle, a stepper motor and two optical encoders a #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:ustation_ry_stage -#+caption: Tilt Stage +#+caption: Tilt Stage. #+attr_latex: :scale 1 :float nil [[file:figs/ustation_ry_stage.png]] #+end_minipage @@ -3861,7 +3862,7 @@ It can also be used to precisely position the acrfull:poi vertically with respec #+attr_latex: :options [t]{0.49\linewidth} #+begin_minipage #+name: fig:ustation_rz_stage -#+caption: Rotation Stage (Spindle) +#+caption: Rotation Stage (Spindle). #+attr_latex: :scale 1 :float nil [[file:figs/ustation_rz_stage.png]] #+end_minipage @@ -3869,7 +3870,7 @@ It can also be used to precisely position the acrfull:poi vertically with respec #+attr_latex: :options [t]{0.49\linewidth} #+begin_minipage #+name: fig:ustation_hexapod_stage -#+caption: Positioning Hexapod +#+caption: Positioning Hexapod. #+attr_latex: :scale 1 :float nil [[file:figs/ustation_hexapod_stage.png]] #+end_minipage @@ -4015,7 +4016,7 @@ Let us consider the motion of a rigid body described at three locations (Figure\ Frame $\{A\}$ represents the initial location, frame $\{B\}$ is an intermediate location, and frame $\{C\}$ represents the rigid body at its final location. #+name: fig:ustation_combined_transformation -#+caption: Motion of a rigid body represented at three locations by frame $\{A\}$, $\{B\}$ and $\{C\}$ +#+caption: Motion of a rigid body represented at three locations by frame $\{A\}$, $\{B\}$ and $\{C\}$. [[file:figs/ustation_combined_transformation.png]] Furthermore, suppose the position vector of a point $P$ of the rigid body is given in the final location, that is ${}^CP$ is given, and the position of this point is to be found in the fixed frame $\{A\}$, that is ${}^AP$. @@ -4052,7 +4053,7 @@ The mobile frame of the translation stage is equal to the fixed frame of the til Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the spindle: $\{B_{R_y}\} = \{A_{R_z}\}$. #+name: fig:ustation_stage_motion -#+caption: Example of the motion induced by the tilt-stage $R_y$. "Rest" position in shown in blue while a arbitrary position in shown in red. Parasitic motions are here magnified for clarity. +#+caption: Example of the motion induced by the tilt-stage $R_y$. Initial position is shown in blue while an arbitrary position is shown in red. Parasitic motions are here magnified for clarity. [[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. @@ -4137,7 +4138,7 @@ 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-DoF joint. One joint DoF (here the tilt angle) can be imposed, 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 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. #+attr_latex: :scale 0.8 [[file:figs/ustation_simscape_stage_example.png]] @@ -4148,7 +4149,7 @@ 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. #+name: fig:ustation_simscape_model -#+caption: 3D view of the micro-station multi-body model +#+caption: 3D view of the micro-station multi-body model. #+attr_latex: :width 0.8\linewidth [[file:figs/ustation_simscape_model.jpg]] @@ -4163,7 +4164,7 @@ The springs and dampers values were first estimated from the joint/stage specifi 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 contrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoF joints are defined are indicated in figures found in Section\nbsp{}ref:ssec:ustation_stages +#+caption: Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoF joints are defined are indicated in figures found in Section\nbsp{}ref:ssec:ustation_stages. #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccccc #+attr_latex: :center t :booktabs t | *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | @@ -4186,7 +4187,7 @@ Even though there is some similarity between the model and the measurements (sim Tuning the numerous model parameters to better match the measurements is a highly non-linear optimization problem that is difficult to solve in practice. #+name: fig:ustation_comp_com_response -#+caption: FRFs between the hammer impacts on the translation stage and the measured stage acceleration expressed at its CoM. Comparison of the measured and extracted FRFs from the multi-body model. Different directions are computed for different stages. +#+caption: FRFs from a hammer impact to the stage acceleration, both expressed at its CoM. The measured FRFs are compared with the multi-body model. Different directions are computed for different stages. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_comp_com_response_rz_x}Spindle, $x$ response} @@ -4223,7 +4224,7 @@ The positioning hexapod top platform was impacted at 10 different points. For each impact position, 10 impacts were performed to average and improve the data quality. #+name: fig:ustation_compliance_meas -#+caption: Schematic of the measurement setup used to estimate the compliance of the micro-station. The top platform of the positioning hexapod is shown with four 3-axis accelerometers (shown in red) are on top. 10 hammer impacts are performed at different locations (shown in blue). +#+caption: Schematic of the measurement setup used to estimate the compliance of the micro-station. Four 3-axis accelerometers (shown in red) are fix on top of the positioning hexapod platform. 10 hammer impacts are performed at different locations (shown in blue). [[file:figs/ustation_compliance_meas.png]] To convert the 12 acceleration signals $a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]$ to the acceleration expressed in the $\{\mathcal{X}\}$ frame $a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]$, a Jacobian matrix $\bm{J}_a$ is written based on the positions and orientations of the accelerometers\nbsp{}eqref:eq:ustation_compliance_acc_jacobian. @@ -4333,7 +4334,7 @@ The obtained ground motion displacement is shown in Figure\nbsp{}ref:fig:ustatio #+attr_latex: :options [b]{0.54\linewidth} #+begin_minipage #+name: fig:ustation_ground_disturbance -#+caption: Measured ground motion +#+caption: Measured ground motion. #+attr_latex: :scale 0.8 :float nil [[file:figs/ustation_ground_disturbance.png]] #+end_minipage @@ -4341,7 +4342,7 @@ The obtained ground motion displacement is shown in Figure\nbsp{}ref:fig:ustatio #+attr_latex: :options [b]{0.44\linewidth} #+begin_minipage #+name: fig:ustation_geophone_picture -#+caption: (3D) L-4C geophone +#+caption: (3D) L-4C geophone. #+attr_latex: :width 0.92\linewidth :float nil [[file:figs/ustation_geophone_picture.jpg]] #+end_minipage @@ -4353,7 +4354,7 @@ A special optical element (called a "straightness interferometer"[fn:ustation_9] A similar setup was used to measure the horizontal deviation (i.e. in the $x$ direction), as well as the pitch and yaw errors of the translation stage. #+name: fig:ustation_errors_ty_setup -#+caption: Experimental setup to measure the straightness (vertical deviation) of the translation stage +#+caption: Experimental setup to measure the straightness (vertical deviation) of the translation stage. [[file:figs/ustation_errors_ty_setup.png]] Six scans were performed between $-4.5\,\text{mm}$ and $4.5\,\text{mm}$. @@ -4389,7 +4390,7 @@ Five capacitive sensors[fn:ustation_8] are pointing at the two spheres, as shown From the 5 measured displacements $[d_1,\,d_2,\,d_3,\,d_4,\,d_5]$, the translations and rotations $[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$ of the target can be estimated. #+name: fig:ustation_rz_meas_lion_setup -#+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}) +#+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-DoF metrology} @@ -4638,7 +4639,7 @@ At NSLS-II, for instance, a system capable of $100\,\upmu\text{m}$ stroke has be Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately $100\,\text{Hz}$ (Figure\nbsp{}ref:fig:nhexa_stages_sapoti). #+name: fig:nhexa_stages_translations -#+caption: Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope\nbsp{}[[cite:&nazaretski15_pushin_limit]] at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] at Sirius facility (\subref{fig:nhexa_stages_sapoti}) +#+caption: Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope\nbsp{}[[cite:&nazaretski15_pushin_limit]] at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] at Sirius facility (\subref{fig:nhexa_stages_sapoti}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_nazaretski} MLL microscope} @@ -4662,7 +4663,7 @@ In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage ab However, attempts to implement real-time feedback using YZ external metrology proved challenging, possibly due to the poor dynamical response of the serial stage configuration. #+name: fig:nhexa_stages_spindle -#+caption: Example of two sample stages for tomography experiments. ID16a endstation\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] at the ESRF (\subref{fig:nhexa_stages_villar}). PtyNAMi microscope\nbsp{}[[cite:&schropp20_ptynam;&schroer17_ptynam]] at PETRA III (\subref{fig:nhexa_stages_schroer}) +#+caption: Example of two sample stages for tomography experiments. ID16a endstation\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] at the ESRF (\subref{fig:nhexa_stages_villar}). PtyNAMi microscope\nbsp{}[[cite:&schropp20_ptynam;&schroer17_ptynam]] at PETRA III (\subref{fig:nhexa_stages_schroer}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_villar} Simplified schematic of ID16a end-station} @@ -4793,7 +4794,7 @@ These examples demonstrate the architecture's versatility in terms of geometry, 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. #+name: fig:nhexa_stewart_examples -#+caption: Two examples of Stewart platform. A Stewart platform based on piezoelectric stack actuators and used for nano-positioning is shown in (\subref{fig:nhexa_stewart_piezo_furutani})\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar]]. A Stewart platform based on voice coil actuators and used for vibration isolation is shown in (\subref{fig:nhexa_stewart_vc_preumont})\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ;&preumont18_vibrat_contr_activ_struc_fourt_edition]] +#+caption: Two examples of Stewart platform. A Stewart platform based on piezoelectric stack actuators and used for nano-positioning is shown in (\subref{fig:nhexa_stewart_piezo_furutani})\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar]]. A Stewart platform based on voice coil actuators and used for vibration isolation is shown in (\subref{fig:nhexa_stewart_vc_preumont})\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ;&preumont18_vibrat_contr_activ_struc_fourt_edition]]. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_stewart_piezo_furutani} Stewart platform for Nano-positioning} @@ -4862,7 +4863,7 @@ The struts' orientations are represented by the unit vectors $\hat{\bm{s}}_i$ an This is summarized in Figure\nbsp{}ref:fig:nhexa_stewart_notations. #+name: fig:nhexa_stewart_notations -#+caption: Frame and key notations for the Stewart platform +#+caption: Frame and key notations for the Stewart platform. #+attr_latex: :scale 0.9 [[file:figs/nhexa_stewart_notations.png]] @@ -4880,7 +4881,7 @@ For each strut $i$ (illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_loop_closu This equation links the pose[fn:nhexa_2] variables ${}^A\bm{P}$ and ${}^A\bm{R}_B$, the position vectors describing the known geometry of the base and the moving platform, $\bm{a}_i$ and $\bm{b}_i$, and the strut vector $l_i {}^A\hat{\bm{s}}_i$: #+name: fig:nhexa_stewart_loop_closure -#+caption: Notations to compute the kinematic loop closure +#+caption: Notations to compute the kinematic loop closure. #+attr_latex: :scale 0.9 [[file:figs/nhexa_stewart_loop_closure.png]] @@ -4933,7 +4934,7 @@ By multiplying both sides by ${}^A\hat{\bm{s}}_i$,\nbsp{}eqref:eq:nhexa_loop_clo {}^A\hat{\bm{s}}_i {}^A\bm{v}_p + \underbrace{{}^A\hat{\bm{s}}_i ({}^A\bm{\omega} \times {}^A\bm{b}_i)}_{=({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) {}^A\bm{\omega}} = \dot{l}_i + \underbrace{{}^A\hat{s}_i l_i \left( {}^A\bm{\omega}_i \times {}^A\hat{\bm{s}}_i \right)}_{=0} \end{equation} -Equation\nbsp{}eqref:eq:nhexa_loop_closure_velocity_bis can be rearranged in matrix form to obtain\nbsp{}eqref:eq:nhexa_jacobian_velocities, with $\dot{\bm{\mathcal{L}}} = [ \dot{l}_1 \ \dots \ \dot{l}_6 ]^{\intercal}$ the vector of strut velocities, and $\dot{\bm{\mathcal{X}}} = [{}^A\bm{v}_p ,\ {}^A\bm{\omega}]^{\intercal}$ the vector of platform velocity and angular velocity. +Equation\nbsp{}eqref:eq:nhexa_loop_closure_velocity_bis can be rearranged in matrix form to obtain\nbsp{}eqref:eq:nhexa_jacobian_velocities, with $\dot{\bm{\mathcal{L}}} = [ \dot{l}_1 \ \dots \ \dot{l}_6 ]^{\intercal}$ the vector of strut velocities, and $\dot{\bm{\mathcal{X}}} = [{}^A\bm{v}_p ,\ {}^A\bm{\omega}]^{\intercal}$ the vector of platform velocity and angular velocity. \begin{equation}\label{eq:nhexa_jacobian_velocities} \boxed{\dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}}} @@ -5143,7 +5144,7 @@ From these parameters, key kinematic properties can be derived: the strut orient #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage #+name: fig:nhexa_stewart_model_def -#+caption: Geometry of the stewart platform +#+caption: Geometry of the stewart platform. #+attr_latex: :float nil :scale 0.9 [[file:figs/nhexa_stewart_model_def.png]] #+end_minipage @@ -5202,7 +5203,7 @@ This modular approach to actuator modeling allows for future refinements as the #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage #+name: fig:nhexa_actuator_model -#+caption: Model of the active platform actuators +#+caption: Model of the active platform actuators. #+attr_latex: :float nil :scale 0.8 [[file:figs/nhexa_actuator_model.png]] #+end_minipage @@ -5238,7 +5239,7 @@ A three-dimensional visualization of the model is presented in Figure\nbsp{}ref: #+attr_latex: :options [b]{0.35\linewidth} #+begin_minipage #+name: fig:nhexa_simscape_screenshot -#+caption: 3D representation of the multi-body model +#+caption: 3D representation of the multi-body model. #+attr_latex: :width 0.8\linewidth :float nil [[file:figs/nhexa_simscape_screenshot.jpg]] #+end_minipage @@ -5267,7 +5268,7 @@ Figure\nbsp{}ref:fig:nhexa_comp_multi_body_analytical presents a comparison betw The close agreement between both approaches across the frequency spectrum validates the multi-body model's accuracy in capturing the system's dynamic behavior. #+name: fig:nhexa_comp_multi_body_analytical -#+caption: Comparison of the analytical transfer functions and the multi-body model +#+caption: Comparison of the analytical transfer functions and the multi-body model. #+attr_latex: :scale 0.8 [[file:figs/nhexa_comp_multi_body_analytical.png]] @@ -5293,7 +5294,7 @@ Each actuator's transfer function to its associated force sensor exhibits altern The inclusion of parallel stiffness introduces an additional complex conjugate zero at low frequency, which was previously observed in the three-degree-of-freedom rotating model. #+name: fig:nhexa_multi_body_plant -#+caption: Bode plot of the transfer functions computed using the active platform multi-body model +#+caption: Bode plot of the transfer functions computed using the active platform multi-body model. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{\mathcal{L}}$} @@ -5375,7 +5376,7 @@ This simplifies the control design because only one controller needs to be tuned Furthermore, at low frequencies, the plant exhibits good decoupling between the struts, allowing for effective independent control of each axis. #+name: fig:nhexa_control_frame -#+caption: Two control strategies +#+caption: Two control strategies. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_control_strut}Control in the frame of the struts. $\bm{J}$ is used to project errors in the frame of the struts} @@ -5410,7 +5411,7 @@ For the conceptual validation of the acrshort:nass, control in the strut space w More sophisticated control strategies will be explored during the detailed design phase. #+name: fig:nhexa_plant_frame -#+caption: Bode plot of the transfer functions computed using the active platform multi-body model +#+caption: Bode plot of the transfer functions computed using the active platform multi-body model. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts} @@ -5435,7 +5436,7 @@ The decentralized Integral Force Feedback (IFF) control strategy is implemented The corresponding block diagram of the control loop is shown in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_schematic, in which the controller $\bm{K}_{\text{IFF}}(s)$ is a diagonal matrix, where each diagonal element is a pure integrator\nbsp{}eqref:eq:nhexa_kiff. #+name: fig:nhexa_decentralized_iff_schematic -#+caption: Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs $\bm{f}^{\prime}$ +#+caption: Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs $\bm{f}^{\prime}$. #+attr_latex: :scale 0.9 [[file:figs/nhexa_decentralized_iff_schematic.png]] @@ -5459,7 +5460,7 @@ The loop-gain is high around the resonance frequencies, indicating that the dece This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability. #+name: fig:nhexa_decentralized_iff_results -#+caption: Decentralized IFF +#+caption: Decentralized IFF. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{n1}}{f_1}(s)$} @@ -5487,7 +5488,7 @@ The Jacobian matrix $\bm{J}^{-1}$ performs (approximate) real-time approximate i A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these errors in the frame of the struts. #+name: fig:nhexa_hac_iff_schematic -#+caption: HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts +#+caption: HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts. #+attr_latex: :scale 0.9 [[file:figs/nhexa_hac_iff_schematic.png]] @@ -5597,7 +5598,7 @@ To further improve the model accuracy, a multi-body model of the micro-station w Furthermore, a multi-body model of the active platform was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure\nbsp{}ref:fig:nass_simscape_model. #+name: fig:nass_simscape_model -#+caption: 3D view of the NASS multi-body model +#+caption: 3D view of the NASS multi-body model. #+attr_latex: :options [h!tbp] #+attr_latex: :width 0.8\linewidth [[file:figs/nass_simscape_model.jpg]] @@ -5620,7 +5621,7 @@ Figure\nbsp{}ref:fig:nass_concept_schematic presents a schematic overview of the This section focuses on the components of the "Instrumentation and Real-Time Control" block. #+name: fig:nass_concept_schematic -#+caption: Schematic of the Nano Active Stabilization System +#+caption: Schematic of the Nano Active Stabilization System. #+attr_latex: :options [h!tbp] [[file:figs/nass_concept_schematic.png]] @@ -5771,7 +5772,7 @@ Adding parallel stiffness (Figure\nbsp{}ref:fig:nass_iff_plant_kp) transforms th Although both cases show significant coupling around the resonances, stability is guaranteed by the collocated arrangement of the actuators and sensors\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with]]. #+name: fig:nass_iff_plant_effect_kp -#+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of $25\,\text{kg}$. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}) +#+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of $25\,\text{kg}$. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_no_kp}without parallel stiffness} @@ -5795,7 +5796,7 @@ The poles and zeros shift in frequency as the payload mass varies. However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining good robustness. #+name: fig:nass_iff_plant_effect_rotation_payload -#+caption: Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload}) +#+caption: Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_effect_rotation}Effect of Spindle rotation} @@ -5835,7 +5836,7 @@ The cut-off frequency of the second-order high-pass filter was tuned to be below The overall gain was then increased to obtain a large loop gain around the resonances to be damped, as illustrated in Figure\nbsp{}ref:fig:nass_iff_loop_gain. #+name: fig:nass_iff_loop_gain -#+caption: Loop gain for the decentralized IFF: $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$ +#+caption: Loop gain for the decentralized IFF: $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$. #+attr_latex: :options [h!tbp] #+attr_latex: :scale 0.8 [[file:figs/nass_iff_loop_gain.png]] @@ -5900,7 +5901,7 @@ This decoupling characteristic ensures consistent performance across various ope This also validates the developed control strategy. #+name: fig:nass_undamped_plant_effect -#+caption: Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass}) +#+caption: Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_Wz}Effect of rotational velocity $\Omega_z$} @@ -5954,7 +5955,7 @@ The only observable difference manifests as additional alternating poles and zer This result confirms effective dynamic decoupling between the active platform and the supporting micro-station structure. #+name: fig:nass_effect_ustation_compliance -#+caption: Effect of the micro-station limited compliance on the plant dynamics +#+caption: Effect of the micro-station limited compliance on the plant dynamics. #+attr_latex: :options [h!tbp] #+attr_latex: :scale 0.8 [[file:figs/nass_effect_ustation_compliance.png]] @@ -5976,7 +5977,7 @@ Figure\nbsp{}ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation The current approach of controlling the position in the strut frame is inadequate for soft active platforms; but even shifting control to a frame matching the payload's acrlong:com would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. #+name: fig:nass_soft_stiff_hexapod -#+caption: Coupling between a stiff active platform ($k_a = 100\,\text{N}/\upmu\text{m}$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance ($k_a = 0.01\,\text{N}/\upmu\text{m}$) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}) +#+caption: Coupling between a stiff active platform ($k_a = 100\,\text{N}/\upmu\text{m}$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance ($k_a = 0.01\,\text{N}/\upmu\text{m}$) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\upmu\text{m}$ - Coupling with the micro-station} @@ -6008,7 +6009,7 @@ First, the decentralized loop gain shown in Figure\nbsp{}ref:fig:nass_hac_loop_g Second, the characteristic loci analysis presented in Figure\nbsp{}ref:fig:nass_hac_loci demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope. #+name: fig:nass_hac_controller -#+caption: High Authority Controller - "Diagonal Loop Gain" (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci}) +#+caption: High Authority Controller - "Diagonal Loop Gain" (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_hac_loop_gain}Loop Gain} @@ -6065,7 +6066,7 @@ It should be noted that the maximum rotational velocity of 360deg/s is primarily For higher mass configurations, rotational velocities are expected to be below 36deg/s. #+name: fig:nass_tomography_hac_iff -#+caption: Simulation of tomography experiments - 360deg/s. Beam size is indicated by the dashed black ellipse +#+caption: Simulation of tomography experiments - 360deg/s. Beam size is indicated by the dashed black ellipse. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,\text{kg}$} @@ -6140,7 +6141,7 @@ 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 a active platform with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise. +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. 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). @@ -6185,7 +6186,7 @@ Subsequently, Stewart proposed a similar design for a flight simulator (shown in Since then, the Stewart platform (sometimes referred to as the Stewart-Gough platform) has been used across diverse applications\nbsp{}[[cite:&dasgupta00_stewar_platf_manip]], including large telescopes\nbsp{}[[cite:&kazezkhan14_dynam_model_stewar_platf_nansh_radio_teles;&yun19_devel_isotr_stewar_platf_teles_secon_mirror]], machine tools\nbsp{}[[cite:&russo24_review_paral_kinem_machin_tools]], and Synchrotron instrumentation\nbsp{}[[cite:&marion04_hexap_esrf;&villar18_nanop_esrf_id16a_nano_imagin_beaml]]. #+name: fig:detail_geometry_stewart_origins -#+caption: Two of the earliest developments of Stewart platforms +#+caption: Two of the earliest developments of Stewart platforms. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_geometry_gough_paper}Tyre test machine proposed by Gough \cite{gough62_univer_tyre_test_machin}} @@ -6216,7 +6217,7 @@ Examples of piezoelectric-actuated Stewart platforms are presented in Figures\nb Although less frequently encountered, magnetostrictive actuators have been successfully implemented in\nbsp{}[[cite:&zhang11_six_dof]] (Figure\nbsp{}ref:fig:detail_kinematics_zhang11). #+name: fig:detail_kinematics_stewart_examples_cubic -#+caption: Some examples of developped Stewart platform with Cubic geometry +#+caption: Some examples of developped Stewart platform with Cubic geometry. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_jpl}California Institute of Technology - USA \cite{spanos95_soft_activ_vibrat_isolat}} @@ -6266,7 +6267,7 @@ The second category comprises non-cubic architectures (Figure\nbsp{}ref:fig:deta The influence of strut orientation and joint positioning on Stewart platform properties is analyzed in Section\nbsp{}ref:sec:detail_kinematics_geometry. #+name: fig:detail_kinematics_stewart_examples_non_cubic -#+caption: Some examples of developped Stewart platform with non-cubic geometry +#+caption: Some examples of developped Stewart platform with non-cubic geometry. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_pph}Naval Postgraduate School - USA \cite{chen03_payload_point_activ_vibrat_isolat}} @@ -6519,7 +6520,7 @@ These trade-offs provide important guidelines when choosing the Stewart platform #+name: tab:detail_kinematics_geometry #+attr_latex: :environment tabularx :width 0.65\linewidth :align Xcc #+attr_latex: :center t :booktabs t :float t -#+caption: Effect of a change in geometry on the manipulator's stiffness and mobility +#+caption: Effect of a change in geometry on the manipulator's stiffness and mobility. | *Struts* | *Vertically Oriented* | *Increased separation* | |-------------------------------+-----------------------+------------------------| | Vertical stiffness | $\nearrow$ | $=$ | @@ -6587,7 +6588,7 @@ The unit vectors corresponding to the edges of the cube are described by equatio \end{equation} #+name: fig:detail_kinematics_cubic_schematic_cases -#+caption: Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic}) +#+caption: Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube} @@ -6676,7 +6677,7 @@ The rotational mobility, illustrated in Figure\nbsp{}ref:fig:detail_kinematics_c Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities. #+name: fig:detail_kinematics_cubic_mobility -#+caption: Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations}) +#+caption: Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_mobility_translations}Mobility in translation} @@ -6701,7 +6702,7 @@ This section examines the dynamics of the cubic architecture in the Cartesian fr When relative motion sensors are integrated in each strut (measuring $\bm{\mathcal{L}}$), the pose $\bm{\mathcal{X}}$ is computed using the Jacobian matrix as shown in Figure\nbsp{}ref:fig:detail_kinematics_centralized_control. #+name: fig:detail_kinematics_centralized_control -#+caption: Typical control architecture in the cartesian frame +#+caption: Typical control architecture in the cartesian frame. [[file:figs/detail_kinematics_centralized_control.png]] ***** Low Frequency and High Frequency Coupling @@ -6724,7 +6725,7 @@ At high frequency, the behavior is governed by the mass matrix (evaluated at fra To achieve a diagonal mass matrix, the acrlong:com of the mobile components must coincide with the $\{B\}$ frame, and the principal axes of inertia must align with the axes of the $\{B\}$ frame. #+name: fig:detail_kinematics_cubic_payload -#+caption: Cubic stewart platform with top cylindrical payload +#+caption: Cubic stewart platform with top cylindrical payload. #+attr_latex: :width 0.5\linewidth [[file:figs/detail_kinematics_cubic_payload.png]] @@ -6761,7 +6762,7 @@ The primary limitation of this approach is that, for many applications including If a design similar to Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload were employed for the active platform, the X-ray beam would intersect with the struts during spindle rotation. #+name: fig:detail_kinematics_cubic_com_cok -#+caption: Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok}) +#+caption: Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_centered_payload}Payload at the cube's center} @@ -6803,7 +6804,7 @@ The first employs a cubic architecture shown in Figure\nbsp{}ref:fig:detail_kine The second uses a non-cubic Stewart platform shown in Figure\nbsp{}ref:fig:detail_kinematics_non_cubic_payload, featuring identical payload and strut dynamics but with struts oriented more vertically to differentiate it from the cubic architecture. #+name: fig:detail_kinematics_non_cubic_payload -#+caption: Stewart platform with non-cubic architecture +#+caption: Stewart platform with non-cubic architecture. #+attr_latex: :width 0.5\linewidth [[file:figs/detail_kinematics_non_cubic_payload.png]] @@ -6817,7 +6818,7 @@ No significant advantage is evident for the cubic architecture (Figure\nbsp{}ref The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes. #+name: fig:detail_kinematics_decentralized_dL -#+caption: Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL}) +#+caption: Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_dL}Non cubic architecture} @@ -6841,7 +6842,7 @@ The results are presented in Figure\nbsp{}ref:fig:detail_kinematics_decentralize The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture. #+name: fig:detail_kinematics_decentralized_fn -#+caption: Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn}) +#+caption: Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_fn}Non cubic architecture} @@ -7052,7 +7053,7 @@ Both platforms take the maximum available size, with joints offset by $15\,\text The positioning angles, as shown in Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_top, are [255, 285, 15, 45, 135, 165] degrees for the top joints and [220, 320, 340, 80, 100, 200] degrees for the bottom joints. #+name: fig:detail_kinematics_nano_hexapod -#+caption: Obtained architecture for the active platform +#+caption: Obtained architecture for the active platform. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_nano_hexapod_iso}Isometric view} @@ -7189,7 +7190,7 @@ The specific design of the acrshort:apa allows for the simultaneous modeling of #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:detail_fem_apa95ml_picture -#+caption: Picture of the APA95ML +#+caption: Picture of the APA95ML. #+attr_latex: :float nil :scale 1 [[file:figs/detail_fem_apa95ml_picture.png]] #+end_minipage @@ -7213,7 +7214,7 @@ The development of the acrfull:fem for the APA95ML required the knowledge of the The finite element mesh, shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_mesh, was then generated. #+name: tab:detail_fem_material_properties -#+caption: Material properties used for FEA. $E$ is the Young's modulus, $\nu$ the Poisson ratio and $\rho$ the material density +#+caption: Material properties used for FEA. $E$ is the Young's modulus, $\nu$ the Poisson ratio and $\rho$ the material density. #+attr_latex: :environment tabularx :width 0.55\linewidth :align Xccc #+attr_latex: :center t :booktabs t | | $E$ | $\nu$ | $\rho$ | @@ -7276,7 +7277,7 @@ Unfortunately, it is difficult to know exactly which material is used for the pi Yet, based on the available properties of the stacks in the data-sheet (summarized in Table\nbsp{}ref:tab:detail_fem_stack_parameters), the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. #+name: tab:detail_fem_stack_parameters -#+caption: Stack Parameters +#+caption: Stack Parameters. #+attr_latex: :environment tabularx :width 0.3\linewidth :align Xc #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | @@ -7293,7 +7294,7 @@ The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are list From these parameters, $g_s = 5.1\,\text{V}/\upmu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained. #+name: tab:detail_fem_piezo_properties -#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities +#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities. #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | *Description* | @@ -7317,7 +7318,7 @@ A value of $23\,\text{N}/\upmu\text{m}$ was found which is close to the specifie The multi-body model predicted a resonant frequency under block-free conditions of $\approx 2\,\text{kHz}$ (Figure\nbsp{}ref:fig:detail_fem_apa95ml_compliance), which is in agreement with the nominal specification. #+name: fig:detail_fem_apa95ml_compliance -#+caption: Estimated compliance of the APA95ML +#+caption: Estimated compliance of the APA95ML. #+attr_latex: :scale 0.8 [[file:figs/detail_fem_apa95ml_compliance.png]] @@ -7365,7 +7366,7 @@ The presence of a non-minimum phase zero in the measured system response (Figure Regarding the amplitude characteristics, the constants $g_a$ and $g_s$ could be further refined through calibration against the experimental data. #+name: fig:detail_fem_apa95ml_comp_plant -#+caption: Comparison of the measured frequency response functions and the finite element model of the APA95ML. Both for the dynamics from $V_a$ to $y$ (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa95ml_comp_plant_sensor}) +#+caption: Comparison of the measured frequency response functions and the finite element model of the APA95ML. Both for the dynamics from $V_a$ to $y$ (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa95ml_comp_plant_sensor}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$} @@ -7487,7 +7488,7 @@ This selection was further reinforced by previous experience with acrshortpl:apa The demonstrated accuracy of the modeling approach for the APA95ML provides confidence in the reliable prediction of the APA300ML's dynamic characteristics, thereby supporting both the selection decision and subsequent dynamical analyses. #+name: tab:detail_fem_piezo_act_models -#+caption: List of some amplified piezoelectric actuators that could be used for the active platform +#+caption: List of some amplified piezoelectric actuators that could be used for the active platform. #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccccc #+attr_latex: :center t :booktabs t :float t | *Specification* | APA150M | *APA300ML* | APA400MML | FPA-0500E-P | FPA-0300E-S | @@ -7539,7 +7540,7 @@ This force is related to the applied voltage $V_a$ through the actuator constant 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 $g_s$. #+name: fig:detail_fem_apa_2dof_model -#+caption: Schematic of the 2DoF model of the Amplified Piezoelectric Actuator +#+caption: Schematic of the 2DoF model of the Amplified Piezoelectric Actuator. [[file:figs/detail_fem_apa_2dof_model.png]] While providing computational efficiency, this simplified model has inherent limitations. @@ -7563,7 +7564,7 @@ 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 DoF 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* | @@ -7578,7 +7579,7 @@ 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 2DoF model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}) +#+caption: Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2DoF model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}from $V_a$ to $d_i$} @@ -7606,7 +7607,7 @@ As demonstrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_effect_electrical_bou The developed models of the acrshort:apa do not represent such behavior, but as this effect is quite small, this validates the simplifying assumption made in the models. #+name: fig:detail_fem_apa95ml_effect_electrical_boundaries -#+caption: Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency +#+caption: Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency. #+attr_latex: :scale 0.8 [[file:figs/detail_fem_apa95ml_effect_electrical_boundaries.png]] @@ -7631,7 +7632,7 @@ The reduction in model order is substantial: while the acrshort:fem implementati These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the active platform. #+name: fig:detail_fem_actuator_fem_vs_perfect_plants -#+caption: Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a active platform having actuators modelled a 2DoF system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). +#+caption: Comparison of the dynamics obtained between an active platform having the actuators modeled with FEM and an active platform having actuators modelled a 2DoF system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7712,7 +7713,7 @@ This effect becomes less significant when using the selected APA300ML actuators A parallel analysis of torsional stiffness revealed similar effects, though these proved less critical for system performance. #+name: fig:detail_fem_joints_bending_stiffness_plants -#+caption: Effect of bending stiffness of the flexible joints on the plant dynamics. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_bending_stiffness_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_bending_stiffness_iff_plant}) +#+caption: Effect of bending stiffness of the flexible joints on the plant dynamics. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_bending_stiffness_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_bending_stiffness_iff_plant}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7730,7 +7731,7 @@ 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. When having an actuator modelled as 1DoF without parallel stiffness to the force sensor (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2DoF 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. When having an actuator modelled as 1DoF without parallel stiffness to the force sensor (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2DoF 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}1DoF actuators} @@ -7771,7 +7772,7 @@ These effects fundamentally limit achievable control bandwidth, making high axia Based on this analysis, an axial stiffness specification of $100\,\text{N}/\upmu\text{m}$ was established for the active platform joints. #+name: fig:detail_fem_joints_axial_stiffness_plants -#+caption: Effect of axial stiffness of the flexible joints on the plant dynamics. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_axial_stiffness_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_axial_stiffness_iff_plant}) +#+caption: Effect of axial stiffness of the flexible joints on the plant dynamics. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_axial_stiffness_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_axial_stiffness_iff_plant}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_axial_stiffness_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7789,7 +7790,7 @@ Based on this analysis, an axial stiffness specification of $100\,\text{N}/\upmu #+end_figure #+name: fig:detail_fem_joints_axial_stiffness_iff_results -#+caption: Effect of axial stiffness of the flexible joints on the attainable damping with decentralized IFF (\subref{fig:detail_fem_joints_axial_stiffness_iff_locus}). Estimation of the coupling of the damped plants using the RGA-number (\subref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}) +#+caption: Effect of axial stiffness of the flexible joints on the attainable damping with decentralized IFF (\subref{fig:detail_fem_joints_axial_stiffness_iff_locus}). Estimation of the coupling of the damped plants using the RGA-number (\subref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root Locus} @@ -7814,7 +7815,7 @@ Critical specifications include sufficient bending stroke to ensure long-term op Based on the dynamic analysis presented in previous sections, quantitative specifications were established and are summarized in Table\nbsp{}ref:tab:detail_fem_joints_specs. #+name: tab:detail_fem_joints_specs -#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model +#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model. #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc #+attr_latex: :center t :booktabs t :float t | | *Specification* | *FEM* | @@ -7872,7 +7873,7 @@ This simplification reduces the total model order to 48 states: 12 for the paylo While additional acrshortpl:dof could potentially capture more dynamic features, the selected configuration preserves essential system characteristics while minimizing computational complexity. #+name: fig:detail_fem_joints_fem_vs_perfect_plants -#+caption: Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a active platform having bottom joint modelled by bending stiffness $k_f$ and axial stiffness $k_a$ and top joints modelled by bending stiffness $k_f$, torsion stiffness $k_t$ and axial stiffness $k_a$. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}). +#+caption: Comparison of the dynamics obtained between an active platform including joints modelled with FEM and an active platform having bottom joint modelled by bending stiffness $k_f$ and axial stiffness $k_a$ and top joints modelled by bending stiffness $k_f$, torsion stiffness $k_t$ and axial stiffness $k_a$. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7940,7 +7941,7 @@ In cases where multiple control objectives must be achieved simultaneously, as i From the literature, three principal approaches for combining sensors have been identified: acrlong:haclac, sensor fusion, and two-sensor control architectures. #+name: fig:detail_control_control_multiple_sensors -#+caption: Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}) +#+caption: Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC-LAC} @@ -8083,7 +8084,7 @@ The sensor dynamics estimate $\hat{G}_i(s)$ may be a simple gain or a more compl #+attr_latex: :scale 1 [[file:figs/detail_control_sensor_model.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_calibrated}Normalized sensors using the inverse of an estimate $\hat{G}} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_calibrated}Normalized sensor using the inverse of an estimate $\hat{G}} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 1 @@ -8144,13 +8145,13 @@ Therefore, by appropriately shaping the norm of the complementary filters, the n ***** Sensor Fusion Robustness In practical systems, sensor normalization is rarely perfect, and condition\nbsp{}eqref:eq:detail_control_sensor_perfect_dynamics is not fully satisfied. -To analyze such imperfections, a multiplicative input uncertainty is incorporated into the sensor dynamics (Figure\nbsp{}ref:fig:detail_control_sensor_model_uncertainty). +To analyze such imperfections, a multiplicative input uncertainty is included into the sensor dynamics (Figure\nbsp{}ref:fig:detail_control_sensor_model_uncertainty). The nominal model is the estimated model used for normalization $\hat{G}_i(s)$, $\Delta_i(s)$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $w_i(s)$ is a weighting transfer function representing the magnitude of uncertainty. Since the nominal sensor dynamics is taken as the normalized filter, the normalized sensor model can be further simplified as shown in Figure\nbsp{}ref:fig:detail_control_sensor_model_uncertainty_simplified. #+name: fig:detail_control_sensor_models_uncertainty -#+caption: Sensor models with dynamical uncertainty +#+caption: Sensor models with dynamical uncertainty. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_uncertainty}Sensor with multiplicative input uncertainty} @@ -8167,7 +8168,7 @@ Since the nominal sensor dynamics is taken as the normalized filter, the normali #+end_subfigure #+end_figure -The sensor fusion architecture incorporating sensor models with dynamical uncertainty is illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_fusion_dynamic_uncertainty. +The sensor fusion architecture including sensor models with dynamical uncertainty is illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_fusion_dynamic_uncertainty. The super sensor dynamics\nbsp{}eqref:eq:detail_control_sensor_super_sensor_dyn_uncertainty is no longer unity but depends on the sensor dynamical uncertainty weights $w_i(s)$ and the complementary filters $H_i(s)$. The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (Figure\nbsp{}ref:fig:detail_control_sensor_uncertainty_set_super_sensor). @@ -8203,7 +8204,7 @@ As it is generally desired to limit the dynamical uncertainty of the super senso As established in Section\nbsp{}ref:ssec:detail_control_sensor_fusion_requirements, the super sensor's noise characteristics and robustness are directly dependent on the complementary filters' norm. A synthesis method enabling precise shaping of these norms would therefore offer substantial practical benefits. This section develops such an approach by formulating the design objective as a standard $\mathcal{H}_\infty$ optimization problem. -The methodology for designing appropriate weighting functions (which specify desired complementary filter shape during synthesis) is examined in detail, and the efficacy of the proposed method is validated with a simple example. +The methodology for designing appropriate weighting functions (which specify desired complementary filter shape during synthesis) is examined in detail, and the efficiency of the proposed method is validated with a simple example. ***** Synthesis Objective @@ -8228,7 +8229,7 @@ The synthesis objective can be expressed as a standard $\mathcal{H}_\infty$ opti \end{equation} #+name: fig:detail_control_sensor_h_infinity_robust_fusion -#+caption: Architecture for the $\mathcal{H}_\infty\text{-synthesis}$ of complementary filters +#+caption: Architecture for the $\mathcal{H}_\infty\text{-synthesis}$ of complementary filters. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_h_infinity_robust_fusion_plant}Generalized plant} @@ -8331,7 +8332,7 @@ The inverse magnitudes of the designed weighting functions, which represent the #+begin_minipage #+name: fig:detail_control_sensor_hinf_filters_results #+attr_latex: :scale 0.8 :float nil -#+caption: Weights and obtained filters +#+caption: Weights and obtained filters. [[file:figs/detail_control_sensor_hinf_filters_results.png]] #+end_minipage @@ -8344,7 +8345,7 @@ This straightforward example demonstrates that the proposed methodology for shap **** Synthesis of a set of Three Complementary Filters <> -Certain applications necessitate the fusion of more than two sensors\nbsp{}[[cite:&stoten01_fusion_kinet_data_using_compos_filter;&carreira15_compl_filter_desig_three_frequen_bands]]. +Some applications require the fusion of more than two sensors\nbsp{}[[cite:&stoten01_fusion_kinet_data_using_compos_filter;&carreira15_compl_filter_desig_three_frequen_bands]]. At LIGO, for example, a super sensor is formed by merging three distinct sensors: a acrshort:lvdt, a seismometer, and a geophone\nbsp{}[[cite:&matichard15_seism_isolat_advan_ligo]]. For merging $n>2$ sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_fusion_three. @@ -8355,7 +8356,7 @@ Previous literature has offered only simple analytical formulas for this purpose This section presents a generalization of the proposed complementary filter synthesis method to address this gap. #+name: fig:detail_control_sensor_fusion_three -#+caption: Possible sensor fusion architecture when more than two sensors are to be merged +#+caption: Possible sensor fusion architecture when more than two sensors are to be merged. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_three_sequential}Sequential fusion} @@ -8422,7 +8423,7 @@ Consider the generalized plant $P_3(s)$ shown in Figure\nbsp{}ref:fig:detail_con \end{equation} #+name: fig:detail_control_sensor_comp_filter_three_hinf -#+caption: Architecture for the $\mathcal{H}_\infty\text{-synthesis}$ of three complementary filters (\subref{fig:detail_control_sensor_comp_filter_three_hinf_fb}). Bode plot of the inverse weighting functions and of the three obtained complementary filters (\subref{fig:detail_control_sensor_three_complementary_filters_results}) +#+caption: Architecture for the $\mathcal{H}_\infty\text{-synthesis}$ of three complementary filters (\subref{fig:detail_control_sensor_comp_filter_three_hinf_fb}). Bode plot of the inverse weighting functions and of the three obtained complementary filters (\subref{fig:detail_control_sensor_three_complementary_filters_results}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_comp_filter_three_hinf_fb}Generalized plant with the synthesized filter} @@ -8498,9 +8499,9 @@ Finally, a comparative analysis with concluding observations is provided in Sect **** 3-DoF Test Model <> -Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis. +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 possesses three acrshortpl:dof and incorporates three parallel struts. +It has three acrshortpl:dof and incorporates three parallel struts. Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform. Two reference frames are defined within this model: frame $\{M\}$ with origin $O_M$ at the acrlong:com of the solid body, and frame $\{K\}$ with origin $O_K$ at the acrlong:cok of the parallel manipulator. @@ -8508,7 +8509,7 @@ Two reference frames are defined within this model: frame $\{M\}$ with origin $O #+attr_latex: :options [b]{0.60\linewidth} #+begin_minipage #+name: fig:detail_control_decoupling_model_test -#+caption: Model used to compare decoupling strategies +#+caption: Model used to compare decoupling strategies. #+attr_latex: :float nil :scale 1 [[file:figs/detail_control_decoupling_model_test.png]] #+end_minipage @@ -8627,7 +8628,7 @@ The resulting plant (Figure\nbsp{}ref:fig:detail_control_jacobian_decoupling_arc - $\bm{\mathcal{X}}_{\{O\}}$ represents translations/rotation of the payload expressed in frame $\{O\}$ #+name: fig:detail_control_jacobian_decoupling_arch -#+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{O\}}$ to $\bm{\mathcal{X}}_{\{O\}}$ +#+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{O\}}$ to $\bm{\mathcal{X}}_{\{O\}}$. [[file:figs/detail_control_decoupling_control_jacobian.png]] The transfer function from $\bm{\mathcal{F}}_{\{O\}$ to $\bm{\mathcal{X}}_{\{O\}}$, denoted $\bm{G}_{\{O\}}(s)$ can be computed using\nbsp{}eqref:eq:detail_control_decoupling_plant_jacobian. @@ -8783,7 +8784,7 @@ This implementation requires knowledge of the system's equations of motion, from The resulting decoupled system features diagonal elements each representing second-order resonant systems that are straightforward to control individually. #+name: fig:detail_control_decoupling_modal -#+caption: Modal Decoupling Architecture +#+caption: Modal Decoupling Architecture. [[file:figs/detail_control_decoupling_modal.png]] ***** Example :ignore: @@ -8819,7 +8820,7 @@ The two computed matrices were implemented in the control architecture of Figure Each of these diagonal elements corresponds to a specific mode, as shown in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test_modal, resulting in a perfectly decoupled system. #+name: fig:detail_control_decoupling_modal_plant_modes -#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to invidiually address different modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal}) +#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to invidiually address different modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space} @@ -8868,7 +8869,7 @@ These singular input and output matrices are then applied to decouple the system \end{equation} #+name: fig:detail_control_decoupling_svd -#+caption: Decoupled plant $\bm{G}_{\text{SVD}}$ using the Singular Value Decomposition +#+caption: Decoupled plant $\bm{G}_{\text{SVD}}$ using the Singular Value Decomposition. [[file:figs/detail_control_decoupling_svd.png]] Implementation of SVD decoupling requires access to the system's acrshort:frf, at least in the vicinity of the desired decoupling frequency. @@ -8911,7 +8912,7 @@ The resulting plant, depicted in Figure\nbsp{}ref:fig:detail_control_decoupling_ Additionally, the diagonal terms manifest as second-order dynamic systems, facilitating straightforward controller design. #+name: fig:detail_control_decoupling_svd_plant -#+caption: Plant dynamics $\bm{G}_{\text{SVD}}(s)$ obtained after decoupling using Singular Value Decomposition +#+caption: Plant dynamics $\bm{G}_{\text{SVD}}(s)$ obtained after decoupling using Singular Value Decomposition. #+attr_latex: :scale 0.8 [[file:figs/detail_control_decoupling_svd_plant.png]] @@ -8967,7 +8968,7 @@ Modal decoupling offers good decoupling across all frequencies, though its effec SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point. #+name: tab:detail_control_decoupling_strategies_comp -#+caption: Comparison of decoupling strategies +#+caption: Comparison of decoupling strategies. #+attr_latex: :environment tabularx :width \linewidth :align lXXX #+attr_latex: :center t :booktabs t :font \scriptsize | | *Jacobian* | *Modal* | *SVD* | @@ -9064,7 +9065,7 @@ With these assumptions, the resulting control architecture is illustrated in Fig The dynamics of this closed-loop system are described by equations\nbsp{}eqref:eq:detail_control_cf_cl_system_y and eqref:eq:detail_control_cf_cl_system_y. #+name: fig:detail_control_cf_arch_class -#+caption: Equivalent classical feedback control architecture +#+caption: Equivalent classical feedback control architecture. [[file:figs/detail_control_cf_arch_class.png]] \begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf} @@ -9143,7 +9144,7 @@ The set of possible plants $\Pi_i$ is described by\nbsp{}eqref:eq:detail_control \end{equation} #+name: fig:detail_control_cf_input_uncertainty_nyquist -#+caption: Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty}) +#+caption: Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_input_uncertainty}Input multiplicative uncertainty} @@ -9206,20 +9207,11 @@ For some applications, first-order complementary filters as shown in Equation\nb \end{align} \end{subequations} -These filters can be transformed into the digital domain using the Bilinear transformation, resulting in the digital filter representations shown in Equation\nbsp{}eqref:eq:detail_control_cf_1st_order_z. - -\begin{subequations}\label{eq:detail_control_cf_1st_order_z} - \begin{align} - H_L(z^{-1}) &= \frac{T_s \omega_0 + T_s \omega_0 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} \\ - H_H(z^{-1}) &= \frac{2 - 2 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} - \end{align} -\end{subequations} - A significant advantage of using analytical formulas for complementary filters is that key parameters such as $\omega_0$ can be tuned in real-time, as illustrated in Figure\nbsp{}ref:fig:detail_control_cf_arch_tunable_params. This real-time tunability allows rapid testing of different control bandwidths to evaluate performance and robustness characteristics. #+name: fig:detail_control_cf_arch_tunable_params -#+caption: Implemented digital complementary filters with parameter $\omega_0$ that can be changed in real-time +#+caption: Implemented digital complementary filters with parameter $\omega_0$ that can be changed in real-time. [[file:figs/detail_control_cf_arch_tunable_params.png]] For many practical applications, first order complementary filters are not sufficient. @@ -9372,7 +9364,7 @@ Performance is evaluated by examining the closed-loop sensitivity and complement It is shown that the sensitivity transfer function achieves the desired $+2$ slope at low frequencies and that the complementary sensitivity transfer function nominally provides the wanted noise filtering. #+name: fig:detail_control_cf_simulation_results -#+caption: Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf}) +#+caption: Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust Stability} @@ -9523,7 +9515,7 @@ When combined with the piezoelectric load (represented as a capacitance $C_p$), \end{equation} #+name: fig:detail_instrumentation_amp_output_impedance -#+caption: Electrical model of a voltage amplifier with output impedance $R_0$ connected to a piezoelectric stack with capacitance $C_p$ +#+caption: Electrical model of a voltage amplifier with output impedance $R_0$ connected to a piezoelectric stack with capacitance $C_p$. [[file:figs/detail_instrumentation_amp_output_impedance.png]] Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases. @@ -9571,7 +9563,7 @@ Note that for the WMA-200, the manufacturer proposed to remove the $50\,\Omega$ The PD200 from PiezoDrive was ultimately selected because it meets all the requirements and is accompanied by clear documentation, particularly regarding noise characteristics and bandwidth specifications. #+name: tab:detail_instrumentation_amp_choice -#+caption: Specifications for the Voltage amplifier and considered commercial products +#+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 | @@ -9628,7 +9620,7 @@ The quantization noise ranges between $\pm q/2$, and its probability density fun Since the integral of this probability density function $p(e)$ equals one, its value is $1/q$ for $-q/2 < e < q/2$, as illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_adc_quantization. #+name: fig:detail_instrumentation_adc_quantization -#+caption: Probability density function $p(e)$ of the ADC quantization error $e$ +#+caption: Probability density function $p(e)$ of the ADC quantization error $e$. [[file:figs/detail_instrumentation_adc_quantization.png]] The variance (or time-average power) of the quantization noise is expressed by\nbsp{}eqref:eq:detail_instrumentation_quant_power. @@ -9679,7 +9671,7 @@ Several sensor technologies are capable of meeting these requirements\nbsp{}[[ci These include optical encoders (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_encoder), capacitive sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_capacitive), and eddy current sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_eddy_current), each with their own advantages and implementation considerations. #+name: fig:detail_instrumentation_sensor_examples -#+caption: Relative motion sensors considered for measuring the active platform strut motion +#+caption: Relative motion sensors considered for measuring the active platform strut motion. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_sensor_encoder}Optical Linear Encoder} @@ -9706,7 +9698,7 @@ From an implementation perspective, capacitive and eddy current sensors offer a In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the acrshort:apa, as shown in Figure\nbsp{}ref:fig:detail_instrumentation_encoder_implementation. #+name: fig:detail_instrumentation_sensor_implementation -#+caption: Implementation of relative displacement sensor to measure the motion of the APA +#+caption: Implementation of relative displacement sensor to measure the motion of the APA. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_encoder_implementation}Optical Encoder} @@ -9731,7 +9723,7 @@ Based on this criterion, an optical encoder with digital output was selected, wh The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table\nbsp{}ref:tab:detail_instrumentation_sensor_specs, alongside alternative options that were considered. #+name: tab:detail_instrumentation_sensor_specs -#+caption: Specifications for the relative displacement sensors and considered commercial products +#+caption: Specifications for the relative displacement sensors and considered commercial products. #+attr_latex: :environment tabularx :width 0.65\linewidth :align Xccc #+attr_latex: :center t :booktabs t :float t | *Specifications* | Renishaw Vionic | LION CPL190 | Cedrat ECP500 | @@ -9759,7 +9751,7 @@ Given that the acrshort:adc can operate at 200kSPS while the real-time controlle This approach is effective because the noise approximates white noise and its amplitude exceeds 1 acrshort:lsb (0.3 mV)\nbsp{}[[cite:&hauser91_princ_overs_d_conver]]. #+name: fig:detail_instrumentation_adc_noise_measured -#+caption: Measured ADC noise (IO318) +#+caption: Measured ADC noise (IO318). #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_adc_noise_measured.png]] @@ -9770,7 +9762,7 @@ The setup is illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_se 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 volage +#+caption: Schematic of the setup to validate the use of the ADC for reading the force sensor volage. [[file:figs/detail_instrumentation_force_sensor_adc_setup.png]] Step signals with an amplitude of $1\,\text{V}$ were generated using the acrshort:dac, and the acrshort:adc signal was recorded. @@ -9850,7 +9842,7 @@ The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:detail_instrumentation_femto_meas_setup -#+caption: Measurement of the instrumentation amplifier input voltage noise +#+caption: Measurement of the instrumentation amplifier input voltage noise. #+attr_latex: :scale 1 :float nil [[file:figs/detail_instrumentation_femto_meas_setup.png]] #+end_minipage @@ -9858,7 +9850,7 @@ The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:detail_instrumentation_femto_input_noise -#+caption: Obtained ASD of the instrumentation amplifier input voltage noise +#+caption: Obtained ASD of the instrumentation amplifier input voltage noise. #+attr_latex: :scale 0.8 :float nil [[file:figs/detail_instrumentation_femto_input_noise.png]] #+end_minipage @@ -9930,7 +9922,7 @@ The noise spectrum of the PD200 amplifiers exhibits several sharp peaks. While the exact cause of these peaks is not fully understood, their amplitudes remain below the specified limits and should not adversely affect system performance. #+name: fig:detail_instrumentation_pd200_noise -#+caption: Measured output voltage noise of the PD200 amplifiers +#+caption: Measured output voltage noise of the PD200 amplifiers. #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_pd200_noise.png]] @@ -9947,7 +9939,7 @@ All six amplifiers demonstrated consistent transfer function characteristics. Th 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. #+name: fig:detail_instrumentation_pd200_tf -#+caption: Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier +#+caption: Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier. #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_pd200_tf.png]] @@ -9965,7 +9957,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:detail_instrumentation_vionic_bench -#+caption: Test bench used to measured the encoder noise +#+caption: Test bench used to measured the encoder noise. #+attr_latex: :width 0.95\linewidth :float nil [[file:figs/detail_instrumentation_vionic_bench.jpg]] #+end_minipage @@ -9973,7 +9965,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:detail_instrumentation_vionic_asd -#+caption: Measured encoder noise ASD +#+caption: Measured encoder noise ASD. #+attr_latex: :scale 0.8 :float nil [[file:figs/detail_instrumentation_vionic_asd.png]] #+end_minipage @@ -9986,7 +9978,7 @@ The total motion induced by all noise sources combined is approximately $1.5\,\t 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 noise budgeting using measured noise of instrumentation. #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_cl_noise_budget.png]] @@ -10026,7 +10018,7 @@ This objective implies that the frequencies of (un-modelled) flexible modes pote Finally, considerations for ease of mounting, alignment, and maintenance were incorporated, specifically ensuring that struts could be easily replaced in the event of failure. #+name: fig:detail_design_nano_hexapod_elements -#+caption: Obtained mechanical design of the Active platform, called the "nano-hexapod" +#+caption: Obtained mechanical design of the Active platform, called the "nano-hexapod". #+attr_latex: :width 0.95\linewidth [[file:figs/detail_design_nano_hexapod_elements.png]] @@ -10155,7 +10147,7 @@ The extent to which these modes might be detrimental is difficult to establish a Finally, the FEA indicated that flexible modes of the top plate itself begin to appear at frequencies above $650\,\text{Hz}$, with the first such mode shown in Figure\nbsp{}ref:fig:detail_design_fem_plate_mode. #+name: fig:detail_design_fem_nano_hexapod -#+caption: Measurement of strut flexible modes. First six modes are "suspension" modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from $205\,\text{Hz}$ to $420\,\text{Hz}$ (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at $650\,\text{Hz}$ (\subref{fig:detail_design_fem_plate_mode}) +#+caption: Measurement of strut flexible modes. First six modes are "suspension" modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from $205\,\text{Hz}$ to $420\,\text{Hz}$ (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at $650\,\text{Hz}$ (\subref{fig:detail_design_fem_plate_mode}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension mode} @@ -10241,14 +10233,14 @@ The multi-body representation corresponding to the 4DoF configuration is shown i This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from acrshort:fea of the joint component. #+name: fig:detail_design_simscape_model_flexible_joint -#+caption: 4DoF multi-body model of the flexible joints +#+caption: 4DoF multi-body model of the flexible joints. #+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 utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a acrshort:fem. +Two distinct representations of the acrshort:apa can be used within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a acrshort:fem. ***** Encoders @@ -10258,7 +10250,7 @@ Therefore, a more sophisticated model of the optical encoder was necessary. The optical encoders operate based on the interaction between an encoder head and a graduated scale or ruler. The optical encoder head contains a light source that illuminates the ruler. -A reference frame $\{E\}$ fixed to the scale, represents the the light position on the scale, as illustrated in Figure\nbsp{}ref:fig:detail_design_simscape_encoder_model. +A reference frame $\{E\}$ fixed to the scale, represents the light position on the scale, as illustrated in Figure\nbsp{}ref:fig:detail_design_simscape_encoder_model. The ruler features a precise grating pattern (in this case, with a $20\,\upmu\text{m}$ pitch), and its position is associated with the reference frame $\{R\}$. The displacement measured by the encoder corresponds to the relative position of the encoder frame $\{E\}$ (specifically, the point where the light interacts with the scale) with respect to the ruler frame $\{R\}$, projected along the measurement direction defined by the scale. @@ -10381,7 +10373,7 @@ This more complex model also captures well capture the axial dynamics of the APA #+name: fig:test_apa_received #+attr_latex: :width 0.7\linewidth -#+caption: Picture of 5 out of the 7 received APA300ML +#+caption: Picture of 5 out of the 7 received APA300ML. [[file:figs/test_apa_received.jpg]] *** Static Measurements @@ -10407,7 +10399,7 @@ The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatnes #+begin_minipage #+name: fig:test_apa_flatness_setup #+attr_latex: :width 0.6\linewidth :float nil -#+caption: Measurement setup for flatness estimation +#+caption: Measurement setup for flatness estimation. [[file:figs/test_apa_flatness_setup.png]] #+end_minipage \hfill @@ -10446,7 +10438,7 @@ This may be because the manufacturer measures the capacitance with large signals #+begin_minipage #+name: fig:test_apa_lcr_meter #+attr_latex: :width 0.8\linewidth :float nil -#+caption: Used LCR meter +#+caption: Used LCR meter. [[file:figs/test_apa_lcr_meter.jpg]] #+end_minipage \hfill @@ -10476,7 +10468,7 @@ The voltage across the two actuator stacks is varied from $-20\,\text{V}$ to $15 Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure\nbsp{}ref:fig:test_apa_stroke_voltage). #+name: fig:test_apa_stroke_bench -#+caption: Bench to measure the APA stroke +#+caption: Bench to measure the APA stroke. #+attr_latex: :width 0.6\linewidth [[file:figs/test_apa_stroke_bench.jpg]] @@ -10492,7 +10484,7 @@ This unit was sent sent back to Cedrat, and a new one was shipped back. From now on, only the six remaining amplified piezoelectric actuators that behave as expected will be used. #+name: fig:test_apa_stroke -#+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of applied voltage (\subref{fig:test_apa_stroke_hysteresis}) +#+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of applied voltage (\subref{fig:test_apa_stroke_hysteresis}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_stroke_voltage}Applied voltage for stroke estimation} @@ -10520,7 +10512,7 @@ Using this setup, the transfer function from the injected force to the measured The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software, and the results are shown in Figure\nbsp{}ref:fig:test_apa_mode_shapes. #+name: fig:test_apa_mode_shapes -#+caption: First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model +#+caption: First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode ($268\,\text{Hz}$)} @@ -10569,7 +10561,7 @@ This could be explained by underestimation of the Young's modulus of the steel u Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades. #+name: fig:test_apa_meas_freq_compare -#+caption: Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$ +#+caption: Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$. #+attr_latex: :scale 0.8 [[file:figs/test_apa_meas_freq_compare.png]] @@ -10582,7 +10574,7 @@ Thus, there is no friction when actuating the APA300ML, and it will be easier to An encoder[fn:test_apa_8] is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the acrshort:apa. #+name: fig:test_bench_apa -#+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML +#+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_bench_picture}Picture of the test bench} @@ -10620,7 +10612,7 @@ For each excitation amplitude, the vertical displacement $d_e$ of the mass is me This is the typical behavior expected from a acrfull:pzt stack actuator, where the hysteresis increases as a function of the applied voltage amplitude\nbsp{}[[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]]. #+name: fig:test_apa_meas_hysteresis -#+caption: Displacement as a function of applied voltage for multiple excitation amplitudes +#+caption: Displacement as a function of applied voltage for multiple excitation amplitudes. #+attr_latex: :scale 0.8 [[file:figs/test_apa_meas_hysteresis.png]] @@ -10644,7 +10636,7 @@ These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measu #+attr_latex: :options [b]{0.57\textwidth} #+begin_minipage #+name: fig:test_apa_meas_stiffness_time -#+caption: Measured displacement when adding (at $t \approx 3\,\text{s}$) and removing (at $t \approx 13\,\text{s}$) the mass +#+caption: Measured displacement when adding (at $t \approx 3\,\text{s}$) and removing (at $t \approx 13\,\text{s}$) the mass. #+attr_latex: :scale 0.8 :float nil [[file:figs/test_apa_meas_stiffness_time.png]] #+end_minipage @@ -10710,7 +10702,7 @@ From this analysis, it can be inferred that the axial stiffness of the shell is All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure\nbsp{}ref:fig:test_apa_frf_encoder and at the force sensor in Figure\nbsp{}ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell. #+name: fig:test_apa_frf_dynamics -#+caption: Measured frequency response function from generated voltage $u$ to the encoder displacement $d_e$ (\subref{fig:test_apa_frf_encoder}) and to the force sensor voltage $V_s$ (\subref{fig:test_apa_frf_force}) for the six APA300ML +#+caption: Measured frequency response function from generated voltage $u$ to the encoder displacement $d_e$ (\subref{fig:test_apa_frf_encoder}) and to the force sensor voltage $V_s$ (\subref{fig:test_apa_frf_force}) for the six APA300ML. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_frf_encoder}FRF from $u$ to $d_e$} @@ -10769,7 +10761,7 @@ The (low frequency) transfer function from $u$ to $V_s$ with and without this re It is confirmed that the added resistor has the effect of adding a high-pass filter with a cut-off frequency of $\approx 0.39\,\text{Hz}$. #+name: fig:test_apa_effect_resistance -#+caption: Transfer function from $u$ to $V_s$ with and without the resistor $R$ in parallel with the piezoelectric stack used as the force sensor +#+caption: Transfer function from $u$ to $V_s$ with and without the resistor $R$ in parallel with the piezoelectric stack used as the force sensor. #+attr_latex: :scale 0.8 [[file:figs/test_apa_effect_resistance.png]] @@ -10801,7 +10793,7 @@ To estimate how the dynamics of the acrshort:apa changes when the Integral Force The transfer function from the "damped" plant input $u\prime$ to the encoder displacement $d_e$ is identified for several IFF controller gains $g$. #+name: fig:test_apa_iff_schematic -#+caption: Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input $u\prime$ +#+caption: Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input $u\prime$. [[file:figs/test_apa_iff_schematic.png]] The identified dynamics were then fitted by second order transfer functions[fn:test_apa_10]. @@ -10814,7 +10806,7 @@ Second using the fitted transfer functions of the damped plants experimentally i The two obtained root loci are compared in Figure\nbsp{}ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were fitted using only a second-order transfer function. #+name: fig:test_apa_iff -#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula} +#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants that match the experimental data (dashed lines)} @@ -10841,7 +10833,7 @@ This two degrees-of-freedom model is developed to accurately represent the APA30 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 -#+caption: Screenshot of the multi-body model +#+caption: Screenshot of the multi-body model. #+attr_latex: :width 0.7\linewidth [[file:figs/test_apa_bench_model.png]] @@ -10862,7 +10854,7 @@ Such a simple model has some limitations: - the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear #+name: fig:test_apa_2dof_model -#+caption: Schematic of the two degrees-of-freedom model of the APA300ML, adapted from\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic]] +#+caption: Schematic of the two degrees-of-freedom model of the APA300ML, adapted from\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic]]. [[file:figs/test_apa_2dof_model.png]] ***** Tuning of the APA model :ignore: @@ -10870,7 +10862,7 @@ 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 degrees-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. @@ -10901,7 +10893,7 @@ 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 DoF 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* | @@ -10924,7 +10916,7 @@ A good match can be observed between the model and the experimental data, both f 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 2DoF 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 2DoF 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 $d_e$} @@ -10984,7 +10976,7 @@ The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are list From these parameters, $g_s = 5.1\,\text{V}/\upmu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained, which are close to the constants identified using the experimentally identified transfer functions. #+name: tab:test_apa_piezo_properties -#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities +#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities. #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | *Description* | @@ -11006,7 +10998,7 @@ It is however surprising that the model is "softer" than the measured system, as Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever). #+name: fig:test_apa_super_element_comp_frf -#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_super_element_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_super_element_comp_frf_force}) +#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_super_element_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_super_element_comp_frf_force}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_super_element_comp_frf_enc}from $u$ to $d_e$} @@ -11059,7 +11051,7 @@ Deviations from these ideal properties will impact the dynamics of the Nano-Hexa During the detailed design phase, specifications in terms of stiffness and stroke were determined and are summarized in Table\nbsp{}ref:tab:test_joints_specs. #+name: tab:test_joints_specs -#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model +#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model. #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc #+attr_latex: :center t :booktabs t :float t | | *Specification* | *FEM* | @@ -11079,7 +11071,7 @@ It serves several functions, as shown in Figure\nbsp{}ref:fig:test_joints_iso, s The rotation axes are represented by the dashed lines that intersect #+name: fig:test_joints_schematic -#+caption: Geometry of the optimized flexible joints +#+caption: Geometry of the optimized flexible joints. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_iso}Isometric view} @@ -11105,7 +11097,7 @@ It serves several functions, as shown in Figure\nbsp{}ref:fig:test_joints_iso, s Sixteen flexible joints have been ordered (shown in Figure\nbsp{}ref:fig:test_joints_received) such that some selection can be made for the twelve that will be used on the nano-hexapod. #+name: fig:test_joints_picture -#+caption: Pictures of the received 16 flexible joints +#+caption: Pictures of the received 16 flexible joints. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_received}15 of the 16 received flexible joints} @@ -11143,7 +11135,7 @@ What is typically observed is shown in Figure\nbsp{}ref:fig:test_joints_profilom It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\upmu\text{m}$, #+name: fig:test_joints_profilometer -#+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image}) +#+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_profilometer_setup}Flexible joint fixed on the profilometer} @@ -11171,7 +11163,7 @@ The measured thickness is less than the specified value of $250\,\upmu\text{m}$, However, what is more important than the true value of the thickness is the consistency between all flexible joints. #+name: fig:test_joints_size_hist -#+caption: Histogram for the (16x2) measured beams' thicknesses +#+caption: Histogram for the (16x2) measured beams' thicknesses. #+attr_latex: :scale 0.8 [[file:figs/test_joints_size_hist.png]] @@ -11180,7 +11172,7 @@ However, what is more important than the true value of the thickness is the cons Using this profilometer allowed to detect flexible joints with manufacturing defects such as non-symmetrical shapes (see Figure\nbsp{}ref:fig:test_joints_bad_shape) or flexible joints with machining chips stuck in the gap (see Figure\nbsp{}ref:fig:test_joints_bad_chips). #+name: fig:test_joints_bad -#+caption: Example of two flexible joints that were considered unsatisfactory after visual inspection +#+caption: Example of two flexible joints that were considered unsatisfactory after visual inspection. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_bad_shape}Non-Symmetrical shape} @@ -11242,7 +11234,7 @@ One part of the flexible joint is fixed to a rigid frame while a (known) force $ The deflection of the joint $d_x$ is measured using a displacement sensor. #+name: fig:test_joints_bench_working_principle -#+caption: Working principle of the test bench used to estimate the bending stiffness $k_{R_y}$ of the flexible joints by measuring $F_x$, $d_x$ and $h$ +#+caption: Working principle of the test bench used to estimate the bending stiffness $k_{R_y}$ of the flexible joints by measuring $F_x$, $d_x$ and $h$. [[file:figs/test_joints_bench_working_principle.png]] ***** Required External Applied Force @@ -11354,7 +11346,7 @@ The most important source of error is the estimation error of the distance betwe An overall accuracy of $\approx 5\,\%$ can be expected with this measurement bench, which should be sufficient for an estimation of the bending stiffness of the flexible joints. #+name: tab:test_joints_error_budget -#+caption: Summary of the error budget for estimating the bending stiffness +#+caption: Summary of the error budget for estimating the bending stiffness. #+attr_latex: :environment tabularx :width 0.35\linewidth :align Xc #+attr_latex: :center t :booktabs t | *Effect* | *Error* | @@ -11383,7 +11375,7 @@ The flexible joint can be rotated by $90^o$ in order to measure the bending stif The obtained design of the measurement bench is shown in Figure\nbsp{}ref:fig:test_joints_bench_overview while a zoom on the flexible joint with the associated important quantities is shown in Figure\nbsp{}ref:fig:test_joints_bench_side. #+name: fig:test_joints_bench -#+caption: 3D view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side}) +#+caption: 3D view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_bench_overview} Schematic of the test bench to measure the bending stiffness of the flexible joints} @@ -11408,7 +11400,7 @@ A picture of the bench used to measure the X-bending stiffness of the flexible j A closer view of the force sensor tip is shown in Figure\nbsp{}ref:fig:test_joints_picture_bench_zoom. #+name: fig:test_joints_picture_bench -#+caption: Manufactured test bench for compliance measurement of the flexible joints +#+caption: Manufactured test bench for compliance measurement of the flexible joints. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_picture_bench_overview}Picture of the measurement bench} @@ -11435,7 +11427,7 @@ The gain mismatch between the two load cells is approximately $4\,\%$ which is h However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\,\text{N}$ and $5\,\text{N}$. #+name: fig:test_joints_force_sensor_calib -#+caption: Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in (\subref{fig:test_joints_force_sensor_calib_picture}). Comparison of the two measured forces and estimated non-linearity are shown in (\subref{fig:test_joints_force_sensor_calib_fit}) +#+caption: Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in (\subref{fig:test_joints_force_sensor_calib_picture}). Comparison of the two measured forces and estimated non-linearity are shown in (\subref{fig:test_joints_force_sensor_calib_fit}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_force_sensor_calib_picture}Zoom on the two load cells in contact} @@ -11555,7 +11547,7 @@ Furthermore, the data obtained from these measurements have provided the necessa The Nano-Hexapod struts (shown in Figure\nbsp{}ref:fig:test_struts_picture_strut) are composed of two flexible joints that are fixed at the two ends of the strut, one Amplified Piezoelectric Actuator[fn:test_struts_5] and one optical encoder[fn:test_struts_6]. #+name: fig:test_struts_picture_strut -#+caption: One strut including two flexible joints, an amplified piezoelectric actuator and an encoder +#+caption: One strut including two flexible joints, an amplified piezoelectric actuator and an encoder. #+attr_latex: :width 0.8\linewidth [[file:figs/test_struts_picture_strut.jpg]] @@ -11591,7 +11583,7 @@ Such flatness was checked using a FARO arm[fn:test_struts_1] (see Figure\nbsp{}r The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes. #+name: fig:test_struts_mounting -#+caption: Strut mounting bench +#+caption: Strut mounting bench. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_bench_first_concept}3D view of the mounting bench} @@ -11631,7 +11623,7 @@ The goal of these "sleeves" is to avoid mechanical stress that could damage the These "sleeves" have one dowel groove (that are fitted to the dowel holes shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_0) that will determine the length of the mounted strut. #+name: fig:test_struts_cylindrical_mounting -#+caption: Preparation of the flexible joints by fixing them in their cylindrical "sleeve" +#+caption: Preparation of the flexible joints by fixing them in their cylindrical "sleeve". #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_cylindrical_mounting_part_top}Cylindral Interface (Top)} @@ -11707,7 +11699,7 @@ The two cylindrical interfaces were fixed (boundary conditions), and the first t The mode shapes are displayed in Figure\nbsp{}ref:fig:test_struts_mode_shapes: an "X-bending" mode at $189\,\text{Hz}$, a "Y-bending" mode at $285\,\text{Hz}$ and a "Z-torsion" mode at $400\,\text{Hz}$. #+name: fig:test_struts_mode_shapes -#+caption: Spurious resonances of the struts estimated from a Finite Element Model +#+caption: Spurious resonances of the struts estimated from a Finite Element Model. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_1}X-bending mode ($189\,\text{Hz}$)} @@ -11739,7 +11731,7 @@ The "Y-bending" mode is measured as shown in Figure\nbsp{}ref:fig:test_struts_me These tests were performed with and without the encoder being fixed to the strut. #+name: fig:test_struts_meas_modes -#+caption: Measurement of strut flexible modes +#+caption: Measurement of strut flexible modes. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_meas_x_bending}X-bending mode} @@ -11765,7 +11757,7 @@ These tests were performed with and without the encoder being fixed to the strut The obtained acrshortpl:frf for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_no_enc when the encoder is not fixed to the strut and in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_enc when the encoder is fixed to the strut. #+name: fig:test_struts_spur_res_frf -#+caption: Measured frequency response functions without the encoder\nbsp{}ref:fig:test_struts_spur_res_frf and with the encoder\nbsp{}ref:fig:test_struts_spur_res_frf_enc +#+caption: Measured frequency response functions without the encoder\nbsp{}ref:fig:test_struts_spur_res_frf and with the encoder\nbsp{}ref:fig:test_struts_spur_res_frf_enc. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_spur_res_frf_no_enc}without encoder} @@ -11788,7 +11780,7 @@ In addition, the computed resonance frequencies from the acrshort:fem are very c This validates the quality of the acrshort:fem. #+name: tab:test_struts_spur_mode_freqs -#+caption: Measured frequency of the flexible modes of the strut +#+caption: Measured frequency of the flexible modes of the strut. #+attr_latex: :environment tabularx :width 0.7\linewidth :align Xccc #+attr_latex: :center t :booktabs t :float t | *Mode* | *FEM with Encoder* | *Exp. with Encoder* | *Exp. without Encoder* | @@ -11835,7 +11827,7 @@ Finally, all measured struts are compared in terms of dynamics in Section\nbsp{} System identification was performed without the encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_front) and with one encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_coder). #+name: fig:test_struts_bench_leg_with_without_enc -#+caption: Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front}) +#+caption: Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_bench_leg_coder}Strut with encoder} @@ -11859,7 +11851,7 @@ Similarly, it has little effect on the transfer function from $u$ to the sensor This means that the encoder should have little effect on the effectiveness of the integral force feedback control strategy. #+name: fig:test_struts_effect_encoder -#+caption: Effect of having the encoder fixed to the struts on the measured dynamics from $u$ to $d_a$ (\subref{fig:test_struts_effect_encoder_int}) and from $u$ to $V_s$ (\subref{fig:test_struts_effect_encoder_iff}). Comparison of the observed dynamics by the encoder and interferometers (\subref{fig:test_struts_comp_enc_int}) +#+caption: Effect of having the encoder fixed to the struts on the measured dynamics from $u$ to $d_a$ (\subref{fig:test_struts_effect_encoder_int}) and from $u$ to $V_s$ (\subref{fig:test_struts_effect_encoder_iff}). Comparison of the observed dynamics by the encoder and interferometers (\subref{fig:test_struts_comp_enc_int}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_effect_encoder_int}$u$ to $d_a$} @@ -11901,7 +11893,7 @@ The obtained dynamics from $u$ to $d_a$ are compared in Figure\nbsp{}ref:fig:tes A very good match can be observed between the struts. #+name: fig:test_struts_comp_plants -#+caption: Comparison of the measured plants +#+caption: Comparison of the measured plants. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$} @@ -11943,7 +11935,7 @@ This misalignment is estimated and measured in Section\nbsp{}ref:ssec:test_strut The struts were then disassembled and reassemble a second time to optimize alignment (Section\nbsp{}ref:sec:test_struts_meas_all_aligned_struts). #+name: fig:test_struts_simscape_model -#+caption: Screenshot of the multi-body model of the strut fixed to the bench +#+caption: Screenshot of the multi-body model of the strut fixed to the bench. #+attr_latex: :width 0.65\linewidth [[file:figs/test_struts_simscape_model.png]] @@ -11994,7 +11986,7 @@ For instance, consider Figure\nbsp{}ref:fig:test_struts_misalign_schematic where In this case, the "x-bending" mode at $200\,\text{Hz}$ (see Figure\nbsp{}ref:fig:test_struts_meas_x_bending) can be expected to have greater impact on the dynamics from the actuator to the encoder. #+name: fig:test_struts_misalign_schematic -#+caption: Mis-alignement between the joints and the APA +#+caption: Mis-alignement between the joints and the APA. #+attr_latex: :width 0.8\linewidth [[file:figs/test_struts_misalign_schematic.png]] @@ -12015,7 +12007,7 @@ A comparison of the experimental acrshortpl:frf in Figure\nbsp{}ref:fig:test_str This similarity suggests that the identified differences in dynamics are caused by misalignment. #+name: fig:test_struts_effect_misalignment -#+caption: Effect of a misalignment between the flexible joints and the APA300ML in the $y$ direction (\subref{fig:test_struts_effect_misalignment_y}) and in the $x$ direction (\subref{fig:test_struts_effect_misalignment_x}) +#+caption: Effect of a misalignment between the flexible joints and the APA300ML in the $y$ direction (\subref{fig:test_struts_effect_misalignment_y}) and in the $x$ direction (\subref{fig:test_struts_effect_misalignment_x}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_effect_misalignment_y}Misalignment along $y$} @@ -12049,7 +12041,7 @@ To check the validity of the measurement, it can be verified that the sum of the Thickness differences for all the struts were found to be between $0.94\,\text{mm}$ and $1.00\,\text{mm}$ which indicate low errors compared to the misalignments found in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment. #+name: tab:test_struts_meas_y_misalignment -#+caption: Measured $y$ misalignment at the top and bottom of the APA. Measurements are in $\text{mm}$ +#+caption: Measured $y$ misalignment at the top and bottom of the APA. Measurements are in $\text{mm}$. #+attr_latex: :environment tabularx :width 0.2\linewidth :align Xcc #+attr_latex: :center t :booktabs t | *Strut* | *Bot* | *Top* | @@ -12070,7 +12062,7 @@ In the next section, the struts are re-assembled with a "positioning pin" to bet With a better alignment, the amplitude of the spurious resonances is expected to decrease, as shown in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y. #+name: fig:test_struts_comp_dy_tuned_model_frf_enc -#+caption: Comparison of the frequency response functions from DAC voltage $u$ to measured displacement $d_e$ by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the $y$ misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the $x$ and $y$ misalignments are tuned +#+caption: Comparison of the frequency response functions from DAC voltage $u$ to measured displacement $d_e$ by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the $y$ misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the $x$ and $y$ misalignments are tuned. #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_dy_tuned_model_frf_enc.png]] @@ -12105,7 +12097,7 @@ The fact that the encoders are fixed to the struts makes the control more challe Therefore, fixing the encoders to the nano-hexapod plates instead may be an interesting option. #+name: fig:test_struts_comp_enc_frf_realign -#+caption: Comparison of the dynamics from $u$ to $d_e$ before and after proper alignment using the dowel pins +#+caption: Comparison of the dynamics from $u$ to $d_e$ before and after proper alignment using the dowel pins. #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_enc_frf_realign.png]] @@ -12143,7 +12135,7 @@ The goal was to fix the six struts to the two nano-hexapod plates (shown in Figu To do so, a precisely machined mounting tool (Figure\nbsp{}ref:fig:test_nhexa_center_part_hexapod_mounting) is used to position the two nano-hexapod plates during the assembly procedure. #+name: fig:test_nhexa_received_parts -#+caption: Nano-Hexapod plates (\subref{fig:test_nhexa_nano_hexapod_plates}) and mounting tool used to position the two plates during assembly (\subref{fig:test_nhexa_center_part_hexapod_mounting}) +#+caption: Nano-Hexapod plates (\subref{fig:test_nhexa_nano_hexapod_plates}) and mounting tool used to position the two plates during assembly (\subref{fig:test_nhexa_center_part_hexapod_mounting}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_nano_hexapod_plates}Top and bottom plates} @@ -12168,7 +12160,7 @@ The two plates were then fixed to the mounting tool, as shown in Figure\nbsp{}re The main goal of this "mounting tool" is to position the flexible joint interfaces (the "V" shapes) of both plates so that a cylinder can rest on the 4 flat interfaces at the same time. #+name: fig:test_nhexa_dimensional_check -#+caption: A FARO arm is used to dimensionally check the received parts (\subref{fig:test_nhexa_plates_tolerances}) and after mounting the two plates with the mounting part (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view}) +#+caption: A FARO arm is used to dimensionally check the received parts (\subref{fig:test_nhexa_plates_tolerances}) and after mounting the two plates with the mounting part (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_plates_tolerances}Dimensional check of the bottom plate} @@ -12208,7 +12200,7 @@ The straightness was found to be better than $15\,\upmu\text{m}$ for all struts[ The encoder rulers and heads were then fixed to the top and bottom plates, respectively (Figure\nbsp{}ref:fig:test_nhexa_mount_encoder), and the encoder heads were aligned to maximize the received contrast. #+name: fig:test_nhexa_mount_encoder -#+caption: Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate (\subref{fig:test_nhexa_mount_encoder_rulers}) while encoders heads are fixed to the bottom plate (\subref{fig:test_nhexa_mount_encoder_heads}) +#+caption: Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate (\subref{fig:test_nhexa_mount_encoder_rulers}) while encoders heads are fixed to the bottom plate (\subref{fig:test_nhexa_mount_encoder_heads}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_mount_encoder_rulers}Encoder rulers} @@ -12234,7 +12226,7 @@ The bottom flexible joint is then fixed. After mounting all six struts, the mounting tool (Figure\nbsp{}ref:fig:test_nhexa_center_part_hexapod_mounting) can be disassembled, and the nano-hexapod as shown in Figure\nbsp{}ref:fig:test_nhexa_nano_hexapod_mounted is fully assembled. #+name: fig:test_nhexa_nano_hexapod_mounted -#+caption: Mounted Nano-Hexapod +#+caption: Mounted Nano-Hexapod. #+attr_org: :width 800px #+attr_latex: :width \linewidth [[file:figs/test_nhexa_mounted_hexapod.jpg]] @@ -12280,7 +12272,7 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{ #+attr_latex: :options [t]{0.45\textwidth} #+begin_minipage #+name: fig:test_nhexa_suspended_table -#+caption: Mounted suspended table. Only 1 or the 15 accelerometer is mounted on top +#+caption: Mounted suspended table. Only 1 or the 15 accelerometer is mounted on top. #+attr_latex: :width 0.99\linewidth :float nil [[file:figs/test_nhexa_suspended_table.jpg]] #+end_minipage @@ -12304,7 +12296,7 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{ #+end_minipage #+name: fig:test_nhexa_table_flexible_modes -#+caption: Three identified flexible modes of the suspended table +#+caption: Three identified flexible modes of the suspended table. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_1}Flexible mode at $701\,\text{Hz}$} @@ -12340,7 +12332,7 @@ The stiffness of the springs in the horizontal plane is set at $0.5\,\text{N/mm} The obtained suspension modes of the multi-body model are compared with the measured modes in Table\nbsp{}ref:tab:test_nhexa_suspended_table_simscape_modes. #+name: tab:test_nhexa_suspended_table_simscape_modes -#+caption: Comparison of suspension modes of the multi-body model and the measured ones +#+caption: Comparison of suspension modes of the multi-body model and the measured ones. #+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcccc #+attr_latex: :center t :booktabs t | Directions | $D_x$, $D_y$ | $R_z$ | $D_z$ | $R_x$, $R_y$ | @@ -12357,7 +12349,7 @@ The Nano-Hexapod was then mounted on top of the suspended table, as shown in Fig All instrumentation (Speedgoat with acrshort:adc, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) were configured and connected to the nano-hexapod using many cables. #+name: fig:test_nhexa_hexa_suspended_table -#+caption: Mounted Nano-Hexapod on top of the suspended table +#+caption: Mounted Nano-Hexapod on top of the suspended table. #+attr_latex: :width 0.7\linewidth [[file:figs/test_nhexa_hexa_suspended_table.jpg]] @@ -12380,7 +12372,7 @@ To facilitate the future analysis of the measured plant dynamics, a basic modal Five 3-axis accelerometers were fixed on the top platform of the nano-hexapod (Figure\nbsp{}ref:fig:test_nhexa_modal_analysis) and the top platform was excited using an instrumented hammer. #+name: fig:test_nhexa_modal_analysis -#+caption: Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis +#+caption: Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis. #+attr_latex: :width 0.7\linewidth [[file:figs/test_nhexa_modal_analysis.jpg]] @@ -12389,7 +12381,7 @@ At around $700\,\text{Hz}$, two flexible modes of the top plate were observed (s These modes are summarized in Table\nbsp{}ref:tab:test_nhexa_hexa_modal_modes_list. #+name: tab:test_nhexa_hexa_modal_modes_list -#+caption: Description of the identified modes of the Nano-Hexapod +#+caption: Description of the identified modes of the Nano-Hexapod. #+attr_latex: :environment tabularx :width 0.6\linewidth :align ccX #+attr_latex: :center t :booktabs t | *Mode* | *Frequency* | *Description* | @@ -12404,7 +12396,7 @@ These modes are summarized in Table\nbsp{}ref:tab:test_nhexa_hexa_modal_modes_li | 8 | $709\,\text{Hz}$ | Second flexible mode of the top platform | #+name: fig:test_nhexa_hexa_flexible_modes -#+caption: Two identified flexible modes of the top plate of the Nano-Hexapod +#+caption: Two identified flexible modes of the top plate of the Nano-Hexapod. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at $692\,\text{Hz}$} @@ -12466,7 +12458,7 @@ To study how the dynamics changes with the payload mass, up to three "cylindrica These three cylindrical masses on top of the nano-hexapod are shown in Figure\nbsp{}ref:fig:test_nhexa_table_mass_3. #+name: fig:test_nhexa_table_mass_3 -#+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,\text{kg}$ +#+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,\text{kg}$. #+attr_org: :width 800px #+attr_latex: :width 0.8\linewidth [[file:figs/test_nhexa_table_mass_3.jpg]] @@ -12513,7 +12505,7 @@ In this section, the dynamics measured in Section\nbsp{}ref:sec:test_nhexa_dynam The nano-hexapod multi-body model was therefore added on top of the vibration table multi-body model, as shown in Figure\nbsp{}ref:fig:test_nhexa_hexa_simscape. #+name: fig:test_nhexa_hexa_simscape -#+caption: 3D representation of the multi-body model with the nano-hexapod on top of the suspended table. Three mass "layers" are here added +#+caption: 3D representation of the multi-body model with the nano-hexapod on top of the suspended table. Three mass "layers" are here added. #+attr_latex: :width 0.8\linewidth [[file:figs/test_nhexa_hexa_simscape.png]] @@ -12538,7 +12530,7 @@ The three resonances that were attributed to "internal" flexible modes of the st At higher frequencies, no resonances can be observed in the model, as the top plate and the encoder supports are modeled as rigid bodies. #+name: fig:test_nhexa_comp_simscape_diag -#+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}) +#+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}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$} @@ -12592,7 +12584,7 @@ One option could be to tune the damping as a function of the mass (similar to wh However, as decentralized IFF will be applied, the damping is actively brought, and the open-loop damping value should have very little impact on the obtained plant dynamics. #+name: fig:test_nhexa_comp_simscape_diag_masses -#+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}) +#+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}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$} @@ -12614,7 +12606,7 @@ 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) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$ +#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all_high_mass.png]] @@ -12671,7 +12663,7 @@ Finally, Section\nbsp{}ref:sec:test_id31_experiments evaluates the NASS's positi These include tomography scans at various speeds and with different payload masses, reflectivity measurements, and combined motion sequences that test the system's full capabilities. #+name: fig:test_id31_micro_station_nano_hexapod -#+caption: Picture of the micro-station without the nano-hexapod (\subref{fig:test_id31_micro_station_cables}) and with the nano-hexapod (\subref{fig:test_id31_fixed_nano_hexapod}) +#+caption: Picture of the micro-station without the nano-hexapod (\subref{fig:test_id31_micro_station_cables}) and with the nano-hexapod (\subref{fig:test_id31_fixed_nano_hexapod}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_micro_station_cables}Micro-station and nano-hexapod cables} @@ -12700,7 +12692,7 @@ 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 5DoF. 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}). The interferometer heads are 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 5DoF. 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}). The interferometer heads are 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} @@ -12796,7 +12788,7 @@ The short-stroke metrology system was placed on top of the main granite using gr Granite is used for its good mechanical and thermal stability. #+name: fig:test_id31_short_stroke_metrology_overview -#+caption: Granite gantry used to fix the short-stroke metrology system +#+caption: Granite gantry used to fix the short-stroke metrology system. #+attr_latex: :width 0.8\linewidth [[file:figs/test_id31_short_stroke_metrology_overview.jpg]] @@ -12925,7 +12917,7 @@ Finally, the Jacobian matrix $\bm{J}$ of the nano-hexapod is used to map $\bm{\e Voltages generated by the force sensor piezoelectric stacks $\bm{V}_s = [V_{s1},\ V_{s2},\ V_{s3},\ V_{s4},\ V_{s5},\ V_{s6}]$ will be used for active damping. #+name: fig:test_id31_block_schematic_plant -#+caption: Schematic of the NASS plant +#+caption: Schematic of the NASS plant. #+attr_latex: :scale 0.9 [[file:figs/test_id31_block_schematic_plant.png]] @@ -12998,7 +12990,7 @@ Between $650\,\text{Hz}$ and $1000\,\text{Hz}$, several modes can be observed, w The flexible modes of the top platform can be passively damped, whereas the modes of the two reference spheres should not be present in the final application. #+name: fig:test_id31_first_id_int_better_rz_align -#+caption: Decrease of the coupling with better Rz alignment +#+caption: Decrease of the coupling with better Rz alignment. #+attr_latex: :scale 0.8 [[file:figs/test_id31_first_id_int_better_rz_align.png]] @@ -13043,7 +13035,7 @@ It is interesting to note that the anti-resonances in the force sensor plant now #+end_figure #+name: fig:test_id31_comp_simscape_diag_masses -#+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 $\epsilon\mathcal{L}$ (\subref{fig:test_id31_comp_simscape_int_diag_masses}) and from $u$ to $V_s$ (\subref{fig:test_id31_comp_simscape_iff_diag_masses}) +#+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 $\epsilon\mathcal{L}$ (\subref{fig:test_id31_comp_simscape_int_diag_masses}) and from $u$ to $V_s$ (\subref{fig:test_id31_comp_simscape_iff_diag_masses}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_simscape_int_diag_masses}from $u$ to $\epsilon\mathcal{L}$} @@ -13131,7 +13123,7 @@ 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) frequency transfer functions 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 modeled (in red) frequency transfer functions 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]] @@ -13150,7 +13142,7 @@ The bode plot of the decentralized IFF controller is shown in Figure\nbsp{}ref:f It can be seen that the loop-gain is larger than $1$ around the suspension modes, which indicates that some damping should be added to the suspension modes. #+name: fig:test_id31_Kiff -#+caption: Bode plot of the decentralized IFF controller (\subref{fig:test_id31_Kiff_bode_plot}). The decentralized controller $K_{\text{IFF}}$ multiplied by the identified dynamics from $u_1$ to $V_{s1}$ for all payloads are shown in (\subref{fig:test_id31_Kiff_loop_gain}) +#+caption: Bode plot of the decentralized IFF controller (\subref{fig:test_id31_Kiff_bode_plot}). The decentralized controller $K_{\text{IFF}}$ multiplied by the identified dynamics from $u_1$ to $V_{s1}$ for all payloads are shown in (\subref{fig:test_id31_Kiff_loop_gain}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_Kiff_bode_plot}Bode plot of $K_{\text{IFF}}$} @@ -13215,7 +13207,7 @@ 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 plants and the damped plant with Decentralized IFF, estimated from the multi-body model (\subref{fig:test_id31_comp_ol_iff_plant_model}). Comparison of measured damped and modeled plants for all considered payloads (\subref{fig:test_id31_hac_plant_effect_mass}). Only "direct" terms ($\epsilon\mathcal{L}_i/u_i^\prime$) are displayed for simplificty +#+caption: Comparison of the open-loop plants and the damped plant with Decentralized IFF, estimated from the multi-body model (\subref{fig:test_id31_comp_ol_iff_plant_model}). Comparison of measured damped and modeled plants for all considered payloads (\subref{fig:test_id31_hac_plant_effect_mass}). Only "direct" terms ($\epsilon\mathcal{L}_i/u_i^\prime$) are displayed for simplificty. #+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} @@ -13271,7 +13263,7 @@ 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) frequency transfer functions 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 modeled (in red) frequency transfer functions 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]] @@ -13309,7 +13301,7 @@ Above the suspension modes of the nano-hexapod, the motion induced by the piezoe This design choice, while beneficial for system simplicity, introduces inherent limitations in the system's ability to handle larger masses at high frequency. #+name: fig:test_id31_hac_rga_number -#+caption: RGA-number for the damped plants - Comparison of all the payload conditions +#+caption: RGA-number for the damped plants - Comparison of all the payload conditions. #+attr_latex: :scale 0.8 [[file:figs/test_id31_hac_rga_number.png]] @@ -13330,7 +13322,7 @@ The closed-loop stability was verified by computing the characteristic Loci (Fig However, small stability margins were observed for the highest mass, indicating that some multivariable effects are in play. #+name: fig:test_id31_hac_loop_gain_loci -#+caption: Robust High Authority Controller. "Decentralized loop-gains" are shown in (\subref{fig:test_id31_hac_loop_gain}) and characteristic loci are shown in (\subref{fig:test_id31_hac_characteristic_loci}) +#+caption: Robust High Authority Controller. "Decentralized loop-gains" are shown in (\subref{fig:test_id31_hac_loop_gain}) and characteristic loci are shown in (\subref{fig:test_id31_hac_characteristic_loci}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_loop_gain}Loop Gains} @@ -13384,7 +13376,7 @@ However, the positioning errors worsen as the payload mass increases, especially However, it was decided that this controller should be tested experimentally and improved only if necessary. #+name: fig:test_id31_hac_tomography_Wz36_simulation -#+caption: Positioning errors in the Y-Z plane during tomography experiments simulated using the multi-body model (in closed-loop) +#+caption: Positioning errors in the Y-Z plane during tomography experiments simulated using the multi-body model (in closed-loop). #+attr_latex: :scale 0.8 [[file:figs/test_id31_hac_tomography_Wz36_simulation.png]] @@ -13432,7 +13424,7 @@ In terms of RMS errors, this corresponds to $30\,\text{nm}$ in $D_y$, $15\,\text Results obtained for all experiments are summarized and compared to the specifications in Section\nbsp{}ref:ssec:test_id31_scans_conclusion. #+name: tab:test_id31_experiments_specifications -#+caption: Specifications for the Nano-Active-Stabilization-System +#+caption: Specifications for the Nano-Active-Stabilization-System. #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :center t :booktabs t | | $D_y$ | $D_z$ | $R_y$ | @@ -13622,7 +13614,7 @@ To eliminate tracking errors, the feedback controller incorporates two integrato Initial testing at $10\,\upmu\text{m/s}$ demonstrated positioning errors well within specifications (indicated by dashed lines in Figure\nbsp{}ref:fig:test_id31_dz_scan_10ums). #+name: fig:test_id31_dz_scan_10ums -#+caption: $D_z$ scan at a velocity of $10\,\upmu \text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry}) +#+caption: $D_z$ scan at a velocity of $10\,\upmu \text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dy}$D_y$} @@ -13650,7 +13642,7 @@ Since detectors typically operate only during the constant velocity phase, these However, performance during acceleration phases could be enhanced through the implementation of feedforward control. #+name: fig:test_id31_dz_scan_100ums -#+caption: $D_z$ scan at a velocity of $100\,\upmu\text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry}) +#+caption: $D_z$ scan at a velocity of $100\,\upmu\text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dy}$D_y$} @@ -13762,7 +13754,7 @@ To avoid high-frequency vibrations typically induced by the stepper motor, the $ The system performance was evaluated at three lateral scanning velocities: $0.1\,\text{mm/s}$, $0.5\,\text{mm/s}$, and $1\,\text{mm/s}$. Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_setpoint presents both the $D_y$ position setpoints and the corresponding measured $D_y$ positions for all tested velocities. #+name: fig:test_id31_diffraction_tomo_setpoint -#+caption: Dy motion for several configured velocities +#+caption: Dy motion for several configured velocities. #+attr_latex: :scale 0.8 [[file:figs/test_id31_diffraction_tomo_setpoint.png]] @@ -14075,7 +14067,7 @@ Designing a future end-station with the understanding that a functional NASS wil One possible configuration, illustrated in Figure\nbsp{}ref:fig:conclusion_nass_architecture, would comprise a long-stroke Stewart platform providing the required mobility without generating high-frequency vibrations; a spindle that needs not deliver exceptional performance but should be stiff and avoid inducing high-frequency vibrations (an air-bearing spindle might not be essential); and a short-stroke Stewart platform for correcting errors from the long-stroke stage and spindle. #+name: fig:conclusion_nass_architecture -#+caption: Proposed alternative configuration for an end-station including the Nano Active Stabilization System +#+caption: Proposed alternative configuration for an end-station including the Nano Active Stabilization System. #+attr_latex: :options [h!tbp] [[file:figs/conclusion_nass_architecture.png]] @@ -14092,7 +14084,7 @@ Specifically, a compact 6DoF stage known as LevCube, providing a mobility of app 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 6DoF 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 6DoF 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}} @@ -14220,8 +14212,8 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:test_id31_8]Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and $100\,\text{nm}$ "resolution" (equal to the vertical beam size). [fn:test_id31_7]The highest rotational velocity of $360\,\text{deg/s}$ could not be tested due to an issue in the Spindle's controller. [fn:test_id31_6]The roundness of the spheres is specified at $50\,\text{nm}$. -[fn:test_id31_5]The "IcePAP"\nbsp{}[[cite:&janvier13_icepap]] which is developed at the ESRF. -[fn:test_id31_4]Note that the eccentricity of the "point of interest" with respect to the Spindle rotation axis has been tuned based on measurements. +[fn:test_id31_5]The "IcePAP"\nbsp{}[[cite:&janvier13_icepap]] which is developed at the ESRF. +[fn:test_id31_4]Note that the eccentricity of the PoI with respect to the Spindle rotation axis has been tuned based on measurements. [fn:test_id31_3]The "PEPU"\nbsp{}[[cite:&hino18_posit_encod_proces_unit]] was used for digital protocol conversion between the interferometers and the Speedgoat. [fn:test_id31_2]M12/F40 model from Attocube. [fn:test_id31_1]Depending on the measuring range, gap can range from $\approx 1\,\upmu\text{m}$ to $\approx 100\,\upmu\text{m}$.