From 89840393eee4e5170fea6cf2766359282f6a66bf Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 15 Apr 2025 21:25:54 +0200 Subject: [PATCH] Add non breaking spaces for references --- phd-thesis.org | 2600 ++++++++++++++++++++++++------------------------ 1 file changed, 1300 insertions(+), 1300 deletions(-) diff --git a/phd-thesis.org b/phd-thesis.org index 990cc71..2f133e0 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -327,26 +327,26 @@ The research presented in this manuscript has been possible thanks to the Fonds In this report, a uniaxial model of the acrfull:nass is developed and used to obtain a first idea of the challenges involved in this complex system. Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, yielding to similar conclusions. -The model is schematically shown in Figure ref:fig:uniaxial_overview_model_sections where the colors represent the parts studied in different sections. +The model is schematically shown in Figure\nbsp{}ref:fig:uniaxial_overview_model_sections where the colors represent the parts studied in different sections. -To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section ref:sec:uniaxial_micro_station_model). +To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section\nbsp{}ref:sec:uniaxial_micro_station_model). Then, a model of the nano-hexapod is added on top of the micro-station. -With the added sample and sensors, this gives a uniaxial dynamical model of the acrshort:nass that will be used for further analysis (Section ref:sec:uniaxial_nano_station_model). +With the added sample and sensors, this gives a uniaxial dynamical model of the acrshort:nass that will be used for further analysis (Section\nbsp{}ref:sec:uniaxial_nano_station_model). -The disturbances affecting position stability are identified experimentally (Section ref:sec:uniaxial_disturbances) and included in the model for dynamical noise budgeting (Section ref:sec:uniaxial_noise_budgeting). +The disturbances affecting position stability are identified experimentally (Section\nbsp{}ref:sec:uniaxial_disturbances) and included in the model for dynamical noise budgeting (Section\nbsp{}ref:sec:uniaxial_noise_budgeting). In all the following analysis, three nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate nano-hexapod design. Three sample masses are also considered to verify the robustness of the applied control strategies with respect to a change of sample. To improve the position stability of the sample, an acrfull:haclac strategy is applied. It consists of first actively damping the plant (the acrshort:lac part), and then applying a position control on the damped plant (the acrshort:hac part). -Three active damping techniques are studied (Section ref:sec:uniaxial_active_damping) which are used to both reduce the effect of disturbances and make the system easier to control afterwards. -Once the system is well damped, a feedback position controller is applied and the obtained performance is analyzed (Section ref:sec:uniaxial_position_control). +Three active damping techniques are studied (Section\nbsp{}ref:sec:uniaxial_active_damping) which are used to both reduce the effect of disturbances and make the system easier to control afterwards. +Once the system is well damped, a feedback position controller is applied and the obtained performance is analyzed (Section\nbsp{}ref:sec:uniaxial_position_control). -Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section ref:sec:uniaxial_support_compliance) and the presence of dynamics between the nano-hexapod and the sample's point of interest (Section ref:sec:uniaxial_payload_dynamics). +Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section\nbsp{}ref:sec:uniaxial_support_compliance) and the presence of dynamics between the nano-hexapod and the sample's point of interest (Section\nbsp{}ref:sec:uniaxial_payload_dynamics). #+name: fig:uniaxial_overview_model_sections -#+caption: Uniaxial Micro-Station model in blue (Section ref:sec:uniaxial_micro_station_model), Nano-Hexapod models in red (Section ref:sec:uniaxial_nano_station_model), Disturbances in yellow (Section ref:sec:uniaxial_disturbances), Active Damping in green (Section ref:sec:uniaxial_active_damping), Position control in purple (Section ref:sec:uniaxial_position_control) and Sample dynamics in cyan (Section ref:sec:uniaxial_payload_dynamics) +#+caption: Uniaxial Micro-Station model in blue (Section\nbsp{}ref:sec:uniaxial_micro_station_model), Nano-Hexapod models in red (Section\nbsp{}ref:sec:uniaxial_nano_station_model), Disturbances in yellow (Section\nbsp{}ref:sec:uniaxial_disturbances), Active Damping in green (Section\nbsp{}ref:sec:uniaxial_active_damping), Position control in purple (Section\nbsp{}ref:sec:uniaxial_position_control) and Sample dynamics in cyan (Section\nbsp{}ref:sec:uniaxial_payload_dynamics) [[file:figs/uniaxial_overview_model_sections.png]] *** Micro Station Model @@ -354,7 +354,7 @@ Two key effects that may limit that positioning performances are then considered **** Introduction :ignore: In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station. -The measurement setup is shown in Figure ref:fig:uniaxial_ustation_first_meas_dynamics where several geophones[fn:uniaxial_1] are fixed to the micro-station and an instrumented hammer is used to inject forces on different stages of the micro-station. +The measurement setup is shown in Figure\nbsp{}ref:fig:uniaxial_ustation_first_meas_dynamics where several geophones[fn:uniaxial_1] are fixed to the micro-station and an instrumented hammer is used to inject forces on different stages of the micro-station. From the measured frequency response functions (FRF), the model can be tuned to approximate the uniaxial dynamics of the micro-station. @@ -365,11 +365,11 @@ From the measured frequency response functions (FRF), the model can be tuned to **** Measured dynamics -The measurement setup is schematically shown in Figure ref:fig:uniaxial_ustation_meas_dynamics_schematic where two vertical hammer hits are performed, one on the Granite (force $F_{g}$) and the other on the micro-hexapod's top platform (force $F_{h}$). +The measurement setup is schematically shown in Figure\nbsp{}ref:fig:uniaxial_ustation_meas_dynamics_schematic where two vertical hammer hits are performed, one on the Granite (force $F_{g}$) and the other on the micro-hexapod's top platform (force $F_{h}$). The vertical inertial motion of the granite $x_{g}$ and the top platform of the micro-hexapod $x_{h}$ are measured using geophones. Three frequency response functions were computed: one from $F_{h}$ to $x_{h}$ (i.e., the compliance of the micro-station), one from $F_{g}$ to $x_{h}$ (or from $F_{h}$ to $x_{g}$) and one from $F_{g}$ to $x_{g}$. -Due to the poor coherence at low frequencies, these frequency response functions will only be shown between 20 and 200Hz (solid lines in Figure ref:fig:uniaxial_comp_frf_meas_model). +Due to the poor coherence at low frequencies, these frequency response functions will only be shown between 20 and 200Hz (solid lines in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model). #+name: fig:micro_station_uniaxial_model #+caption: Schematic of the Micro-Station measurement setup and uniaxial model. @@ -390,7 +390,7 @@ Due to the poor coherence at low frequencies, these frequency response functions #+end_figure **** Uniaxial Model -The uniaxial model of the micro-station is shown in Figure ref:fig:uniaxial_model_micro_station. +The uniaxial model of the micro-station is shown in Figure\nbsp{}ref:fig:uniaxial_model_micro_station. It consists of a mass spring damper system with three degrees of freedom. A mass-spring-damper system represents the granite (with mass $m_g$, stiffness $k_g$ and damping $c_g$). Another mass-spring-damper system represents the different micro-station stages (the $T_y$ stage, the $R_y$ stage and the $R_z$ stage) with mass $m_t$, damping $c_t$ and stiffness $k_t$. @@ -398,7 +398,7 @@ Finally, a third mass-spring-damper system represents the micro-hexapod with mas The masses of the different stages are estimated from the 3D model, while the stiffnesses are from the data-sheet of the manufacturers. The damping coefficients were tuned to match the damping identified from the measurements. -The parameters obtained are summarized in Table ref:tab:uniaxial_ustation_parameters. +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 @@ -414,10 +414,10 @@ Two disturbances are considered which are shown in red: the floor motion $x_f$ a The hammer impacts $F_{h}, F_{g}$ are shown in blue, whereas the measured inertial motions $x_{h}, x_{g}$ are shown in black. **** Comparison of model and measurements -The transfer functions from the forces injected by the hammers to the measured inertial motion of the micro-hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure ref:fig:uniaxial_comp_frf_meas_model. +The transfer functions from the forces injected by the hammers to the measured inertial motion of the micro-hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model. Because the uniaxial model has three degrees of freedom, only three modes with frequencies at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$ are modeled. -Many more modes can be observed in the measurements (see Figure ref:fig:uniaxial_comp_frf_meas_model). +Many more modes can be observed in the measurements (see Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model). However, the goal is not to have a perfect match with the measurement (this would require a much more complex model), but to have a first approximation. More accurate models will be used later on. @@ -429,11 +429,11 @@ More accurate models will be used later on. <> **** Introduction :ignore: -A model of the nano-hexapod and sample is now added on top of the uniaxial model of the micro-station (Figure ref:fig:uniaxial_model_micro_station_nass). +A model of the nano-hexapod and sample is now added on top of the uniaxial model of the micro-station (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). Disturbances (shown in red) are gls:fs the direct forces applied to the sample (for example cable forces), gls:ft representing the vibrations induced when scanning the different stages and gls:xf the floor motion. The control signal is the force applied by the nano-hexapod $f$ and the measurement is the relative motion between the sample and the granite $d$. The sample is here considered as a rigid body and rigidly fixed to the nano-hexapod. -The effect of resonances between the sample's point of interest and the nano-hexapod actuator will be considered in Section ref:sec:uniaxial_payload_dynamics. +The effect of resonances between the sample's point of interest and the nano-hexapod actuator will be considered in Section\nbsp{}ref:sec:uniaxial_payload_dynamics. #+name: fig:uniaxial_model_micro_station_nass_with_tf #+caption: Uniaxial model of the NASS (\subref{fig:uniaxial_model_micro_station_nass}) with the micro-station shown in black, the nano-hexapod represented in blue and the sample represented in green. Disturbances are shown in red. Extracted transfer function from $f$ to $d$ (\subref{fig:uniaxial_plant_first_params}). @@ -454,17 +454,17 @@ The effect of resonances between the sample's point of interest and the nano-hex #+end_figure **** Nano-Hexapod Parameters -The nano-hexapod is represented by a mass spring damper system (shown in blue in Figure ref:fig:uniaxial_model_micro_station_nass). +The nano-hexapod is represented by a mass spring damper system (shown in blue in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). Its mass gls:mn is set to $15\,\text{kg}$ while its stiffness $k_n$ can vary depending on the chosen architecture/technology. The sample is represented by a mass gls:ms that can vary from $1\,\text{kg}$ up to $50\,\text{kg}$. As a first example, the nano-hexapod stiffness of is set at $k_n = 10\,N/\mu m$ and the sample mass is chosen at $m_s = 10\,\text{kg}$. **** Obtained Dynamic Response -The sensitivity to disturbances (i.e., the transfer functions from $x_f,f_t,f_s$ to $d$) can be extracted from the uniaxial model of Figure ref:fig:uniaxial_model_micro_station_nass and are shown in Figure ref:fig:uniaxial_sensitivity_dist_first_params. -The /plant/ (i.e., the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure ref:fig:uniaxial_plant_first_params. +The sensitivity to disturbances (i.e., the transfer functions from $x_f,f_t,f_s$ to $d$) can be extracted from the uniaxial model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass and are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_first_params. +The /plant/ (i.e., the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure\nbsp{}ref:fig:uniaxial_plant_first_params. -For further analysis, 9 "configurations" of the uniaxial NASS model of Figure ref:fig:uniaxial_model_micro_station_nass will be considered: three nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) combined with three sample's masses ($m_s = 1\,kg$, $m_s = 25\,kg$ and $m_s = 50\,kg$). +For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass will be considered: three nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) combined with three sample's masses ($m_s = 1\,kg$, $m_s = 25\,kg$ and $m_s = 50\,kg$). #+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}) @@ -495,9 +495,9 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure re *** Disturbance Identification <> **** Introduction :ignore: -To quantify disturbances (red signals in Figure ref:fig:uniaxial_model_micro_station_nass), three geophones[fn:uniaxial_2] are used. -One is located on the floor, another one on the granite, and the last one on the micro-hexapod's top platform (see Figure ref:fig:uniaxial_ustation_meas_disturbances). -The geophone located on the floor was used to measure the floor motion $x_f$ while the other two geophones were used to measure vibrations introduced by scanning of the $T_y$ stage and $R_z$ stage (see Figure ref:fig:uniaxial_ustation_dynamical_id_setup). +To quantify disturbances (red signals in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), three geophones[fn:uniaxial_2] are used. +One is located on the floor, another one on the granite, and the last one on the micro-hexapod's top platform (see Figure\nbsp{}ref:fig:uniaxial_ustation_meas_disturbances). +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}) @@ -518,13 +518,13 @@ The geophone located on the floor was used to measure the floor motion $x_f$ whi #+end_figure **** Ground Motion -To acquire the geophone signals, the measurement setup shown in Figure ref:fig:uniaxial_geophone_meas_chain is used. +To acquire the geophone signals, the measurement setup shown in Figure\nbsp{}ref:fig:uniaxial_geophone_meas_chain is used. The voltage generated by the geophone is amplified using a low noise voltage amplifier[fn:uniaxial_3] with a gain of 60dB before going to the ADC. This is done to improve the signal-to-noise ratio. To reconstruct the displacement $x_f$ from the measured voltage $\hat{V}_{x_f}$, the transfer function of the measurement chain from $x_f$ to $\hat{V}_{x_f}$ needs to be estimated. -First, the transfer function $G_{geo}$ from the floor motion $x_{f}$ to the generated geophone voltage $V_{x_f}$ is shown in eqref:eq:uniaxial_geophone_tf, with $T_g = 88\,\frac{V}{m/s}$ the sensitivity of the geophone, $f_0 = \frac{\omega_0}{2\pi} = 2\,\text{Hz}$ its resonance frequency and $\xi = 0.7$ its damping ratio. -This model of the geophone was taken from [[cite:&collette12_review]]. +First, the transfer function $G_{geo}$ from the floor motion $x_{f}$ to the generated geophone voltage $V_{x_f}$ is shown in\nbsp{}eqref:eq:uniaxial_geophone_tf, with $T_g = 88\,\frac{V}{m/s}$ the sensitivity of the geophone, $f_0 = \frac{\omega_0}{2\pi} = 2\,\text{Hz}$ its resonance frequency and $\xi = 0.7$ its damping ratio. +This model of the geophone was taken from\nbsp{}[[cite:&collette12_review]]. The gain of the voltage amplifier is $V^{\prime}_{x_f}/V_{x_f} = g_0 = 1000$. \begin{equation}\label{eq:uniaxial_geophone_tf} @@ -535,8 +535,8 @@ G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi #+caption: Measurement setup for one geophone. The inertial displacement $x$ is converted to a voltage $V$ by the geophone. This voltage is amplified by a factor $g_0 = 60\,dB$ using a low-noise voltage amplifier. It is then converted to a digital value $\hat{V}_x$ using a 16bit ADC. [[file:figs/uniaxial_geophone_meas_chain.png]] -The amplitude spectral density of the floor motion $\Gamma_{x_f}$ can be computed from the amplitude spectral density of measured voltage $\Gamma_{\hat{V}_{x_f}}$ using eqref:eq:uniaxial_asd_floor_motion. -The estimated amplitude spectral density $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in Figure ref:fig:uniaxial_asd_floor_motion_id31. +The amplitude spectral density of the floor motion $\Gamma_{x_f}$ can be computed from the amplitude spectral density of measured voltage $\Gamma_{\hat{V}_{x_f}}$ using\nbsp{}eqref:eq:uniaxial_asd_floor_motion. +The estimated amplitude spectral density $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in Figure\nbsp{}ref:fig:uniaxial_asd_floor_motion_id31. \begin{equation}\label{eq:uniaxial_asd_floor_motion} \Gamma_{x_f}(\omega) = \frac{\Gamma_{\hat{V}_{x_f}}(\omega)}{|G_{geo}(j\omega)| \cdot g_0} \quad \left[ m/\sqrt{\text{Hz}} \right] @@ -561,11 +561,11 @@ The estimated amplitude spectral density $\Gamma_{x_f}$ of the floor motion $x_f #+end_figure **** Stage Vibration -To estimate the vibrations induced by scanning the micro-station stages, two geophones are used, as shown in Figure ref:fig:uniaxial_ustation_dynamical_id_setup. +To estimate the vibrations induced by scanning the micro-station stages, two geophones are used, as shown in Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup. The vertical relative velocity between the top platform of the micro hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. The vibrations induced by the $T_y$ stage are not considered here because they have less amplitude than the vibrations induced by the $R_z$ stage and because the $T_y$ stage can be scanned at lower velocities if the induced vibrations are found to be an issue. -The amplitude spectral density of the relative motion with and without the Spindle rotation are compared in Figure ref:fig:uniaxial_asd_vibration_spindle_rotation. +The amplitude spectral density of the relative motion with and without the Spindle rotation are compared in Figure\nbsp{}ref:fig:uniaxial_asd_vibration_spindle_rotation. It is shown that the spindle rotation increases the vibrations above $20\,\text{Hz}$. 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". @@ -573,8 +573,8 @@ The sharp peak observed at $24\,\text{Hz}$ is believed to be induced by electrom #+caption: Amplitude Spectral Density $\Gamma_{R_z}$ of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating [[file:figs/uniaxial_asd_vibration_spindle_rotation.png]] -To compute the equivalent disturbance force $f_t$ (Figure ref:fig:uniaxial_model_micro_station) that induces such motion, the transfer function $G_{f_t}(s)$ from $f_t$ to the relative motion between the micro-hexapod's top platform and the granite $(x_{h} - x_{g})$ is extracted from the model. -The amplitude spectral density $\Gamma_{f_{t}}$ of the disturbance force is them computed from eqref:eq:uniaxial_ft_asd and is shown in Figure ref:fig:uniaxial_asd_disturbance_force. +To compute the equivalent disturbance force $f_t$ (Figure\nbsp{}ref:fig:uniaxial_model_micro_station) that induces such motion, the transfer function $G_{f_t}(s)$ from $f_t$ to the relative motion between the micro-hexapod's top platform and the granite $(x_{h} - x_{g})$ is extracted from the model. +The amplitude spectral density $\Gamma_{f_{t}}$ of the disturbance force is them computed from\nbsp{}eqref:eq:uniaxial_ft_asd and is shown in Figure\nbsp{}ref:fig:uniaxial_asd_disturbance_force. \begin{equation}\label{eq:uniaxial_ft_asd} \Gamma_{f_{t}}(\omega) = \frac{\Gamma_{R_{z}}(\omega)}{|G_{f_t}(j\omega)|} @@ -583,26 +583,26 @@ The amplitude spectral density $\Gamma_{f_{t}}$ of the disturbance force is them *** Open-Loop Dynamic Noise Budgeting <> **** Introduction :ignore: -Now that a model of the acrshort:nass has been obtained (see section ref:sec:uniaxial_nano_station_model) and that the disturbances have been estimated (see section ref:sec:uniaxial_disturbances), it is possible to perform an /open-loop dynamic noise budgeting/. +Now that a model of the acrshort:nass has been obtained (see section\nbsp{}ref:sec:uniaxial_nano_station_model) and that the disturbances have been estimated (see section\nbsp{}ref:sec:uniaxial_disturbances), it is possible to perform an /open-loop dynamic noise budgeting/. -To perform such noise budgeting, the disturbances need to be modeled by their spectral densities (done in section ref:sec:uniaxial_disturbances). -Then, the transfer functions from disturbances to the performance metric (here the distance $d$) are computed (Section ref:ssec:uniaxial_noise_budget_sensitivity). +To perform such noise budgeting, the disturbances need to be modeled by their spectral densities (done in section\nbsp{}ref:sec:uniaxial_disturbances). +Then, the transfer functions from disturbances to the performance metric (here the distance $d$) are computed (Section\nbsp{}ref:ssec:uniaxial_noise_budget_sensitivity). Finally, these two types of information are combined to estimate the corresponding spectral density of the performance metric. -This is very useful to identify what is limiting the performance of the system, or the compare the achievable performance with different system parameters (Section ref:ssec:uniaxial_noise_budget_result). +This is very useful to identify what is limiting the performance of the system, or the compare the achievable performance with different system parameters (Section\nbsp{}ref:ssec:uniaxial_noise_budget_result). **** Sensitivity to disturbances <> -From the uniaxial model of the acrshort:nass (Figure ref:fig:uniaxial_model_micro_station_nass), the transfer function from the disturbances ($f_s$, $x_f$ and $f_t$) to the displacement $d$ are computed. +From the uniaxial model of the acrshort:nass (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), the transfer function from the disturbances ($f_s$, $x_f$ and $f_t$) to the displacement $d$ are computed. This is done for two extreme sample masses $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$ and three nano-hexapod stiffnesses: - $k_n = 0.01\,N/\mu m$ that represents a voice coil actuator with soft flexible guiding - $k_n = 1\,N/\mu m$ that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator - $k_n = 100\,N/\mu m$ that represents a stiff piezoelectric stack actuator -The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the sample mass $m_s = 1\,\text{kg}$ (the same conclusions can be drawn with $m_s = 50\,\text{kg}$): -- The soft nano-hexapod is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs) -- Between the suspension mode of the nano-hexapod (here at 5Hz for the soft nano-hexapod) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out (Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft) -- Above the suspension mode of the nano-hexapod, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to $1$ (Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf) +The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the sample mass $m_s = 1\,\text{kg}$ (the same conclusions can be drawn with $m_s = 50\,\text{kg}$): +- The soft nano-hexapod is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs) +- Between the suspension mode of the nano-hexapod (here at 5Hz for the soft nano-hexapod) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft) +- Above the suspension mode of the nano-hexapod, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to $1$ (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf) #+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}) @@ -631,10 +631,10 @@ The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses **** Open-Loop Dynamic Noise Budgeting <> Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion $d$ for each nano-hexapod and sample configuration. -The Cumulative Amplitude Spectrum of the relative motion $d$ due to both floor motion $x_f$ and stage vibrations $f_t$ are shown in Figure ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three nano-hexapod stiffnesses. +The Cumulative Amplitude Spectrum of the relative motion $d$ due to both floor motion $x_f$ and stage vibrations $f_t$ are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three nano-hexapod stiffnesses. It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft nano-hexapod below $5\,\text{Hz}$. -The total cumulative amplitude spectrum of $d$ for the three nano-hexapod stiffnesses and for the two samples masses are shown in Figure ref:fig:uniaxial_cas_d_disturbances_payload_masses. +The total cumulative amplitude spectrum of $d$ for the three nano-hexapod stiffnesses and for the two samples masses are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. The conclusion is that the sample mass has little effect on the cumulative amplitude spectrum of the relative motion $d$. #+name: fig:uniaxial_cas_d_disturbances @@ -657,25 +657,25 @@ 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 ref:fig:uniaxial_cas_d_disturbances_payload_masses. +The open-loop residual vibrations of $d$ can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. This residual vibration of $d$ is found to be in the order of $100\,nm\,\text{RMS}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$), $200\,nm\,\text{RMS}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$) and $1\,\mu m\,\text{RMS}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$). From this analysis, it may be concluded that the stiffer the nano-hexapod the better. However, what is more important is the /closed-loop/ residual vibration of $d$ (i.e., while the feedback controller is used). -The goal is to obtain a closed-loop residual vibration $\epsilon_d \approx 20\,nm\,\text{RMS}$ (represented by an horizontal dashed black line in Figure ref:fig:uniaxial_cas_d_disturbances_payload_masses). -The bandwidth of the feedback controller leading to a closed-loop residual vibration of $20\,nm\,\text{RMS}$ can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure ref:fig:uniaxial_cas_d_disturbances_payload_masses. +The goal is to obtain a closed-loop residual vibration $\epsilon_d \approx 20\,nm\,\text{RMS}$ (represented by an horizontal dashed black line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses). +The bandwidth of the feedback controller leading to a closed-loop residual vibration of $20\,nm\,\text{RMS}$ can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. # TODO - It would be important to link to a appendix where this is explained in more details, or add some references where this is explained A closed loop bandwidth of $\approx 10\,\text{Hz}$ is found for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$), $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$), and $\approx 100\,\text{Hz}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$). Therefore, while the /open-loop/ vibration is the lowest for the stiff nano-hexapod, it requires the largest feedback bandwidth to meet the specifications. -The advantage of the soft nano-hexapod can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft. +The advantage of the soft nano-hexapod can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft. *** Active Damping <> **** Introduction :ignore: -In this section, three active damping techniques are applied to the nano-hexapod (see Figure ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF) cite:preumont91_activ, Relative Damping Control (RDC) [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb. +In this section, three active damping techniques are applied to the nano-hexapod (see Figure\nbsp{}ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF) cite:preumont91_activ, Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb. -These damping strategies are first described (Section ref:ssec:uniaxial_active_damping_strategies) and are then compared in terms of achievable damping of the nano-hexapod mode (Section ref:ssec:uniaxial_active_damping_achievable_damping), reduction of the effect of disturbances (i.e., $x_f$, $f_t$ and $f_s$) on the displacement $d$ (Sections ref:ssec:uniaxial_active_damping_sensitivity_disturbances). +These damping strategies are first described (Section\nbsp{}ref:ssec:uniaxial_active_damping_strategies) and are then compared in terms of achievable damping of the nano-hexapod mode (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping), reduction of the effect of disturbances (i.e., $x_f$, $f_t$ and $f_s$) on the displacement $d$ (Sections\nbsp{}ref:ssec:uniaxial_active_damping_sensitivity_disturbances). #+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 @@ -704,13 +704,13 @@ These damping strategies are first described (Section ref:ssec:uniaxial_active_d **** Active Damping Strategies <> ***** Integral Force Feedback (IFF) -The Integral Force Feedback strategy consists of using a force sensor in series with the actuator (see Figure ref:fig:uniaxial_active_damping_iff_schematic) and applying an "integral" feedback controller eqref:eq:uniaxial_iff_controller. +The Integral Force Feedback strategy consists of using a force sensor in series with the actuator (see Figure\nbsp{}ref:fig:uniaxial_active_damping_iff_schematic) and applying an "integral" feedback controller\nbsp{}eqref:eq:uniaxial_iff_controller. \begin{equation}\label{eq:uniaxial_iff_controller} \boxed{K_{\text{IFF}}(s) = \frac{g}{s}} \end{equation} -The mechanical equivalent of this IFF strategy is a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain $k/g$ (see Figure ref:fig:uniaxial_active_damping_iff_equiv). +The mechanical equivalent of this 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}) @@ -731,13 +731,13 @@ The mechanical equivalent of this IFF strategy is a dashpot in series with the a #+end_figure ***** Relative Damping Control (RDC) -For the Relative Damping Control strategy, a relative motion sensor that measures the motion of the actuator is used (see Figure ref:fig:uniaxial_active_damping_rdc_schematic) and a "derivative" feedback controller is used eqref:eq:uniaxial_rdc_controller. +For the Relative Damping Control strategy, a relative motion sensor that measures the motion of the actuator is used (see Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_schematic) and a "derivative" feedback controller is used\nbsp{}eqref:eq:uniaxial_rdc_controller. \begin{equation}\label{eq:uniaxial_rdc_controller} \boxed{K_{\text{RDC}}(s) = - g \cdot s} \end{equation} -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 ref:fig:uniaxial_active_damping_rdc_equiv). +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}) @@ -758,14 +758,14 @@ The mechanical equivalent of acrshort:rdc is a dashpot in parallel with the actu #+end_figure ***** Direct Velocity Feedback (DVF) -Finally, the direct velocity feedback strategy consists of using an inertial sensor (usually a geophone) that measures the "absolute" velocity of the body fixed on top of the actuator (see Figure ref:fig:uniaxial_active_damping_dvf_schematic). -This velocity is fed back to the actuator with a "proportional" controller eqref:eq:uniaxial_dvf_controller. +Finally, the direct velocity feedback strategy consists of using an inertial sensor (usually a geophone) that measures the "absolute" velocity of the body fixed on top of the actuator (see Figure\nbsp{}ref:fig:uniaxial_active_damping_dvf_schematic). +This velocity is fed back to the actuator with a "proportional" controller\nbsp{}eqref:eq:uniaxial_dvf_controller. \begin{equation}\label{eq:uniaxial_dvf_controller} \boxed{K_{\text{DVF}}(s) = - g} \end{equation} -This is equivalent to a dashpot (with a damping coefficient equal to the controller gain $g$) between the body (on which the inertial sensor is fixed) and an inertial reference frame (see Figure ref:fig:uniaxial_active_damping_dvf_equiv). +This is equivalent to a dashpot (with a damping coefficient equal to the controller gain $g$) between the body (on which the inertial sensor is fixed) and an inertial reference frame (see Figure\nbsp{}ref:fig:uniaxial_active_damping_dvf_equiv). This is usually referred to as "/sky hook damper/". #+name: fig:uniaxial_active_damping_dvf @@ -788,14 +788,14 @@ This is usually referred to as "/sky hook damper/". **** Plant Dynamics for Active Damping <> -The plant dynamics for all three active damping techniques are shown in Figure ref:fig:uniaxial_plant_active_damping_techniques. +The plant dynamics for all three active damping techniques are shown in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques. All have /alternating poles and zeros/ meaning that the phase does not vary by more than $180\,\text{deg}$ which makes the design of a /robust/ damping controller very easy. -This alternating poles and zeros property is guaranteed for the IFF and RDC cases because the sensors are collocated with the actuator [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7]]. -For the DVF controller, this property is not guaranteed, and may be lost if some flexibility between the nano-hexapod and the sample is considered [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 8.4]]. +This alternating poles and zeros property is guaranteed for the IFF and RDC cases because the sensors are collocated with the actuator\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7]]. +For the DVF controller, this property is not guaranteed, and may be lost if some flexibility between the nano-hexapod and the sample is considered\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 8.4]]. -When the nano-hexapod's suspension modes are at frequencies lower than the resonances of the micro-station (blue and red curves in Figure ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the IFF and DVF transfer functions. -For the stiff nano-hexapod (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure ref:fig:uniaxial_plant_active_damping_techniques. +When the nano-hexapod's suspension modes are at frequencies lower than the resonances of the micro-station (blue and red curves in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the IFF and DVF transfer functions. +For the stiff nano-hexapod (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure\nbsp{}ref:fig:uniaxial_plant_active_damping_techniques. Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used. #+name: fig:uniaxial_plant_active_damping_techniques @@ -828,25 +828,25 @@ Therefore, it is expected that the micro-station dynamics might impact the achie <> To compare the added damping using the three considered active damping strategies, the root locus plot is used. -Indeed, the damping ratio $\xi$ of a pole in the complex plane can be estimated from the angle $\phi$ it makes with the imaginary axis eqref:eq:uniaxial_damping_ratio_angle. +Indeed, the damping ratio $\xi$ of a pole in the complex plane can be estimated from the angle $\phi$ it makes with the imaginary axis\nbsp{}eqref:eq:uniaxial_damping_ratio_angle. Increasing the angle with the imaginary axis therefore means that more damping is added to the considered resonance. -This is illustrated in Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode by the dashed black line indicating the maximum achievable damping. +This is illustrated in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode by the dashed black line indicating the maximum achievable damping. \begin{equation}\label{eq:uniaxial_damping_ratio_angle} \xi = \sin(\phi) \end{equation} -The Root Locus for the three nano-hexapod stiffnesses and the three active damping techniques are shown in Figure ref:fig:uniaxial_root_locus_damping_techniques. +The Root Locus for the three nano-hexapod stiffnesses and the three active damping techniques are shown in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques. All three active damping approaches can lead to /critical damping/ of the nano-hexapod suspension mode (angle $\phi$ can be increased up to 90 degrees). There is even some damping authority on micro-station modes in the following cases: -- IFF with a stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: - This can be understood from the mechanical equivalent of IFF shown in Figure ref:fig:uniaxial_active_damping_iff_equiv considering an high stiffness $k$. +- IFF with a stiff nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: + This can be understood from the mechanical equivalent of IFF shown in Figure\nbsp{}ref:fig:uniaxial_active_damping_iff_equiv considering an high stiffness $k$. The micro-station top platform is connected to an inertial mass (the nano-hexapod) through a damper, which dampens the micro-station suspension suspension mode. -- DVF with a stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: - In that case, the "sky hook damper" (see mechanical equivalent of DVF in Figure ref:fig:uniaxial_active_damping_dvf_equiv) is connected to the micro-station top platform through the stiff nano-hexapod. -- RDC with a soft nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode) :: +- DVF with a stiff nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: + In that case, the "sky hook damper" (see mechanical equivalent of DVF in Figure\nbsp{}ref:fig:uniaxial_active_damping_dvf_equiv) is connected to the micro-station top platform through the stiff nano-hexapod. +- RDC with a soft nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode) :: At the frequency of the micro-station mode, the nano-hexapod top mass behaves as an inertial reference because the suspension mode of the soft nano-hexapod is at much lower frequency. - The micro-station and the nano-hexapod masses are connected through a large damper induced by RDC (see mechanical equivalent in Figure ref:fig:uniaxial_active_damping_rdc_equiv) which allows some damping of the micro-station. + The micro-station and the nano-hexapod masses are connected through a large damper induced by RDC (see mechanical equivalent in Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv) which allows some damping of the micro-station. #+name: fig:uniaxial_root_locus_damping_techniques #+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three nano-hexapod stiffnesses. The Root Loci are zoomed in the suspension mode of the nano-hexapod. @@ -876,7 +876,7 @@ There is even some damping authority on micro-station modes in the following cas #+caption: Root Locus for the three damping techniques applied with the soft nano-hexapod. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod. [[file:figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png]] -The transfer functions from the plant input $f$ to the relative displacement $d$ while active damping is implemented are shown in Figure ref:fig:uniaxial_damped_plant_three_active_damping_techniques. +The transfer functions from the plant input $f$ to the relative displacement $d$ while active damping is implemented are shown in Figure\nbsp{}ref:fig:uniaxial_damped_plant_three_active_damping_techniques. All three active damping techniques yielded similar damped plants. #+name: fig:uniaxial_damped_plant_three_active_damping_techniques @@ -907,15 +907,15 @@ All three active damping techniques yielded similar damped plants. <> Reasonable gains are chosen for the three active damping strategies such that the nano-hexapod suspension mode is well damped. -The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure ref:fig:uniaxial_sensitivity_dist_active_damping. +The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping. The comparison is done with the nano-hexapod having a stiffness $k_n = 1\,N/\mu m$. Several conclusions can be drawn by comparing the obtained sensitivity transfer functions: -- IFF degrades the sensitivity to direct forces on the sample (i.e., the compliance) below the resonance of the nano-hexapod (Figure ref:fig:uniaxial_sensitivity_dist_active_damping_fs). - This is a well-known effect of using IFF for vibration isolation [[cite:&collette15_sensor_fusion_method_high_perfor]]. -- RDC degrades the sensitivity to stage vibrations around the nano-hexapod's resonance as compared to the other two methods (Figure ref:fig:uniaxial_sensitivity_dist_active_damping_ft). - This is because the equivalent damper in parallel with the actuator (see Figure ref:fig:uniaxial_active_damping_rdc_equiv) increases the transmission of the micro-station vibration to the sample which is not the same for the other two active damping strategies. -- both IFF and DVF degrade the sensitivity to floor motion below the resonance of the nano-hexapod (Figure ref:fig:uniaxial_sensitivity_dist_active_damping_xf). +- IFF degrades the sensitivity to direct forces on the sample (i.e., the compliance) below the resonance of the nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_fs). + This is a well-known effect of using IFF for vibration isolation\nbsp{}[[cite:&collette15_sensor_fusion_method_high_perfor]]. +- RDC degrades the sensitivity to stage vibrations around the nano-hexapod's resonance as compared to the other two methods (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_ft). + This is because the equivalent damper in parallel with the actuator (see Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv) increases the transmission of the micro-station vibration to the sample which is not the same for the other two active damping strategies. +- both IFF and DVF degrade the sensitivity to floor motion below the resonance of the nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_xf). #+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}) @@ -941,8 +941,8 @@ Several conclusions can be drawn by comparing the obtained sensitivity transfer #+end_subfigure #+end_figure -From the amplitude spectral density of the disturbances (computed in Section ref:sec:uniaxial_disturbances) and the sensitivity to disturbances estimated using the three active damping strategies, a noise budget can be calculated. -The cumulative amplitude spectrum of the distance $d$ with all three active damping techniques is shown in Figure ref:fig:uniaxial_cas_active_damping and compared with the open-loop case. +From the amplitude spectral density of the disturbances (computed in Section\nbsp{}ref:sec:uniaxial_disturbances) and the sensitivity to disturbances estimated using the three active damping strategies, a noise budget can be calculated. +The cumulative amplitude spectrum of the distance $d$ with all three active damping techniques is shown in Figure\nbsp{}ref:fig:uniaxial_cas_active_damping and compared with the open-loop case. All three active damping methods give similar results. #+name: fig:uniaxial_cas_active_damping @@ -972,15 +972,15 @@ All three active damping methods give similar results. **** Conclusion Three active damping strategies have been studied for the acrfull:nass. -Equivalent mechanical representations were derived in Section ref:ssec:uniaxial_active_damping_strategies which are helpful for understanding the specific effects of each strategy. -The plant dynamics were then compared in Section ref:ssec:uniaxial_active_damping_plants and were found to all have alternating poles and zeros, which helps in the design of the active damping controller. +Equivalent mechanical representations were derived in Section\nbsp{}ref:ssec:uniaxial_active_damping_strategies which are helpful for understanding the specific effects of each strategy. +The plant dynamics were then compared in Section\nbsp{}ref:ssec:uniaxial_active_damping_plants and were found to all have alternating poles and zeros, which helps in the design of the active damping controller. However, this property is not guaranteed for DVF. -The achievable damping of the nano-hexapod suspension mode can be made as large as possible for all three active damping techniques (Section ref:ssec:uniaxial_active_damping_achievable_damping). +The achievable damping of the nano-hexapod suspension mode can be made as large as possible for all three active damping techniques (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping). Even some damping can be applied to some micro-station modes in specific cases. The obtained damped plants were found to be similar. -The damping strategies were then compared in terms of disturbance reduction in Section ref:ssec:uniaxial_active_damping_sensitivity_disturbances. +The damping strategies were then compared in terms of disturbance reduction in Section\nbsp{}ref:ssec:uniaxial_active_damping_sensitivity_disturbances. -The comparison between the three active damping strategies is summarized in Table ref:tab:comp_active_damping. +The comparison between the three active damping strategies is summarized in Table\nbsp{}ref:tab:comp_active_damping. It is difficult to conclude on the best active damping strategy for the acrfull:nass yet. 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 @@ -1003,16 +1003,16 @@ Which one will be used will be determined by the use of more accurate models and *** Position Feedback Controller <> **** Introduction :ignore: -The gls:haclac architecture is shown in Figure ref:fig:uniaxial_hac_lac_architecture. +The gls:haclac architecture is shown in Figure\nbsp{}ref:fig:uniaxial_hac_lac_architecture. This corresponds to a /two step/ control strategy: -- First, an active damping controller $\bm{K}_{\textsc{LAC}}$ is implemented (see Section ref:sec:uniaxial_active_damping). +- First, an active damping controller $\bm{K}_{\textsc{LAC}}$ is implemented (see Section\nbsp{}ref:sec:uniaxial_active_damping). It allows the vibration level to be reduced, and it also makes the damped plant (transfer function from $u^{\prime}$ to $y$) easier to control than the undamped plant (transfer function from $u$ to $y$). - This is called /low authority/ control as it only slightly affects the system poles [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 14.6]]. + This is called /low authority/ control as it only slightly affects the system poles\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 14.6]]. - Then, a position controller $\bm{K}_{\textsc{HAC}}$ is implemented and is used to control the position $d$. This is called /high authority/ control as it usually relocates the system's poles. In this section, Integral Force Feedback is used as the Low Authority Controller (the other two damping strategies would lead to the same conclusions here). -This control architecture applied to the uniaxial model is shown in Figure ref:fig:uniaxial_hac_lac_model. +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 @@ -1034,11 +1034,11 @@ This control architecture applied to the uniaxial model is shown in Figure ref:f **** Damped Plant Dynamics <> -The damped plants obtained for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_hac_iff_damped_plants_masses. -For $k_n = 0.01\,N/\mu m$ and $k_n = 1\,N/\mu m$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). -However, this is not the case for the stiff nano-hexapod ($k_n = 100\,N/\mu m$) where two modes can be seen (Figure ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). +The damped plants obtained for the three nano-hexapod stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses. +For $k_n = 0.01\,N/\mu m$ and $k_n = 1\,N/\mu m$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). +However, this is not the case for the stiff nano-hexapod ($k_n = 100\,N/\mu m$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). This is due to the interaction between the micro-station (modeled modes at 70Hz, 140Hz and 320Hz) and the nano-hexapod. -This effect will be further explained in Section ref:sec:uniaxial_support_compliance. +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 @@ -1070,24 +1070,24 @@ This effect will be further explained in Section ref:sec:uniaxial_support_compli The objective is to design high-authority feedback controllers for the three nano-hexapods. This controller must be robust to the change of sample's mass (from $1\,\text{kg}$ up to $50\,\text{kg}$). -The required feedback bandwidths were estimated in Section ref:sec:uniaxial_noise_budgeting: +The required feedback bandwidths were estimated in Section\nbsp{}ref:sec:uniaxial_noise_budgeting: - $f_b \approx 10\,\text{Hz}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$). - Near this frequency, the plants (shown in Figure ref:fig:uniaxial_hac_iff_damped_plants_masses_soft) are equivalent to a mass line (i.e., slope of $-40\,dB/\text{dec}$ and a phase of -180 degrees). + Near this frequency, the plants (shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft) are equivalent to a mass line (i.e., slope of $-40\,dB/\text{dec}$ and a phase of -180 degrees). The gain of this mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,kg$ up to $65\,kg$). This means that the designed controller will need to have /large gain margins/ to be robust to the change of sample's mass. - $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$). - Similar to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line (Figure ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). + Similar to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). It will probably be easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode. - $\approx 100\,\text{Hz}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$). - Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). + Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). The micro-station is not stiff enough to have a clear stiffness line at this frequency. Therefore, there is both a change of phase and gain depending on the sample mass. This makes the robust design of the controller more complicated. -Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure ref:fig:uniaxial_nyquist_hac). +Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure\nbsp{}ref:fig:uniaxial_nyquist_hac). An arbitrary minimum modulus margin of $0.25$ was chosen when designing the controllers. These high authority controllers are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency, and a low pass filter to increase the robustness to high frequency dynamics. -The controllers used for the three nano-hexapod are shown in Equation eqref:eq:uniaxial_hac_formulas, and the parameters used are summarized in Table ref:tab:uniaxial_feedback_controller_parameters. +The controllers used for the three nano-hexapod are shown in Equation\nbsp{}eqref:eq:uniaxial_hac_formulas, and the parameters used are summarized in Table\nbsp{}ref:tab:uniaxial_feedback_controller_parameters. \begin{subequations} \label{eq:uniaxial_hac_formulas} \begin{align} @@ -1117,10 +1117,10 @@ K_{\text{stiff}}(s) &= g \cdot | *Lag* | $\omega_0 = 5\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_0 = 20\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_i = 0.01\,Hz$ | | *LPF* | $\omega_l = 200\,Hz$ | $\omega_l = 300\,Hz$ | $\omega_l = 500\,Hz$ | -The loop gains corresponding to the designed high authority controllers for the three nano-hexapod are shown in Figure ref:fig:uniaxial_loop_gain_hac. -We can see that for the soft and moderately stiff nano-hexapod (Figures ref:fig:uniaxial_nyquist_hac_vc and ref:fig:uniaxial_nyquist_hac_md), the crossover frequency varies significantly with the sample mass. +The loop gains corresponding to the designed high authority controllers for the three nano-hexapod are shown in Figure\nbsp{}ref:fig:uniaxial_loop_gain_hac. +We can see that for the soft and moderately stiff nano-hexapod (Figures\nbsp{}ref:fig:uniaxial_nyquist_hac_vc and ref:fig:uniaxial_nyquist_hac_md), the crossover frequency varies significantly with the sample mass. This is because the crossover frequency corresponds to the mass line of the plant (whose gain is inversely proportional to the mass). -For the stiff nano-hexapod (Figure ref:fig:uniaxial_nyquist_hac_pz), it was difficult to achieve the desired closed-loop bandwidth of $\approx 100\,\text{Hz}$. +For the stiff nano-hexapod (Figure\nbsp{}ref:fig:uniaxial_nyquist_hac_pz), it was difficult to achieve the desired closed-loop bandwidth of $\approx 100\,\text{Hz}$. A crossover frequency of $\approx 65\,\text{Hz}$ was achieved instead. Note that these controllers were not designed using any optimization methods. @@ -1178,7 +1178,7 @@ The goal is to have a first estimation of the attainable performance. <> The high authority position feedback controllers are then implemented and the closed-loop sensitivities to disturbances are computed. -These are compared with the open-loop and damped plants cases in Figure ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass). +These are compared with the open-loop and damped plants cases in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass). As expected, the sensitivity to disturbances decreased in the controller bandwidth and slightly increased outside this bandwidth. #+name: fig:uniaxial_sensitivity_dist_hac_lac @@ -1206,7 +1206,7 @@ As expected, the sensitivity to disturbances decreased in the controller bandwid #+end_figure The cumulative amplitude spectrum of the motion $d$ is computed for all nano-hexapod configurations, all sample masses and in the open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases. -The results are shown in Figure ref:fig:uniaxial_cas_hac_lac. +The results are shown in Figure\nbsp{}ref:fig:uniaxial_cas_hac_lac. Obtained root mean square values of the distance $d$ are better for the soft nano-hexapod ($\approx 25\,nm$ to $\approx 35\,nm$ depending on the sample's mass) than for the stiffer nano-hexapod (from $\approx 30\,nm$ to $\approx 70\,nm$). #+name: fig:uniaxial_cas_hac_lac @@ -1235,8 +1235,8 @@ Obtained root mean square values of the distance $d$ are better for the soft nan **** Conclusion -On the basis of the open-loop noise budgeting made in Section ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,nm\,\text{RMS}$ was estimated. -To achieve such bandwidth, the acrshort:haclac strategy was followed, which consists of first using an active damping controller (studied in Section ref:sec:uniaxial_active_damping) and then adding a high authority position feedback controller. +On the basis of the open-loop noise budgeting made in Section\nbsp{}ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,nm\,\text{RMS}$ was estimated. +To achieve such bandwidth, the acrshort:haclac strategy was followed, which consists of first using an active damping controller (studied in Section\nbsp{}ref:sec:uniaxial_active_damping) and then adding a high authority position feedback controller. In this section, feedback controllers were designed in such a way that the required closed-loop bandwidth was reached while being robust to changes in the payload mass. The attainable vibration control performances were estimated for the three nano-hexapod stiffnesses and were found to be close to the required values. @@ -1248,13 +1248,13 @@ A slight advantage can be given to the soft nano-hexapod as it requires less fee **** Introduction :ignore: In this section, the impact of the compliance of the support (i.e., the micro-station) on the dynamics of the plant to control is studied. -This is a critical point because the dynamics of the micro-station is complex, depends on the considered direction (see measurements in Figure ref:fig:uniaxial_comp_frf_meas_model) and may vary with position and time. +This is a critical point because the dynamics of the micro-station is complex, depends on the considered direction (see measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model) and may vary with position and time. It would be much better to have a plant dynamics that is not impacted by the micro-station. Therefore, the objective of this section is to obtain some guidance for the design of a nano-hexapod that will not be impacted by the complex micro-station dynamics. -To study this, two models are used (Figure ref:fig:uniaxial_support_compliance_models). -The first one consists of the nano-hexapod directly fixed on top of the granite, thus neglecting any support compliance (Figure ref:fig:uniaxial_support_compliance_nano_hexapod_only). -The second one consists of the nano-hexapod fixed on top of the micro-station having some limited compliance (Figure ref:fig:uniaxial_support_compliance_test_system) +To study this, two models are used (Figure\nbsp{}ref:fig:uniaxial_support_compliance_models). +The first one consists of the nano-hexapod directly fixed on top of the granite, thus neglecting any support compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_nano_hexapod_only). +The second one consists of the nano-hexapod fixed on top of the micro-station having some limited compliance (Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) #+name: fig:uniaxial_support_compliance_models #+caption: Models used to study the effect of limited support compliance @@ -1276,9 +1276,9 @@ The second one consists of the nano-hexapod fixed on top of the micro-station ha **** Neglected support compliance -The limited compliance of the micro-station is first neglected and the uniaxial model shown in Figure ref:fig:uniaxial_support_compliance_nano_hexapod_only is used. +The limited compliance of the micro-station is first neglected and the uniaxial model shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_nano_hexapod_only is used. The nano-hexapod mass (including the payload) is set at $20\,\text{kg}$ and three hexapod stiffnesses are considered, such that their resonance frequencies are at $\omega_{n} = 10\,\text{Hz}$, $\omega_{n} = 70\,\text{Hz}$ and $\omega_{n} = 400\,\text{Hz}$. -Obtained transfer functions from $F$ to $L^\prime$ (shown in Figure ref:fig:uniaxial_effect_support_compliance_neglected) are simple second-order low-pass filters. +Obtained transfer functions from $F$ to $L^\prime$ (shown in Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_neglected) are simple second-order low-pass filters. When neglecting the support compliance, a large feedback bandwidth can be achieved for all three nano-hexapods. #+name: fig:uniaxial_effect_support_compliance_neglected @@ -1307,16 +1307,16 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev **** Effect of support compliance on $L/F$ -Some support compliance is now added and the model shown in Figure ref:fig:uniaxial_support_compliance_test_system is used. -The parameters of the support (i.e., $m_{\mu}$, $c_{\mu}$ and $k_{\mu}$) are chosen to match the vertical mode at $70\,\text{Hz}$ seen on the micro-station (Figure ref:fig:uniaxial_comp_frf_meas_model). +Some support compliance is now added and the model shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system is used. +The parameters of the support (i.e., $m_{\mu}$, $c_{\mu}$ and $k_{\mu}$) are chosen to match the vertical mode at $70\,\text{Hz}$ seen on the micro-station (Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model). The transfer functions from $F$ to $L$ (i.e., control of the relative motion of the nano-hexapod) and from $L$ to $d$ (i.e., control of the position between the nano-hexapod and the fixed granite) can then be computed. -When the relative displacement of the nano-hexapod $L$ is controlled (dynamics shown in Figure ref:fig:uniaxial_effect_support_compliance_dynamics), having a stiff nano-hexapod (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure ref:fig:uniaxial_effect_support_compliance_dynamics_stiff). +When the relative displacement of the nano-hexapod $L$ is controlled (dynamics shown in Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics), having a stiff nano-hexapod (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_stiff). This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages. In such a case, the control of the piezoelectric stage using its integrated metrology (typically capacitive sensors) is quite simple as the plant is not much affected by the dynamics of the support on which it is fixed. # TODO - Add references of such stations with piezo stages on top -If a soft nano-hexapod is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure ref:fig:uniaxial_effect_support_compliance_dynamics_soft) which will impact the control robustness and performance. +If a soft nano-hexapod is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_soft) which will impact the control robustness and performance. #+name: fig:uniaxial_effect_support_compliance_dynamics #+caption: Effect of the support compliance on the transfer functions from $F$ to $L$ @@ -1345,8 +1345,8 @@ If a soft nano-hexapod is used, the support dynamics appears in the dynamics bet **** Effect of support compliance on $d/F$ When the motion to be controlled is the relative displacement $d$ between the granite and the nano-hexapod's top platform (which is the case for the acrshort:nass), the effect of the support compliance on the plant dynamics is opposite to that previously observed. -Indeed, using a "soft" nano-hexapod (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure ref:fig:uniaxial_effect_support_compliance_dynamics_d_soft). -Conversely, if a "stiff" nano-hexapod is used, the support dynamics appears in the plant dynamics (Figure ref:fig:uniaxial_effect_support_compliance_dynamics_d_stiff). +Indeed, using a "soft" nano-hexapod (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_soft). +Conversely, if a "stiff" nano-hexapod is used, the support dynamics appears in the plant dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_stiff). #+name: fig:uniaxial_effect_support_compliance_dynamics_d #+caption: Effect of the support compliance on the transfer functions from $F$ to $d$ @@ -1374,8 +1374,8 @@ Conversely, if a "stiff" nano-hexapod is used, the support dynamics appears in t **** Conclusion -To study the impact of support compliance on plant dynamics, simple models shown in Figure ref:fig:uniaxial_support_compliance_models were used. -Depending on the quantity to be controlled ($L$ or $d$ in Figure ref:fig:uniaxial_support_compliance_test_system) and on the relative location of $\omega_\nu$ (suspension mode of the nano-hexapod) with respect to $\omega_\mu$ (modes of the support), the interaction between the support and the nano-hexapod dynamics can drastically change (observations made are summarized in Table ref:tab:uniaxial_effect_compliance). +To study the impact of support compliance on plant dynamics, simple models shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_models were used. +Depending on the quantity to be controlled ($L$ or $d$ in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) and on the relative location of $\omega_\nu$ (suspension mode of the nano-hexapod) with respect to $\omega_\mu$ (modes of the support), the interaction between the support and the nano-hexapod dynamics can drastically change (observations made are summarized in Table\nbsp{}ref:tab:uniaxial_effect_compliance). For the acrfull:nass, having the suspension mode of the nano-hexapod at lower frequencies than the suspension modes of the micro-station would make the plant less dependent on the micro-station dynamics, and therefore easier to control. Note that the observations made in this section are also affected by the ratio between the support mass $m_{\mu}$ and the nano-hexapod mass $m_n$ (the effect is more pronounced when the ratio $m_n/m_{\mu}$ increases). @@ -1394,9 +1394,9 @@ Note that the observations made in this section are also affected by the ratio b **** Introduction :ignore: -Up to this section, the sample was modeled as a mass rigidly fixed to the nano-hexapod (as shown in Figure ref:fig:uniaxial_paylaod_dynamics_rigid_schematic). +Up to this section, the sample was modeled as a mass rigidly fixed to the nano-hexapod (as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic). However, such a sample may present internal dynamics, and its fixation to the nano-hexapod may have limited stiffness. -To study the effect of the sample dynamics, the models shown in Figure ref:fig:uniaxial_paylaod_dynamics_schematic are used. +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 @@ -1419,11 +1419,11 @@ To study the effect of the sample dynamics, the models shown in Figure ref:fig:u **** Impact on plant dynamics <> -To study the impact of the flexibility between the nano-hexapod and the payload, a first (reference) model with a rigid payload, as shown in Figure ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used. -Then "flexible" payload whose model is shown in Figure ref:fig:uniaxial_paylaod_dynamics_schematic are considered. +To study the impact of the flexibility between the nano-hexapod and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used. +Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are considered. The resonances of the payload are set at $\omega_s = 20\,\text{Hz}$ and at $\omega_s = 200\,\text{Hz}$ while its mass is either $m_s = 1\,\text{kg}$ or $m_s = 50\,\text{kg}$. -The transfer functions from the nano-hexapod force $f$ to the motion of the nano-hexapod top platform are computed for all the above configurations and are compared for a soft Nano-Hexapod ($k_n = 0.01\,N/\mu m$) in Figure ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. +The transfer functions from the nano-hexapod force $f$ to the motion of the nano-hexapod top platform are computed for all the above configurations and are compared for a soft Nano-Hexapod ($k_n = 0.01\,N/\mu m$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern). The frequency of the anti-resonance corresponds to the "free" resonance of the sample $\omega_s = \sqrt{k_s/m_s}$. The flexibility of the sample also changes the high frequency gain (the mass line is shifted from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$). @@ -1446,10 +1446,10 @@ The flexibility of the sample also changes the high frequency gain (the mass lin #+end_subfigure #+end_figure -The same transfer functions are now compared when using a stiff nano-hexapod ($k_n = 100\,N/\mu m$) in Figure ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod. +The same transfer functions are now compared when using a stiff nano-hexapod ($k_n = 100\,N/\mu m$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod. In this case, the sample's resonance $\omega_s$ is smaller than the nano-hexapod resonance $\omega_n$. This changes the zero/pole pattern to a pole/zero pattern (the frequency of the zero still being equal to $\omega_s$). -Even though the added sample's flexibility still shifts the high frequency mass line as for the soft nano-hexapod, the dynamics below the nano-hexapod resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy). +Even though the added sample's flexibility still shifts the high frequency mass line as for the soft nano-hexapod, the dynamics below the nano-hexapod resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy). #+name: fig:uniaxial_payload_dynamics_stiff_nano_hexapod #+caption: Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}) @@ -1474,22 +1474,22 @@ Even though the added sample's flexibility still shifts the high frequency mass Having a flexibility between the measured position (i.e., the top platform of the nano-hexapod) and the point-of-interest to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). -To estimate whether the sample flexibility is critical for the closed-loop position stability of the sample, the model shown in Figure ref:fig:uniaxial_sample_flexibility_control is used. -This is the same model that was used in Section ref:sec:uniaxial_position_control but with an added flexibility between the nano-hexapod and the sample (considered sample modes are at $\omega_s = 20\,\text{Hz}$ and $\omega_n = 200\,\text{Hz}$). +To estimate whether the sample flexibility is critical for the closed-loop position stability of the sample, the model shown in Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_control is used. +This is the same model that was used in Section\nbsp{}ref:sec:uniaxial_position_control but with an added flexibility between the nano-hexapod and the sample (considered sample modes are at $\omega_s = 20\,\text{Hz}$ and $\omega_n = 200\,\text{Hz}$). In this case, the measured (i.e., controlled) distance $d$ is no longer equal to the real performance index (the distance $y$). #+name: fig:uniaxial_sample_flexibility_control #+caption: Uniaxial model considering some flexibility between the nano-hexapod top platform and the sample. In this case, the measured and controlled distance $d$ is different from the distance $y$ which is the real performance index [[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 ref:sec:uniaxial_active_damping. +The system dynamics is computed and IFF is applied using the same gains as those used in Section\nbsp{}ref:sec:uniaxial_active_damping. Due to the collocation between the nano-hexapod and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample. -The High Authority Controllers used in Section ref:sec:uniaxial_position_control are then implemented on the damped plants. +The High Authority Controllers used in Section\nbsp{}ref:sec:uniaxial_position_control are then implemented on the damped plants. The obtained closed-loop systems are stable, indicating good robustness. -Finally, closed-loop noise budgeting is computed for the obtained closed-loop system, and the cumulative amplitude spectrum of $d$ and $y$ are shown in Figure ref:fig:uniaxial_sample_flexibility_noise_budget_y. -The cumulative amplitude spectrum of the measured distance $d$ (Figure ref:fig:uniaxial_sample_flexibility_noise_budget_d) shows that the added flexibility at the sample location has very little effect on the control performance. -However, the cumulative amplitude spectrum of the distance $y$ (Figure ref:fig:uniaxial_sample_flexibility_noise_budget_y) shows that the stability of $y$ is degraded when the sample flexibility is considered and is degraded as $\omega_s$ is lowered. +Finally, closed-loop noise budgeting is computed for the obtained closed-loop system, and the cumulative amplitude spectrum of $d$ and $y$ are shown in Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_y. +The cumulative amplitude spectrum of the measured distance $d$ (Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_d) shows that the added flexibility at the sample location has very little effect on the control performance. +However, the cumulative amplitude spectrum of the distance $y$ (Figure\nbsp{}ref:fig:uniaxial_sample_flexibility_noise_budget_y) shows that the stability of $y$ is degraded when the sample flexibility is considered and is degraded as $\omega_s$ is lowered. What happens is that above $\omega_s$, even though the motion $d$ can be controlled perfectly, the sample's mass is "isolated" from the motion of the nano-hexapod and the control on $y$ is not effective. @@ -1517,11 +1517,11 @@ Payload dynamics is usually a major concern when designing a positioning system. In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the nano-hexapod $\omega_n$. The larger the sample mass, the larger the effect (i.e., change of high frequency gain, appearance of additional resonances and anti-resonances). A zero/pole pattern is observed if $\omega_s > \omega_n$ and a pole/zero pattern if $\omega_s > \omega_n$. -Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in [[cite:&rankers98_machin Section 4.2]]. +Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in\nbsp{}[[cite:&rankers98_machin Section 4.2]]. The general conclusion is that the stiffer the nano-hexapod, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload. This is why high-bandwidth soft positioning stages are usually restricted to constant and calibrated payloads (CD-player, lithography machines, isolation system for gravitational wave detectors, ...), whereas stiff positioning systems are usually used when the control must be robust to a change of payload's mass (stiff piezo nano-positioning stages for instance). -Having some flexibility between the measurement point and the point of interest (i.e., the sample point to be position on the x-ray) also degrades the position stability as shown in Section ref:ssec:uniaxial_payload_dynamics_effect_stability. +Having some flexibility between the measurement point and the point of interest (i.e., the sample point to be position on the x-ray) also degrades the position stability as shown in Section\nbsp{}ref:ssec:uniaxial_payload_dynamics_effect_stability. Therefore, it is important to take special care when designing sampling environments, especially if a soft nano-hexapod is used. *** Conclusion @@ -1529,14 +1529,14 @@ Therefore, it is important to take special care when designing sampling environm # TODO - Make a table summarizing the findings -In this study, a uniaxial model of the nano-active-stabilization-system was tuned from both dynamical measurements (Section ref:sec:uniaxial_micro_station_model) and from disturbances measurements (Section ref:sec:uniaxial_disturbances). +In this study, a uniaxial model of the nano-active-stabilization-system was tuned from both dynamical measurements (Section\nbsp{}ref:sec:uniaxial_micro_station_model) and from disturbances measurements (Section\nbsp{}ref:sec:uniaxial_disturbances). -Three active damping techniques can be used to critically damp the nano-hexapod resonances (Section ref:sec:uniaxial_active_damping). -However, this model does not allow the determination of which one is most suited to this application (a comparison of the three active damping techniques is done in Table ref:tab:comp_active_damping). +Three active damping techniques can be used to critically damp the nano-hexapod resonances (Section\nbsp{}ref:sec:uniaxial_active_damping). +However, this model does not allow the determination of which one is most suited to this application (a comparison of the three active damping techniques is done in Table\nbsp{}ref:tab:comp_active_damping). -Position feedback controllers have been developed for three considered nano-hexapod stiffnesses (Section ref:sec:uniaxial_position_control). +Position feedback controllers have been developed for three considered nano-hexapod stiffnesses (Section\nbsp{}ref:sec:uniaxial_position_control). These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances. -Having a soft nano-hexapod makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section ref:sec:uniaxial_support_compliance) and requires less position feedback bandwidth to fulfill the requirements. +Having a soft nano-hexapod makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section\nbsp{}ref:sec:uniaxial_support_compliance) and requires less position feedback bandwidth to fulfill the requirements. The moderately stiff nano-hexapod ($k_n = 1\,N/\mu m$) is requiring a higher feedback bandwidth, but still gives acceptable results. However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performance. @@ -1546,25 +1546,25 @@ However, the stiff nano-hexapod is the most complex to control and gives the wor An important aspect of the acrfull:nass is that the nano-hexapod continuously rotates around a vertical axis, whereas the external metrology is not. Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance. -To study these effects, a model of a rotating suspended platform is first presented (Section ref:sec:rotating_system_description) +To study these effects, a model of a rotating suspended platform is first presented (Section\nbsp{}ref:sec:rotating_system_description) This model is simple enough to be able to derive its dynamics analytically and to understand its behavior, while still allowing the capture of important physical effects in play. -acrfull:iff is then applied to the rotating platform, and it is shown that the unconditional stability of acrshort:iff is lost due to the gyroscopic effects induced by the rotation (Section ref:sec:rotating_iff_pure_int). +acrfull:iff is then applied to the rotating platform, and it is shown that the unconditional stability of acrshort:iff is lost due to the gyroscopic effects induced by the rotation (Section\nbsp{}ref:sec:rotating_iff_pure_int). Two modifications of the Integral Force Feedback are then proposed. -The first modification involves adding a high-pass filter to the acrshort:iff controller (Section ref:sec:rotating_iff_pseudo_int). +The first modification involves adding a high-pass filter to the acrshort:iff controller (Section\nbsp{}ref:sec:rotating_iff_pseudo_int). It is shown that the acrshort:iff controller is stable for some gain values, and that damping can be added to the suspension modes. The optimal high-pass filter cut-off frequency is computed. -The second modification consists of adding a stiffness in parallel to the force sensors (Section ref:sec:rotating_iff_parallel_stiffness). +The second modification consists of adding a stiffness in parallel to the force sensors (Section\nbsp{}ref:sec:rotating_iff_parallel_stiffness). Under certain conditions, the unconditional stability of the IFF controller is regained. The optimal parallel stiffness is then computed. -This study of adapting acrshort:iff for the damping of rotating platforms has been the subject of two published papers [[cite:&dehaeze20_activ_dampin_rotat_platf_integ_force_feedb;&dehaeze21_activ_dampin_rotat_platf_using]]. +This study of adapting acrshort:iff for the damping of rotating platforms has been the subject of two published papers\nbsp{}[[cite:&dehaeze20_activ_dampin_rotat_platf_integ_force_feedb;&dehaeze21_activ_dampin_rotat_platf_using]]. -It is then shown that acrfull:rdc is less affected by gyroscopic effects (Section ref:sec:rotating_relative_damp_control). -Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, damped plant and closed-loop compliance and transmissibility (Section ref:sec:rotating_comp_act_damp). +It is then shown that acrfull:rdc is less affected by gyroscopic effects (Section\nbsp{}ref:sec:rotating_relative_damp_control). +Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, damped plant and closed-loop compliance and transmissibility (Section\nbsp{}ref:sec:rotating_comp_act_damp). -The previous analysis was applied to three considered nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) and the optimal active damping controller was obtained in each case (Section ref:sec:rotating_nano_hexapod). +The previous analysis was applied to three considered nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) and the optimal active damping controller was obtained in each case (Section\nbsp{}ref:sec:rotating_nano_hexapod). Up until this section, the study was performed on a very simplistic model that only captures the rotation aspect, and the model parameters were not tuned to correspond to the NASS. -In the last section (Section ref:sec:rotating_nass), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. +In the last section (Section\nbsp{}ref:sec:rotating_nass), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. The goal is to determine whether the rotation imposes performance limitation on the NASS. #+name: fig:rotating_overview @@ -1576,7 +1576,7 @@ The goal is to determine whether the rotation imposes performance limitation on <> **** Introduction :ignore: -The system used to study gyroscopic effects consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure ref:fig:rotating_3dof_model_schematic). +The system used to study gyroscopic effects consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure\nbsp{}ref:fig:rotating_3dof_model_schematic). The rotating stage is supposed to be ideal, meaning it induces a perfect rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\s}$. The suspended platform consists of two orthogonal actuators, each represented by three elements in parallel: a spring with a stiffness $k$ in $\si{\newton\per\meter}$, a dashpot with a damping coefficient $c$ in $\si{\newton\per(\meter\per\second)}$ and an ideal force source $F_u, F_v$. A payload with a mass $m$ in $\si{\kilo\gram}$, is mounted on the (rotating) suspended platform. @@ -1590,9 +1590,9 @@ After the dynamics of this system is studied, the objective will be to dampen th [[file:figs/rotating_3dof_model_schematic.png]] **** Equations of motion and transfer functions -To obtain the equations of motion for the system represented in Figure ref:fig:rotating_3dof_model_schematic, the Lagrangian equation eqref:eq:rotating_lagrangian_equations is used. +To obtain the equations of motion for the system represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic, the Lagrangian equation\nbsp{}eqref:eq:rotating_lagrangian_equations is used. $L = T - V$ is the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. -These terms are derived in eqref:eq:rotating_energy_functions_lagrange. +These terms are derived in\nbsp{}eqref:eq:rotating_energy_functions_lagrange. Note that the equation of motion corresponding to constant rotation along $\vec{i}_w$ is disregarded because this motion is imposed by the rotation stage. \begin{equation}\label{eq:rotating_lagrangian_equations} @@ -1606,7 +1606,7 @@ Note that the equation of motion corresponding to constant rotation along $\vec{ \end{aligned} \end{equation} -Substituting equations eqref:eq:rotating_energy_functions_lagrange into equation eqref:eq:rotating_lagrangian_equations for both generalized coordinates gives two coupled differential equations eqref:eq:rotating_eom_coupled_1 and eqref:eq:rotating_eom_coupled_2. +Substituting equations\nbsp{}eqref:eq:rotating_energy_functions_lagrange into equation\nbsp{}eqref:eq:rotating_lagrangian_equations for both generalized coordinates gives two coupled differential equations\nbsp{}eqref:eq:rotating_eom_coupled_1 and eqref:eq:rotating_eom_coupled_2. \begin{subequations} \label{eq:rotating_eom_coupled} \begin{align} @@ -1615,13 +1615,13 @@ Substituting equations eqref:eq:rotating_energy_functions_lagrange into equation \end{align} \end{subequations} -The uniform rotation of the system induces two /gyroscopic effects/ as shown in equation eqref:eq:rotating_eom_coupled: +The uniform rotation of the system induces two /gyroscopic effects/ as shown in equation\nbsp{}eqref:eq:rotating_eom_coupled: - /Centrifugal forces/: that can be seen as an added /negative stiffness/ $- m \Omega^2$ along $\vec{i}_u$ and $\vec{i}_v$ - /Coriolis forces/: that adds /coupling/ between the two orthogonal directions. One can verify that without rotation ($\Omega = 0$), the system becomes equivalent to two /uncoupled/ one degree of freedom mass-spring-damper systems. -To study the dynamics of the system, the two differential equations of motions eqref:eq:rotating_eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\mathbf{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation eqref:eq:rotating_Gd_mimo_tf is obtained. -The four transfer functions in $\mathbf{G}_d$ are shown in equation eqref:eq:rotating_Gd_indiv_el. +To study the dynamics of the system, the two differential equations of motions\nbsp{}eqref:eq:rotating_eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\mathbf{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation\nbsp{}eqref:eq:rotating_Gd_mimo_tf is obtained. +The four transfer functions in $\mathbf{G}_d$ are shown in equation\nbsp{}eqref:eq:rotating_Gd_indiv_el. \begin{equation}\label{eq:rotating_Gd_mimo_tf} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} @@ -1634,8 +1634,8 @@ The four transfer functions in $\mathbf{G}_d$ are shown in equation eqref:eq:rot \end{align} \end{subequations} -To simplify the analysis, the undamped natural frequency $\omega_0$ and the damping ratio $\xi$ defined in eqref:eq:rotating_xi_and_omega are used instead. -The elements of the transfer function matrix $\mathbf{G}_d$ are described by equation eqref:eq:rotating_Gd_w0_xi_k. +To simplify the analysis, the undamped natural frequency $\omega_0$ and the damping ratio $\xi$ defined in\nbsp{}eqref:eq:rotating_xi_and_omega are used instead. +The elements of the transfer function matrix $\mathbf{G}_d$ are described by equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k. \begin{equation} \label{eq:rotating_xi_and_omega} \omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}} \end{equation} @@ -1648,13 +1648,13 @@ The elements of the transfer function matrix $\mathbf{G}_d$ are described by equ \end{subequations} **** System Poles: Campbell Diagram -The poles of $\mathbf{G}_d$ are the complex solutions $p$ of equation eqref:eq:rotating_poles (i.e. the roots of its denominator). +The poles of $\mathbf{G}_d$ are the complex solutions $p$ of equation\nbsp{}eqref:eq:rotating_poles (i.e. the roots of its denominator). \begin{equation}\label{eq:rotating_poles} \left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0 \end{equation} -Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles $[p_{+}, p_{-}]$ are obtained as shown in equation eqref:eq:rotating_pole_values. +Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles $[p_{+}, p_{-}]$ are obtained as shown in equation\nbsp{}eqref:eq:rotating_pole_values. \begin{subequations} \label{eq:rotating_pole_values} \begin{align} @@ -1663,9 +1663,9 @@ Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles $[p_ \end{align} \end{subequations} -The real and complex parts of these two pairs of complex conjugate poles are represented in Figure ref:fig:rotating_campbell_diagram as a function of the rotational speed $\Omega$. -As the rotational speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ goes to lower frequencies (Figure ref:fig:rotating_campbell_diagram_imag). -The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ is positive (Figure ref:fig:rotating_campbell_diagram_real). +The real and complex parts of these two pairs of complex conjugate poles are represented in Figure\nbsp{}ref:fig:rotating_campbell_diagram as a function of the rotational speed $\Omega$. +As the rotational speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ goes to lower frequencies (Figure\nbsp{}ref:fig:rotating_campbell_diagram_imag). +The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ is positive (Figure\nbsp{}ref:fig:rotating_campbell_diagram_real). Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forces exceeds the spring stiffness $k$. #+name: fig:rotating_campbell_diagram @@ -1690,8 +1690,8 @@ Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forc **** System Dynamics: Effect of rotation The system dynamics from actuator forces $[F_u, F_v]$ to the relative motion $[d_u, d_v]$ is identified for several rotating velocities. -Looking at the transfer function matrix $\mathbf{G}_d$ in equation eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. -The bode plots of these two terms are shown in Figure ref:fig:rotating_bode_plot for several rotational speeds $\Omega$. +Looking at the transfer function matrix $\mathbf{G}_d$ in equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. +The bode plots of these two terms are shown in Figure\nbsp{}ref:fig:rotating_bode_plot for several rotational speeds $\Omega$. These plots confirm the expected behavior: the frequencies of the two pairs of complex conjugate poles are further separated as $\Omega$ increases. 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). @@ -1721,7 +1721,7 @@ The goal is now to damp the two suspension modes of the payload using an active As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances cite:collette11_review_activ_vibrat_isolat_strat and to make the plant easier to control for the high authority controller. Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF) cite:lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc, Integral Force Feedback (IFF) cite:preumont91_activ and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb. -In [[cite:&preumont91_activ]], the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system. +In\nbsp{}[[cite:&preumont91_activ]], the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system. When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros, which guarantees the stability of the closed-loop system cite:preumont02_force_feedb_versus_accel_feedb. It was later shown that this property holds for multiple collated actuator/sensor pairs cite:preumont08_trans_zeros_struc_contr_with. @@ -1734,8 +1734,8 @@ However, none of these studies have been applied to rotating systems. In this section, the acrshort:iff strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alter the system dynamics and that IFF cannot be applied as is. **** System and Equations of motion -To apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure ref:fig:rotating_3dof_model_schematic_iff). -Two identical controllers $K_F$ described by eqref:eq:rotating_iff_controller are then used to feedback each of the sensed force to its associated actuator. +To apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_iff). +Two identical controllers $K_F$ described by\nbsp{}eqref:eq:rotating_iff_controller are then used to feedback each of the sensed force to its associated actuator. \begin{equation}\label{eq:rotating_iff_controller} K_{F}(s) = g \cdot \frac{1}{s} @@ -1759,7 +1759,7 @@ Two identical controllers $K_F$ described by eqref:eq:rotating_iff_controller ar #+end_subfigure #+end_figure -The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sensors represented in Figure ref:fig:rotating_3dof_model_schematic_iff are described by equation eqref:eq:rotating_measured_force. +The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sensors represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_iff are described by equation\nbsp{}eqref:eq:rotating_measured_force. \begin{equation}\label{eq:rotating_measured_force} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = @@ -1767,8 +1767,8 @@ The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sen \begin{bmatrix} d_u \\ d_v \end{bmatrix} \end{equation} -The transfer function matrix $\mathbf{G}_{f}$ from actuator forces to measured forces in equation eqref:eq:rotating_Gf_mimo_tf can be obtained by inserting equation eqref:eq:rotating_Gd_w0_xi_k into equation eqref:eq:rotating_measured_force. -Its elements are shown in equation eqref:eq:rotating_Gf. +The transfer function matrix $\mathbf{G}_{f}$ from actuator forces to measured forces in equation\nbsp{}eqref:eq:rotating_Gf_mimo_tf can be obtained by inserting equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k into equation\nbsp{}eqref:eq:rotating_measured_force. +Its elements are shown in equation\nbsp{}eqref:eq:rotating_Gf. \begin{equation}\label{eq:rotating_Gf_mimo_tf} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix} @@ -1781,7 +1781,7 @@ Its elements are shown in equation eqref:eq:rotating_Gf. \end{align} \end{subequations} -The zeros of the diagonal terms of $\mathbf{G}_f$ in equation eqref:eq:rotating_Gf_diag_tf are computed, and neglecting the damping for simplicity, two complex conjugated zeros $z_{c}$ eqref:eq:rotating_iff_zero_cc, and two real zeros $z_{r}$ eqref:eq:rotating_iff_zero_real are obtained. +The zeros of the diagonal terms of $\mathbf{G}_f$ in equation\nbsp{}eqref:eq:rotating_Gf_diag_tf are computed, and neglecting the damping for simplicity, two complex conjugated zeros $z_{c}$\nbsp{}eqref:eq:rotating_iff_zero_cc, and two real zeros $z_{r}$\nbsp{}eqref:eq:rotating_iff_zero_real are obtained. \begin{subequations} \begin{align} @@ -1790,13 +1790,13 @@ The zeros of the diagonal terms of $\mathbf{G}_f$ in equation eqref:eq:rotating_ \end{align} \end{subequations} -It is interesting to see that the frequency of the pair of complex conjugate zeros $z_c$ in equation eqref:eq:rotating_iff_zero_cc always lies between the frequency of the two pairs of complex conjugate poles $p_{-}$ and $p_{+}$ in equation eqref:eq:rotating_pole_values. +It is interesting to see that the frequency of the pair of complex conjugate zeros $z_c$ in equation\nbsp{}eqref:eq:rotating_iff_zero_cc always lies between the frequency of the two pairs of complex conjugate poles $p_{-}$ and $p_{+}$ in equation\nbsp{}eqref:eq:rotating_pole_values. This is what usually gives the unconditional stability of IFF when collocated force sensors are used. -However, for non-null rotational speeds, the two real zeros $z_r$ in equation eqref:eq:rotating_iff_zero_real are inducing a /non-minimum phase behavior/. -This can be seen in the Bode plot of the diagonal terms (Figure ref:fig:rotating_iff_bode_plot_effect_rot) where the low-frequency gain is no longer zero while the phase stays at $\SI{180}{\degree}$. +However, for non-null rotational speeds, the two real zeros $z_r$ in equation\nbsp{}eqref:eq:rotating_iff_zero_real are inducing a /non-minimum phase behavior/. +This can be seen in the Bode plot of the diagonal terms (Figure\nbsp{}ref:fig:rotating_iff_bode_plot_effect_rot) where the low-frequency gain is no longer zero while the phase stays at $\SI{180}{\degree}$. -The low-frequency gain of $\mathbf{G}_f$ increases with the rotational speed $\Omega$ as shown in equation eqref:eq:rotating_low_freq_gain_iff_plan. +The low-frequency gain of $\mathbf{G}_f$ increases with the rotational speed $\Omega$ as shown in equation\nbsp{}eqref:eq:rotating_low_freq_gain_iff_plan. This can be explained as follows: a constant actuator force $F_u$ induces a small displacement of the mass $d_u = \frac{F_u}{k - m\Omega^2}$ (Hooke's law considering the negative stiffness induced by the rotation). This small displacement then increases the centrifugal force $m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u$ which is then measured by the force sensors. @@ -1808,7 +1808,7 @@ This small displacement then increases the centrifugal force $m\Omega^2d_u = \fr \end{equation} **** Effect of rotation speed on IFF plant dynamics -The transfer functions from actuator forces $[F_u,\ F_v]$ to the measured force sensors $[f_u,\ f_v]$ are identified for several rotating velocities and are shown in Figure ref:fig:rotating_iff_bode_plot_effect_rot. +The transfer functions from actuator forces $[F_u,\ F_v]$ to the measured force sensors $[f_u,\ f_v]$ are identified for several rotating velocities and are shown in Figure\nbsp{}ref:fig:rotating_iff_bode_plot_effect_rot. As expected from the derived equations of motion: - when $\Omega < \omega_0$: the low-frequency gain is no longer zero and two (non-minimum phase) real zeros appear at low-frequencies. The low-frequency gain increases with $\Omega$. @@ -1834,8 +1834,8 @@ As expected from the derived equations of motion: #+end_figure **** Decentralized Integral Force Feedback -The control diagram for decentralized acrshort:iff is shown in Figure ref:fig:rotating_iff_diagram. -The decentralized acrshort:iff controller $\bm{K}_F$ corresponds to a diagonal controller with integrators eqref:eq:rotating_Kf_pure_int. +The control diagram for decentralized acrshort:iff is shown in Figure\nbsp{}ref:fig:rotating_iff_diagram. +The decentralized acrshort:iff controller $\bm{K}_F$ corresponds to a diagonal controller with integrators\nbsp{}eqref:eq:rotating_Kf_pure_int. \begin{equation} \label{eq:rotating_Kf_pure_int} \begin{aligned} @@ -1844,12 +1844,12 @@ The decentralized acrshort:iff controller $\bm{K}_F$ corresponds to a diagonal c \end{aligned} \end{equation} -To determine how the acrshort:iff controller affects the poles of the closed-loop system, a Root Locus plot (Figure ref:fig:rotating_root_locus_iff_pure_int) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_{F}$ simultaneously. +To determine how the acrshort:iff controller affects the poles of the closed-loop system, a Root Locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_{F}$ simultaneously. As explained in cite:preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr, the closed-loop poles start at the open-loop poles (shown by crosses) for $g = 0$ and coincide with the transmission zeros (shown by circles) as $g \to \infty$. Whereas collocated IFF is usually associated with unconditional stability cite:preumont91_activ, this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null. -This can be seen in the Root Locus plot (Figure ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. -Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure ref:fig:rotating_iff_bode_plot_effect_rot). +This can be seen in the Root Locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. +Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure\nbsp{}ref:fig:rotating_iff_bode_plot_effect_rot). The control system is thus cancels the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. *** Integral Force Feedback with a High-Pass Filter @@ -1857,7 +1857,7 @@ The control system is thus cancels the spring forces, which makes the suspended **** Introduction :ignore: As explained in the previous section, the instability of the IFF controller applied to the rotating system is due to the high gain of the integrator at low-frequency. -To limit the low-frequency controller gain, a acrfull:hpf can be added to the controller, as shown in equation eqref:eq:rotating_iff_lhf. +To limit the low-frequency controller gain, a acrfull:hpf can be added to the controller, as shown in equation\nbsp{}eqref:eq:rotating_iff_lhf. This is equivalent to slightly shifting the controller pole to the left along the real axis. This modification of the IFF controller is typically performed to avoid saturation associated with the pure integrator cite:preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans. This is however not the reason why this high-pass filter is added here. @@ -1867,13 +1867,13 @@ This is however not the reason why this high-pass filter is added here. \end{equation} **** Modified Integral Force Feedback Controller -The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used eqref:eq:rotating_iff_lhf where $\omega_i$ characterize the frequency down to which the signal is integrated. -The loop gains ($K_F(s)$ times the direct dynamics $f_u/F_u$) with and without the added HPF are shown in Figure ref:fig:rotating_iff_modified_loop_gain. +The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used\nbsp{}eqref:eq:rotating_iff_lhf where $\omega_i$ characterize the frequency down to which the signal is integrated. +The loop gains ($K_F(s)$ times the direct dynamics $f_u/F_u$) with and without the added HPF are shown in Figure\nbsp{}ref:fig:rotating_iff_modified_loop_gain. The effect of the added HPF limits the low-frequency gain to finite values as expected. -The Root Locus plots for the decentralized acrshort:iff with and without the acrshort:hpf are displayed in Figure ref:fig:rotating_iff_root_locus_hpf_large. -With the added acrshort:hpf, the poles of the closed-loop system are shown to be stable up to some value of the gain $g_\text{max}$ given by equation eqref:eq:rotating_gmax_iff_hpf. -It is interesting to note that $g_{\text{max}}$ also corresponds to the controller gain at which the low-frequency loop gain reaches one (for instance the gain $g$ can be increased by a factor $5$ in Figure ref:fig:rotating_iff_modified_loop_gain before the system becomes unstable). +The Root Locus plots for the decentralized acrshort:iff with and without the acrshort:hpf are displayed in Figure\nbsp{}ref:fig:rotating_iff_root_locus_hpf_large. +With the added acrshort:hpf, the poles of the closed-loop system are shown to be stable up to some value of the gain $g_\text{max}$ given by equation\nbsp{}eqref:eq:rotating_gmax_iff_hpf. +It is interesting to note that $g_{\text{max}}$ also corresponds to the controller gain at which the low-frequency loop gain reaches one (for instance the gain $g$ can be increased by a factor $5$ in Figure\nbsp{}ref:fig:rotating_iff_modified_loop_gain before the system becomes unstable). \begin{equation}\label{eq:rotating_gmax_iff_hpf} \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)} @@ -1898,16 +1898,16 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the controll #+end_figure **** Optimal IFF with HPF parameters $\omega_i$ and $g$ -Two parameters can be tuned for the modified controller in equation eqref:eq:rotating_iff_lhf: the gain $g$ and the pole's location $\omega_i$. +Two parameters can be tuned for the modified controller in equation\nbsp{}eqref:eq:rotating_iff_lhf: the gain $g$ and the pole's location $\omega_i$. The optimal values of $\omega_i$ and $g$ are considered here as the values for which the damping of all the closed-loop poles is simultaneously maximized. -To visualize how $\omega_i$ does affect the attainable damping, the Root Locus plots for several $\omega_i$ are displayed in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi. -It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes (see Root locus in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi), the control gain $g$ may be limited to small values due to equation eqref:eq:rotating_gmax_iff_hpf. +To visualize how $\omega_i$ does affect the attainable damping, the Root Locus plots for several $\omega_i$ are displayed in Figure\nbsp{}ref:fig:rotating_root_locus_iff_modified_effect_wi. +It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes (see Root locus in Figure\nbsp{}ref:fig:rotating_root_locus_iff_modified_effect_wi), the control gain $g$ may be limited to small values due to equation\nbsp{}eqref:eq:rotating_gmax_iff_hpf. To study this trade-off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$. -The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:rotating_iff_hpf_optimal_gain). +The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure\nbsp{}ref:fig:rotating_iff_hpf_optimal_gain). -For small values of $\omega_i$, the added damping is limited by the maximum allowed control gain $g_{\text{max}}$ (red curve and dashed red curve superimposed in Figure ref:fig:rotating_iff_hpf_optimal_gain) at which point the pole corresponding to the controller becomes unstable. -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 ref:fig:rotating_iff_root_locus_hpf_large. +For small values of $\omega_i$, the added damping is limited by the maximum allowed control gain $g_{\text{max}}$ (red curve and dashed red curve superimposed in Figure\nbsp{}ref:fig:rotating_iff_hpf_optimal_gain) at which point the pole corresponding to the controller becomes unstable. +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}) @@ -1928,10 +1928,10 @@ For larger values of $\omega_i$, the attainable damping ratio decreases as a fun #+end_figure **** Obtained Damped Plant -To study how the parameter $\omega_i$ affects the damped plant, the obtained damped plants for several $\omega_i$ are compared in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_plant. +To study how the parameter $\omega_i$ affects the damped plant, the obtained damped plants for several $\omega_i$ are compared in Figure\nbsp{}ref:fig:rotating_iff_hpf_damped_plant_effect_wi_plant. It can be seen that the low-frequency coupling increases as $\omega_i$ increases. Therefore, there is a trade-off between achievable damping and added coupling when tuning $\omega_i$. -The same trade-off can be seen between achievable damping and loss of compliance at low-frequency (see Figure ref:fig:rotating_iff_hpf_effect_wi_compliance). +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 @@ -1956,7 +1956,7 @@ The same trade-off can be seen between achievable damping and loss of compliance **** Introduction :ignore: In this section it is proposed to add springs in parallel with the force sensors to counteract the negative stiffness induced by the gyroscopic effects. -Such springs are schematically shown in Figure 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. +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) @@ -1964,7 +1964,7 @@ Such springs are schematically shown in Figure ref:fig:rotating_3dof_model_schem [[file:figs/rotating_3dof_model_schematic_iff_parallel_springs.png]] **** Equations -The forces measured by the two force sensors represented in Figure ref:fig:rotating_3dof_model_schematic_iff_parallel_springs are described by eqref:eq:rotating_measured_force_kp. +The forces measured by the two force sensors represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_iff_parallel_springs are described by\nbsp{}eqref:eq:rotating_measured_force_kp. \begin{equation}\label{eq:rotating_measured_force_kp} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = @@ -1972,14 +1972,14 @@ The forces measured by the two force sensors represented in Figure ref:fig:rotat \begin{bmatrix} d_u \\ d_v \end{bmatrix} \end{equation} -To keep the overall stiffness $k = k_a + k_p$ constant, thus not modifying the open-loop poles as $k_p$ is changed, a scalar parameter $\alpha$ ($0 \le \alpha < 1$) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor as in eqref:eq:rotating_kp_alpha. +To keep the overall stiffness $k = k_a + k_p$ constant, thus not modifying the open-loop poles as $k_p$ is changed, a scalar parameter $\alpha$ ($0 \le \alpha < 1$) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor as in\nbsp{}eqref:eq:rotating_kp_alpha. \begin{equation}\label{eq:rotating_kp_alpha} k_p = \alpha k, \quad k_a = (1 - \alpha) k \end{equation} -After the equations of motion are derived and transformed in the Laplace domain, the transfer function matrix $\mathbf{G}_k$ in Eq. eqref:eq:rotating_Gk_mimo_tf is computed. -Its elements are shown in Eqs. eqref:eq:rotating_Gk_diag and eqref:eq:rotating_Gk_off_diag. +After the equations of motion are derived and transformed in the Laplace domain, the transfer function matrix $\mathbf{G}_k$ in Eq.\nbsp{}eqref:eq:rotating_Gk_mimo_tf is computed. +Its elements are shown in Eqs.\nbsp{}eqref:eq:rotating_Gk_diag and eqref:eq:rotating_Gk_off_diag. \begin{equation}\label{eq:rotating_Gk_mimo_tf} \begin{bmatrix} f_u \\ f_v \end{bmatrix} = @@ -1994,8 +1994,8 @@ Its elements are shown in Eqs. eqref:eq:rotating_Gk_diag and eqref:eq:rotating_G \end{align} \end{subequations} -Comparing $\mathbf{G}_k$ in eqref:eq:rotating_Gk with $\mathbf{G}_f$ in eqref:eq:rotating_Gf shows that while the poles of the system remain the same, the zeros of the diagonal terms change. -The two real zeros $z_r$ in eqref:eq:rotating_iff_zero_real that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in eqref:eq:rotating_kp_cond_cc_zeros holds. +Comparing $\mathbf{G}_k$ in\nbsp{}eqref:eq:rotating_Gk with $\mathbf{G}_f$ in\nbsp{}eqref:eq:rotating_Gf shows that while the poles of the system remain the same, the zeros of the diagonal terms change. +The two real zeros $z_r$ in\nbsp{}eqref:eq:rotating_iff_zero_real that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in\nbsp{}eqref:eq:rotating_kp_cond_cc_zeros holds. Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiffness/ induced by centrifugal forces $m \Omega^2$, the dynamics from the actuator to its collocated force sensor will show /minimum phase behavior/. \begin{equation}\label{eq:rotating_kp_cond_cc_zeros} @@ -2006,11 +2006,11 @@ Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiff **** Effect of parallel stiffness on the IFF plant The IFF plant (transfer function from $[F_u, F_v]$ to $[f_u, f_v]$) is identified without parallel stiffness $k_p = 0$, with a small parallel stiffness $k_p < m \Omega^2$ and with a large parallel stiffness $k_p > m \Omega^2$. -Bode plots of the obtained dynamics are shown in Figure ref:fig:rotating_iff_effect_kp. +Bode plots of the obtained dynamics are shown in Figure\nbsp{}ref:fig:rotating_iff_effect_kp. The two real zeros for $k_p < m \Omega^2$ are transformed into two complex conjugate zeros for $k_p > m \Omega^2$. In that case, the system shows alternating complex conjugate poles and zeros as what is the case in the non-rotating case. -Figure ref:fig:rotating_iff_kp_root_locus shows the Root Locus plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator, as shown in Eq. eqref:eq:rotating_Kf_pure_int. +Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus shows the Root Locus plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator, as shown in Eq.\nbsp{}eqref:eq:rotating_Kf_pure_int. 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 @@ -2034,10 +2034,10 @@ It is shown that if the added stiffness is higher than the maximum negative stif **** Effect of $k_p$ on the attainable damping Even though the parallel stiffness $k_p$ has no impact on the open-loop poles (as the overall stiffness $k$ is kept constant), it has a large impact on the transmission zeros. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros cite:preumont18_vibrat_contr_activ_struc_fourt_edition, the parallel stiffness $k_p$ is expected to have some impact on the attainable damping. -To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure ref:fig:rotating_iff_kp_root_locus_effect_kp. +To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus_effect_kp. The frequencies of the transmission zeros of the system increase with an increase in the parallel stiffness $k_p$ (thus getting closer to the poles), and the associated attainable damping is reduced. Therefore, even though the parallel stiffness $k_p$ should be larger than $m \Omega^2$ for stability reasons, it should not be taken too large as this would limit the attainable damping. -This is confirmed by the Figure 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. +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 @@ -2059,7 +2059,7 @@ This is confirmed by the Figure ref:fig:rotating_iff_kp_optimal_gain where the a **** Damped plant The parallel stiffness are chosen to be $k_p = 2 m \Omega^2$ and the damped plant is computed. -The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in Figure ref:fig:rotating_iff_kp_added_hpf_damped_plant. +The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in Figure\nbsp{}ref:fig:rotating_iff_kp_added_hpf_damped_plant. Even though the two resonances are well damped, the IFF changes the low-frequency behavior of the plant, which is usually not desired. This is because "pure" integrators are used which are inducing large low-frequency loop gains. @@ -2072,9 +2072,9 @@ To lower the low-frequency gain, a high-pass filter is added to the IFF control \end{equation} To determine how the high-pass filter impacts the attainable damping, the controller gain $g$ is kept constant while $\omega_i$ is changed, and the minimum damping ratio of the damped plant is computed. -The obtained damping ratio as a function of $\omega_i/\omega_0$ (where $\omega_0$ is the resonance of the system without rotation) is shown in Figure ref:fig:rotating_iff_kp_added_hpf_effect_damping. -It is shown that the attainable damping ratio reduces as $\omega_i$ is increased (same conclusion than in Section ref:sec:rotating_iff_pseudo_int). -Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compare the obtained damped plant again with the undamped and with the "pure" IFF in Figure ref:fig:rotating_iff_kp_added_hpf_damped_plant. +The obtained damping ratio as a function of $\omega_i/\omega_0$ (where $\omega_0$ is the resonance of the system without rotation) is shown in Figure\nbsp{}ref:fig:rotating_iff_kp_added_hpf_effect_damping. +It is shown that the attainable damping ratio reduces as $\omega_i$ is increased (same conclusion than in Section\nbsp{}ref:sec:rotating_iff_pseudo_int). +Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compare the obtained damped plant again with the undamped and with the "pure" IFF in Figure\nbsp{}ref:fig:rotating_iff_kp_added_hpf_damped_plant. The added high-pass filter gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior. #+name: fig:rotating_iff_optimal_hpf @@ -2099,9 +2099,9 @@ The added high-pass filter gives almost the same damping properties to the suspe <> **** Introduction :ignore: -To apply a "Relative Damping Control" strategy, relative motion sensors are added in parallel with the actuators as shown in Figure ref:fig:rotating_3dof_model_schematic_rdc. +To apply a "Relative Damping Control" strategy, relative motion sensors are added in parallel with the actuators as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_rdc. Two controllers $K_d$ are used to feed back the relative motion to the actuator. -These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter eqref:eq:rotating_rdc_controller. +These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter\nbsp{}eqref:eq:rotating_rdc_controller. \begin{equation}\label{eq:rotating_rdc_controller} K_d(s) = g \cdot \frac{s}{s + \omega_d} @@ -2113,8 +2113,8 @@ K_d(s) = g \cdot \frac{s}{s + \omega_d} [[file:figs/rotating_3dof_model_schematic_rdc.png]] **** Equations of motion -Let's note $\bm{G}_d$ the transfer function between actuator forces and measured relative motion in parallel with the actuators eqref:eq:rotating_rdc_plant_matrix. -The elements of $\bm{G}_d$ were derived in Section ref:sec:rotating_system_description are shown in eqref:eq:rotating_rdc_plant_elements. +Let's note $\bm{G}_d$ the transfer function between actuator forces and measured relative motion in parallel with the actuators\nbsp{}eqref:eq:rotating_rdc_plant_matrix. +The elements of $\bm{G}_d$ were derived in Section\nbsp{}ref:sec:rotating_system_description are shown in\nbsp{}eqref:eq:rotating_rdc_plant_elements. \begin{equation}\label{eq:rotating_rdc_plant_matrix} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} @@ -2127,7 +2127,7 @@ The elements of $\bm{G}_d$ were derived in Section ref:sec:rotating_system_descr \end{align} \end{subequations} -Neglecting the damping for simplicity ($\xi \ll 1$), the direct terms have two complex conjugate zeros between the two pairs of complex conjugate poles eqref:eq:rotating_rdc_zeros_poles. +Neglecting the damping for simplicity ($\xi \ll 1$), the direct terms have two complex conjugate zeros between the two pairs of complex conjugate poles\nbsp{}eqref:eq:rotating_rdc_zeros_poles. Therefore, for $\Omega < \sqrt{k/m}$ (i.e. stable system), the transfer functions for Relative Damping Control have alternating complex conjugate poles and zeros. \begin{equation}\label{eq:rotating_rdc_zeros_poles} @@ -2135,13 +2135,13 @@ Therefore, for $\Omega < \sqrt{k/m}$ (i.e. stable system), the transfer function \end{equation} **** Decentralized Relative Damping Control -The transfer functions from $[F_u,\ F_v]$ to $[d_u,\ d_v]$ were identified for several rotating velocities in Section ref:sec:rotating_system_description and are shown in Figure ref:fig:rotating_bode_plot (page pageref:fig:rotating_bode_plot). +The transfer functions from $[F_u,\ F_v]$ to $[d_u,\ d_v]$ were identified for several rotating velocities in Section\nbsp{}ref:sec:rotating_system_description and are shown in Figure\nbsp{}ref:fig:rotating_bode_plot (page\nbsp{}pageref:fig:rotating_bode_plot). -To see if large damping can be added with Relative Damping Control, the root locus is computed (Figure ref:fig:rotating_rdc_root_locus). +To see if large damping can be added with Relative Damping Control, the root locus is computed (Figure\nbsp{}ref:fig:rotating_rdc_root_locus). The closed-loop system is unconditionally stable as expected and the poles can be damped as much as desired. Let us select a reasonable "Relative Damping Control" gain, and compute the closed-loop damped system. -The open-loop and damped plants are compared in Figure ref:fig:rotating_rdc_damped_plant. +The open-loop and damped plants are compared in Figure\nbsp{}ref:fig:rotating_rdc_damped_plant. The rotating aspect does not add any complexity to the use of Relative Damping Control. It does not increase the low-frequency coupling as compared to the Integral Force Feedback. @@ -2174,7 +2174,7 @@ These values are chosen one the basis of previous discussions about optimal para **** Identify plants :noexport: **** Root Locus -Figure ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications and the relative damping control. +Figure\nbsp{}ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications and the relative damping control. While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not. This means that the closed-loop behavior of both systems will differ when large control gains are used. @@ -2201,7 +2201,7 @@ It is interesting to note that the maximum added damping is very similar for bot #+end_figure **** Obtained Damped Plant -The actively damped plants are computed for the three techniques and compared in Figure ref:fig:rotating_comp_techniques_dampled_plants. +The actively damped plants are computed for the three techniques and compared in Figure\nbsp{}ref:fig:rotating_comp_techniques_dampled_plants. It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, the off-diagonal (coupling) terms are not. The acrshort:iff strategy is adding some coupling at low-frequency, which may negatively impact the positioning performance. @@ -2213,7 +2213,7 @@ The compliance describes the displacement response of the payload to the externa This is a useful metric when disturbances are directly applied to the payload. Here, it is defined as the transfer function from external forces applied on the payload along $\vec{i}_x$ to the displacement of the payload along the same direction. -Very similar results were obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure ref:fig:rotating_comp_techniques_trans_compliance). +Very similar results were obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure\nbsp{}ref:fig:rotating_comp_techniques_trans_compliance). Using IFF degrades the compliance at low frequencies, whereas using relative damping control degrades the transmissibility at high frequencies. This is very well known characteristics of these common active damping techniques that hold when applied to rotating platforms. @@ -2248,7 +2248,7 @@ Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) For the NASS, the maximum rotating velocity is $\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}$ for a suspended mass on top of the nano-hexapod's actuators equal to $m_n + m_s = \SI{16}{\kilo\gram}$. The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \approx \SI{0.6}{\newton\per\mm}$. -The transfer functions from the nano-hexapod actuator force $F_u$ to the displacement of the nano-hexapod in the same direction $d_u$ as well as in the orthogonal direction $d_v$ (coupling) are shown in Figure ref:fig:rotating_nano_hexapod_dynamics for all three considered nano-hexapod stiffnesses. +The transfer functions from the nano-hexapod actuator force $F_u$ to the displacement of the nano-hexapod in the same direction $d_u$ as well as in the orthogonal direction $d_v$ (coupling) are shown in Figure\nbsp{}ref:fig:rotating_nano_hexapod_dynamics for all three considered nano-hexapod stiffnesses. The soft nano-hexapod is the most affected by rotation. This can be seen by the large shift of the resonance frequencies, and by the induced coupling, which is larger than that for the stiffer nano-hexapods. The coupling (or interaction) in a MIMO $2 \times 2$ system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). @@ -2279,13 +2279,13 @@ The coupling (or interaction) in a MIMO $2 \times 2$ system can be visually esti **** Optimal IFF with a High-Pass Filter Integral Force Feedback with an added high-pass filter is applied to the three nano-hexapods. -First, the parameters ($\omega_i$ and $g$) of the IFF controller that yield the best simultaneous damping are determined from Figure ref:fig:rotating_iff_hpf_nass_optimal_gain. +First, the parameters ($\omega_i$ and $g$) of the IFF controller that yield the best simultaneous damping are determined from Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain. The IFF parameters are chosen as follows: -- for $k_n = \SI{0.01}{\N\per\mu\m}$ (Figure ref:fig:rotating_iff_hpf_nass_optimal_gain): $\omega_i$ is chosen such that maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. +- for $k_n = \SI{0.01}{\N\per\mu\m}$ (Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain): $\omega_i$ is chosen such that maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. This is done to have some control robustness. -- for $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$ (Figure ref:fig:rotating_iff_hpf_nass_optimal_gain_md and ref:fig:rotating_iff_hpf_nass_optimal_gain_pz): the largest $\omega_i$ is chosen such that the obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. - Large $\omega_i$ is chosen here to limit the loss of compliance and the increase of coupling at low-frequency as shown in Section ref:sec:rotating_iff_pseudo_int. -The obtained IFF parameters and the achievable damping are visually shown by large dots in Figure ref:fig:rotating_iff_hpf_nass_optimal_gain and are summarized in Table ref:tab:rotating_iff_hpf_opt_iff_hpf_params_nass. +- for $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$ (Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain_md and ref:fig:rotating_iff_hpf_nass_optimal_gain_pz): the largest $\omega_i$ is chosen such that the obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. + Large $\omega_i$ is chosen here to limit the loss of compliance and the increase of coupling at low-frequency as shown in Section\nbsp{}ref:sec:rotating_iff_pseudo_int. +The obtained IFF parameters and the achievable damping are visually shown by large dots in Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain and are summarized in Table\nbsp{}ref:tab:rotating_iff_hpf_opt_iff_hpf_params_nass. #+name: fig:rotating_iff_hpf_nass_optimal_gain #+caption: For each value of $\omega_i$, the maximum damping ratio $\xi$ is computed (blue), and the corresponding controller gain is shown (in red). The chosen controller parameters used for further analysis are indicated by the large dots. @@ -2326,14 +2326,14 @@ For each considered nano-hexapod stiffness, the parallel stiffness $k_p$ is vari To keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased ($k_a = k_n - k_p$, with $k_n$ the total nano-hexapod stiffness). A high-pass filter is also added to limit the low-frequency gain with a cut-off frequency $\omega_i$ equal to one tenth of the system resonance ($\omega_i = \omega_0/10$). -The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure ref:fig:rotating_iff_kp_nass_optimal_gain). +The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure\nbsp{}ref:fig:rotating_iff_kp_nass_optimal_gain). It is shown that the soft nano-hexapod cannot yield good damping because the parallel stiffness cannot be sufficiently large compared to the negative stiffness induced by the rotation. -For the two stiff options, the achievable damping decreases when the parallel stiffness is too high, as explained in Section ref:sec:rotating_iff_parallel_stiffness. -Such behavior can be explained by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chapt 7.2]]. +For the two stiff options, the achievable damping decreases when the parallel stiffness is too high, as explained in Section\nbsp{}ref:sec:rotating_iff_parallel_stiffness. +Such behavior can be explained by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chapt 7.2]]. This distance is larger for stiff nano-hexapod because the open-loop pole will be at higher frequencies while the open-loop zero, whereas depends on the value of the parallel stiffness, can only be made large for stiff nano-hexapods. Let's choose $k_p = 1\,N/mm$, $k_p = 0.01\,N/\mu m$ and $k_p = 1\,N/\mu m$ for the three considered nano-hexapods. -The corresponding optimal controller gains and achievable damping are summarized in Table ref:tab:rotating_iff_kp_opt_iff_kp_params_nass. +The corresponding optimal controller gains and achievable damping are summarized in Table\nbsp{}ref:tab:rotating_iff_kp_opt_iff_kp_params_nass. #+attr_latex: :options [t]{0.49\linewidth} #+begin_minipage @@ -2357,8 +2357,8 @@ The corresponding optimal controller gains and achievable damping are summarized #+end_minipage **** Optimal Relative Motion Control -For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure ref:fig:rotating_rdc_optimal_gain). -The gain is chosen such that 99% of modal damping is obtained (obtained gains are summarized in Table ref:tab:rotating_rdc_opt_params_nass). +For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure\nbsp{}ref:fig:rotating_rdc_optimal_gain). +The gain is chosen such that 99% of modal damping is obtained (obtained gains are summarized in Table\nbsp{}ref:tab:rotating_rdc_opt_params_nass). #+attr_latex: :options [t]{0.49\linewidth} #+begin_minipage @@ -2382,7 +2382,7 @@ The gain is chosen such that 99% of modal damping is obtained (obtained gains ar #+end_minipage **** Comparison of the obtained damped plants -Now that the optimal parameters for the three considered active damping techniques have been determined, the obtained damped plants are computed and compared in Figure ref:fig:rotating_nass_damped_plant_comp. +Now that the optimal parameters for the three considered active damping techniques have been determined, the obtained damped plants are computed and compared in Figure\nbsp{}ref:fig:rotating_nass_damped_plant_comp. Similar to what was concluded in the previous analysis: - acrshort:iff adds more coupling below the resonance frequency as compared to the open-loop and acrshort:rdc cases @@ -2421,7 +2421,7 @@ While quite simplistic, this allowed us to study the effects of rotation and the In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. **** Nano Active Stabilization System model -To have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modeled as shown in Figure ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure ref:fig:rotating_nass_model for a 3D view): +To have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): - the floor whose motion is imposed - a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$) - a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_t = \SI{600}{\kg}$) @@ -2437,8 +2437,8 @@ A payload is rigidly fixed to the nano-hexapod and the $x,y$ motion of the paylo **** System dynamics -The dynamics of the undamped and damped plants are identified using the optimal parameters found in Section ref:sec:rotating_nano_hexapod. -The obtained dynamics are compared in Figure ref:fig:rotating_nass_plant_comp_stiffness in which the direct terms are shown by the solid curves and the coupling terms are shown by the shaded ones. +The dynamics of the undamped and damped plants are identified using the optimal parameters found in Section\nbsp{}ref:sec:rotating_nano_hexapod. +The obtained dynamics are compared in Figure\nbsp{}ref:fig:rotating_nass_plant_comp_stiffness in which the direct terms are shown by the solid curves and the coupling terms are shown by the shaded ones. It can be observed that: - The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod - Damping added using the three proposed techniques is quite high, and the obtained plant is rather easy to control @@ -2471,15 +2471,15 @@ It can be observed that: **** Effect of disturbances -The effect of three disturbances are considered (as for the uniaxial model), floor motion $[x_{f,x},\ x_{f,y}]$ (Figure ref:fig:rotating_nass_effect_floor_motion), micro-Station vibrations $[f_{t,x},\ f_{t,y}]$ (Figure ref:fig:rotating_nass_effect_stage_vibration) and direct forces applied on the sample $[f_{s,x},\ f_{s,y}]$ (Figure ref:fig:rotating_nass_effect_direct_forces). +The effect of three disturbances are considered (as for the uniaxial model), floor motion $[x_{f,x},\ x_{f,y}]$ (Figure\nbsp{}ref:fig:rotating_nass_effect_floor_motion), micro-Station vibrations $[f_{t,x},\ f_{t,y}]$ (Figure\nbsp{}ref:fig:rotating_nass_effect_stage_vibration) and direct forces applied on the sample $[f_{s,x},\ f_{s,y}]$ (Figure\nbsp{}ref:fig:rotating_nass_effect_direct_forces). Note that only the transfer functions from the disturbances in the $x$ direction to the relative position $d_x$ between the sample and the granite in the $x$ direction are displayed because the transfer functions in the $y$ direction are the same due to the system symmetry. Conclusions are similar than those of the uniaxial (non-rotating) model: - Regarding the effect of floor motion and forces applied on the payload: - - The stiffer, the better. This can be seen in Figures ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces where the magnitudes for the stiff hexapod are lower than those for the soft one + - The stiffer, the better. This can be seen in Figures\nbsp{}ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces where the magnitudes for the stiff hexapod are lower than those for the soft one - acrshort:iff degrades the performance at low-frequency compared to acrshort:rdc - Regarding the effect of micro-station vibrations: - - Having a soft nano-hexapod allows filtering of these vibrations between the suspension modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure ref:fig:rotating_nass_effect_stage_vibration_vc). + - Having a soft nano-hexapod allows filtering of these vibrations between the suspension modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure\nbsp{}ref:fig:rotating_nass_effect_stage_vibration_vc). #+name: fig:rotating_nass_effect_floor_motion #+caption: Effect of floor motion $x_{f,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. IFF is shown to increase the sensitivity to floor motion at low-frequency. @@ -2555,27 +2555,27 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: *** Conclusion -In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section ref:sec:rotating_system_description). -Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms (Section ref:sec:rotating_iff_pure_int). +In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section\nbsp{}ref:sec:rotating_system_description). +Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms (Section\nbsp{}ref:sec:rotating_iff_pure_int). Two modifications of the classical acrshort:iff control have been proposed to overcome this issue. The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators. This is equivalent to moving the controller pole to the left along the real axis. -This allows the closed-loop system to be stable up to some value of the controller gain (Section ref:sec:rotating_iff_pseudo_int). +This allows the closed-loop system to be stable up to some value of the controller gain (Section\nbsp{}ref:sec:rotating_iff_pseudo_int). The second proposed modification concerns the mechanical system. Additional springs are added in parallel with the actuators and force sensors. -It was shown that if the stiffness $k_p$ of the additional springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized acrshort:iff regains its unconditional stability property (Section ref:sec:rotating_iff_parallel_stiffness). +It was shown that if the stiffness $k_p$ of the additional springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized acrshort:iff regains its unconditional stability property (Section\nbsp{}ref:sec:rotating_iff_parallel_stiffness). -These two modifications were compared with acrlong:rdc in Section ref:sec:rotating_comp_act_damp. +These two modifications were compared with acrlong:rdc in Section\nbsp{}ref:sec:rotating_comp_act_damp. While having very different implementations, both proposed modifications were found to be very similar with respect to the attainable damping and the obtained closed-loop system behavior. -This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters (Section ref:sec:rotating_nano_hexapod). +This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters (Section\nbsp{}ref:sec:rotating_nano_hexapod). As for the uniaxial model, three nano-hexapod stiffnesses values were considered. The dynamics of the soft nano-hexapod ($k_n = 0.01\,N/\mu m$) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). In addition, the attainable damping ratio of the soft nano-hexapod when using acrshort:iff is limited by gyroscopic effects. -To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been considered (Section ref:sec:rotating_nass). +To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been considered (Section\nbsp{}ref:sec:rotating_nass). Results are similar to those of the uniaxial model except that come complexity is added for the soft nano-hexapod due to the spindle's rotation. For the moderately stiff nano-hexapod ($k_n = 1\,N/\mu m$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that showed better results with the uniaxial model. @@ -2589,7 +2589,7 @@ A multi-body model consisting of several rigid bodies connected by kinematic con Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body. Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station. -The tuning approach for the multi-body model based on measurements is illustrated in Figure ref:fig:modal_vibration_analysis_procedure. +The tuning approach for the multi-body model based on measurements is illustrated in Figure\nbsp{}ref:fig:modal_vibration_analysis_procedure. First, a /response model/ is obtained, which corresponds to a set of frequency response functions computed from experimental measurements. From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes. This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies. @@ -2598,14 +2598,14 @@ This modal model can then be used to tune the spatial model (i.e. the multi-body #+caption: Three models of the same structure. The goal is to tune a spatial model (i.e. mass, stiffness and damping properties) from a response model. The modal model can be used as an intermediate step. [[file:figs/modal_vibration_analysis_procedure.png]] -The measurement setup used to obtain the response model is described in Section ref:sec:modal_meas_setup. +The measurement setup used to obtain the response model is described in Section\nbsp{}ref:sec:modal_meas_setup. This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals. -In Section ref:sec:modal_frf_processing, the obtained frequency response functions between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed. +In Section\nbsp{}ref:sec:modal_frf_processing, the obtained frequency response functions between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed. These measurements are projected at the center of mass of each considered solid body to facilitate the further use of the results. The solid body assumption is then verified, validating the use of the multi-body model. -Finally, the modal analysis is performed in Section ref:sec:modal_analysis. +Finally, the modal analysis is performed in Section\nbsp{}ref:sec:modal_analysis. This shows how complex the micro-station dynamics is, and the necessity of having a model representing its complex dynamics. *** Measurement Setup @@ -2613,17 +2613,17 @@ This shows how complex the micro-station dynamics is, and the necessity of havin **** Introduction :ignore: In order to perform an experimental modal analysis, a suitable measurement setup is essential. -This includes using appropriate instrumentation (presented in Section ref:ssec:modal_instrumentation) and properly preparing the structure to be measured (Section ref:ssec:modal_test_preparation). -Then, the locations of the measured motions (Section ref:ssec:modal_accelerometers) and the locations of the hammer impacts (Section ref:ssec:modal_hammer_impacts) have to be chosen carefully. -The obtained force and acceleration signals are described in Section ref:ssec:modal_measured_signals, and the quality of the measured data is assessed. +This includes using appropriate instrumentation (presented in Section\nbsp{}ref:ssec:modal_instrumentation) and properly preparing the structure to be measured (Section\nbsp{}ref:ssec:modal_test_preparation). +Then, the locations of the measured motions (Section\nbsp{}ref:ssec:modal_accelerometers) and the locations of the hammer impacts (Section\nbsp{}ref:ssec:modal_hammer_impacts) have to be chosen carefully. +The obtained force and acceleration signals are described in Section\nbsp{}ref:ssec:modal_measured_signals, and the quality of the measured data is assessed. **** Instrumentation <> Three types of equipment are essential for a good modal analysis. First, /accelerometers/ are used to measure the response of the structure. -Here, 3-axis accelerometers[fn:modal_1] shown in figure ref:fig:modal_accelero_M393B05 are used. -These accelerometers were glued to the micro-station using a thin layer of wax for best results [[cite:&ewins00_modal chapt. 3.5.7]]. +Here, 3-axis accelerometers[fn:modal_1] shown in figure\nbsp{}ref:fig:modal_accelero_M393B05 are used. +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 @@ -2649,11 +2649,11 @@ These accelerometers were glued to the micro-station using a thin layer of wax f #+end_subfigure #+end_figure -Then, an /instrumented hammer/[fn:modal_2] (figure ref:fig:modal_instrumented_hammer) is used to apply forces to the structure in a controlled manner. +Then, an /instrumented hammer/[fn:modal_2] (figure\nbsp{}ref:fig:modal_instrumented_hammer) is used to apply forces to the structure in a controlled manner. Tests were conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one). The softer tip was found to give best results as it injects more energy in the low-frequency range where the coherence was low, such that the overall coherence was improved. -Finally, an /acquisition system/[fn:modal_3] (figure ref:fig:modal_oros) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise. +Finally, an /acquisition system/[fn:modal_3] (figure\nbsp{}ref:fig:modal_oros) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise. **** Structure Preparation and Test Planing <> @@ -2668,7 +2668,7 @@ The top part representing the active stabilization stage was disassembled as the To perform the modal analysis from the measured responses, the $n \times n$ frequency response function matrix $\mathbf{H}$ needs to be measured, where $n$ is the considered number of degrees of freedom. The $H_{jk}$ element of this acrfull:frf matrix corresponds to the frequency response function from a force $F_k$ applied at acrfull:dof $k$ to the displacement of the structure $X_j$ at acrshort:dof $j$. Measuring this acrshort:frf matrix is time consuming as it requires to make $n \times n$ measurements. -However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$ [[cite:&ewins00_modal chapt. 5.2]]. +However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$\nbsp{}[[cite:&ewins00_modal chapt. 5.2]]. Therefore, a minimum set of $n$ frequency response functions is required. This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof. @@ -2680,11 +2680,11 @@ In this modal analysis, it is chosen to measure the response of the structure at The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured. A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod. -The positions of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers. -Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure ref:fig:modal_accelerometer_pictures. +The positions of the accelerometers are visually shown on a CAD model in Figure\nbsp{}ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table\nbsp{}ref:tab:modal_position_accelerometers. +Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure\nbsp{}ref:fig:modal_accelerometer_pictures. As all key stages of the micro-station are expected to behave as solid bodies, only 6 acrshort:dof can be considered for each solid body. -However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption). +However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section\nbsp{}ref:ssec:modal_solid_body_assumption). #+attr_latex: :options [b]{0.63\linewidth} #+begin_minipage @@ -2751,9 +2751,9 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrsh <> The selected location of the hammer impact corresponds to the location of accelerometer number $11$ fixed to the translation stage. -It was chosen to match the location of one accelerometer, because a /point measurement/ (i.e. a measurement of $H_{kk}$) is necessary to be able to reconstruct the full acrshort:frf matrix [[cite:ewins00_modal]]. +It was chosen to match the location of one accelerometer, because a /point measurement/ (i.e. a measurement of $H_{kk}$) is necessary to be able to reconstruct the full acrshort:frf matrix\nbsp{}[[cite:ewins00_modal]]. -The impacts were performed in three directions, as shown in figures ref:fig:modal_impact_x, ref:fig:modal_impact_y and ref:fig:modal_impact_z. +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 @@ -2782,11 +2782,11 @@ The impacts were performed in three directions, as shown in figures ref:fig:moda **** Force and Response signals <> -The force sensor of the instrumented hammer and the accelerometer signals are shown in the time domain in Figure ref:fig:modal_raw_meas. +The force sensor of the instrumented hammer and the accelerometer signals are shown in the time domain in Figure\nbsp{}ref:fig:modal_raw_meas. Sharp "impacts" can be observed for the force sensor, indicating wide frequency band excitation. For the accelerometer, a much more complex signal can be observed, indicating complex dynamics. -The "normalized" acrfull:asd of the two signals were computed and shown in Figure ref:fig:modal_asd_acc_force. +The "normalized" acrfull:asd of the two signals were computed and shown in Figure\nbsp{}ref:fig:modal_asd_acc_force. Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer). These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the micro-hexapod). Similar results were obtained for all measured frequency response functions. @@ -2809,8 +2809,8 @@ Similar results were obtained for all measured frequency response functions. #+end_subfigure #+end_figure -The frequency response function from the applied force to the measured acceleration is then computed and shown Figure ref:fig:modal_frf_acc_force. -The quality of the obtained data can be estimated using the /coherence/ function (Figure ref:fig:modal_coh_acc_force). +The frequency response function from the applied force to the measured acceleration is then computed and shown Figure\nbsp{}ref:fig:modal_frf_acc_force. +The quality of the obtained data can be estimated using the /coherence/ function (Figure\nbsp{}ref:fig:modal_coh_acc_force). Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest. #+name: fig:modal_frf_coh_acc_force @@ -2840,7 +2840,7 @@ After all measurements are conducted, a $n \times p \times q$ acrlongpl:frf matr - $p = 3$: number of input force excitation - $q = 801$: number of frequency points $\omega_{i}$ -For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that links the 3 force inputs to the 69 output accelerations eqref:eq:modal_frf_matrix_raw. +For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that links the 3 force inputs to the 69 output accelerations\nbsp{}eqref:eq:modal_frf_matrix_raw. \begin{equation}\label{eq:modal_frf_matrix_raw} \mathbf{H}(\omega_i) = \begin{bmatrix} @@ -2857,24 +2857,24 @@ However, for the multi-body model, only 6 solid bodies are considered, namely: t Therefore, only $6 \times 6 = 36$ degrees of freedom are of interest. Therefore, the objective of this section is to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36. -The coordinate transformation from accelerometers acrshort:dof to the solid body 6 acrshortpl:dof (three translations and three rotations) is performed in Section ref:ssec:modal_acc_to_solid_dof. +The coordinate transformation from accelerometers acrshort:dof to the solid body 6 acrshortpl:dof (three translations and three rotations) is performed in Section\nbsp{}ref:ssec:modal_acc_to_solid_dof. The $69 \times 3 \times 801$ frequency response matrix is then reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to its center of mass. -To validate this reduction of acrshort:dof and the solid body assumption, the frequency response function at the accelerometer location are "reconstructed" from the reduced frequency response matrix and are compared with the initial measurements in Section ref:ssec:modal_solid_body_assumption. +To validate this reduction of acrshort:dof and the solid body assumption, the frequency response function at the accelerometer location are "reconstructed" from the reduced frequency response matrix and are compared with the initial measurements in Section\nbsp{}ref:ssec:modal_solid_body_assumption. **** From accelerometer DOFs to solid body DOFs <> -Let us consider the schematic shown in Figure ref:fig:modal_local_to_global_coordinates where the motion of a solid body is measured at 4 distinct locations (in $x$, $y$ and $z$ directions). +Let us consider the schematic shown in Figure\nbsp{}ref:fig:modal_local_to_global_coordinates where the motion of a solid body is measured at 4 distinct locations (in $x$, $y$ and $z$ directions). 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 [[file:figs/modal_local_to_global_coordinates.png]] -The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$. +The motion of the rigid body of figure\nbsp{}ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$. -The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix [[cite:&ewins00_modal chapt. 4.3.2]]. +The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation\nbsp{}eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation\nbsp{}eqref:eq:modal_rotation_matrix\nbsp{}[[cite:&ewins00_modal chapt. 4.3.2]]. \begin{equation}\label{eq:modal_compute_point_response} \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\ @@ -2888,7 +2888,7 @@ The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\del \end{bmatrix} \end{equation} -Writing this in matrix form for the four points gives eqref:eq:modal_cart_to_acc. +Writing this in matrix form for the four points gives\nbsp{}eqref:eq:modal_cart_to_acc. \begin{equation}\label{eq:modal_cart_to_acc} \left[\begin{array}{c} @@ -2907,12 +2907,12 @@ Writing this in matrix form for the four points gives eqref:eq:modal_cart_to_acc \end{array}\right] \end{equation} -Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:modal_5]. -The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined by inverting equation eqref:eq:modal_cart_to_acc. +Provided that the four sensors are properly located, the system of equation\nbsp{}eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:modal_5]. +The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined by inverting equation\nbsp{}eqref:eq:modal_cart_to_acc. Note that this matrix inversion is equivalent to resolving a mean square problem. Therefore, having more accelerometers permits better approximation of the motion of a solid body. -From the CAD model, the position of the center of mass of each solid body is computed (see Table ref:tab:modal_com_solid_bodies). +From the CAD model, the position of the center of mass of each solid body is computed (see Table\nbsp{}ref:tab:modal_com_solid_bodies). The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined. #+name: tab:modal_com_solid_bodies @@ -2928,7 +2928,7 @@ The position of each accelerometer with respect to the center of mass of the cor | Spindle | $0$ | $0$ | $-580\,\text{mm}$ | | Hexapod | $-4\,\text{mm}$ | $6\,\text{mm}$ | $-319\,\text{mm}$ | -Using eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$. +Using\nbsp{}eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H}_\text{CoM}$\nbsp{}eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$. \begin{equation}\label{eq:modal_frf_matrix_com} \mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix} @@ -2947,11 +2947,11 @@ Using eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H}_\tex **** Verification of solid body assumption <> -From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location. +From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation\nbsp{}eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location. In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements $\mathbf{H}$. This is what is done here to check whether the solid body assumption is correct in the frequency band of interest. -The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure ref:fig:modal_comp_acc_solid_body_frf). +The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure\nbsp{}ref:fig:modal_comp_acc_solid_body_frf). The original frequency response functions and those computed from the CoM responses match well in the frequency range of interest. Similar results were obtained for the other solid bodies, indicating that the solid body assumption is valid and that a multi-body model can be used to represent the dynamics of the micro-station. This also validates the reduction in the number of degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof). @@ -2968,18 +2968,18 @@ The goal here is to extract the modal parameters describing the modes of the mic This is performed from the acrshort:frf matrix previously extracted from the measurements. In order to perform the modal parameter extraction, the order of the modal model has to be estimated (i.e. the number of modes in the frequency band of interest). -This is achived using the acrfull:mif in section ref:ssec:modal_number_of_modes. +This is achived using the acrfull:mif in section\nbsp{}ref:ssec:modal_number_of_modes. -In section ref:ssec:modal_parameter_extraction, the modal parameter extraction is performed. +In section\nbsp{}ref:ssec:modal_parameter_extraction, the modal parameter extraction is performed. The graphical display of the mode shapes can be computed from the modal model, which is quite useful for physical interpretation of the modes. -To validate the quality of the modal model, the full acrshort:frf matrix is computed from the modal model and compared to the initial measured acrshort:frf (section ref:ssec:modal_model_validity). +To validate the quality of the modal model, the full acrshort:frf matrix is computed from the modal model and compared to the initial measured acrshort:frf (section\nbsp{}ref:ssec:modal_model_validity). **** Number of modes determination <> The acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$). -The complex modal indication function is defined in equation eqref:eq:modal_cmif where the diagonal matrix $\Sigma$ is obtained from a acrlong:svd of the acrshort:frf matrix as shown in equation eqref:eq:modal_svd. +The complex modal indication function is defined in equation\nbsp{}eqref:eq:modal_cmif where the diagonal matrix $\Sigma$ is obtained from a acrlong:svd of the acrshort:frf matrix as shown in equation\nbsp{}eqref:eq:modal_svd. \begin{equation} \label{eq:modal_cmif} [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^{\intercal} [\Sigma(\omega)]_{n\times p} \end{equation} @@ -2991,9 +2991,9 @@ The complex modal indication function is defined in equation eqref:eq:modal_cmif The acrshort:mif therefore yields to $p$ values that are also frequency dependent. A peak in the acrshort:mif plot indicates the presence of a mode. Repeated modes can also be detected when multiple singular values have peaks at the same frequency. -The obtained acrshort:mif is shown on Figure ref:fig:modal_indication_function. +The obtained acrshort:mif is shown on Figure\nbsp{}ref:fig:modal_indication_function. A total of 16 modes were found between 0 and $200\,\text{Hz}$. -The obtained natural frequencies and associated modal damping are summarized in Table ref:tab:modal_obtained_modes_freqs_damps. +The obtained natural frequencies and associated modal damping are summarized in Table\nbsp{}ref:tab:modal_obtained_modes_freqs_damps. #+attr_latex: :options [b]{0.70\linewidth} #+begin_minipage @@ -3040,7 +3040,7 @@ However, there are multiple levels of complexity, from fitting of a single reson Here, the last method is used because it provides a unique and consistent model. It takes into account the fact that the properties of all individual curves are related by being from the same structure: all acrshort:frf plots on a given structure should indicate the same values for the natural frequencies and damping factor of each mode. -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 ref:fig:modal_mode_animations). +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 @@ -3067,8 +3067,8 @@ From the obtained modal parameters, the mode shapes are computed and can be disp #+end_figure These animations are useful for visually obtaining a better understanding of the system's dynamic behavior. -For instance, the mode shape of the first mode at $11\,\text{Hz}$ (figure ref:fig:modal_mode1_animation) indicates an issue with the lower granite. -It turns out that four /Airloc Levelers/ are used to level the lower granite (figure ref:fig:modal_airloc). +For instance, the mode shape of the first mode at $11\,\text{Hz}$ (figure\nbsp{}ref:fig:modal_mode1_animation) indicates an issue with the lower granite. +It turns out that four /Airloc Levelers/ are used to level the lower granite (figure\nbsp{}ref:fig:modal_airloc). These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it has a low frequency "tilt mode". The levelers were then better adjusted. @@ -3078,13 +3078,13 @@ The levelers were then better adjusted. [[file:figs/modal_airlock_picture.jpg]] The modal parameter extraction is made using a proprietary software[fn:modal_4]. -For each mode $r$ (from $1$ to the number of considered modes $m=16$), it outputs the frequency $\omega_r$, the damping ratio $\xi_r$, the eigenvectors $\{\phi_{r}\}$ (vector of complex numbers with a size equal to the number of measured acrshort:dof $n=69$, see equation eqref:eq:modal_eigenvector) and a scaling factor $a_r$. +For each mode $r$ (from $1$ to the number of considered modes $m=16$), it outputs the frequency $\omega_r$, the damping ratio $\xi_r$, the eigenvectors $\{\phi_{r}\}$ (vector of complex numbers with a size equal to the number of measured acrshort:dof $n=69$, see equation\nbsp{}eqref:eq:modal_eigenvector) and a scaling factor $a_r$. \begin{equation}\label{eq:modal_eigenvector} \{\phi_i\} = \begin{Bmatrix} \phi_{i, 1_x} & \phi_{i, 1_y} & \phi_{i, 1_z} & \phi_{i, 2_x} & \dots & \phi_{i, 23_z} \end{Bmatrix}^{\intercal} \end{equation} -The eigenvalues $s_r$ and $s_r^*$ can then be computed from equation eqref:eq:modal_eigenvalues. +The eigenvalues $s_r$ and $s_r^*$ can then be computed from equation\nbsp{}eqref:eq:modal_eigenvalues. \begin{equation}\label{eq:modal_eigenvalues} s_r = \omega_r (-\xi_r + i \sqrt{1 - \xi_r^2}), \quad s_r^* = \omega_r (-\xi_r - i \sqrt{1 - \xi_r^2}) @@ -3096,7 +3096,7 @@ The eigenvalues $s_r$ and $s_r^*$ can then be computed from equation eqref:eq:mo To check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ is first synthesized from the modal parameters. Then, the elements of this acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ that were already measured can be compared to the measured acrshort:frf matrix $\mathbf{H}$. -In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ are first organized in matrix from as shown in equation eqref:eq:modal_eigvector_matrix. +In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ are first organized in matrix from as shown in equation\nbsp{}eqref:eq:modal_eigvector_matrix. \begin{equation}\label{eq:modal_eigvector_matrix} \Phi = \begin{bmatrix} & & & & &\\ @@ -3105,21 +3105,21 @@ In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ a \end{bmatrix}_{n \times 2m} \end{equation} -The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be obtained using eqref:eq:modal_synthesized_frf. +The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be obtained using\nbsp{}eqref:eq:modal_synthesized_frf. \begin{equation}\label{eq:modal_synthesized_frf} [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^{\intercal} \end{equation} -With $\mathbf{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response of the different modes eqref:eq:modal_modal_resp. +With $\mathbf{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response of the different modes\nbsp{}eqref:eq:modal_modal_resp. \begin{equation}\label{eq:modal_modal_resp} \mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m} \end{equation} -A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal. +A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal. Whether the obtained match is good or bad is quite arbitrary. However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective. -This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function from the force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction. +This can be seen in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function from the force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction. #+name: fig:modal_comp_acc_frf_modal #+caption: Comparison of the measured FRF with the FRF synthesized from the modal model. @@ -3174,20 +3174,20 @@ Therefore, a multi-body model is a good candidate to accurately represent the mi In this report, the development of such a multi-body model is presented. First, each stage of the micro-station is described. -The kinematics of the micro-station (i.e. how the motion of the stages are combined) is presented in Section ref:sec:ustation_kinematics. +The kinematics of the micro-station (i.e. how the motion of the stages are combined) is presented in Section\nbsp{}ref:sec:ustation_kinematics. -Then, the multi-body model is presented and tuned to match the measured dynamics of the micro-station (Section ref:sec:ustation_modeling). +Then, the multi-body model is presented and tuned to match the measured dynamics of the micro-station (Section\nbsp{}ref:sec:ustation_modeling). Disturbances affecting the positioning accuracy also need to be modeled properly. -To do so, the effects of these disturbances were first measured experimental and then injected into the multi-body model (Section ref:sec:ustation_disturbances). +To do so, the effects of these disturbances were first measured experimental and then injected into the multi-body model (Section\nbsp{}ref:sec:ustation_disturbances). -To validate the accuracy of the micro-station model, "real world" experiments are simulated and compared with measurements in Section ref:sec:ustation_experiments. +To validate the accuracy of the micro-station model, "real world" experiments are simulated and compared with measurements in Section\nbsp{}ref:sec:ustation_experiments. *** Micro-Station Kinematics <> **** Introduction :ignore: -The micro-station consists of 4 stacked positioning stages (Figure ref:fig:ustation_cad_view). +The micro-station consists of 4 stacked positioning stages (Figure\nbsp{}ref:fig:ustation_cad_view). From bottom to top, the stacked stages are the translation stage $D_y$, the tilt stage $R_y$, the rotation stage (Spindle) $R_z$ and the positioning hexapod. Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex. @@ -3288,14 +3288,14 @@ Finally, the motions of all stacked stages are combined, and the sample's motion The /pose/ of a solid body relative to a specific frame can be described by six independent parameters. Three parameters are typically used to describe its position, and three other parameters describe its orientation. -The /position/ of a point $P$ with respect to a frame $\{A\}$ can be described by a $3 \times 1$ position vector eqref:eq:ustation_position. +The /position/ of a point $P$ with respect to a frame $\{A\}$ can be described by a $3 \times 1$ position vector\nbsp{}eqref:eq:ustation_position. The name of the frame is usually added as a leading superscript: ${}^AP$ which reads as vector $P$ in frame $\{A\}$. \begin{equation}\label{eq:ustation_position} {}^AP = \begin{bmatrix} P_x\\ P_y\\ P_z \end{bmatrix} \end{equation} -A pure translation of a solid body (i.e., of a frame $\{B\}$ attached to the solid body) can be described by the position ${}^AP_{O_B}$ as shown in Figure ref:fig:ustation_translation. +A pure translation of a solid body (i.e., of a frame $\{B\}$ attached to the solid body) can be described by the position ${}^AP_{O_B}$ as shown in Figure\nbsp{}ref:fig:ustation_translation. #+name: fig:ustation_transformation_schematics #+caption: Rigid body motion representation. (\subref{fig:ustation_translation}) pure translation. (\subref{fig:ustation_rotation}) pure rotation. (\subref{fig:ustation_transformation}) combined rotation and translation. @@ -3325,20 +3325,20 @@ The /orientation/ of a rigid body is the same at all its points (by definition). Hence, the orientation of a rigid body can be viewed as that of a moving frame attached to the rigid body. It can be represented in several different ways: the rotation matrix, the screw axis representation, and the Euler angles are common descriptions. -The rotation matrix ${}^A\bm{R}_B$ is a $3 \times 3$ matrix containing the Cartesian unit vectors $[{}^A\hat{\bm{x}}_B,\ {}^A\hat{\bm{y}}_B,\ {}^A\hat{\bm{z}}_B]$ of frame $\{\bm{B}\}$ represented in frame $\{\bm{A}\}$ eqref:eq:ustation_rotation_matrix. +The rotation matrix ${}^A\bm{R}_B$ is a $3 \times 3$ matrix containing the Cartesian unit vectors $[{}^A\hat{\bm{x}}_B,\ {}^A\hat{\bm{y}}_B,\ {}^A\hat{\bm{z}}_B]$ of frame $\{\bm{B}\}$ represented in frame $\{\bm{A}\}$\nbsp{}eqref:eq:ustation_rotation_matrix. \begin{equation}\label{eq:ustation_rotation_matrix} {}^A\bm{R}_B = \left[ {}^A\hat{\bm{x}}_B | {}^A\hat{\bm{y}}_B | {}^A\hat{\bm{z}}_B \right] \end{equation} -Consider a pure rotation of a rigid body ($\{\bm{A}\}$ and $\{\bm{B}\}$ are coincident at their origins, as shown in Figure ref:fig:ustation_rotation). -The rotation matrix can be used to express the coordinates of a point $P$ in a fixed frame $\{A\}$ (i.e. ${}^AP$) from its coordinate in the moving frame $\{B\}$ using Equation eqref:eq:ustation_rotation. +Consider a pure rotation of a rigid body ($\{\bm{A}\}$ and $\{\bm{B}\}$ are coincident at their origins, as shown in Figure\nbsp{}ref:fig:ustation_rotation). +The rotation matrix can be used to express the coordinates of a point $P$ in a fixed frame $\{A\}$ (i.e. ${}^AP$) from its coordinate in the moving frame $\{B\}$ using Equation\nbsp{}eqref:eq:ustation_rotation. \begin{equation} \label{eq:ustation_rotation} {}^AP = {}^A\bm{R}_B {}^BP \end{equation} -For rotations along $x$, $y$ or $z$ axis, the formulas of the corresponding rotation matrices are given in Equation eqref:eq:ustation_rotation_matrices_xyz. +For rotations along $x$, $y$ or $z$ axis, the formulas of the corresponding rotation matrices are given in Equation\nbsp{}eqref:eq:ustation_rotation_matrices_xyz. \begin{subequations}\label{eq:ustation_rotation_matrices_xyz} \begin{align} @@ -3349,13 +3349,13 @@ For rotations along $x$, $y$ or $z$ axis, the formulas of the corresponding rota \end{subequations} Sometimes, it is useful to express a rotation as a combination of three rotations described by $\bm{R}_x$, $\bm{R}_y$ and $\bm{R}_z$. -The order of rotation is very important[fn:ustation_5], therefore, in this study, rotations are expressed as three successive rotations about the coordinate axes of the moving frame eqref:eq:ustation_rotation_combination. +The order of rotation is very important[fn:ustation_5], therefore, in this study, rotations are expressed as three successive rotations about the coordinate axes of the moving frame\nbsp{}eqref:eq:ustation_rotation_combination. \begin{equation}\label{eq:ustation_rotation_combination} {}^A\bm{R}_B(\alpha, \beta, \gamma) = \bm{R}_u(\alpha) \bm{R}_v(\beta) \bm{R}_c(\gamma) \end{equation} -Such rotation can be parameterized by three Euler angles $(\alpha,\ \beta,\ \gamma)$, which can be computed from a given rotation matrix using equations eqref:eq:ustation_euler_angles. +Such rotation can be parameterized by three Euler angles $(\alpha,\ \beta,\ \gamma)$, which can be computed from a given rotation matrix using equations\nbsp{}eqref:eq:ustation_euler_angles. \begin{subequations}\label{eq:ustation_euler_angles} \begin{align} @@ -3373,15 +3373,15 @@ Therefore, the pose of a rigid body can be fully determined by: 1. The position vector of point $O_B$ with respect to frame $\{A\}$ which is denoted ${}^AP_{O_B}$ 2. The orientation of the rigid body, or the moving frame $\{B\}$ attached to it with respect to the fixed frame $\{A\}$, that is represented by ${}^A\bm{R}_B$. -The position of any point $P$ of the rigid body with respect to the fixed frame $\{\bm{A}\}$, which is denoted ${}^A\bm{P}$ may be determined thanks to the /Chasles' theorem/, which states that if the pose of a rigid body $\{{}^A\bm{R}_B, {}^AP_{O_B}\}$ is given, then the position of any point $P$ of this rigid body with respect to $\{\bm{A}\}$ is given by Equation eqref:eq:ustation_chasles_therorem. +The position of any point $P$ of the rigid body with respect to the fixed frame $\{\bm{A}\}$, which is denoted ${}^A\bm{P}$ may be determined thanks to the /Chasles' theorem/, which states that if the pose of a rigid body $\{{}^A\bm{R}_B, {}^AP_{O_B}\}$ is given, then the position of any point $P$ of this rigid body with respect to $\{\bm{A}\}$ is given by Equation\nbsp{}eqref:eq:ustation_chasles_therorem. \begin{equation} \label{eq:ustation_chasles_therorem} {}^AP = {}^A\bm{R}_B {}^BP + {}^AP_{O_B} \end{equation} -While equation eqref:eq:ustation_chasles_therorem can describe the motion of a rigid body, it can be written in a more convenient way using $4 \times 4$ homogeneous transformation matrices and $4 \times 1$ homogeneous coordinates. +While equation\nbsp{}eqref:eq:ustation_chasles_therorem can describe the motion of a rigid body, it can be written in a more convenient way using $4 \times 4$ homogeneous transformation matrices and $4 \times 1$ homogeneous coordinates. The homogeneous transformation matrix is composed of the rotation matrix ${}^A\bm{R}_B$ representing the orientation and the position vector ${}^AP_{O_B}$ representing the translation. -It is partitioned as shown in Equation eqref:eq:ustation_homogeneous_transformation_parts. +It is partitioned as shown in Equation\nbsp{}eqref:eq:ustation_homogeneous_transformation_parts. \begin{equation}\label{eq:ustation_homogeneous_transformation_parts} {}^A\bm{T}_B = @@ -3394,7 +3394,7 @@ It is partitioned as shown in Equation eqref:eq:ustation_homogeneous_transformat \end{array} \right] \end{equation} -Then, ${}^AP$ can be computed from ${}^BP$ and the homogeneous transformation matrix using eqref:eq:ustation_homogeneous_transformation. +Then, ${}^AP$ can be computed from ${}^BP$ and the homogeneous transformation matrix using\nbsp{}eqref:eq:ustation_homogeneous_transformation. \begin{equation}\label{eq:ustation_homogeneous_transformation} \left[ \begin{array}{c} \\ {}^AP \\ \cr \hline 1 \end{array} \right] @@ -3410,7 +3410,7 @@ Then, ${}^AP$ can be computed from ${}^BP$ and the homogeneous transformation ma \end{equation} One key advantage of homogeneous transformation is that it can easily be generalized for consecutive transformations. -Let us consider the motion of a rigid body described at three locations (Figure ref:fig:ustation_combined_transformation). +Let us consider the motion of a rigid body described at three locations (Figure\nbsp{}ref:fig:ustation_combined_transformation). 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 @@ -3421,14 +3421,14 @@ Furthermore, suppose the position vector of a point $P$ of the rigid body is giv Since the locations of the rigid body are known relative to each other, ${}^CP$ can be transformed to ${}^BP$ using ${}^B\bm{T}_C$ using ${}^BP = {}^B\bm{T}_C {}^CP$. Similarly, ${}^BP$ can be transformed into ${}^AP$ using ${}^AP = {}^A\bm{T}_B {}^BP$. -Combining the two relations, Equation eqref:eq:ustation_consecutive_transformations is obtained. +Combining the two relations, Equation\nbsp{}eqref:eq:ustation_consecutive_transformations is obtained. This shows that combining multiple transformations is equivalent as to compute $4 \times 4$ matrix multiplications. \begin{equation}\label{eq:ustation_consecutive_transformations} {}^AP = \underbrace{{}^A\bm{T}_B {}^B\bm{T}_C}_{{}^A\bm{T}_C} {}^CP \end{equation} -Another key advantage of homogeneous transformation is the easy inverse transformation, which can be computed using Equation eqref:eq:ustation_inverse_homogeneous_transformation. +Another key advantage of homogeneous transformation is the easy inverse transformation, which can be computed using Equation\nbsp{}eqref:eq:ustation_inverse_homogeneous_transformation. \begin{equation}\label{eq:ustation_inverse_homogeneous_transformation} {}^B\bm{T}_A = {}^A\bm{T}_B^{-1} = @@ -3446,7 +3446,7 @@ Another key advantage of homogeneous transformation is the easy inverse transfor Each stage is described by two frames; one is attached to the fixed platform $\{A\}$ while the other is fixed to the mobile platform $\{B\}$. At "rest" position, the two have the same pose and coincide with the point of interest ($O_A = O_B$). -An example of the tilt stage is shown in Figure ref:fig:ustation_stage_motion. +An example of the tilt stage is shown in Figure\nbsp{}ref:fig:ustation_stage_motion. The mobile frame of the translation stage is equal to the fixed frame of the tilt stage: $\{B_{D_y}\} = \{A_{R_y}\}$. Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the spindle: $\{B_{R_y}\} = \{A_{R_z}\}$. @@ -3454,8 +3454,8 @@ Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the #+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. [[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 ref:ssec:ustation_kinematics. -As any motion stage induces parasitic motion in all 6 DoF, the transformation matrix representing its induced motion can be written as in eqref:eq:ustation_translation_stage_errors. +The motion induced by a positioning stage can be described by a homogeneous transformation matrix from frame $\{A\}$ to frame $\{B\}$ as explain in Section\nbsp{}ref:ssec:ustation_kinematics. +As any motion stage induces parasitic motion in all 6 DoF, the transformation matrix representing its induced motion can be written as in\nbsp{}eqref:eq:ustation_translation_stage_errors. \begin{equation}\label{eq:ustation_translation_stage_errors} {}^A\bm{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) = @@ -3468,7 +3468,7 @@ As any motion stage induces parasitic motion in all 6 DoF, the transformation ma \end{array} \right] \end{equation} -The homogeneous transformation matrix corresponding to the micro-station $\bm{T}_{\mu\text{-station}}$ is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation eqref:eq:ustation_transformation_station. +The homogeneous transformation matrix corresponding to the micro-station $\bm{T}_{\mu\text{-station}}$ is simply equal to the matrix multiplication of the homogeneous transformation matrices of the individual stages as shown in Equation\nbsp{}eqref:eq:ustation_transformation_station. \begin{equation}\label{eq:ustation_transformation_station} \bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}} @@ -3477,7 +3477,7 @@ The homogeneous transformation matrix corresponding to the micro-station $\bm{T} $\bm{T}_{\mu\text{-station}}$ represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite. If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), $\bm{T}_{\mu\text{-station}}$ then represents the pose setpoint of the sample with respect to the granite. -The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation eqref:eq:ustation_transformation_matrices_stages. +The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation\nbsp{}eqref:eq:ustation_transformation_matrices_stages. The setpoints are $D_y$ for the translation stage, $\theta_y$ for the tilt-stage, $\theta_z$ for the spindle, $[D_{\mu x},\ D_{\mu y}, D_{\mu z}]$ for the micro-hexapod translations and $[\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]$ for the micro-hexapod rotations. \begin{equation}\label{eq:ustation_transformation_matrices_stages} @@ -3517,13 +3517,13 @@ The setpoints are $D_y$ for the translation stage, $\theta_y$ for the tilt-stage In this section, the multi-body model of the micro-station is presented. Such model consists of several rigid bodies connected by springs and dampers. -The inertia of the solid bodies and the stiffness properties of the guiding mechanisms were first estimated based on the CAD model and data-sheets (Section ref:ssec:ustation_model_simscape). +The inertia of the solid bodies and the stiffness properties of the guiding mechanisms were first estimated based on the CAD model and data-sheets (Section\nbsp{}ref:ssec:ustation_model_simscape). -The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section ref:ssec:ustation_model_comp_dynamics). +The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section\nbsp{}ref:ssec:ustation_model_comp_dynamics). # TODO - Add reference to uniaxial model As the dynamics of the nano-hexapod is impacted by the micro-station compliance, the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. -To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section ref:ssec:ustation_model_compliance). +To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_model_compliance). **** Multi-Body Model <> @@ -3541,10 +3541,10 @@ External forces can be used to model disturbances, and "sensors" can be used to [[file:figs/ustation_simscape_stage_example.png]] Therefore, the micro-station is modeled by several solid bodies connected by joints. -A typical stage (here the tilt-stage) is modeled as shown in Figure ref:fig:ustation_simscape_stage_example where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoF joint. +A typical stage (here the tilt-stage) is modeled as shown in Figure\nbsp{}ref:fig:ustation_simscape_stage_example where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoF joint. One DoF of the 6-DoF joint is "imposed" by a setpoint (i.e. modeled as infinitely stiff), while the other 5 are each modeled by a spring and damper. Additional forces can be used to model disturbances induced by the stage motion. -The obtained 3D representation of the multi-body model is shown in Figure ref:fig:ustation_simscape_model. +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 @@ -3559,10 +3559,10 @@ Finally, the positioning hexapod has 6-DoF. The total number of "free" degrees of freedom is 27, so the model has 54 states. The springs and dampers values were first estimated from the joint/stage specifications and were later fined-tuned based on the measurements. -The spring values are summarized in Table ref:tab:ustation_6dof_stiffness_values. +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 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 \linewidth :align Xcccccc #+attr_latex: :center t :booktabs t | *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | @@ -3579,7 +3579,7 @@ The spring values are summarized in Table ref:tab:ustation_6dof_stiffness_values The dynamics of the micro-station was measured by placing accelerometers on each stage and by impacting the translation stage with an instrumented hammer in three directions. The obtained FRFs were then projected at the CoM of each stage. -To gain a first insight into the accuracy of the obtained model, the FRFs from the hammer impacts to the acceleration of each stage were extracted from the multi-body model and compared with the measurements in Figure ref:fig:ustation_comp_com_response. +To gain a first insight into the accuracy of the obtained model, the FRFs from the hammer impacts to the acceleration of each stage were extracted from the multi-body model and compared with the measurements in Figure\nbsp{}ref:fig:ustation_comp_com_response. Even though there is some similarity between the model and the measurements (similar overall shapes and amplitudes), it is clear that the multi-body model does not accurately represent the complex micro-station dynamics. Tuning the numerous model parameters to better match the measurements is a highly non-linear optimization problem that is difficult to solve in practice. @@ -3616,7 +3616,7 @@ As discussed in the previous section, the dynamics of the micro-station is compl When considering the NASS, the most important dynamical characteristics of the micro-station is its compliance, as it can affect the plant dynamics. Therefore, the adopted strategy is to accurately model the micro-station compliance. -The micro-station compliance was experimentally measured using the setup illustrated in Figure ref:fig:ustation_compliance_meas. +The micro-station compliance was experimentally measured using the setup illustrated in Figure\nbsp{}ref:fig:ustation_compliance_meas. Four 3-axis accelerometers were fixed to the micro-hexapod top platform. The micro-hexapod top platform was impacted at 10 different points. For each impact position, 10 impacts were performed to average and improve the data quality. @@ -3625,7 +3625,7 @@ For each impact position, 10 impacts were performed to average and improve the d #+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). [[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 frame $\{\mathcal{X}\}$ $a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]$, a Jacobian matrix $\bm{J}_a$ is written based on the positions and orientations of the accelerometers eqref:eq:ustation_compliance_acc_jacobian. +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 frame $\{\mathcal{X}\}$ $a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]$, a Jacobian matrix $\bm{J}_a$ is written based on the positions and orientations of the accelerometers\nbsp{}eqref:eq:ustation_compliance_acc_jacobian. \begin{equation}\label{eq:ustation_compliance_acc_jacobian} \bm{J}_a = \begin{bmatrix} @@ -3644,13 +3644,13 @@ To convert the 12 acceleration signals $a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z \end{bmatrix} \end{equation} -Then, the acceleration in the cartesian frame can be computed using eqref:eq:ustation_compute_cart_acc. +Then, the acceleration in the cartesian frame can be computed using\nbsp{}eqref:eq:ustation_compute_cart_acc. \begin{equation}\label{eq:ustation_compute_cart_acc} a_{\mathcal{X}} = \bm{J}_a^\dagger \cdot a_{\mathcal{L}} \end{equation} -Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is computed eqref:eq:ustation_compliance_force_jacobian and used to convert the individual hammer forces $F_{\mathcal{L}}$ to force and torques $F_{\mathcal{X}}$ applied at the center of the micro-hexapod top plate (defined by frame $\{\mathcal{X}\}$ in Figure ref:fig:ustation_compliance_meas). +Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is computed\nbsp{}eqref:eq:ustation_compliance_force_jacobian and used to convert the individual hammer forces $F_{\mathcal{L}}$ to force and torques $F_{\mathcal{X}}$ applied at the center of the micro-hexapod top plate (defined by frame $\{\mathcal{X}\}$ in Figure\nbsp{}ref:fig:ustation_compliance_meas). \begin{equation}\label{eq:ustation_compliance_force_jacobian} \bm{J}_F = \begin{bmatrix} @@ -3667,7 +3667,7 @@ Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is \end{bmatrix} \end{equation} -The equivalent forces and torques applied at center of $\{\mathcal{X}\}$ are then computed using eqref:eq:ustation_compute_cart_force. +The equivalent forces and torques applied at center of $\{\mathcal{X}\}$ are then computed using\nbsp{}eqref:eq:ustation_compute_cart_force. \begin{equation}\label{eq:ustation_compute_cart_force} F_{\mathcal{X}} = \bm{J}_F^{\intercal} \cdot F_{\mathcal{L}} @@ -3677,7 +3677,7 @@ Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 ac These FRFs were then used for comparison with the multi-body model. The compliance of the micro-station multi-body model was extracted by computing the transfer function from forces/torques applied on the hexapod's top platform to the "absolute" motion of the top platform. -These results are compared with the measurements in Figure ref:fig:ustation_frf_compliance_model. +These results are compared with the measurements in Figure\nbsp{}ref:fig:ustation_frf_compliance_model. Considering the complexity of the micro-station compliance dynamics, the model compliance matches sufficiently well for the current application. #+name: fig:ustation_frf_compliance_model @@ -3709,10 +3709,10 @@ Such disturbances include ground motions and vibrations induce by scanning the t In the multi-body model, stage vibrations are modeled as internal forces applied in the stage joint. In practice, disturbance forces cannot be directly measured. -Instead, the vibrations of the micro-station's top platform induced by the disturbances were measured (Section ref:ssec:ustation_disturbances_meas). +Instead, the vibrations of the micro-station's top platform induced by the disturbances were measured (Section\nbsp{}ref:ssec:ustation_disturbances_meas). -To estimate the equivalent disturbance force that induces such vibration, the transfer functions from disturbance sources (i.e. forces applied in the stages' joint) to the displacements of the micro-station's top platform with respect to the granite are extracted from the multi-body model (Section ref:ssec:ustation_disturbances_sensitivity). -Finally, the obtained disturbance sources are compared in Section ref:ssec:ustation_disturbances_results. +To estimate the equivalent disturbance force that induces such vibration, the transfer functions from disturbance sources (i.e. forces applied in the stages' joint) to the displacements of the micro-station's top platform with respect to the granite are extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_disturbances_sensitivity). +Finally, the obtained disturbance sources are compared in Section\nbsp{}ref:ssec:ustation_disturbances_results. **** Disturbance measurements <> @@ -3725,9 +3725,9 @@ Therefore, from a control perspective, they are not important. ***** Ground Motion -The ground motion was measured by using a sensitive 3-axis geophone shown in Figure ref:fig:ustation_geophone_picture placed on the ground. +The ground motion was measured by using a sensitive 3-axis geophone shown in Figure\nbsp{}ref:fig:ustation_geophone_picture placed on the ground. The generated voltages were recorded with a high resolution DAC, and converted to displacement using the Geophone sensitivity transfer function. -The obtained ground motion displacement is shown in Figure ref:fig:ustation_ground_disturbance. +The obtained ground motion displacement is shown in Figure\nbsp{}ref:fig:ustation_ground_disturbance. #+attr_latex: :options [b]{0.54\linewidth} #+begin_minipage @@ -3747,7 +3747,7 @@ The obtained ground motion displacement is shown in Figure ref:fig:ustation_grou ***** Ty Stage -To measure the positioning errors of the translation stage, the setup shown in Figure ref:fig:ustation_errors_ty_setup is used. +To measure the positioning errors of the translation stage, the setup shown in Figure\nbsp{}ref:fig:ustation_errors_ty_setup is used. A special optical element (called a "straightness interferometer"[fn:ustation_9]) is fixed on top of the micro-station, while a laser source[fn:ustation_10] and a straightness reflector are fixed on the ground. 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. @@ -3756,9 +3756,9 @@ A similar setup was used to measure the horizontal deviation (i.e. in the $x$ di [[file:figs/ustation_errors_ty_setup.png]] Six scans were performed between $-4.5\,mm$ and $4.5\,mm$. -The results for each individual scan are shown in Figure ref:fig:ustation_errors_dy_vertical. +The results for each individual scan are shown in Figure\nbsp{}ref:fig:ustation_errors_dy_vertical. The measurement axis may not be perfectly aligned with the translation stage axis; this, a linear fit is removed from the measurement. -The remaining vertical displacement is shown in Figure ref:fig:ustation_errors_dy_vertical_remove_mean. +The remaining vertical displacement is shown in Figure\nbsp{}ref:fig:ustation_errors_dy_vertical_remove_mean. A vertical error of $\pm300\,nm$ induced by the translation stage is expected. Similar result is obtained for the $x$ lateral direction. @@ -3782,9 +3782,9 @@ Similar result is obtained for the $x$ lateral direction. ***** Spindle -To measure the positioning errors induced by the Spindle, a "Spindle error analyzer"[fn:ustation_7] is used as shown in Figure ref:fig:ustation_rz_meas_lion_setup. +To measure the positioning errors induced by the Spindle, a "Spindle error analyzer"[fn:ustation_7] is used as shown in Figure\nbsp{}ref:fig:ustation_rz_meas_lion_setup. A specific target is fixed on top of the micro-station, which consists of two sphere with 1 inch diameter precisely aligned with the spindle rotation axis. -Five capacitive sensors[fn:ustation_8] are pointing at the two spheres, as shown in Figure ref:fig:ustation_rz_meas_lion_zoom. +Five capacitive sensors[fn:ustation_8] are pointing at the two spheres, as shown in Figure\nbsp{}ref:fig:ustation_rz_meas_lion_zoom. 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 @@ -3806,12 +3806,12 @@ From the 5 measured displacements $[d_1,\,d_2,\,d_3,\,d_4,\,d_5]$, the translati #+end_figure A measurement was performed during a constant rotational velocity of the spindle of 60rpm and during 10 turns. -The obtained results are shown in Figure ref:fig:ustation_errors_spindle. -A large fraction of the radial (Figure ref:fig:ustation_errors_spindle_radial) and tilt (Figure ref:fig:ustation_errors_spindle_tilt) errors is linked to the fact that the two spheres are not perfectly aligned with the rotation axis of the Spindle. +The obtained results are shown in Figure\nbsp{}ref:fig:ustation_errors_spindle. +A large fraction of the radial (Figure\nbsp{}ref:fig:ustation_errors_spindle_radial) and tilt (Figure\nbsp{}ref:fig:ustation_errors_spindle_tilt) errors is linked to the fact that the two spheres are not perfectly aligned with the rotation axis of the Spindle. This is displayed by the dashed circle. After removing the best circular fit from the data, the vibrations induced by the Spindle may be viewed as stochastic disturbances. However, some misalignment between the "point-of-interest" of the sample and the rotation axis will be considered because the alignment is not perfect in practice. -The vertical motion induced by scanning the spindle is in the order of $\pm 30\,nm$ (Figure ref:fig:ustation_errors_spindle_axial). +The vertical motion induced by scanning the spindle is in the order of $\pm 30\,nm$ (Figure\nbsp{}ref:fig:ustation_errors_spindle_axial). #+name: fig:ustation_errors_spindle #+caption: Measurement of the radial (\subref{fig:ustation_errors_spindle_radial}), axial (\subref{fig:ustation_errors_spindle_axial}) and tilt (\subref{fig:ustation_errors_spindle_tilt}) Spindle errors during a 60rpm spindle rotation. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis. @@ -3840,9 +3840,9 @@ The vertical motion induced by scanning the spindle is in the order of $\pm 30\, **** Sensitivity to disturbances <> -To compute the disturbance source (i.e. forces) that induced the measured vibrations in Section ref:ssec:ustation_disturbances_meas, the transfer function from the disturbance sources to the stage vibration (i.e. the "sensitivity to disturbances") needs to be estimated. -This is achieved using the multi-body model presented in Section ref:sec:ustation_modeling. -The obtained transfer functions are shown in Figure ref:fig:ustation_model_sensitivity. +To compute the disturbance source (i.e. forces) that induced the measured vibrations in Section\nbsp{}ref:ssec:ustation_disturbances_meas, the transfer function from the disturbance sources to the stage vibration (i.e. the "sensitivity to disturbances") needs to be estimated. +This is achieved using the multi-body model presented in Section\nbsp{}ref:sec:ustation_modeling. +The obtained transfer functions are shown in Figure\nbsp{}ref:fig:ustation_model_sensitivity. #+name: fig:ustation_model_sensitivity #+caption: Extracted transfer functions from disturbances to relative motion between the micro-station's top platform and the granite. The considered disturbances are the ground motion (\subref{fig:ustation_model_sensitivity_ground_motion}), the translation stage vibrations (\subref{fig:ustation_model_sensitivity_ty}), and the spindle vibrations (\subref{fig:ustation_model_sensitivity_rz}). @@ -3871,8 +3871,8 @@ The obtained transfer functions are shown in Figure ref:fig:ustation_model_sensi **** Obtained disturbance sources <> -From the measured effect of disturbances in Section ref:ssec:ustation_disturbances_meas and the sensitivity to disturbances extracted from the multi-body model in Section ref:ssec:ustation_disturbances_sensitivity, the power spectral density of the disturbance sources (i.e. forces applied in the stage's joint) can be estimated. -The obtained power spectral density of the disturbances are shown in Figure ref:fig:ustation_dist_sources. +From the measured effect of disturbances in Section\nbsp{}ref:ssec:ustation_disturbances_meas and the sensitivity to disturbances extracted from the multi-body model in Section\nbsp{}ref:ssec:ustation_disturbances_sensitivity, the power spectral density of the disturbance sources (i.e. forces applied in the stage's joint) can be estimated. +The obtained power spectral density of the disturbances are shown in Figure\nbsp{}ref:fig:ustation_dist_sources. #+name: fig:ustation_dist_sources #+caption: Measured spectral density of the micro-station disturbance sources. Ground motion (\subref{fig:ustation_dist_source_ground_motion}), translation stage (\subref{fig:ustation_dist_source_translation_stage}) and spindle (\subref{fig:ustation_dist_source_spindle}). @@ -3898,10 +3898,10 @@ The obtained power spectral density of the disturbances are shown in Figure ref: #+end_subfigure #+end_figure -The disturbances are characterized by their power spectral densities, as shown in Figure ref:fig:ustation_dist_sources. +The disturbances are characterized by their power spectral densities, as shown in Figure\nbsp{}ref:fig:ustation_dist_sources. However, to perform time domain simulations, disturbances must be represented by a time domain signal. -To generate stochastic time-domain signals with a specific power spectral density, the discrete inverse Fourier transform is used, as explained in [[cite:&preumont94_random_vibrat_spect_analy chap. 12.11]]. -Examples of the obtained time-domain disturbance signals are shown in Figure ref:fig:ustation_dist_sources_time. +To generate stochastic time-domain signals with a specific power spectral density, the discrete inverse Fourier transform is used, as explained in\nbsp{}[[cite:&preumont94_random_vibrat_spect_analy chap. 12.11]]. +Examples of the obtained time-domain disturbance signals are shown in Figure\nbsp{}ref:fig:ustation_dist_sources_time. #+name: fig:ustation_dist_sources_time #+caption: Generated time domain disturbance signals. Ground motion (\subref{fig:ustation_dist_source_ground_motion_time}), translation stage (\subref{fig:ustation_dist_source_translation_stage_time}) and spindle (\subref{fig:ustation_dist_source_spindle_time}). @@ -3933,18 +3933,18 @@ Examples of the obtained time-domain disturbance signals are shown in Figure ref To fully validate the micro-station multi-body model, two time-domain simulations corresponding to typical use cases were performed. -First, a tomography experiment (i.e. a constant Spindle rotation) was performed and was compared with experimental measurements (Section ref:sec:ustation_experiments_tomography). -Second, a constant velocity scans with the translation stage was performed and also compared with the experimental data (Section ref:sec:ustation_experiments_ty_scans). +First, a tomography experiment (i.e. a constant Spindle rotation) was performed and was compared with experimental measurements (Section\nbsp{}ref:sec:ustation_experiments_tomography). +Second, a constant velocity scans with the translation stage was performed and also compared with the experimental data (Section\nbsp{}ref:sec:ustation_experiments_ty_scans). **** Tomography Experiment <> To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of 60rpm. -Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section ref:sec:ustation_disturbances. +Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section\nbsp{}ref:sec:ustation_disturbances. A radial offset of $\approx 1\,\mu m$ between the "point-of-interest" and the spindle's rotation axis is introduced to represent what is experimentally observed. During the 10 second simulation (i.e. 10 spindle turns), the position of the "point-of-interest" with respect to the granite was recorded. -Results are shown in Figure ref:fig:ustation_errors_model_spindle. -A good correlation with the measurements is observed both for radial errors (Figure ref:fig:ustation_errors_model_spindle_radial) and axial errors (Figure ref:fig:ustation_errors_model_spindle_axial). +Results are shown in Figure\nbsp{}ref:fig:ustation_errors_model_spindle. +A good correlation with the measurements is observed both for radial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_radial) and axial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_axial). #+name: fig:ustation_errors_model_spindle #+caption: Simulation results for a tomography experiment at constant velocity of 60rpm. The comparison is made with measurements for both radial (\subref{fig:ustation_errors_model_spindle_radial}) and axial errors (\subref{fig:ustation_errors_model_spindle_axial}). @@ -3970,7 +3970,7 @@ A good correlation with the measurements is observed both for radial errors (Fig A second experiment was performed in which the translation stage was scanned at constant velocity. The translation stage setpoint is configured to have a "triangular" shape with stroke of $\pm 4.5\, mm$. Both ground motion and translation stage vibrations were included in the simulation. -Similar to what was performed for the tomography simulation, the PoI position with respect to the granite was recorded and compared with the experimental measurements in Figure ref:fig:ustation_errors_model_dy_vertical. +Similar to what was performed for the tomography simulation, the PoI position with respect to the granite was recorded and compared with the experimental measurements in Figure\nbsp{}ref:fig:ustation_errors_model_dy_vertical. A similar error amplitude was observed, thus indicating that the multi-body model with the included disturbances accurately represented the micro-station behavior in typical scientific experiments. #+name: fig:ustation_errors_model_dy_vertical @@ -3983,10 +3983,10 @@ A similar error amplitude was observed, thus indicating that the multi-body mode In this study, a multi-body model of the micro-station was developed. It was difficult to match the measured dynamics obtained from the modal analysis of the micro-station. However, the most important dynamical characteristic to be modeled is the compliance, as it affects the dynamics of the NASS. -After tuning the model parameters, a good match with the measured compliance was obtained (Figure ref:fig:ustation_frf_compliance_model). +After tuning the model parameters, a good match with the measured compliance was obtained (Figure\nbsp{}ref:fig:ustation_frf_compliance_model). The disturbances affecting the sample position should also be well modeled. -After experimentally estimating the disturbances (Section ref:sec:ustation_disturbances), the multi-body model was finally validated by performing a tomography simulation (Figure ref:fig:ustation_errors_model_spindle) as well as a simulation in which the translation stage was scanned (Figure ref:fig:ustation_errors_model_dy_vertical). +After experimentally estimating the disturbances (Section\nbsp{}ref:sec:ustation_disturbances), the multi-body model was finally validated by performing a tomography simulation (Figure\nbsp{}ref:fig:ustation_errors_model_spindle) as well as a simulation in which the translation stage was scanned (Figure\nbsp{}ref:fig:ustation_errors_model_dy_vertical). ** Nano Hexapod - Multi Body Model <> @@ -3994,14 +3994,14 @@ After experimentally estimating the disturbances (Section ref:sec:ustation_distu Building upon the validated multi-body model of the micro-station presented in previous sections, this section focuses on the development and integration of an active vibration platform model. -A review of existing active vibration platforms is given in Section ref:sec:nhexa_platform_review, leading to the selection of the Stewart platform architecture. -This parallel manipulator architecture, described in Section ref:sec:nhexa_stewart_platform, requires specialized analytical tools for kinematic analysis. +A review of existing active vibration platforms is given in Section\nbsp{}ref:sec:nhexa_platform_review, leading to the selection of the Stewart platform architecture. +This parallel manipulator architecture, described in Section\nbsp{}ref:sec:nhexa_stewart_platform, requires specialized analytical tools for kinematic analysis. However, the complexity of its dynamic behavior poses significant challenges for purely analytical approaches. -Consequently, a multi-body modeling approach was adopted (Section ref:sec:nhexa_model), facilitating seamless integration with the existing micro-station model. +Consequently, a multi-body modeling approach was adopted (Section\nbsp{}ref:sec:nhexa_model), facilitating seamless integration with the existing micro-station model. The control of the Stewart platform introduces additional complexity due to its multi-input multi-output (MIMO) nature. -Section ref:sec:nhexa_control explores how the High Authority Control/Low Authority Control (HAC-LAC) strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform. +Section\nbsp{}ref:sec:nhexa_control explores how the High Authority Control/Low Authority Control (HAC-LAC) strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform. This adaptation requires fundamental decisions regarding both the control architecture (centralized versus decentralized) and the control frame (Cartesian versus strut space). Through careful analysis of system interactions and plant characteristics in different frames, a control architecture combining decentralized Integral Force Feedback for active damping with a centralized high authority controller for positioning was developed, with both controllers implemented in the frame of the struts. @@ -4014,12 +4014,12 @@ These models were chosen for their ease of analysis, and despite their simplicit However, the development of the Nano Active Stabilization System (NASS) now requires the use of a more accurate model that will be integrated with the multi-body representation of the micro-station. To develop this model, the architecture of the active platform must first be determined. -The selection of an appropriate architecture begins with a review of existing positioning stages that incorporate active platforms similar to NASS (Section ref:ssec:nhexa_sample_stages). +The selection of an appropriate architecture begins with a review of existing positioning stages that incorporate active platforms similar to NASS (Section\nbsp{}ref:ssec:nhexa_sample_stages). This review reveals two distinctive features of the NASS that set it apart from existing systems: the fact that the active platform is continuously rotating and its requirement to accommodate variable payload masses. In existing systems, the sample mass is typically negligible compared to the stage mass, whereas in NASS, the sample mass significantly influences the system's dynamic behavior. These distinctive requirements drive the selection of the active platform architecture. -In Section ref:ssec:nhexa_active_platforms, different active platform configurations, including serial and parallel configurations, are evaluated, ultimately leading to the choice of a Stewart platform architecture. +In Section\nbsp{}ref:ssec:nhexa_active_platforms, different active platform configurations, including serial and parallel configurations, are evaluated, ultimately leading to the choice of a Stewart platform architecture. **** Sample Stages with Active Control <> @@ -4029,11 +4029,11 @@ To overcome this limitation, external metrology systems have been implemented to A review of existing sample stages with active vibration control reveals various approaches to implementing such feedback systems. In many cases, sample position control is limited to translational degrees of freedom. -At NSLS-II, for instance, a system capable of $100\,\mu m$ stroke has been developed for payloads up to 500g, utilizing interferometric measurements for position feedback (Figure ref:fig:nhexa_stages_nazaretski). -Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately 100 Hz (Figure ref:fig:nhexa_stages_sapoti). +At NSLS-II, for instance, a system capable of $100\,\mu m$ stroke has been developed for payloads up to 500g, utilizing interferometric measurements for position feedback (Figure\nbsp{}ref:fig:nhexa_stages_nazaretski). +Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately 100 Hz (Figure\nbsp{}ref:fig:nhexa_stages_sapoti). #+name: fig:nhexa_stages_translations -#+caption: Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope [[cite:&nazaretski15_pushin_limit]] at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline [[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} @@ -4051,13 +4051,13 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac #+end_figure The integration of $R_z$ rotational capability, which is necessary for tomography experiments, introduces additional complexity. -At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section ref:sec:nhexa_stewart_platform) using piezoelectric actuators has been positioned below the spindle (Figure ref:fig:nhexa_stages_villar). +At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section\nbsp{}ref:sec:nhexa_stewart_platform) using piezoelectric actuators has been positioned below the spindle (Figure\nbsp{}ref:fig:nhexa_stages_villar). While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to $50\,\mu m$ due to the inherent constraints of piezoelectric actuators. -In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\mu m$ stroke (Figure ref:fig:nhexa_stages_schroer). +In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\mu m$ stroke (Figure\nbsp{}ref:fig:nhexa_stages_schroer). 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 [[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] at the ESRF (\subref{fig:nhexa_stages_villar}). PtyNAMi microscope [[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} @@ -4074,7 +4074,7 @@ However, attempts to implement real-time feedback using YZ external metrology pr #+end_subfigure #+end_figure -Table ref:tab:nhexa_sample_stages provides an overview of existing end-stations that incorporate feedback loops based on online metrology for sample positioning. +Table\nbsp{}ref:tab:nhexa_sample_stages provides an overview of existing end-stations that incorporate feedback loops based on online metrology for sample positioning. Although direct performance comparisons between these systems are challenging due to their varying experimental requirements, scanning velocities, and specific use cases, several distinctive characteristics of the NASS can be identified. #+name: tab:nhexa_sample_stages @@ -4084,40 +4084,40 @@ Although direct performance comparisons between these systems are challenging du | *Stacked Stages* | *Specifications* | *Measured DoFs* | *Bandwidth* | *Reference* | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | light | Interferometers | 3 PID, n/a | APS | -| *XYZ stage (piezo)* | $D_{xyz}: 0.05\,mm$ | $D_{xyz}$ | | [[cite:&nazaretski15_pushin_limit]] | +| *XYZ stage (piezo)* | $D_{xyz}: 0.05\,mm$ | $D_{xyz}$ | |\nbsp{}[[cite:&nazaretski15_pushin_limit]] | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | light | Capacitive sensors | $\approx 10\,\text{Hz}$ | ESRF | | Spindle | $R_z: \pm 90\,\text{deg}$ | $D_{xyz},\ R_{xy}$ | | ID16a | -| *Hexapod (piezo)* | $D_{xyz}: 0.05\,mm$ | | | [[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] | +| *Hexapod (piezo)* | $D_{xyz}: 0.05\,mm$ | | |\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] | | | $R_{xy}: 500\,\mu\text{rad}$ | | | | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | light | Interferometers | n/a | PETRA III | | *XYZ stage (piezo)* | $D_{xyz}: 0.1\,mm$ | $D_{yz}$ | | P06 | -| Spindle | $R_z: 180\,\text{deg}$ | | | [[cite:&schroer17_ptynam;&schropp20_ptynam]] | +| Spindle | $R_z: 180\,\text{deg}$ | | |\nbsp{}[[cite:&schroer17_ptynam;&schropp20_ptynam]] | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | light | Interferometers | PID, n/a | PSI | | Spindle | $R_z: \pm 182\,\text{deg}$ | $D_{yz},\ R_x$ | | OMNY | -| *Tripod (piezo)* | $D_{xyz}: 0.4\,mm$ | | | [[cite:&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage]] | +| *Tripod (piezo)* | $D_{xyz}: 0.4\,mm$ | | |\nbsp{}[[cite:&holler17_omny_pin_versat_sampl_holder;&holler18_omny_tomog_nano_cryo_stage]] | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | light | Interferometers | n/a | Soleil | | (XY stage) | | $D_{xyz},\ R_{xy}$ | | Nanoprobe | -| Spindle | $R_z: 360\,\text{deg}$ | | | [[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&engblom18_nanop_resul]] | +| Spindle | $R_z: 360\,\text{deg}$ | | |\nbsp{}[[cite:&stankevic17_inter_charac_rotat_stages_x_ray_nanot;&engblom18_nanop_resul]] | | *XYZ linear motors* | $D_{xyz}: 0.4\,mm$ | | | | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | up to 0.5kg | Interferometers | n/a | NSLS | | Spindle | $R_z: 360\,\text{deg}$ | $D_{xyz}$ | | SRX | -| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,mm$ | | | [[cite:&nazaretski22_new_kirkp_baez_based_scann]] | +| *XYZ stage (piezo)* | $D_{xyz}: 0.1\,mm$ | | |\nbsp{}[[cite:&nazaretski22_new_kirkp_baez_based_scann]] | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | up to 0.35kg | Interferometers | $\approx 100\,\text{Hz}$ | Diamond, I14 | -| *Parallel XYZ VC* | $D_{xyz}: 3\,mm$ | $D_{xyz}$ | | [[cite:&kelly22_delta_robot_long_travel_nano]] | +| *Parallel XYZ VC* | $D_{xyz}: 3\,mm$ | $D_{xyz}$ | |\nbsp{}[[cite:&kelly22_delta_robot_long_travel_nano]] | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | light | Capacitive sensors | $\approx 100\,\text{Hz}$ | LNLS | | *Parallel XYZ VC* | $D_{xyz}: 3\,mm$ | and interferometers | | CARNAUBA | -| (Spindle) | $R_z: \pm 110 \,\text{deg}$ | $D_{xyz}$ | | [[cite:&geraldes23_sapot_carnaub_sirius_lnls]] | +| (Spindle) | $R_z: \pm 110 \,\text{deg}$ | $D_{xyz}$ | |\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] | |---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------| | Sample | up to 50kg | $D_{xyz},\ R_{xy}$ | | ESRF | | *Active Platform* | | | | ID31 | -| (Micro-Hexapod) | | | | [[cite:&dehaeze18_sampl_stabil_for_tomog_exper;&dehaeze21_mechat_approac_devel_nano_activ_stabil_system]] | +| (Micro-Hexapod) | | | |\nbsp{}[[cite:&dehaeze18_sampl_stabil_for_tomog_exper;&dehaeze21_mechat_approac_devel_nano_activ_stabil_system]] | | Spindle | $R_z: 360\,\text{deg}$ | | | | | Tilt-Stage | $R_y: \pm 3\,\text{deg}$ | | | | | Translation Stage | $D_y: \pm 10\,mm$ | | | | @@ -4152,19 +4152,19 @@ To this purpose, the design includes force sensors for active damping. Compliant mechanisms must also be used to eliminate friction and backlash, which would otherwise compromise the nano-positioning capabilities. Two primary categories of positioning platform architectures are considered: serial and parallel mechanisms. -Serial robots, characterized by open-loop kinematic chains, typically dedicate one actuator per degree of freedom as shown in Figure ref:fig:nhexa_serial_architecture_kenton. +Serial robots, characterized by open-loop kinematic chains, typically dedicate one actuator per degree of freedom as shown in Figure\nbsp{}ref:fig:nhexa_serial_architecture_kenton. While offering large workspaces and high maneuverability, serial mechanisms suffer from several inherent limitations. -These include low structural stiffness, cumulative positioning errors along the kinematic chain, high mass-to-payload ratios due to actuator placement, and limited payload capacity [[cite:&taghirad13_paral]]. -These limitations generally make serial architectures unsuitable for nano-positioning applications, except when handling very light samples, as was used in [[cite:&nazaretski15_pushin_limit]] and shown in Figure ref:fig:nhexa_stages_nazaretski. +These include low structural stiffness, cumulative positioning errors along the kinematic chain, high mass-to-payload ratios due to actuator placement, and limited payload capacity\nbsp{}[[cite:&taghirad13_paral]]. +These limitations generally make serial architectures unsuitable for nano-positioning applications, except when handling very light samples, as was used in\nbsp{}[[cite:&nazaretski15_pushin_limit]] and shown in Figure\nbsp{}ref:fig:nhexa_stages_nazaretski. In contrast, parallel mechanisms, which connect the mobile platform to the fixed base through multiple parallel struts, offer several advantages for precision positioning. -Their closed-loop kinematic structure provides inherently higher structural stiffness, as the platform is simultaneously supported by multiple struts [[cite:&taghirad13_paral]]. +Their closed-loop kinematic structure provides inherently higher structural stiffness, as the platform is simultaneously supported by multiple struts\nbsp{}[[cite:&taghirad13_paral]]. Although parallel mechanisms typically exhibit limited workspace compared to serial architectures, this limitation is not critical for NASS given its modest stroke requirements. Numerous parallel kinematic architectures have been developed cite:dong07_desig_precis_compl_paral_posit to address various positioning requirements, with designs varying based on the desired degrees of freedom and specific application constraints. -Furthermore, hybrid architectures combining both serial and parallel elements have been proposed [[cite:&shen19_dynam_analy_flexur_nanop_stage]], as illustrated in Figure ref:fig:nhexa_serial_parallel_examples, offering potential compromises between the advantages of both approaches. +Furthermore, hybrid architectures combining both serial and parallel elements have been proposed\nbsp{}[[cite:&shen19_dynam_analy_flexur_nanop_stage]], as illustrated in Figure\nbsp{}ref:fig:nhexa_serial_parallel_examples, offering potential compromises between the advantages of both approaches. #+name: fig:nhexa_serial_parallel_examples -#+caption: Examples of an XYZ serial positioning stage [[cite:&kenton12_desig_contr_three_axis_serial]] (\subref{fig:nhexa_serial_architecture_kenton}) and of a 5-DoF hybrid (parallel/serial) positioning platform [[cite:&shen19_dynam_analy_flexur_nanop_stage]] (\subref{fig:nhexa_parallel_architecture_shen}). +#+caption: Examples of an XYZ serial positioning stage\nbsp{}[[cite:&kenton12_desig_contr_three_axis_serial]] (\subref{fig:nhexa_serial_architecture_kenton}) and of a 5-DoF hybrid (parallel/serial) positioning platform\nbsp{}[[cite:&shen19_dynam_analy_flexur_nanop_stage]] (\subref{fig:nhexa_parallel_architecture_shen}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_serial_architecture_kenton} Serial positioning stage} @@ -4183,12 +4183,12 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha After evaluating the different options, the Stewart platform architecture was selected for several reasons. In addition to providing control over all required degrees of freedom, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints. -Stewart platforms have been implemented in a wide variety of configurations, as illustrated in Figure ref:fig:nhexa_stewart_examples, which shows two distinct implementations: one utilizing piezoelectric actuators for nano-positioning applications, and another based on voice coil actuators for vibration isolation. +Stewart platforms have been implemented in a wide variety of configurations, as illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_examples, which shows two distinct implementations: one utilizing piezoelectric actuators for nano-positioning applications, and another based on voice coil actuators for vibration isolation. These examples demonstrate the architecture's versatility in terms of geometry, actuator selection, and scale, all of which can be optimized for specific applications. -Furthermore, the successful implementation of Integral Force Feedback (IFF) control on Stewart platforms has been well documented [[cite:&abu02_stiff_soft_stewar_platf_activ;&hanieh03_activ_stewar;&preumont07_six_axis_singl_stage_activ]], and the extensive body of research on this architecture enables thorough optimization specifically for the NASS. +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}) [[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}) [[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} @@ -4209,7 +4209,7 @@ Furthermore, the successful implementation of Integral Force Feedback (IFF) cont <> **** Introduction :ignore: -The Stewart platform, first introduced by Stewart in 1965 [[cite:&stewart65_platf_with_six_degrees_freed]] for flight simulation applications, represents a significant milestone in parallel manipulator design. +The Stewart platform, first introduced by Stewart in 1965\nbsp{}[[cite:&stewart65_platf_with_six_degrees_freed]] for flight simulation applications, represents a significant milestone in parallel manipulator design. This mechanical architecture has evolved far beyond its original purpose, and has been applied across diverse field, from precision positioning systems to robotic surgery. The fundamental design consists of two platforms connected by six adjustable struts in parallel, creating a fully parallel manipulator capable of six degrees of freedom motion. @@ -4226,13 +4226,13 @@ While Stewart platforms excel in precision and stiffness, they typically exhibit However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of $10\,\mu m$. This section provides a comprehensive analysis of the Stewart platform's properties, focusing on aspects crucial for precision positioning applications. -The analysis encompasses the platform's kinematic relationships (Section ref:ssec:nhexa_stewart_platform_kinematics), the use of the Jacobian matrix (Section ref:ssec:nhexa_stewart_platform_jacobian), static behavior (Section ref:ssec:nhexa_stewart_platform_static), and dynamic characteristics (Section ref:ssec:nhexa_stewart_platform_dynamics). +The analysis encompasses the platform's kinematic relationships (Section\nbsp{}ref:ssec:nhexa_stewart_platform_kinematics), the use of the Jacobian matrix (Section\nbsp{}ref:ssec:nhexa_stewart_platform_jacobian), static behavior (Section\nbsp{}ref:ssec:nhexa_stewart_platform_static), and dynamic characteristics (Section\nbsp{}ref:ssec:nhexa_stewart_platform_dynamics). These theoretical foundations form the basis for subsequent design decisions and control strategies, which will be elaborated in later sections. **** Mechanical Architecture <> -The Stewart platform consists of two rigid platforms connected by six parallel struts (Figure ref:fig:nhexa_stewart_architecture). +The Stewart platform consists of two rigid platforms connected by six parallel struts (Figure\nbsp{}ref:fig:nhexa_stewart_architecture). Each strut is modelled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom[fn:nhexa_1]. @@ -4253,7 +4253,7 @@ For the nano-hexapod, frames $\{A\}$ and $\{B\}$ are chosen to be located at the The location of the joints and the orientation and length of the struts are crucial for subsequent kinematic, static, and dynamic analyses of the Stewart platform. The center of rotation for the joint fixed to the base is noted $\bm{a}_i$, while $\bm{b}_i$ is used for the top platform joints. The struts' orientations are represented by the unit vectors $\hat{\bm{s}}_i$ and their lengths are represented by the scalars $l_i$. -This is summarized in Figure ref:fig:nhexa_stewart_notations. +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 @@ -4264,7 +4264,7 @@ This is summarized in Figure ref:fig:nhexa_stewart_notations. ***** Loop Closure The foundation of the kinematic analysis lies in the geometric constraints imposed by each strut, which can be expressed using loop closure equations. -For each strut $i$ (illustrated in Figure ref:fig:nhexa_stewart_loop_closure), the loop closure equation eqref:eq:nhexa_loop_closure can be written. +For each strut $i$ (illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_loop_closure), the loop closure equation\nbsp{}eqref:eq:nhexa_loop_closure can be written. \begin{equation}\label{eq:nhexa_loop_closure} {}^A\bm{P}_B = {}^A\bm{a}_i + l_i{}^A\hat{\bm{s}}_i - \underbrace{{}^B\bm{b}_i}_{{}^A\bm{R}_B {}^B\bm{b}_i} \quad \text{for } i=1 \text{ to } 6 @@ -4279,8 +4279,8 @@ This equation links the pose[fn:nhexa_2] variables ${}^A\bm{P}$ and ${}^A\bm{R}_ ***** Inverse Kinematics The inverse kinematic problem involves determining the required strut lengths $\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^{\intercal}$ for a desired platform pose $\bm{\mathcal{X}}$ (i.e. position ${}^A\bm{P}$ and orientation ${}^A\bm{R}_B$). -This problem can be solved analytically using the loop closure equations eqref:eq:nhexa_loop_closure. -The obtained strut lengths are given by eqref:eq:nhexa_inverse_kinematics. +This problem can be solved analytically using the loop closure equations\nbsp{}eqref:eq:nhexa_loop_closure. +The obtained strut lengths are given by\nbsp{}eqref:eq:nhexa_inverse_kinematics. \begin{equation}\label{eq:nhexa_inverse_kinematics} l_i = \sqrt{{}^A\bm{P}^{\intercal} {}^A\bm{P} + {}^B\bm{b}_i^{\intercal} {}^B\bm{b}_i + {}^A\bm{a}_i^{\intercal} {}^A\bm{a}_i - 2 {}^A\bm{P}^{\intercal} {}^A\bm{a}_i + 2 {}^A\bm{P}^{\intercal} \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^{\intercal} {}^A\bm{a}_i} @@ -4309,9 +4309,9 @@ While the previously derived kinematic relationships are essential for position ***** Jacobian Computation -As discussed in Section ref:ssec:nhexa_stewart_platform_kinematics, the strut lengths $\bm{\mathcal{L}}$ and the platform pose $\bm{\mathcal{X}}$ are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts. +As discussed in Section\nbsp{}ref:ssec:nhexa_stewart_platform_kinematics, the strut lengths $\bm{\mathcal{L}}$ and the platform pose $\bm{\mathcal{X}}$ are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts. -By taking the time derivative of the position loop close eqref:eq:nhexa_loop_closure, equation eqref:eq:nhexa_loop_closure_velocity is obtained[fn:nhexa_3]. +By taking the time derivative of the position loop close\nbsp{}eqref:eq:nhexa_loop_closure, equation\nbsp{}eqref:eq:nhexa_loop_closure_velocity is obtained[fn:nhexa_3]. \begin{equation}\label{eq:nhexa_loop_closure_velocity} {}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0} @@ -4321,19 +4321,19 @@ Moreover, we have: - ${}^A\dot{\bm{R}}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{R}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{b}_i$ in which ${}^A\bm{\omega}$ denotes the angular velocity of the moving platform expressed in the fixed frame $\{\bm{A}\}$. - $l_i {}^A\dot{\hat{\bm{s}}}_i = l_i \left( {}^A\bm{\omega}_i \times \hat{\bm{s}}_i \right)$ in which ${}^A\bm{\omega}_i$ is the angular velocity of strut $i$ express in fixed frame $\{\bm{A}\}$. -By multiplying both sides by ${}^A\hat{\bm{s}}_i$, eqref:eq:nhexa_loop_closure_velocity_bis is obtained. +By multiplying both sides by ${}^A\hat{\bm{s}}_i$,\nbsp{}eqref:eq:nhexa_loop_closure_velocity_bis is obtained. \begin{equation}\label{eq:nhexa_loop_closure_velocity_bis} {}^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 eqref:eq:nhexa_loop_closure_velocity_bis can be rearranged in matrix form to obtain 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}}}} \end{equation} -The matrix $\bm{J}$ is called the Jacobian matrix and is defined by eqref:eq:nhexa_jacobian, with ${}^A\hat{\bm{s}}_i$ the orientation of the struts expressed in $\{A\}$ and ${}^A\bm{b}_i$ the position of the joints with respect to $O_B$ and express in $\{A\}$. +The matrix $\bm{J}$ is called the Jacobian matrix and is defined by\nbsp{}eqref:eq:nhexa_jacobian, with ${}^A\hat{\bm{s}}_i$ the orientation of the struts expressed in $\{A\}$ and ${}^A\bm{b}_i$ the position of the joints with respect to $O_B$ and express in $\{A\}$. \begin{equation}\label{eq:nhexa_jacobian} \bm{J} = \begin{bmatrix} @@ -4351,13 +4351,13 @@ However, $\bm{J}$ needs to be recomputed for every Stewart platform pose because ***** Approximate solution to the Forward and Inverse Kinematic problems -For small displacements $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^{\intercal}$ around an operating point $\bm{\mathcal{X}}_0$ (for which the Jacobian was computed), the associated joint displacement $\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^{\intercal}$ can be computed using the Jacobian eqref:eq:nhexa_inverse_kinematics_approximate. +For small displacements $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^{\intercal}$ around an operating point $\bm{\mathcal{X}}_0$ (for which the Jacobian was computed), the associated joint displacement $\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^{\intercal}$ can be computed using the Jacobian\nbsp{}eqref:eq:nhexa_inverse_kinematics_approximate. \begin{equation}\label{eq:nhexa_inverse_kinematics_approximate} \boxed{\delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}} \end{equation} -Similarly, for small joint displacements $\delta\bm{\mathcal{L}}$, it is possible to find the induced small displacement of the mobile platform eqref:eq:nhexa_forward_kinematics_approximate. +Similarly, for small joint displacements $\delta\bm{\mathcal{L}}$, it is possible to find the induced small displacement of the mobile platform\nbsp{}eqref:eq:nhexa_forward_kinematics_approximate. \begin{equation}\label{eq:nhexa_forward_kinematics_approximate} \boxed{\delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}}} @@ -4369,10 +4369,10 @@ While this approximation offers limited value for inverse kinematics, which can ***** Range validity of the approximate inverse kinematics The accuracy of the Jacobian-based forward kinematics solution was estimated by a simple analysis. -For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation eqref:eq:nhexa_inverse_kinematics. -These strut lengths are then used with the Jacobian to estimate the platform pose eqref:eq:nhexa_forward_kinematics_approximate, from which the error between the estimated and true poses can be calculated, both in terms of position $\epsilon_D$ and orientation $\epsilon_R$. +For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation\nbsp{}eqref:eq:nhexa_inverse_kinematics. +These strut lengths are then used with the Jacobian to estimate the platform pose\nbsp{}eqref:eq:nhexa_forward_kinematics_approximate, from which the error between the estimated and true poses can be calculated, both in terms of position $\epsilon_D$ and orientation $\epsilon_R$. -For motion strokes from $1\,\mu m$ to $10\,mm$, the errors are estimated for all direction of motion, and the worst case errors are shown in Figure ref:fig:nhexa_forward_kinematics_approximate_errors. +For motion strokes from $1\,\mu m$ to $10\,mm$, the errors are estimated for all direction of motion, and the worst case errors are shown in Figure\nbsp{}ref:fig:nhexa_forward_kinematics_approximate_errors. The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\mu m$ as the size of the Stewart platform is here $\approx 100\,mm$), the Jacobian approximation provides excellent accuracy. Since the maximum required stroke of the nano-hexapod ($\approx 100\,\mu m$) is three orders of magnitude smaller than its overall size ($\approx 100\,mm$), the Jacobian matrix can be considered constant throughout the workspace. @@ -4397,12 +4397,12 @@ Thus, the principle of virtual work can be expressed as: \delta W = \bm{f}^{\intercal} \delta \bm{\mathcal{L}} - \bm{\mathcal{F}}^{\intercal} \delta \bm{\mathcal{X}} = 0 \end{equation} -Using the Jacobian relationship that links virtual displacements eqref:eq:nhexa_inverse_kinematics_approximate, this equation becomes: +Using the Jacobian relationship that links virtual displacements\nbsp{}eqref:eq:nhexa_inverse_kinematics_approximate, this equation becomes: \begin{equation} \left( \bm{f}^{\intercal} \bm{J} - \bm{\mathcal{F}}^{\intercal} \right) \delta \bm{\mathcal{X}} = 0 \end{equation} -Because this equation must hold for any virtual displacement $\delta \bm{\mathcal{X}}$, the force mapping relationships eqref:eq:nhexa_jacobian_forces can be derived. +Because this equation must hold for any virtual displacement $\delta \bm{\mathcal{X}}$, the force mapping relationships\nbsp{}eqref:eq:nhexa_jacobian_forces can be derived. \begin{equation}\label{eq:nhexa_jacobian_forces} \bm{f}^{\intercal} \bm{J} - \bm{\mathcal{F}}^{\intercal} = 0 \quad \Rightarrow \quad \boxed{\bm{\mathcal{F}} = \bm{J}^{\intercal} \bm{f}} \quad \text{and} \quad \boxed{\bm{f} = \bm{J}^{-\intercal} \bm{\mathcal{F}}} @@ -4426,7 +4426,7 @@ These individual relationships can be combined into a matrix form using the diag \bm{f} = \bm{\mathcal{K}} \cdot \delta \bm{\mathcal{L}}, \quad \bm{\mathcal{K}} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] \end{equation} -By applying the force mapping relationships eqref:eq:nhexa_jacobian_forces derived in the previous section and the Jacobian relationship for small displacements eqref:eq:nhexa_forward_kinematics_approximate, the relationship between applied wrench $\bm{\mathcal{F}}$ and resulting platform displacement $\delta \bm{\mathcal{X}}$ is obtained eqref:eq:nhexa_stiffness_matrix. +By applying the force mapping relationships\nbsp{}eqref:eq:nhexa_jacobian_forces derived in the previous section and the Jacobian relationship for small displacements\nbsp{}eqref:eq:nhexa_forward_kinematics_approximate, the relationship between applied wrench $\bm{\mathcal{F}}$ and resulting platform displacement $\delta \bm{\mathcal{X}}$ is obtained\nbsp{}eqref:eq:nhexa_stiffness_matrix. \begin{equation}\label{eq:nhexa_stiffness_matrix} \bm{\mathcal{F}} = \underbrace{\bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J}}_{\bm{K}} \cdot \delta \bm{\mathcal{X}} @@ -4468,19 +4468,19 @@ The primary forces acting on the system are actuator forces $\bm{f}$, elastic fo \Sigma \bm{\mathcal{F}} = \bm{J}^{\intercal} (\bm{f} - \bm{\mathcal{K}} \bm{\mathcal{L}} - s \bm{\mathcal{C}} \bm{\mathcal{L}}), \quad \bm{\mathcal{K}} = \text{diag}(k_1\,\dots\,k_6),\ \bm{\mathcal{C}} = \text{diag}(c_1\,\dots\,c_6) \end{equation} -Combining these forces and using eqref:eq:nhexa_forward_kinematics_approximate yields the complete dynamic equation eqref:eq:nhexa_dynamical_equations. +Combining these forces and using\nbsp{}eqref:eq:nhexa_forward_kinematics_approximate yields the complete dynamic equation\nbsp{}eqref:eq:nhexa_dynamical_equations. \begin{equation}\label{eq:nhexa_dynamical_equations} \bm{M} s^2 \bm{\mathcal{X}} = \bm{\mathcal{F}} - \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J} \bm{\mathcal{X}} - \bm{J}^{\intercal} \bm{\mathcal{C}} \bm{J} s \bm{\mathcal{X}} \end{equation} -The transfer function matrix in the Cartesian frame becomes eqref:eq:nhexa_transfer_function_cart. +The transfer function matrix in the Cartesian frame becomes\nbsp{}eqref:eq:nhexa_transfer_function_cart. \begin{equation}\label{eq:nhexa_transfer_function_cart} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{\intercal} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J} )^{-1} \end{equation} -Through coordinate transformation using the Jacobian matrix, the dynamics in the actuator space is obtained eqref:eq:nhexa_transfer_function_struts. +Through coordinate transformation using the Jacobian matrix, the dynamics in the actuator space is obtained\nbsp{}eqref:eq:nhexa_transfer_function_struts. \begin{equation}\label{eq:nhexa_transfer_function_struts} \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-\intercal} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1} @@ -4488,8 +4488,8 @@ Through coordinate transformation using the Jacobian matrix, the dynamics in the Although this simplified model provides useful insights, real Stewart platforms exhibit more complex behaviors. Several factors can significantly increase the model complexity, such as: -- Strut dynamics, including mass distribution and internal resonances [[cite:&afzali-far16_inert_matrix_hexap_strut_joint_space;&chen04_decoup_contr_flexur_joint_hexap]] -- Joint compliance and friction effects [[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]] +- Strut dynamics, including mass distribution and internal resonances\nbsp{}[[cite:&afzali-far16_inert_matrix_hexap_strut_joint_space;&chen04_decoup_contr_flexur_joint_hexap]] +- Joint compliance and friction effects\nbsp{}[[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]] - Supporting structure dynamics and payload dynamics, which are both very critical for NASS These additional effects render analytical modeling impractical for complete system analysis. @@ -4517,19 +4517,19 @@ To overcome these limitations, a flexible multi-body approach was developed that Through this multi-body modeling approach, each component model (including joints, actuators, and sensors) can be progressively refined. The analysis is structured as follows. -First, the multi-body model is developed, and the geometric parameters, inertial properties, and actuator characteristics are established (Section ref:ssec:nhexa_model_def). -The model is then validated through comparison with the analytical equations in a simplified configuration (Section ref:ssec:nhexa_model_validation). -Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section ref:ssec:nhexa_model_dynamics). +First, the multi-body model is developed, and the geometric parameters, inertial properties, and actuator characteristics are established (Section\nbsp{}ref:ssec:nhexa_model_def). +The model is then validated through comparison with the analytical equations in a simplified configuration (Section\nbsp{}ref:ssec:nhexa_model_validation). +Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section\nbsp{}ref:ssec:nhexa_model_dynamics). **** Model Definition <> ***** Geometry -The Stewart platform's geometry is defined by two principal coordinate frames (Figure ref:fig:nhexa_stewart_model_def): a fixed base frame $\{F\}$ and a moving platform frame $\{M\}$. +The Stewart platform's geometry is defined by two principal coordinate frames (Figure\nbsp{}ref:fig:nhexa_stewart_model_def): a fixed base frame $\{F\}$ and a moving platform frame $\{M\}$. The joints connecting the actuators to these frames are located at positions ${}^F\bm{a}_i$ and ${}^M\bm{b}_i$ respectively. The point of interest, denoted by frame $\{A\}$, is situated $150\,mm$ above the moving platform frame $\{M\}$. -The geometric parameters of the nano-hexapod are summarized in Table ref:tab:nhexa_stewart_model_geometry. +The geometric parameters of the nano-hexapod are summarized in Table\nbsp{}ref:tab:nhexa_stewart_model_geometry. These parameters define the positions of all connection points in their respective coordinate frames. From these parameters, key kinematic properties can be derived: the strut orientations $\hat{\bm{s}}_i$, strut lengths $l_i$, and the system's Jacobian matrix $\bm{J}$. @@ -4585,12 +4585,12 @@ These joints are considered massless and exhibit no stiffness along their degree ***** Actuators -The actuator model comprises several key elements (Figure ref:fig:nhexa_actuator_model). +The actuator model comprises several key elements (Figure\nbsp{}ref:fig:nhexa_actuator_model). At its core, each actuator is modeled as a prismatic joint with internal stiffness $k_a$ and damping $c_a$, driven by a force source $f$. Similarly to what was found using the rotating 3-DoF model, a parallel stiffness $k_p$ is added in parallel with the force sensor to ensure stability when considering spindle rotation effects. Each actuator is equipped with two sensors: a force sensor providing measurements $f_n$ and a relative motion sensor that measures the strut length $l_i$. -The actuator parameters used in the conceptual phase are listed in Table ref:tab:nhexa_actuator_parameters. +The actuator parameters used in the conceptual phase are listed in Table\nbsp{}ref:tab:nhexa_actuator_parameters. This modular approach to actuator modeling allows for future refinements as the design evolves, enabling the incorporation of additional dynamic effects or sensor characteristics as needed. @@ -4620,9 +4620,9 @@ This modular approach to actuator modeling allows for future refinements as the **** Validation of the multi-body model <> -The developed multi-body model of the Stewart platform is represented schematically in Figure ref:fig:nhexa_stewart_model_input_outputs, highlighting the key inputs and outputs: actuator forces $\bm{f}$, force sensor measurements $\bm{f}_n$, and relative displacement measurements $\bm{\mathcal{L}}$. +The developed multi-body model of the Stewart platform is represented schematically in Figure\nbsp{}ref:fig:nhexa_stewart_model_input_outputs, highlighting the key inputs and outputs: actuator forces $\bm{f}$, force sensor measurements $\bm{f}_n$, and relative displacement measurements $\bm{\mathcal{L}}$. The frames $\{F\}$ and $\{M\}$ serve as interfaces for integration with other elements in the multi-body system. -A three-dimensional visualization of the model is presented in Figure ref:fig:nhexa_simscape_screenshot. +A three-dimensional visualization of the model is presented in Figure\nbsp{}ref:fig:nhexa_simscape_screenshot. #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage @@ -4640,13 +4640,13 @@ A three-dimensional visualization of the model is presented in Figure ref:fig:nh [[file:figs/nhexa_simscape_screenshot.jpg]] #+end_minipage -The validation of the multi-body model was performed using the simplest Stewart platform configuration, enabling direct comparison with the analytical transfer functions derived in Section ref:ssec:nhexa_stewart_platform_dynamics. +The validation of the multi-body model was performed using the simplest Stewart platform configuration, enabling direct comparison with the analytical transfer functions derived in Section\nbsp{}ref:ssec:nhexa_stewart_platform_dynamics. This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness $k_a = 1\,\text{N}/\mu\text{m}$ and damping $c_a = 10\,\text{N}/({\text{m}/\text{s}})$. -The geometric parameters remain as specified in Table ref:tab:nhexa_actuator_parameters. +The geometric parameters remain as specified in Table\nbsp{}ref:tab:nhexa_actuator_parameters. While the moving platform itself is considered massless, a $10\,\text{kg}$ cylindrical payload is mounted on top with a radius of $r = 110\,mm$ and a height $h = 300\,mm$. -For the analytical model, the stiffness, damping, and mass matrices are defined in eqref:eq:nhexa_analytical_matrices. +For the analytical model, the stiffness, damping, and mass matrices are defined in\nbsp{}eqref:eq:nhexa_analytical_matrices. \begin{subequations}\label{eq:nhexa_analytical_matrices} \begin{align} @@ -4656,11 +4656,11 @@ For the analytical model, the stiffness, damping, and mass matrices are defined \end{align} \end{subequations} -The transfer functions from the actuator forces to the strut displacements are computed using these matrices according to equation eqref:eq:nhexa_transfer_function_struts. +The transfer functions from the actuator forces to the strut displacements are computed using these matrices according to equation\nbsp{}eqref:eq:nhexa_transfer_function_struts. These analytical transfer functions are then compared with those extracted from the multi-body model. The developed multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform. -Figure ref:fig:nhexa_comp_multi_body_analytical presents a comparison between the analytical and multi-body transfer functions, specifically showing the response from the first actuator force to all six strut displacements. +Figure\nbsp{}ref:fig:nhexa_comp_multi_body_analytical presents a comparison between the analytical and multi-body transfer functions, specifically showing the response from the first actuator force to all six strut displacements. The close agreement between both approaches across the frequency spectrum validates the multi-body model's accuracy in capturing the system's dynamic behavior. #+name: fig:nhexa_comp_multi_body_analytical @@ -4671,10 +4671,10 @@ The close agreement between both approaches across the frequency spectrum valida <> Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics was performed. -The model parameters were set according to the specifications outlined in Section ref:ssec:nhexa_model_def, with a payload mass of $10\,kg$. +The model parameters were set according to the specifications outlined in Section\nbsp{}ref:ssec:nhexa_model_def, with a payload mass of $10\,kg$. The transfer functions from actuator forces $\bm{f}$ to both strut displacements $\bm{\mathcal{L}}$ and force measurements $\bm{f}_n$ were derived from the multi-body model. -The transfer functions relating actuator forces to strut displacements are presented in Figure ref:fig:nhexa_multi_body_plant_dL. +The transfer functions relating actuator forces to strut displacements are presented in Figure\nbsp{}ref:fig:nhexa_multi_body_plant_dL. Due to the system's symmetrical design and identical strut configurations, all diagonal terms (transfer functions from force $f_i$ to displacement $l_i$ of the same strut) exhibit identical behavior. While the system has six degrees of freedom, only four distinct resonance frequencies were observed in the frequency response. This reduction from six to four observable modes is attributed to the system's symmetry, where two pairs of resonances occur at identical frequencies. @@ -4684,7 +4684,7 @@ At low frequencies, well below the first resonance, the plant demonstrates good In the mid-frequency range, the system exhibits coupled dynamics through its resonant modes, reflecting the complex interactions between the platform's degrees of freedom. At high frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: $\bm{G}(j\omega) \xrightarrow[\omega \to \infty]{} \bm{J} \bm{M}^{-\intercal} \bm{J}^{\intercal} \frac{-1}{\omega^2}$ -The force sensor transfer functions, shown in Figure ref:fig:nhexa_multi_body_plant_fm, display characteristics typical of collocated actuator-sensor pairs. +The force sensor transfer functions, shown in Figure\nbsp{}ref:fig:nhexa_multi_body_plant_fm, display characteristics typical of collocated actuator-sensor pairs. Each actuator's transfer function to its associated force sensor exhibits alternating complex conjugate poles and zeros. 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. @@ -4713,7 +4713,7 @@ Through comparison with analytical solutions in a simplified configuration, the A key advantage of this modeling approach lies in its flexibility for future refinements. While the current implementation employs idealized joints for the conceptual design phase, the framework readily accommodates the incorporation of joint stiffness and other non-ideal effects. -The joint stiffness, which is known to impact the performance of decentralized IFF control strategy [[cite:&preumont07_six_axis_singl_stage_activ]], will be studied and optimized during the detailed design phase. +The joint stiffness, which is known to impact the performance of decentralized IFF control strategy\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]], will be studied and optimized during the detailed design phase. The validated multi-body model will serve as a valuable tool for predicting system behavior and evaluating control performance throughout the design process. *** Control of Stewart Platforms @@ -4723,10 +4723,10 @@ The validated multi-body model will serve as a valuable tool for predicting syst The control of Stewart platforms presents distinct challenges compared to the uniaxial model due to their multi-input multi-output nature. Although the uniaxial model demonstrated the effectiveness of the HAC-LAC strategy, its extension to Stewart platforms requires careful considerations discussed in this section. -First, the distinction between centralized and decentralized control approaches is discussed in Section ref:ssec:nhexa_control_centralized_decentralized. -The impact of the control space selection - either Cartesian or strut space - is then analyzed in Section ref:ssec:nhexa_control_space, highlighting the trade-offs between direction-specific tuning and implementation simplicity. +First, the distinction between centralized and decentralized control approaches is discussed in Section\nbsp{}ref:ssec:nhexa_control_centralized_decentralized. +The impact of the control space selection - either Cartesian or strut space - is then analyzed in Section\nbsp{}ref:ssec:nhexa_control_space, highlighting the trade-offs between direction-specific tuning and implementation simplicity. -Building on these analyses, a decentralized active damping strategy using Integral Force Feedback is developed in Section ref:ssec:nhexa_control_iff, followed by the implementation of a centralized High Authority Control for positioning in Section ref:ssec:nhexa_control_hac_lac. +Building on these analyses, a decentralized active damping strategy using Integral Force Feedback is developed in Section\nbsp{}ref:ssec:nhexa_control_iff, followed by the implementation of a centralized High Authority Control for positioning in Section\nbsp{}ref:ssec:nhexa_control_hac_lac. This architecture, while simple, will be used to demonstrate the feasibility of the NASS concept and will provide a foundation for more sophisticated control strategies to be developed during the detailed design phase. **** Centralized and Decentralized Control @@ -4734,7 +4734,7 @@ This architecture, while simple, will be used to demonstrate the feasibility of In the control of MIMO systems, and more specifically of Stewart platforms, a fundamental architectural decision lies in the choice between centralized and decentralized control strategies. -In decentralized control, each actuator operates based on feedback from its associated sensor only, creating independent control loops, as illustrated in Figure ref:fig:nhexa_stewart_decentralized_control. +In decentralized control, each actuator operates based on feedback from its associated sensor only, creating independent control loops, as illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_decentralized_control. While mechanical coupling between the struts exists, control decisions are made locally, with each controller processing information from a single sensor-actuator pair. This approach offers simplicity in implementation and reduces computational requirements. @@ -4745,8 +4745,8 @@ The choice between these approaches depends significantly on the degree of inter For instance, when using external metrology systems that measure the platform's global position, centralized control becomes necessary because each sensor measurement depends on all actuator inputs. In the context of the nano-hexapod, two distinct control strategies were examined during the conceptual phase: -- Decentralized Integral Force Feedback (IFF), which utilizes collocated force sensors to implement independent control loops for each strut (Section ref:ssec:nhexa_control_iff) -- High-Authority Control (HAC), which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section ref:ssec:nhexa_control_hac_lac) +- Decentralized Integral Force Feedback (IFF), which utilizes collocated force sensors to implement independent control loops for each strut (Section\nbsp{}ref:ssec:nhexa_control_iff) +- High-Authority Control (HAC), which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section\nbsp{}ref:ssec:nhexa_control_hac_lac) #+name: fig:nhexa_stewart_decentralized_control #+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity. @@ -4760,11 +4760,11 @@ This choice affects both the control design and the obtained performance. ***** Control in the Strut space -In this approach, as illustrated in Figure ref:fig:nhexa_control_strut, the control is performed in the space of the struts. +In this approach, as illustrated in Figure\nbsp{}ref:fig:nhexa_control_strut, the control is performed in the space of the struts. The Jacobian matrix is used to solve the inverse kinematics in real-time by mapping position errors from Cartesian space $\bm{\epsilon}_{\mathcal{X}}$ to strut space $\bm{\epsilon}_{\mathcal{L}}$. A diagonal controller then processes these strut-space errors to generate force commands for each actuator. -The main advantage of this approach emerges from the plant characteristics in the strut space, as shown in Figure ref:fig:nhexa_plant_frame_struts. +The main advantage of this approach emerges from the plant characteristics in the strut space, as shown in Figure\nbsp{}ref:fig:nhexa_plant_frame_struts. The diagonal terms of the plant (transfer functions from force to displacement of the same strut, as measured by the external metrology) are identical due to the system's symmetry. 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. @@ -4791,10 +4791,10 @@ Furthermore, at low frequencies, the plant exhibits good decoupling between the ***** Control in Cartesian Space -Alternatively, control can be implemented directly in Cartesian space, as illustrated in Figure ref:fig:nhexa_control_cartesian. -Here, the controller processes Cartesian errors $\bm{\epsilon}_{\mathcal{X}}$ to generate forces and torques $\bm{\mathcal{F}}$, which are then mapped to actuator forces using the transpose of the inverse Jacobian matrix eqref:eq:nhexa_jacobian_forces. +Alternatively, control can be implemented directly in Cartesian space, as illustrated in Figure\nbsp{}ref:fig:nhexa_control_cartesian. +Here, the controller processes Cartesian errors $\bm{\epsilon}_{\mathcal{X}}$ to generate forces and torques $\bm{\mathcal{F}}$, which are then mapped to actuator forces using the transpose of the inverse Jacobian matrix\nbsp{}eqref:eq:nhexa_jacobian_forces. -The plant behavior in Cartesian space, illustrated in Figure ref:fig:nhexa_plant_frame_cartesian, reveals interesting characteristics. +The plant behavior in Cartesian space, illustrated in Figure\nbsp{}ref:fig:nhexa_plant_frame_cartesian, reveals interesting characteristics. Some degrees of freedom, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics. A key advantage of this approach is that the control performance can be tuned individually for each direction. This is particularly valuable when performance requirements differ between degrees of freedom - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational degrees of freedom can tolerate larger errors than others. @@ -4825,9 +4825,9 @@ More sophisticated control strategies will be explored during the detailed desig **** Active Damping with Decentralized IFF <> -The decentralized Integral Force Feedback (IFF) control strategy is implemented using independent control loops for each strut, similarly to what is shown in Figure ref:fig:nhexa_stewart_decentralized_control, but using force sensors instead of relative motion sensors. +The decentralized Integral Force Feedback (IFF) control strategy is implemented using independent control loops for each strut, similarly to what is shown in Figure\nbsp{}ref:fig:nhexa_stewart_decentralized_control, but using force sensors instead of relative motion sensors. -The corresponding block diagram of the control loop is shown in Figure 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 eqref:eq:nhexa_kiff. +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 $\bm{f}^{\prime}$ @@ -4844,11 +4844,11 @@ The corresponding block diagram of the control loop is shown in Figure ref:fig:n In this section, the stiffness in parallel with the force sensor was omitted since the Stewart platform is not subjected to rotation. The effect of this parallel stiffness is examined in the next section when the platform is integrated into the complete NASS. -Root Locus analysis, shown in Figure ref:fig:nhexa_decentralized_iff_root_locus, reveals the evolution of the closed-loop poles as the controller gain $g$ varies from $0$ to $\infty$. -A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability [[cite:&preumont08_trans_zeros_struc_contr_with]]. +Root Locus analysis, shown in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_root_locus, reveals the evolution of the closed-loop poles as the controller gain $g$ varies from $0$ to $\infty$. +A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with]]. This property is particularly valuable because the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone. -The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)$), presented in Figure ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$. +The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)$), presented in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$. The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes. This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability. @@ -4874,9 +4874,9 @@ This high gain, combined with the bounded phase, enables effective damping of th <> The design of the High Authority Control positioning loop is now examined. -The complete HAC-IFF control architecture is illustrated in Figure ref:fig:nhexa_hac_iff_schematic, where the reference signal $\bm{r}_{\mathcal{X}}$ represents the desired pose, and $\bm{\mathcal{X}}$ is the measured pose by the external metrology system. +The complete HAC-IFF control architecture is illustrated in Figure\nbsp{}ref:fig:nhexa_hac_iff_schematic, where the reference signal $\bm{r}_{\mathcal{X}}$ represents the desired pose, and $\bm{\mathcal{X}}$ is the measured pose by the external metrology system. -Following the analysis from Section ref:ssec:nhexa_control_space, the control is implemented in the strut space. +Following the analysis from Section\nbsp{}ref:ssec:nhexa_control_space, the control is implemented in the strut space. The Jacobian matrix $\bm{J}^{-1}$ performs (approximate) real-time approximate inverse kinematics to map position errors from Cartesian space $\bm{\epsilon}_{\mathcal{X}}$ to strut space $\bm{\epsilon}_{\mathcal{L}}$. A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these errors in the frame of the struts. @@ -4885,8 +4885,8 @@ A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these [[file:figs/nhexa_hac_iff_schematic.png]] The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions. -Figure ref:fig:nhexa_decentralized_hac_iff_plant_undamped shows the original transfer functions from actuator forces $\bm{f}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, which are characterized by pronounced resonant peaks. -When the decentralized IFF is implemented, the transfer functions from modified inputs $\bm{f}^{\prime}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$ exhibit significantly attenuated resonances (Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped). +Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_plant_undamped shows the original transfer functions from actuator forces $\bm{f}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, which are characterized by pronounced resonant peaks. +When the decentralized IFF is implemented, the transfer functions from modified inputs $\bm{f}^{\prime}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$ exhibit significantly attenuated resonances (Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_plant_damped). This damping of structural resonances serves two purposes: it reduces vibrations near resonances and simplifies the design of the high authority controller by providing simpler plant dynamics. #+name: fig:nhexa_decentralized_hac_iff_plant @@ -4907,9 +4907,9 @@ This damping of structural resonances serves two purposes: it reduces vibrations #+end_subfigure #+end_figure -Based upon the damped plant dynamics shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, a high authority controller was designed with the structure given in eqref:eq:nhexa_khac. +Based upon the damped plant dynamics shown in Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_plant_damped, a high authority controller was designed with the structure given in\nbsp{}eqref:eq:nhexa_khac. The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter for robustness against unmodeled high-frequency dynamics. -The loop gain of an individual control channel is shown in Figure ref:fig:nhexa_decentralized_hac_iff_loop_gain. +The loop gain of an individual control channel is shown in Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_loop_gain. \begin{equation}\label{eq:nhexa_khac} \bm{K}_{\text{HAC}}(s) = \begin{bmatrix} @@ -4921,8 +4921,8 @@ The loop gain of an individual control channel is shown in Figure ref:fig:nhexa_ The stability of the MIMO feedback loop is analyzed through the /characteristic loci/ method. Such characteristic loci represent the eigenvalues of the loop gain matrix $\bm{G}(j\omega)\bm{K}(j\omega)$ plotted in the complex plane as the frequency varies from $0$ to $\infty$. -For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point [[cite:&skogestad07_multiv_feedb_contr]]. -As shown in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, all loci remain to the right of the $-1$ point, validating the stability of the closed-loop system. +For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point\nbsp{}[[cite:&skogestad07_multiv_feedb_contr]]. +As shown in Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_root_locus, all loci remain to the right of the $-1$ point, validating the stability of the closed-loop system. Additionally, the distance of the loci from the $-1$ point provides information about stability margins of the coupled system. #+name: fig:nhexa_decentralized_hac_iff_results @@ -4984,7 +4984,7 @@ The previous chapters have established crucial foundational elements for the dev The uniaxial model study demonstrated that very stiff nano-hexapod configurations should be avoided due to their high coupling with the micro-station dynamics. A rotating three-degree-of-freedom model revealed that soft nano-hexapod designs prove unsuitable due to gyroscopic effect induced by the spindle rotation. To further improve the model accuracy, a multi-body model of the micro-station was developed, which was carefully tuned using experimental modal analysis. -Furthermore, a multi-body model of the nano-hexapod was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure ref:fig:nass_simscape_model. +Furthermore, a multi-body model of the nano-hexapod was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure\nbsp{}ref:fig:nass_simscape_model. #+name: fig:nass_simscape_model #+caption: 3D view of the NASS multi-body model @@ -4994,8 +4994,8 @@ Furthermore, a multi-body model of the nano-hexapod was created, that can then b Building upon these foundations, this chapter presents the validation of the NASS concept. The investigation begins with the previously established nano-hexapod model with actuator stiffness $k_a = 1\,N/\mu m$. -A thorough examination of the control kinematics is presented in Section ref:sec:nass_kinematics, detailing how both external metrology and nano-hexapod internal sensors are used in the control architecture. -The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section ref:sec:nass_active_damping), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section ref:sec:nass_hac). +A thorough examination of the control kinematics is presented in Section\nbsp{}ref:sec:nass_kinematics, detailing how both external metrology and nano-hexapod internal sensors are used in the control architecture. +The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section\nbsp{}ref:sec:nass_active_damping), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section\nbsp{}ref:sec:nass_hac). The robustness of the proposed control scheme was evaluated under various operational conditions. Particular attention was paid to system performance under changing payload masses and varying spindle rotational velocities. @@ -5006,7 +5006,7 @@ This chapter concludes the conceptual design phase, with the simulation of tomog <> **** Introduction :ignore: -Figure ref:fig:nass_concept_schematic presents a schematic overview of the NASS. +Figure\nbsp{}ref:fig:nass_concept_schematic presents a schematic overview of the NASS. This section focuses on the components of the "Instrumentation and Real-Time Control" block. #+name: fig:nass_concept_schematic @@ -5017,12 +5017,12 @@ This section focuses on the components of the "Instrumentation and Real-Time Con As established in the previous section on Stewart platforms, the proposed control strategy combines Decentralized Integral Force Feedback with a High Authority Controller performed in the frame of the struts. For the Nano Active Stabilization System, computing the positioning errors in the frame of the struts involves three key steps. -First, desired sample pose with respect to a fixed reference frame is computed using the micro-station kinematics as detailed in Section ref:ssec:nass_ustation_kinematics. +First, desired sample pose with respect to a fixed reference frame is computed using the micro-station kinematics as detailed in Section\nbsp{}ref:ssec:nass_ustation_kinematics. This fixed frame is located at the X-ray beam focal point, as it is where the point of interest needs to be positioned. -Second, it measures the actual sample pose relative to the same fix frame, described in Section ref:ssec:nass_sample_pose_error. -Finally, it determines the sample pose error and maps these errors to the nano-hexapod struts, as explained in Section ref:ssec:nass_error_struts. +Second, it measures the actual sample pose relative to the same fix frame, described in Section\nbsp{}ref:ssec:nass_sample_pose_error. +Finally, it determines the sample pose error and maps these errors to the nano-hexapod struts, as explained in Section\nbsp{}ref:ssec:nass_error_struts. -The complete control architecture is described in Section ref:ssec:nass_control_architecture. +The complete control architecture is described in Section\nbsp{}ref:ssec:nass_control_architecture. **** Micro Station Kinematics <> @@ -5031,7 +5031,7 @@ The micro-station kinematics enables the computation of the desired sample pose These reference signals consist of the desired lateral position $r_{D_y}$, tilt angle $r_{R_y}$, and spindle angle $r_{R_z}$. The micro-hexapod pose is defined by six parameters: three translations ($r_{D_{\mu x}}$, $r_{D_{\mu y}}$, $r_{D_{\mu z}}$) and three rotations ($r_{\theta_{\mu x}}$, $r_{\theta_{\mu y}}$, $r_{\theta_{\mu z}}$). -Using these reference signals, the desired sample position relative to the fixed frame is expressed through the homogeneous transformation matrix $\bm{T}_{\mu\text{-station}}$, as defined in equation eqref:eq:nass_sample_ref. +Using these reference signals, the desired sample position relative to the fixed frame is expressed through the homogeneous transformation matrix $\bm{T}_{\mu\text{-station}}$, as defined in equation\nbsp{}eqref:eq:nass_sample_ref. \begin{equation}\label{eq:nass_sample_ref} \bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}} @@ -5077,7 +5077,7 @@ Due to the system's symmetry, this metrology provides measurements for five degr The sixth degree of freedom ($R_z$) is still required to compute the errors in the frame of the nano-hexapod struts (i.e. to compute the nano-hexapod inverse kinematics). This $R_z$ rotation is estimated by combining measurements from the spindle encoder and the nano-hexapod's internal metrology, which consists of relative motion sensors in each strut (note that the micro-hexapod is not used for $R_z$ rotation, and is therefore ignored for $R_z$ estimation). -The measured sample pose is represented by the homogeneous transformation matrix $\bm{T}_{\text{sample}}$, as shown in equation eqref:eq:nass_sample_pose. +The measured sample pose is represented by the homogeneous transformation matrix $\bm{T}_{\text{sample}}$, as shown in equation\nbsp{}eqref:eq:nass_sample_pose. \begin{equation}\label{eq:nass_sample_pose} \bm{T}_{\text{sample}} = @@ -5095,14 +5095,14 @@ The measured sample pose is represented by the homogeneous transformation matrix The homogeneous transformation formalism enables straightforward computation of the sample position error. This computation involves the previously computed homogeneous $4 \times 4$ matrices: $\bm{T}_{\mu\text{-station}}$ representing the desired pose, and $\bm{T}_{\text{sample}}$ representing the measured pose. -Their combination yields $\bm{T}_{\text{error}}$, which expresses the position error of the sample in the frame of the rotating nano-hexapod, as shown in equation eqref:eq:nass_transformation_error. +Their combination yields $\bm{T}_{\text{error}}$, which expresses the position error of the sample in the frame of the rotating nano-hexapod, as shown in equation\nbsp{}eqref:eq:nass_transformation_error. \begin{equation}\label{eq:nass_transformation_error} \bm{T}_{\text{error}} = \bm{T}_{\mu\text{-station}}^{-1} \cdot \bm{T}_{\text{sample}} \end{equation} The known structure of the homogeneous transformation matrix facilitates efficient real-time inverse computation. -From $\bm{T}_{\text{error}}$, the position and orientation errors $\bm{\epsilon}_{\mathcal{X}} = [\epsilon_{D_x},\ \epsilon_{D_y},\ \epsilon_{D_z},\ \epsilon_{R_x},\ \epsilon_{R_y},\ \epsilon_{R_z}]$ of the sample are extracted using equation eqref:eq:nass_compute_errors: +From $\bm{T}_{\text{error}}$, the position and orientation errors $\bm{\epsilon}_{\mathcal{X}} = [\epsilon_{D_x},\ \epsilon_{D_y},\ \epsilon_{D_z},\ \epsilon_{R_x},\ \epsilon_{R_y},\ \epsilon_{R_z}]$ of the sample are extracted using equation\nbsp{}eqref:eq:nass_compute_errors: \begin{equation}\label{eq:nass_compute_errors} \begin{align} @@ -5115,7 +5115,7 @@ From $\bm{T}_{\text{error}}$, the position and orientation errors $\bm{\epsilon} \end{align} \end{equation} -Finally, these errors are mapped to the strut space using the nano-hexapod Jacobian matrix eqref:eq:nass_inverse_kinematics. +Finally, these errors are mapped to the strut space using the nano-hexapod Jacobian matrix\nbsp{}eqref:eq:nass_inverse_kinematics. \begin{equation}\label{eq:nass_inverse_kinematics} \bm{\epsilon}_{\mathcal{L}} = \bm{J} \cdot \bm{\epsilon}_{\mathcal{X}} @@ -5124,15 +5124,15 @@ Finally, these errors are mapped to the strut space using the nano-hexapod Jacob **** Control Architecture - Summary <> -The complete control architecture is summarized in Figure ref:fig:nass_control_architecture. +The complete control architecture is summarized in Figure\nbsp{}ref:fig:nass_control_architecture. The sample pose is measured using external metrology for five degrees of freedom, while the sixth degree of freedom (Rz) is estimated by combining measurements from the nano-hexapod encoders and spindle encoder. The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and micro-hexapod. The position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the nano-hexapod frame. Then, the Jacobian matrix $\bm{J}$ maps these errors to individual strut coordinates. -For control purposes, force sensors mounted on each strut are used in a decentralized manner for active damping, as detailed in Section ref:sec:nass_active_damping. -Then, the high authority controller uses the computed errors in the frame of the struts to provides real-time stabilization of the sample position (Section ref:sec:nass_hac). +For control purposes, force sensors mounted on each strut are used in a decentralized manner for active damping, as detailed in Section\nbsp{}ref:sec:nass_active_damping. +Then, the high authority controller uses the computed errors in the frame of the struts to provides real-time stabilization of the sample position (Section\nbsp{}ref:sec:nass_hac). #+name: fig:nass_control_architecture #+caption: Control architecture for the NASS. Physical systems are shown in blue, control kinematics elements in red, decentralized Integral Force Feedback controller in yellow, and centralized high authority controller in green. @@ -5153,12 +5153,12 @@ The payloads used for validation have a cylindrical shape with 250 mm height and <> Transfer functions from actuator forces $f_i$ to force sensor measurements $f_{mi}$ are computed using the multi-body model. -Figure ref:fig:nass_iff_plant_effect_kp examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity $\Omega_z = 360\,\text{deg/s}$ and with a payload mass of 25 kg. +Figure\nbsp{}ref:fig:nass_iff_plant_effect_kp examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity $\Omega_z = 360\,\text{deg/s}$ and with a payload mass of 25 kg. -Without parallel stiffness (Figure ref:fig:nass_iff_plant_no_kp), the plant dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model. -Adding parallel stiffness (Figure ref:fig:nass_iff_plant_kp) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation. +Without parallel stiffness (Figure\nbsp{}ref:fig:nass_iff_plant_no_kp), the plant dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model. +Adding parallel stiffness (Figure\nbsp{}ref:fig:nass_iff_plant_kp) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation. -Although both cases show significant coupling around the resonances, stability is guaranteed by the collocated arrangement of the actuators and sensors [[cite:&preumont08_trans_zeros_struc_contr_with]]. +Although both cases show significant coupling around the resonances, stability is guaranteed by the collocated arrangement of the actuators and sensors\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with]]. #+name: fig:nass_iff_plant_effect_kp #+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of 25kg. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}) @@ -5178,9 +5178,9 @@ Although both cases show significant coupling around the resonances, stability i #+end_subfigure #+end_figure -The effect of rotation, as shown in Figure ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,N/\mu m$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). +The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,N/\mu m$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). -Figure ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics. +Figure\nbsp{}ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics. The poles and zeros shift in frequency as the payload mass varies. However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining robust stability properties. @@ -5209,9 +5209,9 @@ The previous analysis using the 3DoF rotating model showed that decentralized In This finding was also confirmed with the multi-body model of the NASS: the system was unstable when using pure integrators and without parallel stiffness. This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors. -However, as illustrated in Figure ref:fig:nass_iff_plant_kp, adding parallel stiffness increases the low frequency gain. +However, as illustrated in Figure\nbsp{}ref:fig:nass_iff_plant_kp, adding parallel stiffness increases the low frequency gain. Using pure integrators would result in high loop gain at low frequencies, adversely affecting the damped plant dynamics, which is undesirable. -To resolve this issue, a second-order high-pass filter is introduced to limit the low frequency gain, as shown in Equation eqref:eq:nass_kiff. +To resolve this issue, a second-order high-pass filter is introduced to limit the low frequency gain, as shown in Equation\nbsp{}eqref:eq:nass_kiff. \begin{equation}\label{eq:nass_kiff} \bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix} @@ -5222,14 +5222,14 @@ To resolve this issue, a second-order high-pass filter is introduced to limit th \end{equation} The cut-off frequency of the second-order high-pass filter was tuned to be below the frequency of the complex conjugate zero for the highest mass, which is at $5\,\text{Hz}$. -The overall gain was then increased to obtain a large loop gain around the resonances to be damped, as illustrated in Figure ref:fig:nass_iff_loop_gain. +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: $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$ #+attr_latex: :options [h!tbp] [[file:figs/nass_iff_loop_gain.png]] -To verify stability, the root loci for the three payload configurations were computed, as shown in Figure ref:fig:nass_iff_root_locus. +To verify stability, the root loci for the three payload configurations were computed, as shown in Figure\nbsp{}ref:fig:nass_iff_root_locus. The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability of the applied decentralized IFF. #+name: fig:nass_iff_root_locus @@ -5264,11 +5264,11 @@ The implementation of high-bandwidth position control for the nano-hexapod prese The plant dynamics exhibits complex behavior influenced by multiple factors, including payload mass, rotational velocity, and the mechanical coupling between the nano-hexapod and the micro-station. This section presents the development and validation of a centralized control strategy designed to achieve precise sample positioning during high-speed tomography experiments. -First, a comprehensive analysis of the plant dynamics is presented in Section ref:ssec:nass_hac_plant, examining the effects of spindle rotation, payload mass variation, and the implementation of Integral Force Feedback (IFF). -Section ref:ssec:nass_hac_stiffness validates previous modeling predictions that both overly stiff and compliant nano-hexapod configurations lead to degraded performance. -Building upon these findings, Section ref:ssec:nass_hac_controller presents the design of a robust high-authority controller that maintains stability across varying payload masses while achieving the desired control bandwidth. +First, a comprehensive analysis of the plant dynamics is presented in Section\nbsp{}ref:ssec:nass_hac_plant, examining the effects of spindle rotation, payload mass variation, and the implementation of Integral Force Feedback (IFF). +Section\nbsp{}ref:ssec:nass_hac_stiffness validates previous modeling predictions that both overly stiff and compliant nano-hexapod configurations lead to degraded performance. +Building upon these findings, Section\nbsp{}ref:ssec:nass_hac_controller presents the design of a robust high-authority controller that maintains stability across varying payload masses while achieving the desired control bandwidth. -The performance of the developed control strategy was validated through simulations of tomography experiments in Section ref:ssec:nass_hac_tomography. +The performance of the developed control strategy was validated through simulations of tomography experiments in Section\nbsp{}ref:ssec:nass_hac_tomography. These simulations included realistic disturbance sources and were used to evaluate the system performance against the stringent positioning requirements imposed by future beamline specifications. Particular attention was paid to the system's behavior under maximum rotational velocity conditions and its ability to accommodate varying payload masses, demonstrating the practical viability of the proposed control approach. @@ -5276,10 +5276,10 @@ Particular attention was paid to the system's behavior under maximum rotational <> The plant dynamics from force inputs $\bm{f}$ to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were first extracted from the multi-body model without the implementation of the decentralized IFF. -The influence of spindle rotation on plant dynamics was investigated, and the results are presented in Figure ref:fig:nass_undamped_plant_effect_Wz. +The influence of spindle rotation on plant dynamics was investigated, and the results are presented in Figure\nbsp{}ref:fig:nass_undamped_plant_effect_Wz. While rotational motion introduces coupling effects at low frequencies, these effects remain minimal at operational velocities, owing to the high stiffness characteristics of the nano-hexapod assembly. -Payload mass emerged as a significant parameter affecting system behavior, as illustrated in Figure ref:fig:nass_undamped_plant_effect_mass. +Payload mass emerged as a significant parameter affecting system behavior, as illustrated in Figure\nbsp{}ref:fig:nass_undamped_plant_effect_mass. As expected, increasing the payload mass decreased the resonance frequencies while amplifying coupling at low frequency. These mass-dependent dynamic changes present considerable challenges for control system design, particularly for configurations with high payload masses. @@ -5308,11 +5308,11 @@ This also validates the developed control strategy. The Decentralized Integral Force Feedback was implemented in the multi-body model, and transfer functions from force inputs $\bm{f}^\prime$ of the damped plant to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were extracted from this model. -The effectiveness of the IFF implementation was first evaluated with a $1\,\text{kg}$ payload, as demonstrated in Figure ref:fig:nass_comp_undamped_damped_plant_m1. +The effectiveness of the IFF implementation was first evaluated with a $1\,\text{kg}$ payload, as demonstrated in Figure\nbsp{}ref:fig:nass_comp_undamped_damped_plant_m1. The results indicate successful damping of the nano-hexapod resonance modes, although a minor increase in low-frequency coupling was observed. This trade-off was considered acceptable, given the overall improvement in system behavior. -The benefits of IFF implementation were further assessed across the full range of payload configurations, and the results are presented in Figure ref:fig:nass_hac_plants. +The benefits of IFF implementation were further assessed across the full range of payload configurations, and the results are presented in Figure\nbsp{}ref:fig:nass_hac_plants. For all tested payloads ($1\,\text{kg}$, $25\,\text{kg}$ and $50\,\text{kg}$), the decentralized IFF significantly damped the nano-hexapod modes and therefore simplified the system dynamics. More importantly, in the vicinity of the desired high authority control bandwidth (i.e. between $10\,\text{Hz}$ and $50\,\text{Hz}$), the damped dynamics (shown in red) exhibited minimal gain and phase variations with frequency. For the undamped plants (shown in blue), achieving robust control with bandwidth above 10Hz while maintaining stability across different payload masses would be practically impossible. @@ -5338,7 +5338,7 @@ For the undamped plants (shown in blue), achieving robust control with bandwidth The coupling between the nano-hexapod and the micro-station was evaluated through a comparative analysis of plant dynamics under two mounting conditions. In the first configuration, the nano-hexapod was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance. -As illustrated in Figure ref:fig:nass_effect_ustation_compliance, the complex dynamics of the micro-station were found to have little impact on the plant dynamics. +As illustrated in Figure\nbsp{}ref:fig:nass_effect_ustation_compliance, the complex dynamics of the micro-station were found to have little impact on the plant dynamics. The only observable difference manifests as additional alternating poles and zeros above 100Hz, a frequency range sufficiently beyond the control bandwidth to avoid interference with the system performance. This result confirms effective dynamic decoupling between the nano-hexapod and the supporting micro-station structure. @@ -5355,12 +5355,12 @@ These models suggest that a moderate stiffness of approximately $1\,N/\mu m$ wou For the stiff nano-hexapod analysis, a system with an actuator stiffness of $100\,N/\mu m$ was simulated with a $25\,\text{kg}$ payload. The transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station. -As shown in Figure ref:fig:nass_stiff_nano_hexapod_coupling_ustation, significant coupling was observed between the nano-hexapod and micro-station dynamics. +As shown in Figure\nbsp{}ref:fig:nass_stiff_nano_hexapod_coupling_ustation, significant coupling was observed between the nano-hexapod and micro-station dynamics. This coupling introduces complex behavior that is difficult to model and predict accurately, thus corroborating the predictions of the simplified uniaxial model. The soft nano-hexapod configuration was evaluated using a stiffness of $0.01\,N/\mu m$ with a $25\,\text{kg}$ payload. The dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s. -Figure ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies. +Figure\nbsp{}ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies. The current approach of controlling the position in the strut frame is inadequate for soft nano-hexapods; but even shifting control to a frame matching the payload's center of mass would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. #+name: fig:nass_soft_stiff_hexapod @@ -5384,16 +5384,16 @@ The current approach of controlling the position in the strut frame is inadequat **** Controller design <> -A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation. -The controller structure is defined in Equation eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes. +A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure\nbsp{}ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation. +The controller structure is defined in Equation\nbsp{}eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes. \begin{equation}\label{eq:nass_robust_hac} K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right) \end{equation} The controller performance was evaluated through two complementary analyses. -First, the decentralized loop gain shown in Figure ref:fig:nass_hac_loop_gain, confirms the achievement of the desired 10Hz bandwidth. -Second, the characteristic loci analysis presented in Figure ref:fig:nass_hac_loci demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope. +First, the decentralized loop gain shown in Figure\nbsp{}ref:fig:nass_hac_loop_gain, confirms the achievement of the desired 10Hz bandwidth. +Second, the characteristic loci analysis presented in Figure\nbsp{}ref:fig:nass_hac_loci demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope. #+name: fig:nass_hac_controller #+caption: High Authority Controller - "Diagonal Loop Gain" (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci}) @@ -5424,7 +5424,7 @@ The primary requirement stipulates that the point of interest must remain within The simulation included two principal disturbance sources: ground motion and spindle vibrations. Additional noise sources, including measurement noise and electrical noise from DAC and voltage amplifiers, were not included in this analysis, as these parameters will be optimized during the detailed design phase. -Figure ref:fig:nass_tomo_1kg_60rpm presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (1kg). +Figure\nbsp{}ref:fig:nass_tomo_1kg_60rpm presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (1kg). The results demonstrate the system's capability to maintain the sample's position within the specified beam dimensions, thus validating the fundamental concept of the stabilization system. #+name: fig:nass_tomo_1kg_60rpm @@ -5446,7 +5446,7 @@ The results demonstrate the system's capability to maintain the sample's positio #+end_figure The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25 and 50kg payloads, complementing the initial 1kg test case. -As illustrated in Figure ref:fig:nass_tomography_hac_iff, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis. +As illustrated in Figure\nbsp{}ref:fig:nass_tomography_hac_iff, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis. While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions. It should be noted that the maximum rotational velocity of 360deg/s is primarily intended for lightweight payload applications. @@ -5542,17 +5542,17 @@ As the project advanced to the detailed design phase, a rigorous analysis of how In this chapter, the nano-hexapod geometry is optimized through careful analysis of how design parameters influence critical performance aspects: attainable workspace, mechanical stiffness, strut-to-strut coupling for decentralized control strategies, and dynamic response in Cartesian coordinates. -The chapter begins with a comprehensive review of existing Stewart platform designs in Section ref:sec:detail_kinematics_stewart_review, surveying various approaches to geometry, actuation, sensing, and joint design from the literature. -Section ref:sec:detail_kinematics_geometry develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process. -Section ref:sec:detail_kinematics_cubic examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application. -Finally, Section ref:sec:detail_kinematics_nano_hexapod presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. +The chapter begins with a comprehensive review of existing Stewart platform designs in Section\nbsp{}ref:sec:detail_kinematics_stewart_review, surveying various approaches to geometry, actuation, sensing, and joint design from the literature. +Section\nbsp{}ref:sec:detail_kinematics_geometry develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process. +Section\nbsp{}ref:sec:detail_kinematics_cubic examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application. +Finally, Section\nbsp{}ref:sec:detail_kinematics_nano_hexapod presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. *** Review of Stewart platforms <> -The first parallel platform similar to the Stewart platform was built in 1954 by Gough [[cite:&gough62_univer_tyre_test_machin]], for a tyre test machine (shown in Figure ref:fig:detail_geometry_gough_paper). -Subsequently, Stewart proposed a similar design for a flight simulator (shown in Figure ref:fig:detail_geometry_stewart_flight_simulator) in a 1965 publication [[cite:&stewart65_platf_with_six_degrees_freed]]. -Since then, the Stewart platform (sometimes referred to as the Stewart-Gough platform) has been utilized across diverse applications [[cite:&dasgupta00_stewar_platf_manip]], including large telescopes [[cite:&kazezkhan14_dynam_model_stewar_platf_nansh_radio_teles;&yun19_devel_isotr_stewar_platf_teles_secon_mirror]], machine tools [[cite:&russo24_review_paral_kinem_machin_tools]], and Synchrotron instrumentation [[cite:&marion04_hexap_esrf;&villar18_nanop_esrf_id16a_nano_imagin_beaml]]. +The first parallel platform similar to the Stewart platform was built in 1954 by Gough\nbsp{}[[cite:&gough62_univer_tyre_test_machin]], for a tyre test machine (shown in Figure\nbsp{}ref:fig:detail_geometry_gough_paper). +Subsequently, Stewart proposed a similar design for a flight simulator (shown in Figure\nbsp{}ref:fig:detail_geometry_stewart_flight_simulator) in a 1965 publication\nbsp{}[[cite:&stewart65_platf_with_six_degrees_freed]]. +Since then, the Stewart platform (sometimes referred to as the Stewart-Gough platform) has been utilized 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 @@ -5572,20 +5572,20 @@ Since then, the Stewart platform (sometimes referred to as the Stewart-Gough pla #+end_subfigure #+end_figure -# TODO - Section ref:sec:nhexa_stewart_platform +# TODO - Section\nbsp{}ref:sec:nhexa_stewart_platform As explained in the conceptual phase, Stewart platforms comprise the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors. -# TODO - ref:sec:detail_fem_joint +# TODO -\nbsp{}ref:sec:detail_fem_joint The specific geometry (i.e., position of joints and orientation of the struts) can be selected based on the application requirements, resulting in numerous designs throughout the literature. This discussion focuses primarily on Stewart platforms designed for nano-positioning and vibration control, which necessitates the use of flexible joints. The implementation of these flexible joints, will be discussed when designing the nano-hexapod flexible joints. -Long stroke Stewart platforms are not addressed here as their design presents different challenges, such as singularity-free workspace and complex kinematics [[cite:&merlet06_paral_robot]]. +Long stroke Stewart platforms are not addressed here as their design presents different challenges, such as singularity-free workspace and complex kinematics\nbsp{}[[cite:&merlet06_paral_robot]]. In terms of actuation, mainly two types are used: voice coil actuators and piezoelectric actuators. -Voice coil actuators, providing stroke ranges from $0.5\,mm$ to $10\,mm$, are commonly implemented in cubic architectures (as illustrated in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_pph) and are mainly used for vibration isolation [[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax;&thayer98_stewar;&mcinroy99_dynam;&preumont07_six_axis_singl_stage_activ]]. -For applications requiring short stroke (typically smaller than $500\,\mu m$), piezoelectric actuators present an interesting alternative, as shown in [[cite:&agrawal04_algor_activ_vibrat_isolat_spacec;&furutani04_nanom_cuttin_machin_using_stewar;&yang19_dynam_model_decoup_contr_flexib]]. -Examples of piezoelectric-actuated Stewart platforms are presented in Figures ref:fig:detail_kinematics_ulb_pz, ref:fig:detail_kinematics_uqp and ref:fig:detail_kinematics_yang19. -Although less frequently encountered, magnetostrictive actuators have been successfully implemented in [[cite:&zhang11_six_dof]] (Figure ref:fig:detail_kinematics_zhang11). +Voice coil actuators, providing stroke ranges from $0.5\,mm$ to $10\,mm$, are commonly implemented in cubic architectures (as illustrated in Figures\nbsp{}ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_pph) and are mainly used for vibration isolation\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax;&thayer98_stewar;&mcinroy99_dynam;&preumont07_six_axis_singl_stage_activ]]. +For applications requiring short stroke (typically smaller than $500\,\mu m$), piezoelectric actuators present an interesting alternative, as shown in\nbsp{}[[cite:&agrawal04_algor_activ_vibrat_isolat_spacec;&furutani04_nanom_cuttin_machin_using_stewar;&yang19_dynam_model_decoup_contr_flexib]]. +Examples of piezoelectric-actuated Stewart platforms are presented in Figures\nbsp{}ref:fig:detail_kinematics_ulb_pz, ref:fig:detail_kinematics_uqp and ref:fig:detail_kinematics_yang19. +Although less frequently encountered, magnetostrictive actuators have been successfully implemented in\nbsp{}[[cite:&zhang11_six_dof]] (Figure\nbsp{}ref:fig:detail_kinematics_zhang11). #+name: fig:detail_kinematics_stewart_examples_cubic #+caption: Some examples of developped Stewart platform with Cubic geometry @@ -5619,23 +5619,23 @@ Although less frequently encountered, magnetostrictive actuators have been succe #+end_subfigure #+end_figure -The sensors integrated in these platforms are selected based on specific control requirements, as different sensors offer distinct advantages and limitations [[cite:&hauge04_sensor_contr_space_based_six]]. +The sensors integrated in these platforms are selected based on specific control requirements, as different sensors offer distinct advantages and limitations\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six]]. Force sensors are typically integrated within the struts in a collocated arrangement with actuators to enhance control robustness. -Stewart platforms incorporating force sensors are frequently utilized for vibration isolation [[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax]] and active damping applications [[cite:&geng95_intel_contr_system_multip_degree;&abu02_stiff_soft_stewar_platf_activ]], as exemplified in Figure ref:fig:detail_kinematics_ulb_pz. +Stewart platforms incorporating force sensors are frequently utilized for vibration isolation\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax]] and active damping applications\nbsp{}[[cite:&geng95_intel_contr_system_multip_degree;&abu02_stiff_soft_stewar_platf_activ]], as exemplified in Figure\nbsp{}ref:fig:detail_kinematics_ulb_pz. -Inertial sensors (accelerometers and geophones) are commonly employed in vibration isolation applications [[cite:&chen03_payload_point_activ_vibrat_isolat;&chi15_desig_exper_study_vcm_based]]. -These sensors are predominantly aligned with the struts [[cite:&hauge04_sensor_contr_space_based_six;&li01_simul_fault_vibrat_isolat_point;&thayer02_six_axis_vibrat_isolat_system;&zhang11_six_dof;&jiao18_dynam_model_exper_analy_stewar;&tang18_decen_vibrat_contr_voice_coil]], although they may also be fixed to the top platform [[cite:&wang16_inves_activ_vibrat_isolat_stewar]]. +Inertial sensors (accelerometers and geophones) are commonly employed in vibration isolation applications\nbsp{}[[cite:&chen03_payload_point_activ_vibrat_isolat;&chi15_desig_exper_study_vcm_based]]. +These sensors are predominantly aligned with the struts\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six;&li01_simul_fault_vibrat_isolat_point;&thayer02_six_axis_vibrat_isolat_system;&zhang11_six_dof;&jiao18_dynam_model_exper_analy_stewar;&tang18_decen_vibrat_contr_voice_coil]], although they may also be fixed to the top platform\nbsp{}[[cite:&wang16_inves_activ_vibrat_isolat_stewar]]. -For high-precision positioning applications, various displacement sensors are implemented, including LVDTs [[cite:&thayer02_six_axis_vibrat_isolat_system;&kim00_robus_track_contr_desig_dof_paral_manip;&li01_simul_fault_vibrat_isolat_point;&thayer98_stewar]], capacitive sensors [[cite:&ting07_measur_calib_stewar_microm_system;&ting13_compos_contr_desig_stewar_nanos_platf]], eddy current sensors [[cite:&chen03_payload_point_activ_vibrat_isolat;&furutani04_nanom_cuttin_machin_using_stewar]], and strain gauges [[cite:&du14_piezo_actuat_high_precis_flexib]]. -Notably, some designs incorporate external sensing methodologies rather than integrating sensors within the struts [[cite:&li01_simul_fault_vibrat_isolat_point;&chen03_payload_point_activ_vibrat_isolat;&ting13_compos_contr_desig_stewar_nanos_platf]]. -A recent design [[cite:&naves21_desig_optim_large_strok_flexur_mechan]], although not strictly speaking a Stewart platform, has demonstrated the use of 3-phase rotary motors with rotary encoders for achieving long-stroke and highly repeatable positioning, as illustrated in Figure ref:fig:detail_kinematics_naves. +For high-precision positioning applications, various displacement sensors are implemented, including LVDTs\nbsp{}[[cite:&thayer02_six_axis_vibrat_isolat_system;&kim00_robus_track_contr_desig_dof_paral_manip;&li01_simul_fault_vibrat_isolat_point;&thayer98_stewar]], capacitive sensors\nbsp{}[[cite:&ting07_measur_calib_stewar_microm_system;&ting13_compos_contr_desig_stewar_nanos_platf]], eddy current sensors\nbsp{}[[cite:&chen03_payload_point_activ_vibrat_isolat;&furutani04_nanom_cuttin_machin_using_stewar]], and strain gauges\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]]. +Notably, some designs incorporate external sensing methodologies rather than integrating sensors within the struts\nbsp{}[[cite:&li01_simul_fault_vibrat_isolat_point;&chen03_payload_point_activ_vibrat_isolat;&ting13_compos_contr_desig_stewar_nanos_platf]]. +A recent design\nbsp{}[[cite:&naves21_desig_optim_large_strok_flexur_mechan]], although not strictly speaking a Stewart platform, has demonstrated the use of 3-phase rotary motors with rotary encoders for achieving long-stroke and highly repeatable positioning, as illustrated in Figure\nbsp{}ref:fig:detail_kinematics_naves. Two primary categories of Stewart platform geometry can be identified. -The first is cubic architecture (examples presented in Figure ref:fig:detail_kinematics_stewart_examples_cubic), wherein struts are positioned along six sides of a cube (and therefore oriented orthogonally to each other). +The first is cubic architecture (examples presented in Figure\nbsp{}ref:fig:detail_kinematics_stewart_examples_cubic), wherein struts are positioned along six sides of a cube (and therefore oriented orthogonally to each other). This architecture represents the most prevalent configuration for vibration isolation applications in the literature. -Its distinctive properties will be examined in Section ref:sec:detail_kinematics_cubic. -The second category comprises non-cubic architectures (Figure ref:fig:detail_kinematics_stewart_examples_non_cubic), where strut orientation and joint positioning can be optimized according to defined performance criteria. -The influence of strut orientation and joint positioning on Stewart platform properties is analyzed in Section ref:sec:detail_kinematics_geometry. +Its distinctive properties will be examined in Section\nbsp{}ref:sec:detail_kinematics_cubic. +The second category comprises non-cubic architectures (Figure\nbsp{}ref:fig:detail_kinematics_stewart_examples_non_cubic), where strut orientation and joint positioning can be optimized according to defined performance criteria. +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 @@ -5673,7 +5673,7 @@ The influence of strut orientation and joint positioning on Stewart platform pro <> **** Introduction :ignore: -# TODO - Section ref:sec:nhexa_stewart_platform (page pageref:sec:nhexa_stewart_platform), +# TODO - Section\nbsp{}ref:sec:nhexa_stewart_platform (page\nbsp{}pageref:sec:nhexa_stewart_platform), As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility (or workspace), the force authority, and the dynamics of the manipulator. It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications. @@ -5690,8 +5690,8 @@ This represents a six-dimensional property which is difficult to represent. Depending on the applications, only the translation mobility (i.e., fixed orientation workspace) or the rotation mobility may be represented. This approach is equivalent to projecting the six-dimensional value into a three-dimensional space, which is easier to represent. -Mobility of parallel manipulators is inherently difficult to study as the translational and orientation workspace are coupled [[cite:&merlet02_still]]. -The analysis is significantly simplified when considering small motions, as the Jacobian matrix can be used to link the strut motion to the motion of frame $\{B\}$ with respect to $\{A\}$ through eqref:eq:detail_kinematics_jacobian, which is a linear equation. +Mobility of parallel manipulators is inherently difficult to study as the translational and orientation workspace are coupled\nbsp{}[[cite:&merlet02_still]]. +The analysis is significantly simplified when considering small motions, as the Jacobian matrix can be used to link the strut motion to the motion of frame $\{B\}$ with respect to $\{A\}$ through\nbsp{}eqref:eq:detail_kinematics_jacobian, which is a linear equation. \begin{equation}\label{eq:detail_kinematics_jacobian} \begin{bmatrix} \delta l_1 \\ \delta l_2 \\ \delta l_3 \\ \delta l_4 \\ \delta l_5 \\ \delta l_6 \end{bmatrix} = \underbrace{\begin{bmatrix} @@ -5712,7 +5712,7 @@ More specifically, the XYZ mobility only depends on the $\hat{\bm{s}}_i$ (orient For simplicity, only translations are first considered (i.e., the Stewart platform is considered to have fixed orientation). In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions. The faces are therefore perpendicular to the strut direction. -The obtained mobility for the Stewart platform geometry shown in Figure ref:fig:detail_kinematics_mobility_trans_arch is computed and represented in Figure ref:fig:detail_kinematics_mobility_trans_result. +The obtained mobility for the Stewart platform geometry shown in Figure\nbsp{}ref:fig:detail_kinematics_mobility_trans_arch is computed and represented in Figure\nbsp{}ref:fig:detail_kinematics_mobility_trans_result. #+name: fig:detail_kinematics_mobility_trans #+caption: One Stewart platform geometry (\subref{fig:detail_kinematics_mobility_trans_arch}) and its associated translational mobility (\subref{fig:detail_kinematics_mobility_trans_result}). A sphere with radius equal to the strut stroke is contained in the translational mobility shape. @@ -5732,11 +5732,11 @@ The obtained mobility for the Stewart platform geometry shown in Figure ref:fig: #+end_subfigure #+end_figure -With the previous interpretations of the 12 faces making the translational mobility 3D shape, it can be concluded that for a strut stroke of $\pm d$, a sphere with radius $d$ is contained in the 3D shape and touches it in directions defined by the strut axes, as illustrated in Figure ref:fig:detail_kinematics_mobility_trans_result. +With the previous interpretations of the 12 faces making the translational mobility 3D shape, it can be concluded that for a strut stroke of $\pm d$, a sphere with radius $d$ is contained in the 3D shape and touches it in directions defined by the strut axes, as illustrated in Figure\nbsp{}ref:fig:detail_kinematics_mobility_trans_result. This means that the mobile platform can be translated in any direction with a stroke equal to the strut stroke. -To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared with struts oriented vertically (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts) and struts oriented horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). -The vertically oriented struts configuration leads to greater stroke in the horizontal direction and reduced stroke in the vertical direction (Figure ref:fig:detail_kinematics_mobility_translation_strut_orientation). +To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared with struts oriented vertically (Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_vert_struts) and struts oriented horizontally (Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_hori_struts). +The vertically oriented struts configuration leads to greater stroke in the horizontal direction and reduced stroke in the vertical direction (Figure\nbsp{}ref:fig:detail_kinematics_mobility_translation_strut_orientation). Conversely, horizontal oriented struts configuration provides more stroke in the vertical direction. It may seem counterintuitive that less stroke is available in the direction of the struts. @@ -5769,13 +5769,13 @@ The amplification factor increases when the struts have a high angle with the di ***** Mobility in rotation -As shown by equation eqref:eq:detail_kinematics_jacobian, the rotational mobility depends both on the orientation of the struts and on the location of the top joints. +As shown by equation\nbsp{}eqref:eq:detail_kinematics_jacobian, the rotational mobility depends both on the orientation of the struts and on the location of the top joints. Similarly to the translational case, to increase the rotational mobility in one direction, it is advantageous to have the struts more perpendicular to the rotational direction. -For instance, having the struts more vertical (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts) provides less rotational stroke along the vertical direction than having the struts oriented more horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). +For instance, having the struts more vertical (Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_vert_struts) provides less rotational stroke along the vertical direction than having the struts oriented more horizontally (Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_hori_struts). -Two cases are considered with the same strut orientation but with different top joint positions: struts positioned close to each other (Figure ref:fig:detail_kinematics_stewart_mobility_close_struts) and struts positioned further apart (Figure ref:fig:detail_kinematics_stewart_mobility_space_struts). -The mobility for pure rotations is compared in Figure ref:fig:detail_kinematics_mobility_angle_strut_distance. +Two cases are considered with the same strut orientation but with different top joint positions: struts positioned close to each other (Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_close_struts) and struts positioned further apart (Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_space_struts). +The mobility for pure rotations is compared in Figure\nbsp{}ref:fig:detail_kinematics_mobility_angle_strut_distance. Having struts further apart decreases the "lever arm" and therefore reduces the rotational mobility. #+name: fig:detail_kinematics_stewart_mobility_rotation_examples @@ -5806,23 +5806,23 @@ Having struts further apart decreases the "lever arm" and therefore reduces the It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex. For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke. -This analysis is conducted in Section ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the nano-hexapod geometry. +This analysis is conducted in Section\nbsp{}ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the nano-hexapod geometry. **** Stiffness <> ***** Introduction :ignore: The stiffness matrix defines how the top platform of the Stewart platform (i.e. frame $\{B\}$) deforms with respect to its fixed base (i.e. frame $\{A\}$) due to static forces/torques applied between frames $\{A\}$ and $\{B\}$. -It depends on the Jacobian matrix (i.e., the geometry) and the strut axial stiffness as shown in equation eqref:eq:detail_kinematics_stiffness_matrix. +It depends on the Jacobian matrix (i.e., the geometry) and the strut axial stiffness as shown in equation\nbsp{}eqref:eq:detail_kinematics_stiffness_matrix. The contribution of joints stiffness is not considered here, as the joints were optimized after the geometry was fixed. -However, theoretical frameworks for evaluating flexible joint contribution to the stiffness matrix have been established in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]]. +However, theoretical frameworks for evaluating flexible joint contribution to the stiffness matrix have been established in the literature \nbsp{}[[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]]. \begin{equation}\label{eq:detail_kinematics_stiffness_matrix} \bm{K} = \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J} \end{equation} It is assumed that the stiffness of all struts is the same: $\bm{\mathcal{K}} = k \cdot \mathbf{I}_6$. -In that case, the obtained stiffness matrix linearly depends on the strut stiffness $k$, and is structured as shown in equation eqref:eq:detail_kinematics_stiffness_matrix_simplified. +In that case, the obtained stiffness matrix linearly depends on the strut stiffness $k$, and is structured as shown in equation\nbsp{}eqref:eq:detail_kinematics_stiffness_matrix_simplified. \begin{equation}\label{eq:detail_kinematics_stiffness_matrix_simplified} \bm{K} = k \bm{J}^{\intercal} \bm{J} = @@ -5837,43 +5837,43 @@ In that case, the obtained stiffness matrix linearly depends on the strut stiffn ***** Translation Stiffness -As shown by equation eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the $3 \times 3$ top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal}$. +As shown by equation\nbsp{}eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the $3 \times 3$ top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal}$. In the extreme case where all struts are vertical ($s_i = [0\ 0\ 1]$), a vertical stiffness of $6k$ is achieved, but with null stiffness in the horizontal directions. If two struts are aligned with the X axis, two struts with the Y axis, and two struts with the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} = 2 \bm{I}_3$, resulting in well-distributed stiffness along all directions. -This configuration corresponds to the cubic architecture presented in Section ref:sec:detail_kinematics_cubic. +This configuration corresponds to the cubic architecture presented in Section\nbsp{}ref:sec:detail_kinematics_cubic. -When the struts are oriented more vertically, as shown in Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts, the vertical stiffness increases while the horizontal stiffness decreases. +When the struts are oriented more vertically, as shown in Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_vert_struts, the vertical stiffness increases while the horizontal stiffness decreases. Additionally, $R_x$ and $R_y$ stiffness increases while $R_z$ stiffness decreases. -The opposite conclusions apply if struts are oriented more horizontally, illustrated in Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts. +The opposite conclusions apply if struts are oriented more horizontally, illustrated in Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_hori_struts. ***** Rotational Stiffness The rotational stiffnesses depend both on the orientation of the struts and on the location of the top joints with respect to the considered center of rotation (i.e., the location of frame $\{A\}$). With the same orientation but increased distances to the frame $\{A\}$ by a factor of 2, the rotational stiffness is increased by a factor of 4. -Therefore, the compact Stewart platform depicted in Figure ref:fig:detail_kinematics_stewart_mobility_close_struts has less rotational stiffness than the Stewart platform shown in Figure ref:fig:detail_kinematics_stewart_mobility_space_struts. +Therefore, the compact Stewart platform depicted in Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_close_struts has less rotational stiffness than the Stewart platform shown in Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_space_struts. ***** Diagonal Stiffness Matrix Having a diagonal stiffness matrix $\bm{K}$ can be beneficial for control purposes as it would make the plant in the Cartesian frame decoupled at low frequency. This property depends on both the geometry and the chosen $\{A\}$ frame. For specific geometry and choice of $\{A\}$ frame, it is possible to achieve a diagonal $K$ matrix. -This is discussed in Section ref:ssec:detail_kinematics_cubic_static. +This is discussed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_static. **** Dynamical properties <> The dynamical equations (both in the Cartesian frame and in the frame of the struts) for the Stewart platform were derived during the conceptual phase with simplifying assumptions (massless struts and perfect joints). The dynamics depends both on the geometry (Jacobian matrix) and on the payload being placed on top of the platform. -# Section ref:ssec:nhexa_stewart_platform_dynamics (page pageref:ssec:nhexa_stewart_platform_dynamics). +# Section\nbsp{}ref:ssec:nhexa_stewart_platform_dynamics (page\nbsp{}pageref:ssec:nhexa_stewart_platform_dynamics). -Under very specific conditions, the equations of motion in the Cartesian frame, given by equation eqref:eq:detail_kinematics_transfer_function_cart, can be decoupled. -These conditions are studied in Section ref:ssec:detail_kinematics_cubic_dynamic. +Under very specific conditions, the equations of motion in the Cartesian frame, given by equation\nbsp{}eqref:eq:detail_kinematics_transfer_function_cart, can be decoupled. +These conditions are studied in Section\nbsp{}ref:ssec:detail_kinematics_cubic_dynamic. \begin{equation}\label{eq:detail_kinematics_transfer_function_cart} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{\intercal} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J} )^{-1} \end{equation} -In the frame of the struts, the equations of motion eqref:eq:detail_kinematics_transfer_function_struts are well decoupled at low frequency. +In the frame of the struts, the equations of motion\nbsp{}eqref:eq:detail_kinematics_transfer_function_struts are well decoupled at low frequency. This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is well decoupled and SISO control may be applied for each strut, independently of the payload being used. \begin{equation}\label{eq:detail_kinematics_transfer_function_struts} @@ -5881,11 +5881,11 @@ This is why most Stewart platforms are controlled in the frame of the struts: be \end{equation} Coupling between sensors (force sensors, relative position sensors or inertial sensors) in different struts may also be important for decentralized control. -In section ref:ssec:detail_kinematics_decentralized_control, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts. +In section\nbsp{}ref:ssec:detail_kinematics_decentralized_control, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts. **** Conclusion -The effects of two changes in the manipulator's geometry, namely the position and orientation of the struts, are summarized in Table ref:tab:detail_kinematics_geometry. +The effects of two changes in the manipulator's geometry, namely the position and orientation of the struts, are summarized in Table\nbsp{}ref:tab:detail_kinematics_geometry. These results could have been easily deduced based on mechanical principles, but thanks to the kinematic analysis, they can be quantified. These trade-offs provide important guidelines when choosing the Stewart platform geometry. @@ -5909,12 +5909,12 @@ These trade-offs provide important guidelines when choosing the Stewart platform <> **** Introduction :ignore: -The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart [[cite:&stewart65_platf_with_six_degrees_freed]]. -This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure ref:fig:detail_kinematics_cubic_architecture_example. +The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart\nbsp{}[[cite:&stewart65_platf_with_six_degrees_freed]]. +This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_architecture_example. -Typically, the struts have similar length to the cube's edges, as illustrated in Figure ref:fig:detail_kinematics_cubic_architecture_example. -Practical implementations of such configurations can be observed in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_uqp. -It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure ref:fig:detail_kinematics_cubic_architecture_example_small), as exemplified in Figure ref:fig:detail_kinematics_ulb_pz. +Typically, the struts have similar length to the cube's edges, as illustrated in Figure\nbsp{}ref:fig:detail_kinematics_cubic_architecture_example. +Practical implementations of such configurations can be observed in Figures\nbsp{}ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_uqp. +It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure\nbsp{}ref:fig:detail_kinematics_cubic_architecture_example_small), as exemplified in Figure\nbsp{}ref:fig:detail_kinematics_ulb_pz. #+name: fig:detail_kinematics_cubic_architecture_examples #+caption: Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}). @@ -5934,21 +5934,21 @@ It is also possible to implement designs with strut lengths smaller than the cub #+end_subfigure #+end_figure -Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption [[cite:&geng94_six_degree_of_freed_activ;&preumont07_six_axis_singl_stage_activ;&jafari03_orthog_gough_stewar_platf_microm]]: simplified kinematics relationships and dynamical analysis [[cite:&geng94_six_degree_of_freed_activ]]; uniform stiffness in all directions [[cite:&hanieh03_activ_stewar]]; uniform mobility [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt.8.5.2]]; and minimization of the cross coupling between actuators and sensors in different struts [[cite:&preumont07_six_axis_singl_stage_activ]]. -This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control [[cite:&geng94_six_degree_of_freed_activ;&thayer02_six_axis_vibrat_isolat_system]]. +Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption\nbsp{}[[cite:&geng94_six_degree_of_freed_activ;&preumont07_six_axis_singl_stage_activ;&jafari03_orthog_gough_stewar_platf_microm]]: simplified kinematics relationships and dynamical analysis\nbsp{}[[cite:&geng94_six_degree_of_freed_activ]]; uniform stiffness in all directions\nbsp{}[[cite:&hanieh03_activ_stewar]]; uniform mobility\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt.8.5.2]]; and minimization of the cross coupling between actuators and sensors in different struts\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. +This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control\nbsp{}[[cite:&geng94_six_degree_of_freed_activ;&thayer02_six_axis_vibrat_isolat_system]]. These properties are examined in this section to assess their relevance for the nano-hexapod. -The mobility and stiffness properties of the cubic configuration are analyzed in Section ref:ssec:detail_kinematics_cubic_static. -Dynamical decoupling is investigated in Section ref:ssec:detail_kinematics_cubic_dynamic, while decentralized control, crucial for the NASS, is examined in Section ref:ssec:detail_kinematics_decentralized_control. -Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section ref:ssec:detail_kinematics_cubic_design. +The mobility and stiffness properties of the cubic configuration are analyzed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_static. +Dynamical decoupling is investigated in Section\nbsp{}ref:ssec:detail_kinematics_cubic_dynamic, while decentralized control, crucial for the NASS, is examined in Section\nbsp{}ref:ssec:detail_kinematics_decentralized_control. +Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_design. The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod. **** Static Properties <> ***** Stiffness matrix for the Cubic architecture -Consider the cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_schematic_full. -The unit vectors corresponding to the edges of the cube are described by equation eqref:eq:detail_kinematics_cubic_s. +Consider the cubic architecture shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_schematic_full. +The unit vectors corresponding to the edges of the cube are described by equation\nbsp{}eqref:eq:detail_kinematics_cubic_s. \begin{equation}\label{eq:detail_kinematics_cubic_s} \hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad @@ -5977,7 +5977,7 @@ The unit vectors corresponding to the edges of the cube are described by equatio #+end_subfigure #+end_figure -Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation eqref:eq:detail_kinematics_cubic_vertices. +Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation\nbsp{}eqref:eq:detail_kinematics_cubic_vertices. \begin{equation}\label{eq:detail_kinematics_cubic_vertices} \tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad @@ -5985,7 +5985,7 @@ Coordinates of the cube's vertices relevant for the top joints, expressed with r \tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix} \end{equation} -In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation eqref:eq:detail_kinematics_cubic_stiffness. +In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation\nbsp{}eqref:eq:detail_kinematics_cubic_stiffness. Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size $H_c$. \begin{equation}\label{eq:detail_kinematics_cubic_stiffness} @@ -5999,8 +5999,8 @@ Translation stiffness is twice the stiffness of the struts, and rotational stiff \end{bmatrix} \end{equation} -However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure ref:fig:detail_kinematics_cubic_schematic). -In that case, the location of the top joints can be expressed by equation eqref:eq:detail_kinematics_cubic_edges, yet the computed stiffness matrix remains identical to Equation eqref:eq:detail_kinematics_cubic_stiffness. +However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure\nbsp{}ref:fig:detail_kinematics_cubic_schematic). +In that case, the location of the top joints can be expressed by equation\nbsp{}eqref:eq:detail_kinematics_cubic_edges, yet the computed stiffness matrix remains identical to Equation\nbsp{}eqref:eq:detail_kinematics_cubic_stiffness. \begin{equation}\label{eq:detail_kinematics_cubic_edges} \bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i @@ -6014,7 +6014,7 @@ This specific location where the stiffness matrix is diagonal is referred to as When the reference frames $\{A\}$ and $\{B\}$ are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix. -Considering a vertical shift as shown in Figure ref:fig:detail_kinematics_cubic_schematic, the stiffness matrix transforms into that shown in Equation eqref:eq:detail_kinematics_cubic_stiffness_off_centered. +Considering a vertical shift as shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_schematic, the stiffness matrix transforms into that shown in Equation\nbsp{}eqref:eq:detail_kinematics_cubic_stiffness_off_centered. Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered $\{B\}$ frame. \begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered} @@ -6033,19 +6033,19 @@ Therefore, the stiffness characteristics of the cubic architecture are only dist This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform. It should be noted that for the stiffness matrix to be diagonal, the cube's center doesn't need to coincide with the geometric center of the Stewart platform. -This observation leads to the interesting alternative architectures presented in Section ref:ssec:detail_kinematics_cubic_design. +This observation leads to the interesting alternative architectures presented in Section\nbsp{}ref:ssec:detail_kinematics_cubic_design. ***** Uniform Mobility The translational mobility of the Stewart platform with constant orientation was analyzed. Considering limited actuator stroke (elongation of each strut), the maximum achievable positions in XYZ space were estimated. -The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_translations. +The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure\nbsp{}ref:fig:detail_kinematics_cubic_mobility_translations. The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke. -These findings suggest that the mobility pattern is more subtle than sometimes described in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions. -This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure ref:fig:detail_kinematics_mobility_trans. +These findings suggest that the mobility pattern is more subtle than sometimes described in the literature\nbsp{}[[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions. +This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure\nbsp{}ref:fig:detail_kinematics_mobility_trans. -The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibits greater achievable angular stroke in the $R_x$ and $R_y$ directions compared to the $R_z$ direction. +The rotational mobility, illustrated in Figure\nbsp{}ref:fig:detail_kinematics_cubic_mobility_rotations, exhibits greater achievable angular stroke in the $R_x$ and $R_y$ directions compared to the $R_z$ direction. 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 @@ -6071,7 +6071,7 @@ Furthermore, an inverse relationship exists between the cube's dimension and rot ***** Introduction :ignore: This section examines the dynamics of the cubic architecture in the Cartesian frame which corresponds to the transfer function from forces and torques $\bm{\mathcal{F}}$ to translations and rotations $\bm{\mathcal{X}}$ of the top platform. -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 ref:fig:detail_kinematics_centralized_control. +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 @@ -6079,16 +6079,16 @@ When relative motion sensors are integrated in each strut (measuring $\bm{\mathc ***** Low frequency and High frequency coupling -As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation eqref:eq:detail_kinematics_transfer_function_cart. -At low frequency, the behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq. +As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation\nbsp{}eqref:eq:detail_kinematics_transfer_function_cart. +At low frequency, the behavior of the platform depends on the stiffness matrix\nbsp{}eqref:eq:detail_kinematics_transfer_function_cart_low_freq. \begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1} \end{equation} -In Section ref:ssec:detail_kinematics_cubic_static, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame $\{B\}$ is positioned at the cube's center. +In Section\nbsp{}ref:ssec:detail_kinematics_cubic_static, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame $\{B\}$ is positioned at the cube's center. In this case, the "Cartesian" plant is decoupled at low frequency. -At high frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$) eqref:eq:detail_kinematics_transfer_function_high_freq. +At high frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$)\nbsp{}eqref:eq:detail_kinematics_transfer_function_high_freq. \begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1} @@ -6101,10 +6101,10 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m #+attr_latex: :width 0.6\linewidth [[file:figs/detail_kinematics_cubic_payload.png]] -To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure ref:fig:detail_kinematics_cubic_payload). +To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure\nbsp{}ref:fig:detail_kinematics_cubic_payload). Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames. -When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com). -Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_cok). +When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_com). +Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_cok). #+name: fig:detail_kinematics_cubic_cart_coupling #+caption: Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the $\{B\}$ frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}). @@ -6126,12 +6126,12 @@ Conversely, when positioned at the center of stiffness, coupling occurred at hig ***** Payload's CoM at the cube's center -An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components [[cite:&li01_simul_fault_vibrat_isolat_point]]. -This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure ref:fig:detail_kinematics_cubic_centered_payload). -This approach was physically implemented in several studies [[cite:&mcinroy99_dynam;&jafari03_orthog_gough_stewar_platf_microm]], as shown in Figure ref:fig:detail_kinematics_uw_gsp. -The resulting dynamics are indeed well-decoupled (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com_cok), taking advantage from diagonal stiffness and mass matrices. +An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components\nbsp{}[[cite:&li01_simul_fault_vibrat_isolat_point]]. +This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload). +This approach was physically implemented in several studies\nbsp{}[[cite:&mcinroy99_dynam;&jafari03_orthog_gough_stewar_platf_microm]], as shown in Figure\nbsp{}ref:fig:detail_kinematics_uw_gsp. +The resulting dynamics are indeed well-decoupled (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_com_cok), taking advantage from diagonal stiffness and mass matrices. The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform. -If a design similar to Figure ref:fig:detail_kinematics_cubic_centered_payload were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation. +If a design similar to Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation. #+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}) @@ -6165,15 +6165,15 @@ While this configuration offers powerful control advantages, it requires positio The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control. Two sensor types integrated in the struts are considered: displacement sensors and force sensors. -The control architecture is illustrated in Figure ref:fig:detail_kinematics_decentralized_control, where $\bm{K}_{\mathcal{L}}$ represents a diagonal transfer function matrix. +The control architecture is illustrated in Figure\nbsp{}ref:fig:detail_kinematics_decentralized_control, where $\bm{K}_{\mathcal{L}}$ represents a diagonal transfer function matrix. #+name: fig:detail_kinematics_decentralized_control #+caption: Decentralized control in the frame of the struts. [[file:figs/detail_kinematics_decentralized_control.png]] The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms. -The first employs a cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_payload. -The second uses a non-cubic Stewart platform shown in Figure 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. +The first employs a cubic architecture shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_payload. +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 @@ -6182,11 +6182,11 @@ The second uses a non-cubic Stewart platform shown in Figure ref:fig:detail_kine ***** Relative Displacement Sensors -The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure ref:fig:detail_kinematics_decentralized_dL. -As anticipated from the equations of motion from $\bm{f}$ to $\bm{\mathcal{L}}$ eqref:eq:detail_kinematics_transfer_function_struts, the $6 \times 6$ plant is decoupled at low frequency. +The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure\nbsp{}ref:fig:detail_kinematics_decentralized_dL. +As anticipated from the equations of motion from $\bm{f}$ to $\bm{\mathcal{L}}$\nbsp{}eqref:eq:detail_kinematics_transfer_function_struts, the $6 \times 6$ plant is decoupled at low frequency. At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal. -No significant advantage is evident for the cubic architecture (Figure ref:fig:detail_kinematics_cubic_decentralized_dL) compared to the non-cubic architecture (Figure ref:fig:detail_kinematics_non_cubic_decentralized_dL). +No significant advantage is evident for the cubic architecture (Figure\nbsp{}ref:fig:detail_kinematics_cubic_decentralized_dL) compared to the non-cubic architecture (Figure\nbsp{}ref:fig:detail_kinematics_non_cubic_decentralized_dL). 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 @@ -6210,7 +6210,7 @@ The resonance frequencies differ between the two cases because the more vertical ***** Force Sensors Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms. -The results are presented in Figure ref:fig:detail_kinematics_decentralized_fn. +The results are presented in Figure\nbsp{}ref:fig:detail_kinematics_decentralized_fn. 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 @@ -6240,27 +6240,27 @@ Both the cubic and non-cubic configurations exhibited similar coupling character <> ***** Introduction :ignore: -As demonstrated in Section ref:ssec:detail_kinematics_cubic_dynamic, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. -As shown in Section ref:ssec:detail_kinematics_cubic_static, the stiffness matrix is diagonal when the considered $\{B\}$ frame is located at the cube's center. +As demonstrated in Section\nbsp{}ref:ssec:detail_kinematics_cubic_dynamic, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. +As shown in Section\nbsp{}ref:ssec:detail_kinematics_cubic_static, the stiffness matrix is diagonal when the considered $\{B\}$ frame is located at the cube's center. However, the $\{B\}$ frame is typically positioned above the top platform where forces are applied and displacements are measured. This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration. -Three key parameters define the geometry of the cubic Stewart platform: $H$, the height of the Stewart platform (distance from fixed base to mobile platform); $H_c$, the height of the cube, as shown in Figure ref:fig:detail_kinematics_cubic_schematic_full; and $H_{CoM}$, the height of the center of mass relative to the mobile platform (coincident with the cube's center). +Three key parameters define the geometry of the cubic Stewart platform: $H$, the height of the Stewart platform (distance from fixed base to mobile platform); $H_c$, the height of the cube, as shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_schematic_full; and $H_{CoM}$, the height of the center of mass relative to the mobile platform (coincident with the cube's center). Depending on the cube's size $H_c$ in relation to $H$ and $H_{CoM}$, different designs emerge. In the following examples, $H = 100\,mm$ and $H_{CoM} = 20\,mm$. ***** Small cube -When the cube size $H_c$ is smaller than twice the height of the CoM $H_{CoM}$ eqref:eq:detail_kinematics_cube_small, the resulting design is shown in Figure ref:fig:detail_kinematics_cubic_above_small. +When the cube size $H_c$ is smaller than twice the height of the CoM $H_{CoM}$\nbsp{}eqref:eq:detail_kinematics_cube_small, the resulting design is shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_above_small. \begin{equation}\label{eq:detail_kinematics_cube_small} H_c < 2 H_{CoM} \end{equation} -# TODO - Add link to Figure ref:fig:nhexa_stewart_piezo_furutani (page pageref:fig:nhexa_stewart_piezo_furutani) -This configuration is similar to that described in [[cite:&furutani04_nanom_cuttin_machin_using_stewar]], although they do not explicitly identify it as a cubic configuration. +# TODO - Add link to Figure\nbsp{}ref:fig:nhexa_stewart_piezo_furutani (page\nbsp{}pageref:fig:nhexa_stewart_piezo_furutani) +This configuration is similar to that described in\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar]], although they do not explicitly identify it as a cubic configuration. Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other. This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness. @@ -6291,13 +6291,13 @@ This approach yields a compact architecture, but the small cube size may result ***** Medium sized cube -Increasing the cube's size such that eqref:eq:detail_kinematics_cube_medium is verified produces an architecture with intersecting struts (Figure ref:fig:detail_kinematics_cubic_above_medium). +Increasing the cube's size such that\nbsp{}eqref:eq:detail_kinematics_cube_medium is verified produces an architecture with intersecting struts (Figure\nbsp{}ref:fig:detail_kinematics_cubic_above_medium). \begin{equation}\label{eq:detail_kinematics_cube_medium} 2 H_{CoM} < H_c < 2 (H_{CoM} + H) \end{equation} -This configuration resembles the design proposed in [[cite:&yang19_dynam_model_decoup_contr_flexib]] (Figure ref:fig:detail_kinematics_yang19), although their design is not strictly cubic. +This configuration resembles the design proposed in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]] (Figure\nbsp{}ref:fig:detail_kinematics_yang19), although their design is not strictly cubic. #+name: fig:detail_kinematics_cubic_above_medium #+caption: Cubic architecture with cube's center above the top platform. A cube height of 140mm is used. @@ -6325,7 +6325,7 @@ This configuration resembles the design proposed in [[cite:&yang19_dynam_model_d ***** Large cube -When the cube's height exceeds twice the sum of the platform height and CoM height eqref:eq:detail_kinematics_cube_large, the architecture shown in Figure ref:fig:detail_kinematics_cubic_above_large is obtained. +When the cube's height exceeds twice the sum of the platform height and CoM height\nbsp{}eqref:eq:detail_kinematics_cube_large, the architecture shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_above_large is obtained. \begin{equation}\label{eq:detail_kinematics_cube_large} 2 (H_{CoM} + H) < H_c @@ -6357,7 +6357,7 @@ When the cube's height exceeds twice the sum of the platform height and CoM heig ***** Platform size -For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom joints $\bm{a}_i$) and are positioned on a circle with radius $R_{b_i}$ (resp. $R_{a_i}$) described by Equation eqref:eq:detail_kinematics_cube_joints. +For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom joints $\bm{a}_i$) and are positioned on a circle with radius $R_{b_i}$ (resp. $R_{a_i}$) described by Equation\nbsp{}eqref:eq:detail_kinematics_cube_joints. \begin{subequations}\label{eq:detail_kinematics_cube_joints} \begin{align} @@ -6366,8 +6366,8 @@ For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom join \end{align} \end{subequations} -Since the rotational stiffness for the cubic architecture scales with the square of the cube's height eqref:eq:detail_kinematics_cubic_stiffness, the cube's size can be determined based on rotational stiffness requirements. -Subsequently, using Equation eqref:eq:detail_kinematics_cube_joints, the dimensions of the top and bottom platforms can be calculated. +Since the rotational stiffness for the cubic architecture scales with the square of the cube's height\nbsp{}eqref:eq:detail_kinematics_cubic_stiffness, the cube's size can be determined based on rotational stiffness requirements. +Subsequently, using Equation\nbsp{}eqref:eq:detail_kinematics_cube_joints, the dimensions of the top and bottom platforms can be calculated. **** Conclusion @@ -6423,7 +6423,7 @@ Regarding dynamical properties, particularly for control in the frame of the str Consequently, the geometry was selected according to practical constraints. The height between the two plates is maximized and set at $95\,mm$. Both platforms utilize the maximum available size, with joints offset by $15\,mm$ from the plate surfaces and positioned along circles with radii of $120\,mm$ for the fixed joints and $110\,mm$ for the mobile joints. -The positioning angles, as shown in Figure ref:fig:detail_kinematics_nano_hexapod_top, are $[255,\ 285,\ 15,\ 45,\ 135,\ 165]$ degrees for the top joints and $[220,\ 320,\ 340,\ 80,\ 100,\ 200]$ degrees for the bottom joints. +The positioning angles, as shown in Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_top, are $[255,\ 285,\ 15,\ 45,\ 135,\ 165]$ degrees for the top joints and $[220,\ 320,\ 340,\ 80,\ 100,\ 200]$ degrees for the bottom joints. #+name: fig:detail_kinematics_nano_hexapod #+caption: Obtained architecture for the Nano Hexapod @@ -6443,12 +6443,12 @@ The positioning angles, as shown in Figure ref:fig:detail_kinematics_nano_hexapo #+end_subfigure #+end_figure -The resulting geometry is illustrated in Figure ref:fig:detail_kinematics_nano_hexapod. +The resulting geometry is illustrated in Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod. While minor refinements may occur during detailed mechanical design to address manufacturing and assembly considerations, the fundamental geometry will remain consistent with this configuration. -This geometry serves as the foundation for estimating required actuator stroke (Section ref:ssec:detail_kinematics_nano_hexapod_actuator_stroke), determining flexible joint stroke requirements (Section ref:ssec:detail_kinematics_nano_hexapod_joint_stroke), performing noise budgeting for instrumentation selection, and developing control strategies. +This geometry serves as the foundation for estimating required actuator stroke (Section\nbsp{}ref:ssec:detail_kinematics_nano_hexapod_actuator_stroke), determining flexible joint stroke requirements (Section\nbsp{}ref:ssec:detail_kinematics_nano_hexapod_joint_stroke), performing noise budgeting for instrumentation selection, and developing control strategies. # TODO - Add link to sections -Implementing a cubic architecture as proposed in Section ref:ssec:detail_kinematics_cubic_design was considered. +Implementing a cubic architecture as proposed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_design was considered. However, positioning the cube's center $150\,mm$ above the top platform would have resulted in platform dimensions exceeding the maximum available size. Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the nano-hexapod, ensuring that its center of mass coincides with the cube's center. Given the impracticality of consistently aligning the center of mass with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. @@ -6465,7 +6465,7 @@ Calculations based on the selected geometry indicate that an actuator stroke of This specification will be used during the actuator selection process. # TODO - Add link to section -Figure ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of $\pm 94\,\mu m$. +Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of $\pm 94\,\mu m$. The diagram confirms that the required workspace fits within the system's capabilities. #+name: fig:detail_kinematics_nano_hexapod_mobility @@ -6505,8 +6505,8 @@ This led to a practical design approach where struts were oriented more vertical During the nano-hexapod's detailed design phase, a hybrid modeling approach combining finite element analysis with multi-body dynamics was developed. This methodology, utilizing reduced-order flexible bodies, was created to enable both detailed component optimization and efficient system-level simulation, addressing the impracticality of a full FEM for real-time control scenarios. -The theoretical foundations and implementation are presented in Section ref:sec:detail_fem_super_element, where experimental validation was performed using an Amplified Piezoelectric Actuator. -The framework was then applied to optimize two critical nano-hexapod elements: the actuators (Section ref:sec:detail_fem_actuator) and the flexible joints (Section ref:sec:detail_fem_joint). +The theoretical foundations and implementation are presented in Section\nbsp{}ref:sec:detail_fem_super_element, where experimental validation was performed using an Amplified Piezoelectric Actuator. +The framework was then applied to optimize two critical nano-hexapod elements: the actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and the flexible joints (Section\nbsp{}ref:sec:detail_fem_joint). Through this approach, system-level dynamic behavior under closed-loop control conditions could be successfully predicted while detailed component-level optimization was facilitated. *** Reduced order flexible bodies @@ -6515,13 +6515,13 @@ Through this approach, system-level dynamic behavior under closed-loop control c Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models. These components are traditionally analyzed using Finite Element Analysis (FEA) software. -However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models [[cite:&hatch00_vibrat_matlab_ansys]]. -This combined multibody-FEA modeling approach presents significant advantages, as it enables the accurate FE modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system [[cite:&rankers98_machin]]. +However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models\nbsp{}[[cite:&hatch00_vibrat_matlab_ansys]]. +This combined multibody-FEA modeling approach presents significant advantages, as it enables the accurate FE modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system\nbsp{}[[cite:&rankers98_machin]]. The investigation of this hybrid modeling approach is structured in three sections. -First, the fundamental principles and methodological approaches of this modeling framework are introduced (Section ref:ssec:detail_fem_super_element_theory). -It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section ref:ssec:detail_fem_super_element_example). -Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section ref:ssec:detail_fem_super_element_validation). +First, the fundamental principles and methodological approaches of this modeling framework are introduced (Section\nbsp{}ref:ssec:detail_fem_super_element_theory). +It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section\nbsp{}ref:ssec:detail_fem_super_element_example). +Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section\nbsp{}ref:ssec:detail_fem_super_element_validation). **** Procedure <> @@ -6537,9 +6537,9 @@ Initially, the component is modeled in a finite element software with appropriat Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component. These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames. -Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method [[cite:&craig68_coupl_subst_dynam_analy]] (also known as the "fixed-interface method"), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics. +Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method\nbsp{}[[cite:&craig68_coupl_subst_dynam_analy]] (also known as the "fixed-interface method"), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics. This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF. -The number of degrees of freedom in the reduced model is determined by eqref:eq:detail_fem_model_order where $n$ represents the number of defined frames and $p$ denotes the number of additional modes to be modeled. +The number of degrees of freedom in the reduced model is determined by\nbsp{}eqref:eq:detail_fem_model_order where $n$ represents the number of defined frames and $p$ denotes the number of additional modes to be modeled. The outcome of this procedure is an $m \times m$ set of reduced mass and stiffness matrices, $m$ being the total retained number of degrees of freedom, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. \begin{equation}\label{eq:detail_fem_model_order} @@ -6551,10 +6551,10 @@ m = 6 \times n + p ***** Introduction :ignore: The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons. -Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section ref:sec:detail_fem_actuator. -Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure ref:fig:detail_fem_apa95ml_picture) was available in the laboratory for experimental testing. +Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section\nbsp{}ref:sec:detail_fem_actuator. +Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_picture) was available in the laboratory for experimental testing. -The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure ref:fig:detail_fem_apa95ml_picture) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement in the vertical direction [[cite:&claeyssen07_amplif_piezoel_actuat]]. +The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure\nbsp{}ref:fig:detail_fem_apa95ml_picture) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement in the vertical direction\nbsp{}[[cite:&claeyssen07_amplif_piezoel_actuat]]. The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework. The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation. @@ -6581,8 +6581,8 @@ The specific design of the APA allows for the simultaneous modeling of a mechani ***** Finite Element Model -The development of the finite element model for the APA95ML required the knowledge of the material properties, as summarized in Table ref:tab:detail_fem_material_properties. -The finite element mesh, shown in Figure ref:fig:detail_fem_apa95ml_mesh, was then generated. +The development of the finite element model for the APA95ML required the knowledge of the material properties, as summarized in Table\nbsp{}ref:tab:detail_fem_material_properties. +The 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 modal reduction model. $E$ is the Young's modulus, $\nu$ the Poisson ratio and $\rho$ the material density @@ -6624,28 +6624,28 @@ This block has several interface frames corresponding to the ones defined in the Frame $\{4\}$ was connected to the "world" frame, while frame $\{6\}$ was coupled to a vertically guided payload. In this example, two piezoelectric stacks were used for actuation while one piezoelectric stack was used as a force sensor. Therefore, a force source $F_a$ operating between frames $\{3\}$ and $\{2\}$ was used, while a displacement sensor $d_L$ between frames $\{1\}$ and $\{7\}$ was used for the sensor stack. -This is illustrated in Figure ref:fig:detail_fem_apa_model_schematic. +This is illustrated in Figure\nbsp{}ref:fig:detail_fem_apa_model_schematic. However, to have access to the physical voltage input of the actuators stacks $V_a$ and to the generated voltage by the force sensor $V_s$, conversion between the electrical and mechanical domains need to be determined. ***** Sensor and Actuator "constants" -To link the electrical domain to the mechanical domain, an "actuator constant" $g_a$ and a "sensor constant" $g_s$ were introduced as shown in Figure ref:fig:detail_fem_apa_model_schematic. +To link the electrical domain to the mechanical domain, an "actuator constant" $g_a$ and a "sensor constant" $g_s$ were introduced as shown in Figure\nbsp{}ref:fig:detail_fem_apa_model_schematic. -From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:detail_fem_dl_to_vs. +From\nbsp{}[[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by\nbsp{}eqref:eq:detail_fem_dl_to_vs. \begin{equation}\label{eq:detail_fem_dl_to_vs} V_s = g_s \cdot d_L, \quad g_s = \frac{d_{33}}{\epsilon^T s^D n} \end{equation} -From [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:detail_fem_va_to_fa. +From\nbsp{}[[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by\nbsp{}eqref:eq:detail_fem_va_to_fa. \begin{equation}\label{eq:detail_fem_va_to_fa} F_a = g_a \cdot V_a, \quad g_a = d_{33} n k_a, \quad k_a = \frac{c^{E} A}{L} \end{equation} Unfortunately, it is difficult to know exactly which material is used for the piezoelectric stacks[fn:detail_fem_1]. -Yet, based on the available properties of the stacks in the data-sheet (summarized in Table ref:tab:detail_fem_stack_parameters), the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. +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 @@ -6661,7 +6661,7 @@ Yet, based on the available properties of the stacks in the data-sheet (summariz | Length | $mm$ | 20 | | Stack Area | $mm^2$ | 10x10 | -The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:detail_fem_piezo_properties. +The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table\nbsp{}ref:tab:detail_fem_piezo_properties. From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained. #+name: tab:detail_fem_piezo_properties @@ -6682,11 +6682,11 @@ From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained. Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications. -The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure ref:fig:detail_fem_apa95ml_compliance), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames. +The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure\nbsp{}ref:fig:detail_fem_apa95ml_compliance), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames. The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML. A value of $23\,N/\mu m$ was found which is close to the specified stiffness in the datasheet of $k = 21\,N/\mu m$. -The multi-body model predicted a resonant frequency under block-free conditions of $\approx 2\,\text{kHz}$ (Figure ref:fig:detail_fem_apa95ml_compliance), which is in agreement with the nominal specification. +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 @@ -6695,7 +6695,7 @@ The multi-body model predicted a resonant frequency under block-free conditions In order to estimate the stroke of the APA95ML, the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, was first determined. This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of $1.5$ was derived. -The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ relative to their length (here equal to $20\,mm$), produce an individual nominal stroke of $20\,\mu m$ (see data-sheet of the piezoelectric stacks on Table ref:tab:detail_fem_stack_parameters, page pageref:tab:detail_fem_stack_parameters). +The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ relative to their length (here equal to $20\,mm$), produce an individual nominal stroke of $20\,\mu m$ (see data-sheet of the piezoelectric stacks on Table\nbsp{}ref:tab:detail_fem_stack_parameters, page\nbsp{}pageref:tab:detail_fem_stack_parameters). As three stacks are used, the horizontal displacement is $60\,\mu m$. Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of $90\,\mu m$ which falls within the manufacturer-specified range of $80\,\mu m$ and $120\,\mu m$. @@ -6708,7 +6708,7 @@ The high degree of concordance observed across multiple performance metrics prov Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation. The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element. -The test bench illustrated in Figure ref:fig:detail_fem_apa95ml_bench_schematic was used, which consists of a $5.7\,kg$ granite suspended on top of the APA95ML. +The test bench illustrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_bench_schematic was used, which consists of a $5.7\,kg$ granite suspended on top of the APA95ML. The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement $y$. A digital-to-analog converter (DAC) was used to generate the control signal $u$, which was subsequently conditioned through a voltage amplifier with a gain of $20$, ultimately yielding the effective voltage $V_a$ across the two piezoelectric stacks. Measurement of the sensor stack voltage $V_s$ was performed using an analog-to-digital converter (ADC). @@ -6721,17 +6721,17 @@ Measurement of the sensor stack voltage $V_s$ was performed using an analog-to-d ***** Comparison of the dynamics Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML. -The identification procedure required careful choice of the excitation signal [[cite:&pintelon12_system_ident, chap. 5]]. +The identification procedure required careful choice of the excitation signal\nbsp{}[[cite:&pintelon12_system_ident, chap. 5]]. During all this experimental work, random noise excitation was predominantly employed. The designed excitation signal is then generated and both input and output signals are synchronously acquired. From the obtained input and output data, the frequency response functions were derived. -To improve the quality of the obtained frequency domain data, averaging and windowing were used [[cite:&pintelon12_system_ident, chap. 13]]. +To improve the quality of the obtained frequency domain data, averaging and windowing were used\nbsp{}[[cite:&pintelon12_system_ident, chap. 13]]. -The obtained frequency response functions from $V_a$ to $V_s$ and to $y$ are compared with the theoretical predictions derived from the multi-body model in Figure ref:fig:detail_fem_apa95ml_comp_plant. +The obtained frequency response functions from $V_a$ to $V_s$ and to $y$ are compared with the theoretical predictions derived from the multi-body model in Figure\nbsp{}ref:fig:detail_fem_apa95ml_comp_plant. The difference in phase between the model and the measurements can be attributed to the sampling time of $0.1\,ms$ and to additional delays induced by electronic instrumentation related to the interferometer. -The presence of a non-minimum phase zero in the measured system response (Figure ref:fig:detail_fem_apa95ml_comp_plant_sensor), shall be addressed during the experimental phase. +The presence of a non-minimum phase zero in the measured system response (Figure\nbsp{}ref:fig:detail_fem_apa95ml_comp_plant_sensor), shall be addressed during the experimental phase. Regarding the amplitude characteristics, the constants $g_a$ and $g_s$ could be further refined through calibration against the experimental data. @@ -6758,15 +6758,15 @@ Regarding the amplitude characteristics, the constants $g_a$ and $g_s$ could be To further validate this modeling methodology, its ability to predict closed-loop behavior was verified experimentally. Integral Force Feedback (IFF) was implemented using the force sensor stack, and the measured dynamics of the damped system were compared with model predictions across multiple feedback gains. -The IFF controller implementation, defined in equation ref:eq:detail_fem_iff_controller, incorporated a tunable gain parameter $g$ and was designed to provide integral action near the system resonances and to limit the low frequency gain using an high pass filter. +The IFF controller implementation, defined in equation\nbsp{}ref:eq:detail_fem_iff_controller, incorporated a tunable gain parameter $g$ and was designed to provide integral action near the system resonances and to limit the low frequency gain using an high pass filter. \begin{equation}\label{eq:detail_fem_iff_controller} K_{\text{IFF}}(s) = \frac{g}{s + 2\cdot 2\pi} \cdot \frac{s}{s + 0.5 \cdot 2\pi} \end{equation} -The theoretical damped dynamics of the closed-loop system was estimated using the model by computed the root locus plot shown in Figure ref:fig:detail_fem_apa95ml_iff_root_locus. +The theoretical damped dynamics of the closed-loop system was estimated using the model by computed the root locus plot shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_iff_root_locus. For experimental validation, six gain values were tested: $g = [0,\,10,\,50,\,100,\,500,\,1000]$. -The measured frequency responses for each gain configuration were compared with model predictions, as presented in Figure ref:fig:detail_fem_apa95ml_damped_plants. +The measured frequency responses for each gain configuration were compared with model predictions, as presented in Figure\nbsp{}ref:fig:detail_fem_apa95ml_damped_plants. The close agreement between experimental measurements and theoretical predictions across all gain configurations demonstrates the model's capability to accurately predict both open-loop and closed-loop system dynamics. @@ -6793,7 +6793,7 @@ The close agreement between experimental measurements and theoretical prediction The experimental validation with an Amplified Piezoelectric Actuator confirms that this methodology accurately predicts both open-loop and closed-loop dynamic behaviors. This verification establishes its effectiveness for component design and system analysis applications. -The approach will be especially beneficial for optimizing actuators (Section ref:sec:detail_fem_actuator) and flexible joints (Section ref:sec:detail_fem_joint) for the nano-hexapod. +The approach will be especially beneficial for optimizing actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and flexible joints (Section\nbsp{}ref:sec:detail_fem_joint) for the nano-hexapod. *** Actuator Selection <> @@ -6812,7 +6812,7 @@ The system's geometric constraints limit the actuator height to 50mm, given the Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility. An actuator stroke of $\approx 100\,\mu m$ is therefore required. -Three actuator technologies were evaluated (examples of such actuators are shown in Figure ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators. +Three actuator technologies were evaluated (examples of such actuators are shown in Figure\nbsp{}ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators. Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity. #+name: fig:detail_fem_actuator_pictures @@ -6839,22 +6839,22 @@ Variable reluctance actuators were not considered despite their superior efficie #+end_subfigure #+end_figure -Voice coil actuators (shown in Figure ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,N/\mu m$), would require forces in the order of $100\,N$ to achieve the specified $100\,\mu m$ displacement. +Voice coil actuators (shown in Figure\nbsp{}ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,N/\mu m$), would require forces in the order of $100\,N$ to achieve the specified $100\,\mu m$ displacement. While these actuators offer excellent linearity and long strokes capabilities, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability. Their advantages (linearity and long stroke) were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions. -Conventional piezoelectric stack actuators (shown in Figure ref:fig:detail_fem_piezo_picture) present two significant limitations for the current application. +Conventional piezoelectric stack actuators (shown in Figure\nbsp{}ref:fig:detail_fem_piezo_picture) present two significant limitations for the current application. Their stroke is inherently limited to approximately $0.1\,\%$ of their length, meaning that even with the maximum allowable height of $50\,mm$, the achievable stroke would only be $50\,\mu m$, insufficient for the application. Additionally, their extremely high stiffness, typically around $100\,N/\mu m$, exceeds the desired specifications by two orders of magnitude. Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through a specific mechanical design. The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile. -Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF [[cite:&souleille18_concep_activ_mount_space_applic;&verma20_dynam_stabil_thin_apert_light]]. -Moreover, using APA for active damping has been successfully demonstrated in similar applications [[cite:&hanieh03_activ_stewar]]. +Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic;&verma20_dynam_stabil_thin_apert_light]]. +Moreover, using APA for active damping has been successfully demonstrated in similar applications\nbsp{}[[cite:&hanieh03_activ_stewar]]. -Several specific APA models were evaluated against the established specifications (Table ref:tab:detail_fem_piezo_act_models). +Several specific APA models were evaluated against the established specifications (Table\nbsp{}ref:tab:detail_fem_piezo_act_models). The APA300ML emerged as the optimal choice. -This selection was further reinforced by previous experience with APAs from the same manufacturer[fn:detail_fem_2], and particularly by the successful validation of the modeling methodology with a similar actuator (Section ref:ssec:detail_fem_super_element_example). +This selection was further reinforced by previous experience with APAs from the same manufacturer[fn:detail_fem_2], and particularly by the successful validation of the modeling methodology with a similar actuator (Section\nbsp{}ref:ssec:detail_fem_super_element_example). 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 @@ -6872,8 +6872,8 @@ The demonstrated accuracy of the modeling approach for the APA95ML provides conf **** APA300ML - Reduced Order Flexible Body <> -The validation of the APA300ML started by incorporating a "reduced order flexible body" into the multi-body model as explained in Section ref:sec:detail_fem_super_element. -The FEA model was developed with particular attention to the placement of reference frames, as illustrated in Figure ref:fig:detail_fem_apa300ml_frames. +The validation of the APA300ML started by incorporating a "reduced order flexible body" into the multi-body model as explained in Section\nbsp{}ref:sec:detail_fem_super_element. +The FEA model was developed with particular attention to the placement of reference frames, as illustrated in Figure\nbsp{}ref:fig:detail_fem_apa300ml_frames. Seven distinct frames were defined, with blue frames designating the force sensor stack interfaces for strain measurement, red frames denoting the actuator stack interfaces for force application and green frames for connecting to other elements. 120 additional modes were added during the modal reduction for a total order of 162. While this high order provides excellent accuracy for validation purposes, it proves computationally intensive for simulations. @@ -6896,14 +6896,14 @@ While this high order provides excellent accuracy for validation purposes, it pr #+end_subfigure #+end_figure -The sensor and actuator "constants" ($g_s$ and $g_a$) derived in Section ref:ssec:detail_fem_super_element_example for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks. +The sensor and actuator "constants" ($g_s$ and $g_a$) derived in Section\nbsp{}ref:ssec:detail_fem_super_element_example for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks. **** Simpler 2DoF Model of the APA300ML <> -To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model, adapted from [[cite:&souleille18_concep_activ_mount_space_applic]], was developed. +To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model, adapted from\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic]], was developed. -This model, illustrated in Figure ref:fig:detail_fem_apa_2dof_model, comprises three components. +This model, illustrated in Figure\nbsp{}ref:fig:detail_fem_apa_2dof_model, comprises three components. The mechanical shell is characterized by its axial stiffness $k_1$ and damping $c_1$. The actuator is modelled with stiffness $k_a$ and damping $c_a$, incorporating a force source $f$. This force is related to the applied voltage $V_a$ through the actuator constant $g_a$. @@ -6930,7 +6930,7 @@ The stack parameters ($k_a$, $c_a$, $k_e$, $c_e$) were then derived from the fir Given that identical piezoelectric stacks are used for both sensing and actuation, the relationships $k_e = 2k_a$ and $c_e = 2c_a$ were enforced, reflecting the series configuration of the dual actuator stacks. Finally, the sensitivities $g_s$ and $g_a$ were adjusted to match the DC gains of the respective transfer functions. -The resulting parameters, listed in Table ref:tab:detail_fem_apa300ml_2dof_parameters, yield dynamic behavior that closely matches the high-order finite element model, as demonstrated in Figure ref:fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof. +The resulting parameters, listed in Table\nbsp{}ref:tab:detail_fem_apa300ml_2dof_parameters, yield dynamic behavior that closely matches the high-order finite element model, as demonstrated in Figure\nbsp{}ref:fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof. 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 @@ -6969,11 +6969,11 @@ While higher-order modes and non-axial flexibility are not captured, the model a **** Electrical characteristics of the APA <> -The behavior of piezoelectric actuators is characterized by coupled constitutive equations that establish relationships between electrical properties (charges, voltages) and mechanical properties (stress, strain) [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapter 5.5]]. +The behavior of piezoelectric actuators is characterized by coupled constitutive equations that establish relationships between electrical properties (charges, voltages) and mechanical properties (stress, strain)\nbsp{}[[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapter 5.5]]. To evaluate the impact of electrical boundary conditions on the system dynamics, experimental measurements were conducted using the APA95ML, comparing the transfer function from $V_a$ to $y$ under two distinct configurations. With the force sensor stack in open-circuit condition (analogous to voltage measurement with high input impedance) and in short-circuit condition (similar to charge measurement with low output impedance). -As demonstrated in Figure ref:fig:detail_fem_apa95ml_effect_electrical_boundaries, short-circuiting the force sensor stack results in a minor decrease in resonance frequency. +As demonstrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_effect_electrical_boundaries, short-circuiting the force sensor stack results in a minor decrease in resonance frequency. The developed models of the APA do not represent such behavior, but as this effect is quite small, this validates the simplifying assumption made in the models. #+name: fig:detail_fem_apa95ml_effect_electrical_boundaries @@ -6990,11 +6990,11 @@ These aspects will be addressed in the instrumentation chapter. The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation. The dynamics predicted using the flexible body model align well with the design requirements established during the conceptual phase. -The dynamics from $\bm{u}$ to $\bm{V}_s$ exhibits the desired alternating pole-zero pattern (Figure ref:fig:detail_fem_actuator_fem_vs_perfect_hac_plant), a critical characteristic for implementing robust decentralized Integral Force Feedback. -Additionally, the model predicts no problematic high-frequency modes in the dynamics from $\bm{u}$ to $\bm{\epsilon}_{\mathcal{L}}$ (Figure ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant), maintaining consistency with earlier conceptual simulations. +The dynamics from $\bm{u}$ to $\bm{V}_s$ exhibits the desired alternating pole-zero pattern (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_hac_plant), a critical characteristic for implementing robust decentralized Integral Force Feedback. +Additionally, the model predicts no problematic high-frequency modes in the dynamics from $\bm{u}$ to $\bm{\epsilon}_{\mathcal{L}}$ (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant), maintaining consistency with earlier conceptual simulations. These findings suggest that the control performance targets established during the conceptual phase remain achievable with the selected actuator. -Comparative analysis between the high-order FEM implementation and the simplified 2DoF model (Figure ref:fig:detail_fem_actuator_fem_vs_perfect_plants) demonstrates remarkable agreement in the frequency range of interest. +Comparative analysis between the high-order FEM implementation and the simplified 2DoF model (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_plants) demonstrates remarkable agreement in the frequency range of interest. This validates the use of the simplified model for time-domain simulations. The reduction in model order is substantial: while the FEM implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete nano-hexapod. @@ -7024,11 +7024,11 @@ These results validate both the selection of the APA300ML and the effectiveness High-precision position control at the nanometer scale requires systems to be free from friction and backlash, as these nonlinear phenomena severely limit achievable positioning accuracy. This fundamental requirement prevents the use of conventional joints, necessitating instead the implementation of flexible joints that achieve motion through elastic deformation. -For Stewart platforms requiring nanometric precision, numerous flexible joint designs have been developed and successfully implemented, as illustrated in Figure ref:fig:detail_fem_joints_examples. +For Stewart platforms requiring nanometric precision, numerous flexible joint designs have been developed and successfully implemented, as illustrated in Figure\nbsp{}ref:fig:detail_fem_joints_examples. For design simplicity and component standardization, identical joints are employed at both ends of the nano-hexapod struts. #+name: fig:detail_fem_joints_examples -#+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical "universal" flexible joint used in [[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in [[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_wire}) "Thin" flexible joints having "notch curves" are also used [[cite:&du14_piezo_actuat_high_precis_flexib]]. +#+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical "universal" flexible joint used in\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_wire}) "Thin" flexible joints having "notch curves" are also used\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]]. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_preumont}} @@ -7051,33 +7051,33 @@ For design simplicity and component standardization, identical joints are employ #+end_subfigure #+end_figure -While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other degrees of freedom, practical implementations exhibit parasitic stiffness that can impact control performance [[cite:&mcinroy02_model_desig_flexur_joint_stewar]]. -This section examines how these non-ideal characteristics affect system behavior, focusing particularly on bending/torsional stiffness (Section ref:ssec:detail_fem_joint_bending) and axial compliance (Section ref:ssec:detail_fem_joint_axial). +While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other degrees of freedom, practical implementations exhibit parasitic stiffness that can impact control performance\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]]. +This section examines how these non-ideal characteristics affect system behavior, focusing particularly on bending/torsional stiffness (Section\nbsp{}ref:ssec:detail_fem_joint_bending) and axial compliance (Section\nbsp{}ref:ssec:detail_fem_joint_axial). The analysis of bending and axial stiffness effects enables the establishment of comprehensive specifications for the flexible joints. -These specifications guide the development and optimization of a flexible joint design through finite element analysis (Section ref:ssec:detail_fem_joint_specs). -The validation process, detailed in Section ref:ssec:detail_fem_joint_validation, begins with the integration of the joints as "reduced order flexible bodies" in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints. +These specifications guide the development and optimization of a flexible joint design through finite element analysis (Section\nbsp{}ref:ssec:detail_fem_joint_specs). +The validation process, detailed in Section\nbsp{}ref:ssec:detail_fem_joint_validation, begins with the integration of the joints as "reduced order flexible bodies" in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints. **** Bending and Torsional Stiffness <> -The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction [[cite:&mcinroy02_model_desig_flexur_joint_stewar]] and can affect system dynamics. +The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]] and can affect system dynamics. To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of $1\,N/\mu m$) without parallel stiffness to the force sensors. Flexible joint bending stiffness was varied from 0 (ideal case) to $500\,Nm/\text{rad}$. Analysis of the plant dynamics reveals two significant effects. -For the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$, bending stiffness increases low-frequency coupling, though this remains small for realistic stiffness values (Figure ref:fig:detail_fem_joints_bending_stiffness_hac_plant). -In [[cite:&mcinroy02_model_desig_flexur_joint_stewar]], it is established that forces remain effectively aligned with the struts when the flexible joint bending stiffness is much small than the actuator stiffness multiplied by the square of the strut length. +For the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$, bending stiffness increases low-frequency coupling, though this remains small for realistic stiffness values (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_hac_plant). +In\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]], it is established that forces remain effectively aligned with the struts when the flexible joint bending stiffness is much small than the actuator stiffness multiplied by the square of the strut length. For the nano-hexapod, this corresponds to having the bending stiffness much lower than 9000 Nm/rad. This condition is more readily satisfied with the relatively stiff actuators selected, and could be problematic for softer Stewart platforms. -For the force sensor plant, bending stiffness introduces complex conjugate zeros at low frequency (Figure ref:fig:detail_fem_joints_bending_stiffness_iff_plant). -This behavior resembles having parallel stiffness to the force sensor as was the case with the APA300ML (see Figure ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant). +For the force sensor plant, bending stiffness introduces complex conjugate zeros at low frequency (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_plant). +This behavior resembles having parallel stiffness to the force sensor as was the case with the APA300ML (see Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant). However, this time the parallel stiffness does not comes from the considered strut, but from the bending stiffness of the flexible joints of the other five struts. -This characteristic impacts the achievable damping using decentralized Integral Force Feedback [[cite:&preumont07_six_axis_singl_stage_activ]]. -This is confirmed by the Root Locus plot in Figure ref:fig:detail_fem_joints_bending_stiffness_iff_locus_1dof. -This effect becomes less significant when using the selected APA300ML actuators (Figure ref:fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml), which already incorporate parallel stiffness by design which is higher than the one induced by flexible joint stiffness. +This characteristic impacts the achievable damping using decentralized Integral Force Feedback\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. +This is confirmed by the Root Locus plot in Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_locus_1dof. +This effect becomes less significant when using the selected APA300ML actuators (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml), which already incorporate parallel stiffness by design which is higher than the one induced by flexible joint stiffness. A parallel analysis of torsional stiffness revealed similar effects, though these proved less critical for system performance. @@ -7121,17 +7121,17 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes <> The limited axial stiffness ($k_a$) of flexible joints introduces an additional compliance between the actuation point and the measurement point. -As explained in [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapter 6]] and in [[cite:&rankers98_machin]] (effect called "actuator flexibility"), such intermediate flexibility invariably degrades control performance. +As explained in\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapter 6]] and in\nbsp{}[[cite:&rankers98_machin]] (effect called "actuator flexibility"), such intermediate flexibility invariably degrades control performance. Therefore, determining the minimum acceptable axial stiffness that maintains nano-hexapod performance becomes crucial. The analysis incorporates the strut mass (112g per APA300ML) to accurately model internal resonance effects. A parametric study was conducted by varying the axial stiffness from $1\,N/\mu m$ (matching actuator stiffness) to $1000\,N/\mu m$ (approximating rigid behavior). -The resulting frequency responses (Figure ref:fig:detail_fem_joints_axial_stiffness_plants) reveal distinct effects on system dynamics. +The resulting frequency responses (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_plants) reveal distinct effects on system dynamics. -The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both frequency response data (Figure ref:fig:detail_fem_joints_axial_stiffness_iff_plant) and root locus analysis (Figure ref:fig:detail_fem_joints_axial_stiffness_iff_locus). +The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both frequency response data (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_plant) and root locus analysis (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_locus). However, the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ demonstrates significant effects: internal strut modes appear at high frequencies, introducing substantial cross-coupling between axes. -This coupling is quantified through RGA analysis of the damped system (Figure ref:fig:detail_fem_joints_axial_stiffness_rga_hac_plant), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance. +This coupling is quantified through RGA analysis of the damped system (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_rga_hac_plant), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance. Above this resonance frequency, two critical limitations emerge. First, the system exhibits strong coupling between control channels, making decentralized control strategies ineffective. @@ -7181,7 +7181,7 @@ Based on this analysis, an axial stiffness specification of $100\,N/\mu m$ was e The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics. Critical specifications include sufficient bending stroke to ensure long-term operation below yield stress, high axial stiffness for precise positioning, low bending and torsional stiffnesses to minimize parasitic forces, adequate load capacity, and well-defined rotational axes. -Based on the dynamic analysis presented in previous sections, quantitative specifications were established and are summarized in Table ref:tab:detail_fem_joints_specs. +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 @@ -7195,14 +7195,14 @@ Based on the dynamic analysis presented in previous sections, quantitative speci | Torsion Stiffness $k_t$ | $< 500\,Nm/\text{rad}$ | 260 | | Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | -Among various possible flexible joint architectures, the design shown in Figure ref:fig:detail_fem_joints_design was selected for three key advantages. +Among various possible flexible joint architectures, the design shown in Figure\nbsp{}ref:fig:detail_fem_joints_design was selected for three key advantages. First, the geometry creates coincident $x$ and $y$ rotation axes, ensuring well-defined kinematic behavior, important for the precise definition of the nano-hexapod Jacobian matrix. Second, the design allows easy tuning of different directional stiffnesses through a limited number of geometric parameters. Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational degrees of freedom. The joint geometry was optimized through parametric finite element analysis. The optimization process revealed an inherent trade-off between maximizing axial stiffness and achieving sufficiently low bending/torsional stiffness, while maintaining material stresses within acceptable limits. -The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through finite element analysis and summarized in Table ref:tab:detail_fem_joints_specs. +The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through finite element analysis and summarized in Table\nbsp{}ref:tab:detail_fem_joints_specs. #+name: fig:detail_fem_joints_design #+caption: Designed flexible joints. @@ -7226,8 +7226,8 @@ The final design, featuring a neck dimension of 0.25mm, achieves mechanical prop <> The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from finite element analysis. -This high-fidelity representation was created by defining two interface frames (Figure ref:fig:detail_fem_joints_frames) and extracting six additional modes, resulting in reduced-order mass and stiffness matrices of dimension $18 \times 18$. -The computed transfer functions from actuator forces to both force sensor measurements ($\bm{f}$ to $\bm{f}_m$) and external metrology ($\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$) demonstrate dynamics consistent with predictions from earlier analyses (Figure ref:fig:detail_fem_joints_fem_vs_perfect_plants), thereby validating the joint design. +This high-fidelity representation was created by defining two interface frames (Figure\nbsp{}ref:fig:detail_fem_joints_frames) and extracting six additional modes, resulting in reduced-order mass and stiffness matrices of dimension $18 \times 18$. +The computed transfer functions from actuator forces to both force sensor measurements ($\bm{f}$ to $\bm{f}_m$) and external metrology ($\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$) demonstrate dynamics consistent with predictions from earlier analyses (Figure\nbsp{}ref:fig:detail_fem_joints_fem_vs_perfect_plants), thereby validating the joint design. #+name: fig:detail_fem_joints_frames #+caption: Defined frames for the reduced order flexible body. The two flat interfaces are considered rigid, and are linked to the two frames $\{F\}$ and $\{M\}$ both located at the center of the rotation. @@ -7286,13 +7286,13 @@ Control was implemented in the frame of the struts, leveraging the inherent low- For these decoupled plants, open-loop shaping techniques were employed to tune the individual controllers. While these initial strategies proved effective in validating the NASS concept, this work explores alternative approaches with the potential to further enhance the performance. -Section ref:sec:detail_control_sensor examines different methods for combining multiple sensors, with particular emphasis on sensor fusion techniques that utilize complementary filters. +Section\nbsp{}ref:sec:detail_control_sensor examines different methods for combining multiple sensors, with particular emphasis on sensor fusion techniques that utilize complementary filters. A novel approach for designing these filters is proposed, which allows optimization of the sensor fusion effectiveness. -Section ref:sec:detail_control_decoupling presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. +Section\nbsp{}ref:sec:detail_control_decoupling presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. Each method is evaluated in terms of its theoretical foundations, implementation requirements, and performance characteristics, providing insights into their respective advantages for different applications. -Finally, Section ref:sec:detail_control_cf addresses the challenge of controller design for decoupled plants. +Finally, Section\nbsp{}ref:sec:detail_control_cf addresses the challenge of controller design for decoupled plants. A method for directly shaping closed-loop transfer functions using complementary filters is proposed, offering an intuitive approach to achieving desired performance specifications while ensuring robustness to plant uncertainty. *** Multiple Sensor Control @@ -7303,7 +7303,7 @@ A method for directly shaping closed-loop transfer functions using complementary The literature review of Stewart platforms revealed a wide diversity of designs with various sensor and actuator configurations. Control objectives (such as active damping, vibration isolation, or precise positioning) directly dictate sensor selection, whether inertial, force, or relative position sensors. -In cases where multiple control objectives must be achieved simultaneously, as is the case for the Nano Active Stabilization System (NASS) where the Stewart platform must both position the sample and provide isolation from micro-station vibrations, combining multiple sensors within the control architecture has been demonstrated to yield significant performance benefits [[cite:&hauge04_sensor_contr_space_based_six]]. +In cases where multiple control objectives must be achieved simultaneously, as is the case for the Nano Active Stabilization System (NASS) where the Stewart platform must both position the sample and provide isolation from micro-station vibrations, combining multiple sensors within the control architecture has been demonstrated to yield significant performance benefits\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six]]. From the literature, three principal approaches for combining sensors have been identified: High Authority Control-Low Authority Control (HAC-LAC), sensor fusion, and two-sensor control architectures. #+name: fig:detail_control_control_multiple_sensors @@ -7332,38 +7332,38 @@ From the literature, three principal approaches for combining sensors have been #+end_subfigure #+end_figure -The HAC-LAC approach employs a dual-loop control strategy in which two control loops utilize different sensors for distinct purposes (Figure ref:fig:detail_control_sensor_arch_hac_lac). -In [[cite:&li01_simul_vibrat_isolat_point_contr]], vibration isolation is provided by accelerometers collocated with the voice coil actuators, while external rotational sensors are utilized to achieve pointing control. -In [[cite:&geng95_intel_contr_system_multip_degree]], force sensors collocated with the magnetostrictive actuators are used for active damping using decentralized IFF, and subsequently accelerometers are employed for adaptive vibration isolation. -Similarly, in [[cite:&wang16_inves_activ_vibrat_isolat_stewar]], piezoelectric actuators with collocated force sensors are used in a decentralized manner to provide active damping while accelerometers are implemented in an adaptive feedback loop to suppress periodic vibrations. -In [[cite:&xie17_model_contr_hybrid_passiv_activ]], force sensors are integrated in the struts for decentralized force feedback while accelerometers fixed to the top platform are employed for centralized control. +The HAC-LAC approach employs a dual-loop control strategy in which two control loops utilize different sensors for distinct purposes (Figure\nbsp{}ref:fig:detail_control_sensor_arch_hac_lac). +In\nbsp{}[[cite:&li01_simul_vibrat_isolat_point_contr]], vibration isolation is provided by accelerometers collocated with the voice coil actuators, while external rotational sensors are utilized to achieve pointing control. +In\nbsp{}[[cite:&geng95_intel_contr_system_multip_degree]], force sensors collocated with the magnetostrictive actuators are used for active damping using decentralized IFF, and subsequently accelerometers are employed for adaptive vibration isolation. +Similarly, in\nbsp{}[[cite:&wang16_inves_activ_vibrat_isolat_stewar]], piezoelectric actuators with collocated force sensors are used in a decentralized manner to provide active damping while accelerometers are implemented in an adaptive feedback loop to suppress periodic vibrations. +In\nbsp{}[[cite:&xie17_model_contr_hybrid_passiv_activ]], force sensors are integrated in the struts for decentralized force feedback while accelerometers fixed to the top platform are employed for centralized control. -The second approach, sensor fusion (illustrated in Figure ref:fig:detail_control_sensor_arch_sensor_fusion), involves filtering signals from two sensors using complementary filters[fn:detail_control_1] and summing them to create an improved sensor signal. -In [[cite:&hauge04_sensor_contr_space_based_six]], geophones (used at low frequency) are merged with force sensors (used at high frequency). +The second approach, sensor fusion (illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_arch_sensor_fusion), involves filtering signals from two sensors using complementary filters[fn:detail_control_1] and summing them to create an improved sensor signal. +In\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six]], geophones (used at low frequency) are merged with force sensors (used at high frequency). It is demonstrated that combining both sensors using sensor fusion can improve performance compared to using only one of the two sensors. -In [[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]], sensor fusion architecture is implemented with an accelerometer and a force sensor. +In\nbsp{}[[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]], sensor fusion architecture is implemented with an accelerometer and a force sensor. This implementation is shown to simultaneously achieve high damping of structural modes (through the force sensors) while maintaining very low vibration transmissibility (through the accelerometers). -In [[cite:&beijen14_two_sensor_contr_activ_vibrat]], the performance of sensor fusion is compared with the more general case of "two-sensor control" (illustrated in Figure ref:fig:detail_control_sensor_arch_two_sensor_control). +In\nbsp{}[[cite:&beijen14_two_sensor_contr_activ_vibrat]], the performance of sensor fusion is compared with the more general case of "two-sensor control" (illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_arch_two_sensor_control). It is highlighted that "two-sensor control" provides greater control freedom, potentially enhancing performance. -In [[cite:&thayer02_six_axis_vibrat_isolat_system]], the use of force sensors and geophones is compared for vibration isolation purposes. +In\nbsp{}[[cite:&thayer02_six_axis_vibrat_isolat_system]], the use of force sensors and geophones is compared for vibration isolation purposes. Geophones are shown to provide better isolation performance than load cells but suffer from poor robustness. Conversely, the controller based on force sensors exhibited inferior performance (due to the presence of a pair of low frequency zeros), but demonstrated better robustness properties. A "two-sensor control" approach was proven to perform better than controllers based on individual sensors while maintaining better robustness. A Linear Quadratic Regulator (LQG) was employed to optimize the two-input/one-output controller. Beyond these three main approaches, other control architectures have been proposed for different purposes. -For instance, in [[cite:&yang19_dynam_model_decoup_contr_flexib]], a first control loop utilizes force sensors and relative motion sensors to compensate for parasitic stiffness of the flexible joints. +For instance, in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]], a first control loop utilizes force sensors and relative motion sensors to compensate for parasitic stiffness of the flexible joints. Subsequently, the system is decoupled in the modal space (facilitated by the removal of parasitic stiffness) and accelerometers are employed for vibration isolation. The HAC-LAC architecture was previously investigated during the conceptual phase and successfully implemented to validate the NASS concept, demonstrating excellent performance. At the other end of the spectrum, the two-sensor approach yields greater control design freedom but introduces increased complexity in tuning, and thus was not pursued in this study. This work instead focuses on sensor fusion, which represents a promising middle ground between the proven HAC-LAC approach and the more complex two-sensor control strategy. -A review of sensor fusion is first presented in Section ref:ssec:detail_control_sensor_review. -Then, in Section ref:ssec:detail_control_sensor_fusion_requirements, both the robustness of the fusion and the noise characteristics of the resulting "fused sensor" are derived and expressed as functions of the complementary filters' norms. -A synthesis method for shaping complementary filters is proposed in Section ref:ssec:detail_control_sensor_hinf_method. -The investigation is then extended beyond the conventional two-sensor scenario, demonstrating how the proposed complementary filter synthesis can be generalized for applications requiring the fusion of three or more sensors (Section ref:ssec:detail_control_sensor_hinf_three_comp_filters). +A review of sensor fusion is first presented in Section\nbsp{}ref:ssec:detail_control_sensor_review. +Then, in Section\nbsp{}ref:ssec:detail_control_sensor_fusion_requirements, both the robustness of the fusion and the noise characteristics of the resulting "fused sensor" are derived and expressed as functions of the complementary filters' norms. +A synthesis method for shaping complementary filters is proposed in Section\nbsp{}ref:ssec:detail_control_sensor_hinf_method. +The investigation is then extended beyond the conventional two-sensor scenario, demonstrating how the proposed complementary filter synthesis can be generalized for applications requiring the fusion of three or more sensors (Section\nbsp{}ref:ssec:detail_control_sensor_hinf_three_comp_filters). **** Review of Sensor Fusion <> @@ -7371,50 +7371,50 @@ The investigation is then extended beyond the conventional two-sensor scenario, Measuring a physical quantity using sensors is always subject to several limitations. First, the accuracy of the measurement is affected by various noise sources, such as electrical noise from the conditioning electronics. Second, the frequency range in which the measurement is relevant is bounded by the bandwidth of the sensor. -One way to overcome these limitations is to combine several sensors using a technique called "sensor fusion" [[cite:&bendat57_optim_filter_indep_measur_two]]. +One way to overcome these limitations is to combine several sensors using a technique called "sensor fusion"\nbsp{}[[cite:&bendat57_optim_filter_indep_measur_two]]. Fortunately, a wide variety of sensors exists, each with different characteristics. By carefully selecting the sensors to be fused, a "super sensor" is obtained that combines the benefits of the individual sensors. -In some applications, sensor fusion is employed to increase measurement bandwidth [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim]]. -For instance, in [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel]], the bandwidth of a position sensor is extended by fusing it with an accelerometer that provides high-frequency motion information. -In other applications, sensor fusion is utilized to obtain an estimate of the measured quantity with reduced noise [[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system;&plummer06_optim_compl_filter_their_applic_motion_measur;&robert12_introd_random_signal_applied_kalman]]. -More recently, the fusion of sensors measuring different physical quantities has been proposed to enhance control properties [[cite:&collette15_sensor_fusion_method_high_perfor;&yong16_high_speed_vertic_posit_stage]]. -In [[cite:&collette15_sensor_fusion_method_high_perfor]], an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator to improve the stability margins of the feedback controller. +In some applications, sensor fusion is employed to increase measurement bandwidth\nbsp{}[[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim]]. +For instance, in\nbsp{}[[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel]], the bandwidth of a position sensor is extended by fusing it with an accelerometer that provides high-frequency motion information. +In other applications, sensor fusion is utilized to obtain an estimate of the measured quantity with reduced noise\nbsp{}[[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system;&plummer06_optim_compl_filter_their_applic_motion_measur;&robert12_introd_random_signal_applied_kalman]]. +More recently, the fusion of sensors measuring different physical quantities has been proposed to enhance control properties\nbsp{}[[cite:&collette15_sensor_fusion_method_high_perfor;&yong16_high_speed_vertic_posit_stage]]. +In\nbsp{}[[cite:&collette15_sensor_fusion_method_high_perfor]], an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator to improve the stability margins of the feedback controller. Beyond Stewart platforms, practical applications of sensor fusion are numerous. -It is widely implemented for attitude estimation in autonomous vehicles such as unmanned aerial vehicles [[cite:&baerveldt97_low_cost_low_weigh_attit;&corke04_inert_visual_sensin_system_small_auton_helic;&jensen13_basic_uas]] and underwater vehicles [[cite:&pascoal99_navig_system_desig_using_time;&batista10_optim_posit_veloc_navig_filter_auton_vehic]]. -Sensor fusion offers significant benefits for high-performance positioning control as demonstrated in [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim;&yong16_high_speed_vertic_posit_stage]]. -It has also been identified as a key technology for improving the performance of active vibration isolation systems [[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]]. -Emblematic examples include the isolation stages of gravitational wave detectors [[cite:&collette15_sensor_fusion_method_high_perfor;&heijningen18_low]] such as those employed at LIGO [[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system]] and Virgo [[cite:&lucia18_low_frequen_optim_perfor_advan]]. +It is widely implemented for attitude estimation in autonomous vehicles such as unmanned aerial vehicles\nbsp{}[[cite:&baerveldt97_low_cost_low_weigh_attit;&corke04_inert_visual_sensin_system_small_auton_helic;&jensen13_basic_uas]] and underwater vehicles\nbsp{}[[cite:&pascoal99_navig_system_desig_using_time;&batista10_optim_posit_veloc_navig_filter_auton_vehic]]. +Sensor fusion offers significant benefits for high-performance positioning control as demonstrated in\nbsp{}[[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim;&yong16_high_speed_vertic_posit_stage]]. +It has also been identified as a key technology for improving the performance of active vibration isolation systems\nbsp{}[[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]]. +Emblematic examples include the isolation stages of gravitational wave detectors\nbsp{}[[cite:&collette15_sensor_fusion_method_high_perfor;&heijningen18_low]] such as those employed at LIGO\nbsp{}[[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system]] and Virgo\nbsp{}[[cite:&lucia18_low_frequen_optim_perfor_advan]]. -Two principal methods are employed to perform sensor fusion: using complementary filters [[cite:&anderson53_instr_approac_system_steer_comput]] or using Kalman filtering [[cite:&brown72_integ_navig_system_kalman_filter]]. -For sensor fusion applications, these methods share many relationships [[cite:&brown72_integ_navig_system_kalman_filter;&higgins75_compar_compl_kalman_filter;&robert12_introd_random_signal_applied_kalman;&fonseca15_compl]]. -However, Kalman filtering requires assumptions about the probabilistic characteristics of sensor noise [[cite:&robert12_introd_random_signal_applied_kalman]], whereas complementary filters do not impose such requirements. -Furthermore, complementary filters offer advantages over Kalman filtering for sensor fusion through their general applicability, low computational cost [[cite:&higgins75_compar_compl_kalman_filter]], and intuitive nature, as their effects can be readily interpreted in the frequency domain. +Two principal methods are employed to perform sensor fusion: using complementary filters\nbsp{}[[cite:&anderson53_instr_approac_system_steer_comput]] or using Kalman filtering\nbsp{}[[cite:&brown72_integ_navig_system_kalman_filter]]. +For sensor fusion applications, these methods share many relationships\nbsp{}[[cite:&brown72_integ_navig_system_kalman_filter;&higgins75_compar_compl_kalman_filter;&robert12_introd_random_signal_applied_kalman;&fonseca15_compl]]. +However, Kalman filtering requires assumptions about the probabilistic characteristics of sensor noise\nbsp{}[[cite:&robert12_introd_random_signal_applied_kalman]], whereas complementary filters do not impose such requirements. +Furthermore, complementary filters offer advantages over Kalman filtering for sensor fusion through their general applicability, low computational cost\nbsp{}[[cite:&higgins75_compar_compl_kalman_filter]], and intuitive nature, as their effects can be readily interpreted in the frequency domain. A set of filters is considered complementary if the sum of their transfer functions equals one at all frequencies. -In early implementations of complementary filtering, analog circuits were used to physically realize the filters [[cite:&anderson53_instr_approac_system_steer_comput]]. -While analog complementary filters remain in use today [[cite:&yong16_high_speed_vertic_posit_stage;&moore19_capac_instr_sensor_fusion_high_bandw_nanop]], digital implementation is now more common as it provides greater flexibility. +In early implementations of complementary filtering, analog circuits were used to physically realize the filters\nbsp{}[[cite:&anderson53_instr_approac_system_steer_comput]]. +While analog complementary filters remain in use today\nbsp{}[[cite:&yong16_high_speed_vertic_posit_stage;&moore19_capac_instr_sensor_fusion_high_bandw_nanop]], digital implementation is now more common as it provides greater flexibility. Various design methods have been developed to optimize complementary filters. -The most straightforward approach utilizes analytical formulas, which depending on the application may be first order [[cite:&corke04_inert_visual_sensin_system_small_auton_helic;&yeh05_model_contr_hydraul_actuat_two;&yong16_high_speed_vertic_posit_stage]], second order [[cite:&baerveldt97_low_cost_low_weigh_attit;&stoten01_fusion_kinet_data_using_compos_filter;&jensen13_basic_uas]], or higher orders [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&stoten01_fusion_kinet_data_using_compos_filter;&collette15_sensor_fusion_method_high_perfor;&matichard15_seism_isolat_advan_ligo]]. -Since the characteristics of the super sensor depend on proper complementary filter design [[cite:&dehaeze19_compl_filter_shapin_using_synth]], several optimization techniques have emerged—ranging from optimizing parameters for analytical formulas [[cite:&jensen13_basic_uas;&min15_compl_filter_desig_angle_estim;&fonseca15_compl]] to employing convex optimization tools [[cite:&hua04_polyp_fir_compl_filter_contr_system;&hua05_low_ligo]] such as linear matrix inequalities [[cite:&pascoal99_navig_system_desig_using_time]]. -As demonstrated in [[cite:&plummer06_optim_compl_filter_their_applic_motion_measur]], complementary filter design can be linked to the standard mixed-sensitivity control problem, allowing powerful classical control theory tools to be applied. -For example, in [[cite:&jensen13_basic_uas]], two gains of a Proportional Integral (PI) controller are optimized to minimize super sensor noise. +The most straightforward approach utilizes analytical formulas, which depending on the application may be first order\nbsp{}[[cite:&corke04_inert_visual_sensin_system_small_auton_helic;&yeh05_model_contr_hydraul_actuat_two;&yong16_high_speed_vertic_posit_stage]], second order\nbsp{}[[cite:&baerveldt97_low_cost_low_weigh_attit;&stoten01_fusion_kinet_data_using_compos_filter;&jensen13_basic_uas]], or higher orders\nbsp{}[[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&stoten01_fusion_kinet_data_using_compos_filter;&collette15_sensor_fusion_method_high_perfor;&matichard15_seism_isolat_advan_ligo]]. +Since the characteristics of the super sensor depend on proper complementary filter design\nbsp{}[[cite:&dehaeze19_compl_filter_shapin_using_synth]], several optimization techniques have emerged—ranging from optimizing parameters for analytical formulas\nbsp{}[[cite:&jensen13_basic_uas;&min15_compl_filter_desig_angle_estim;&fonseca15_compl]] to employing convex optimization tools\nbsp{}[[cite:&hua04_polyp_fir_compl_filter_contr_system;&hua05_low_ligo]] such as linear matrix inequalities\nbsp{}[[cite:&pascoal99_navig_system_desig_using_time]]. +As demonstrated in\nbsp{}[[cite:&plummer06_optim_compl_filter_their_applic_motion_measur]], complementary filter design can be linked to the standard mixed-sensitivity control problem, allowing powerful classical control theory tools to be applied. +For example, in\nbsp{}[[cite:&jensen13_basic_uas]], two gains of a Proportional Integral (PI) controller are optimized to minimize super sensor noise. All these complementary filter design methods share the common objective of creating a super sensor with desired characteristics, typically in terms of noise and dynamics. -As reported in [[cite:&zimmermann92_high_bandw_orien_measur_contr;&plummer06_optim_compl_filter_their_applic_motion_measur]], phase shifts and magnitude bumps in the super sensor dynamics may occur if complementary filters are poorly designed or if sensors are improperly calibrated. +As reported in\nbsp{}[[cite:&zimmermann92_high_bandw_orien_measur_contr;&plummer06_optim_compl_filter_their_applic_motion_measur]], phase shifts and magnitude bumps in the super sensor dynamics may occur if complementary filters are poorly designed or if sensors are improperly calibrated. Therefore, the robustness of the fusion must be considered when designing complementary filters. Despite the numerous design methods proposed in the literature, a simple approach that specifies desired super sensor characteristics while ensuring good fusion robustness has been lacking. -Fortunately, both fusion robustness and super sensor characteristics can be linked to complementary filter magnitude [[cite:&dehaeze19_compl_filter_shapin_using_synth]]. +Fortunately, both fusion robustness and super sensor characteristics can be linked to complementary filter magnitude\nbsp{}[[cite:&dehaeze19_compl_filter_shapin_using_synth]]. Based on this relationship, the present work introduces an approach to designing complementary filters using $\mathcal{H}_\infty\text{-synthesis}$, which enables intuitive shaping of complementary filter magnitude in a straightforward manner. **** Sensor Fusion and Complementary Filters Requirements <> ***** Sensor Fusion Architecture :ignore: -A general sensor fusion architecture using complementary filters is shown in Figure ref:fig:detail_control_sensor_fusion_overview, where multiple sensors (in this case two) measure the same physical quantity $x$. +A general sensor fusion architecture using complementary filters is shown in Figure\nbsp{}ref:fig:detail_control_sensor_fusion_overview, where multiple sensors (in this case two) measure the same physical quantity $x$. The sensor output signals $\hat{x}_1$ and $\hat{x}_2$ represent estimates of $x$. These estimates are filtered by complementary filters and combined to form a new estimate $\hat{x}$. @@ -7422,7 +7422,7 @@ These estimates are filtered by complementary filters and combined to form a new #+caption: Schematic of a sensor fusion architecture using complementary filters. [[file:figs/detail_control_sensor_fusion_overview.png]] -The complementary property of filters $H_1(s)$ and $H_2(s)$ requires that the sum of their transfer functions equals one at all frequencies eqref:eq:detail_control_sensor_comp_filter. +The complementary property of filters $H_1(s)$ and $H_2(s)$ requires that the sum of their transfer functions equals one at all frequencies\nbsp{}eqref:eq:detail_control_sensor_comp_filter. \begin{equation}\label{eq:detail_control_sensor_comp_filter} H_1(s) + H_2(s) = 1 @@ -7431,11 +7431,11 @@ The complementary property of filters $H_1(s)$ and $H_2(s)$ requires that the su ***** Sensor Models and Sensor Normalization To analyze sensor fusion architectures, appropriate sensor models are required. -The model shown in Figure ref:fig:detail_control_sensor_model consists of a linear time invariant (LTI) system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing sensor noise. +The model shown in Figure\nbsp{}ref:fig:detail_control_sensor_model consists of a linear time invariant (LTI) system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing sensor noise. The model input $x$ is the measured physical quantity, and its output $\tilde{x}_i$ is the "raw" output of the sensor. Prior to filtering the sensor outputs $\tilde{x}_i$ with complementary filters, the sensors are typically normalized to simplify the fusion process. -This normalization involves using an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$, and filtering the sensor output by the inverse of this estimate $\hat{G}_i^{-1}(s)$, as shown in Figure ref:fig:detail_control_sensor_model_calibrated. +This normalization involves using an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$, and filtering the sensor output by the inverse of this estimate $\hat{G}_i^{-1}(s)$, as shown in Figure\nbsp{}ref:fig:detail_control_sensor_model_calibrated. It is assumed that the sensor inverse $\hat{G}_i^{-1}(s)$ is proper and stable. This approach ensures that the units of the estimates $\hat{x}_i$ match the units of the physical quantity $x$. The sensor dynamics estimate $\hat{G}_i(s)$ may be a simple gain or a more complex transfer function. @@ -7458,10 +7458,10 @@ The sensor dynamics estimate $\hat{G}_i(s)$ may be a simple gain or a more compl #+end_subfigure #+end_figure -Two normalized sensors are then combined to form a super sensor as shown in Figure ref:fig:detail_control_sensor_fusion_super_sensor. +Two normalized sensors are then combined to form a super sensor as shown in Figure\nbsp{}ref:fig:detail_control_sensor_fusion_super_sensor. The two sensors measure the same physical quantity $x$ with dynamics $G_1(s)$ and $G_2(s)$, and with uncorrelated noises $n_1$ and $n_2$. The signals from both normalized sensors are fed into two complementary filters $H_1(s)$ and $H_2(s)$ and then combined to yield an estimate $\hat{x}$ of $x$. -The super sensor output $\hat{x}$ is therefore described by eqref:eq:detail_control_sensor_comp_filter_estimate. +The super sensor output $\hat{x}$ is therefore described by\nbsp{}eqref:eq:detail_control_sensor_comp_filter_estimate. \begin{equation}\label{eq:detail_control_sensor_comp_filter_estimate} \hat{x} = \Big( H_1(s) \hat{G}_1^{-1}(s) G_1(s) + H_2(s) \hat{G}_2^{-1}(s) G_2(s) \Big) x + H_1(s) \hat{G}_1^{-1}(s) G_1(s) n_1 + H_2(s) \hat{G}_2^{-1}(s) G_2(s) n_2 @@ -7473,14 +7473,14 @@ The super sensor output $\hat{x}$ is therefore described by eqref:eq:detail_cont ***** Noise Sensor Filtering -First, consider the case where all sensors are perfectly normalized eqref:eq:detail_control_sensor_perfect_dynamics. +First, consider the case where all sensors are perfectly normalized\nbsp{}eqref:eq:detail_control_sensor_perfect_dynamics. The effects of imperfect normalization will be addressed subsequently. \begin{equation}\label{eq:detail_control_sensor_perfect_dynamics} \frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) = 1 \end{equation} -In that case, the super sensor output $\hat{x}$ equals $x$ plus the filtered noise from both sensors eqref:eq:detail_control_sensor_estimate_perfect_dyn. +In that case, the super sensor output $\hat{x}$ equals $x$ plus the filtered noise from both sensors\nbsp{}eqref:eq:detail_control_sensor_estimate_perfect_dyn. From this equation, it is evident that the complementary filters $H_1(s)$ and $H_2(s)$ operate solely on the sensor noise. Thus, this sensor fusion architecture allows filtering of sensor noise without introducing distortion in the measured physical quantity. This fundamental property necessitates that the two filters are complementary. @@ -7489,13 +7489,13 @@ This fundamental property necessitates that the two filters are complementary. \hat{x} = x + H_1(s) n_1 + H_2(s) n_2 \end{equation} -The estimation error $\epsilon_x$, defined as the difference between the sensor output $\hat{x}$ and the measured quantity $x$, is computed for the super sensor eqref:eq:detail_control_sensor_estimate_error. +The estimation error $\epsilon_x$, defined as the difference between the sensor output $\hat{x}$ and the measured quantity $x$, is computed for the super sensor\nbsp{}eqref:eq:detail_control_sensor_estimate_error. \begin{equation}\label{eq:detail_control_sensor_estimate_error} \epsilon_x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2 \end{equation} -As shown in eqref:eq:detail_control_sensor_noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\epsilon_x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$. +As shown in\nbsp{}eqref:eq:detail_control_sensor_noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\epsilon_x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$. \begin{equation}\label{eq:detail_control_sensor_noise_filtering_psd} \Phi_{\epsilon_x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega) @@ -7510,11 +7510,11 @@ 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 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 ref:fig:detail_control_sensor_model_uncertainty). +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). 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 ref:fig:detail_control_sensor_model_uncertainty_simplified. +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 @@ -7534,9 +7534,9 @@ 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 ref:fig:detail_control_sensor_fusion_dynamic_uncertainty. -The super sensor dynamics 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 ref:fig:detail_control_sensor_uncertainty_set_super_sensor). +The sensor fusion architecture incorporating 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). \begin{equation}\label{eq:detail_control_sensor_super_sensor_dyn_uncertainty} \frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s) @@ -7567,15 +7567,15 @@ As it is generally desired to limit the dynamical uncertainty of the super senso <> ***** Introduction :ignore: -As established in Section ref:ssec:detail_control_sensor_fusion_requirements, the super sensor's noise characteristics and robustness are directly dependent on the complementary filters' norm. +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. ***** Synthesis Objective -The primary objective is to shape the norms of two filters $H_1(s)$ and $H_2(s)$ while ensuring they maintain their complementary property as defined in eqref:eq:detail_control_sensor_comp_filter. -This is equivalent to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ that satisfy conditions eqref:eq:detail_control_sensor_hinf_cond_complementarity, eqref:eq:detail_control_sensor_hinf_cond_h1, and eqref:eq:detail_control_sensor_hinf_cond_h2. +The primary objective is to shape the norms of two filters $H_1(s)$ and $H_2(s)$ while ensuring they maintain their complementary property as defined in\nbsp{}eqref:eq:detail_control_sensor_comp_filter. +This is equivalent to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ that satisfy conditions\nbsp{}eqref:eq:detail_control_sensor_hinf_cond_complementarity, eqref:eq:detail_control_sensor_hinf_cond_h1, and eqref:eq:detail_control_sensor_hinf_cond_h2. Weighting transfer functions $W_1(s)$ and $W_2(s)$ are strategically selected to define the maximum desired norm of the complementary filters during the synthesis process. \begin{subequations}\label{eq:detail_control_sensor_comp_filter_problem_form} @@ -7588,7 +7588,7 @@ Weighting transfer functions $W_1(s)$ and $W_2(s)$ are strategically selected to ***** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis -The synthesis objective can be expressed as a standard $\mathcal{H}_\infty$ optimization problem by considering the generalized plant $P(s)$ illustrated in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant and mathematically described by eqref:eq:detail_control_sensor_generalized_plant. +The synthesis objective can be expressed as a standard $\mathcal{H}_\infty$ optimization problem by considering the generalized plant $P(s)$ illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant and mathematically described by\nbsp{}eqref:eq:detail_control_sensor_generalized_plant. \begin{equation}\label{eq:detail_control_sensor_generalized_plant} \begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) \\ 1 & 0 \end{bmatrix} @@ -7612,13 +7612,13 @@ The synthesis objective can be expressed as a standard $\mathcal{H}_\infty$ opti #+end_subfigure #+end_figure -Applying standard $\mathcal{H}_\infty\text{-synthesis}$ to the generalized plant $P(s)$ is equivalent to finding a stable filter $H_2(s)$ that, based on input $v$, generates an output signal $u$ such that the $\mathcal{H}_\infty$ norm of the system shown in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_fb from $w$ to $[z_1, \ z_2]$ does not exceed unity, as expressed in eqref:eq:detail_control_sensor_hinf_syn_obj. +Applying standard $\mathcal{H}_\infty\text{-synthesis}$ to the generalized plant $P(s)$ is equivalent to finding a stable filter $H_2(s)$ that, based on input $v$, generates an output signal $u$ such that the $\mathcal{H}_\infty$ norm of the system shown in Figure\nbsp{}ref:fig:detail_control_sensor_h_infinity_robust_fusion_fb from $w$ to $[z_1, \ z_2]$ does not exceed unity, as expressed in\nbsp{}eqref:eq:detail_control_sensor_hinf_syn_obj. \begin{equation}\label{eq:detail_control_sensor_hinf_syn_obj} \left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 \end{equation} -By defining $H_1(s)$ as the complement of $H_2(s)$ eqref:eq:detail_control_sensor_definition_H1, the $\mathcal{H}_\infty\text{-synthesis}$ objective becomes equivalent to eqref:eq:detail_control_sensor_hinf_problem, ensuring that conditions eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2 are satisfied. +By defining $H_1(s)$ as the complement of $H_2(s)$\nbsp{}eqref:eq:detail_control_sensor_definition_H1, the $\mathcal{H}_\infty\text{-synthesis}$ objective becomes equivalent to eqref:eq:detail_control_sensor_hinf_problem, ensuring that conditions\nbsp{}eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2 are satisfied. \begin{equation}\label{eq:detail_control_sensor_definition_H1} H_1(s) \triangleq 1 - H_2(s) @@ -7628,10 +7628,10 @@ By defining $H_1(s)$ as the complement of $H_2(s)$ eqref:eq:detail_control_senso \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 \end{equation} -Therefore, applying $\mathcal{H}_\infty\text{-synthesis}$ to the standard plant $P(s)$ generates two filters, $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$, that are complementary as required by eqref:eq:detail_control_sensor_comp_filter_problem_form, with norms bounded by the specified constraints in eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2. +Therefore, applying $\mathcal{H}_\infty\text{-synthesis}$ to the standard plant $P(s)$ generates two filters, $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$, that are complementary as required by\nbsp{}eqref:eq:detail_control_sensor_comp_filter_problem_form, with norms bounded by the specified constraints in\nbsp{}eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2. -It should be noted that there exists only an implication (not an equivalence) between the $\mathcal{H}_\infty$ norm condition in eqref:eq:detail_control_sensor_hinf_problem and the initial synthesis objectives in eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2. -Consequently, the optimization may be somewhat conservative with respect to the set of filters on which it operates [[cite:&skogestad07_multiv_feedb_contr,Chap. 2.8.3]]. +It should be noted that there exists only an implication (not an equivalence) between the $\mathcal{H}_\infty$ norm condition in\nbsp{}eqref:eq:detail_control_sensor_hinf_problem and the initial synthesis objectives in\nbsp{}eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2. +Consequently, the optimization may be somewhat conservative with respect to the set of filters on which it operates\nbsp{}[[cite:&skogestad07_multiv_feedb_contr,Chap. 2.8.3]]. ***** Weighting Functions Design @@ -7641,17 +7641,17 @@ The proper design of these weighting functions is essential for the successful i Three key considerations should guide the design of weighting functions. First, only proper and stable transfer functions should be employed. Second, the order of the weighting functions should remain reasonably small to minimize computational costs associated with solving the optimization problem and to facilitate practical implementation of the filters (as the order of the synthesized filters equals the sum of the weighting functions' orders). -Third, the fundamental limitations imposed by the complementary property eqref:eq:detail_control_sensor_comp_filter must be respected, which implies that $|H_1(j\omega)|$ and $|H_2(j\omega)|$ cannot both be made small at the same frequency. +Third, the fundamental limitations imposed by the complementary property\nbsp{}eqref:eq:detail_control_sensor_comp_filter must be respected, which implies that $|H_1(j\omega)|$ and $|H_2(j\omega)|$ cannot both be made small at the same frequency. When designing complementary filters, it is typically desirable to specify their slopes, "blending" frequency, and maximum gains at low and high frequencies. -To facilitate the expression of these specifications, formula eqref:eq:detail_control_sensor_weight_formula is proposed for the design of weighting functions. +To facilitate the expression of these specifications, formula\nbsp{}eqref:eq:detail_control_sensor_weight_formula is proposed for the design of weighting functions. The parameters in this formula are $G_0 = \lim_{\omega \to 0} |W(j\omega)|$ (the low-frequency gain), $G_\infty = \lim_{\omega \to \infty} |W(j\omega)|$ (the high-frequency gain), $G_c = |W(j\omega_c)|$ (the gain at a specific frequency $\omega_c$ in $\si{rad/s}$), and $n$ (the slope between high and low frequency, which also corresponds to the order of the weighting function). -The typical magnitude response of a weighting function generated using eqref:eq:detail_control_sensor_weight_formula is illustrated in Figure ref:fig:detail_control_sensor_weight_formula. +The typical magnitude response of a weighting function generated using\nbsp{}eqref:eq:detail_control_sensor_weight_formula is illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_weight_formula. #+attr_latex: :options []{0.49\linewidth} #+begin_minipage #+name: fig:detail_control_sensor_weight_formula -#+caption: Magnitude of a weighting function generated using eqref:eq:detail_control_sensor_weight_formula, $G_0 = 10^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$. +#+caption: Magnitude of a weighting function generated using\nbsp{}eqref:eq:detail_control_sensor_weight_formula, $G_0 = 10^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$. #+attr_latex: :width 0.95\linewidth :float nil [[file:figs/detail_control_sensor_weight_formula.png]] #+end_minipage @@ -7676,9 +7676,9 @@ Consider the design of two complementary filters $H_1(s)$ and $H_2(s)$ with the - The slope of $|H_2(j\omega)|$ should be $-3$ above $\SI{10}{Hz}$, with a high-frequency gain of $10^{-3}$ The first step involves translating these requirements by appropriately designing the weighting functions. -The formula proposed in eqref:eq:detail_control_sensor_weight_formula is employed for this purpose. -The parameters used are summarized in Table ref:tab:detail_control_sensor_weights_params. -The inverse magnitudes of the designed weighting functions, which represent the maximum allowable norms of the complementary filters, are depicted by the dashed lines in Figure ref:fig:detail_control_sensor_hinf_filters_results. +The formula proposed in\nbsp{}eqref:eq:detail_control_sensor_weight_formula is employed for this purpose. +The parameters used are summarized in Table\nbsp{}ref:tab:detail_control_sensor_weights_params. +The inverse magnitudes of the designed weighting functions, which represent the maximum allowable norms of the complementary filters, are depicted by the dashed lines in Figure\nbsp{}ref:fig:detail_control_sensor_hinf_filters_results. #+attr_latex: :options [b]{0.44\linewidth} #+begin_minipage @@ -7702,23 +7702,23 @@ The inverse magnitudes of the designed weighting functions, which represent the [[file:figs/detail_control_sensor_hinf_filters_results.png]] #+end_minipage -Standard $\mathcal{H}_\infty\text{-synthesis}$ is then applied to the generalized plant shown in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant. +Standard $\mathcal{H}_\infty\text{-synthesis}$ is then applied to the generalized plant shown in Figure\nbsp{}ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant. This yields the filter $H_2(s)$ that minimizes the $\mathcal{H}_\infty$ norm from input $w$ to outputs $[z_1,\ z_2]^{\intercal}$. The resulting $\mathcal{H}_\infty$ norm is found to be close to unity, indicating successful synthesis: the norms of the complementary filters remain below the specified upper bounds. -This is confirmed by the Bode plots of the obtained complementary filters in Figure ref:fig:detail_control_sensor_hinf_filters_results. +This is confirmed by the Bode plots of the obtained complementary filters in Figure\nbsp{}ref:fig:detail_control_sensor_hinf_filters_results. This straightforward example demonstrates that the proposed methodology for shaping complementary filters is both simple and effective. **** Synthesis of a set of three complementary filters <> -Certain applications necessitate the fusion of more than two sensors [[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]]. -At LIGO, for example, a super sensor is formed by merging three distinct sensors: an LVDT, a seismometer, and a geophone [[cite:&matichard15_seism_isolat_advan_ligo]]. +Certain applications necessitate the fusion of more than two sensors\nbsp{}[[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]]. +At LIGO, for example, a super sensor is formed by merging three distinct sensors: an LVDT, a seismometer, and a geophone\nbsp{}[[cite:&matichard15_seism_isolat_advan_ligo]]. -For merging $n>2$ sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure ref:fig:detail_control_sensor_fusion_three. -Fusion can be implemented either "sequentially," utilizing $n-1$ sets of two complementary filters (Figure ref:fig:detail_control_sensor_fusion_three_sequential), or "in parallel," employing a single set of $n$ complementary filters (Figure ref:fig:detail_control_sensor_fusion_three_parallel). +For merging $n>2$ sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_fusion_three. +Fusion can be implemented either "sequentially," utilizing $n-1$ sets of two complementary filters (Figure\nbsp{}ref:fig:detail_control_sensor_fusion_three_sequential), or "in parallel," employing a single set of $n$ complementary filters (Figure\nbsp{}ref:fig:detail_control_sensor_fusion_three_parallel). While conventional sensor fusion synthesis techniques can be applied to the sequential approach, parallel architecture implementation requires a novel synthesis method for multiple complementary filters. -Previous literature has offered only simple analytical formulas for this purpose [[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]]. +Previous literature has offered only simple analytical formulas for this purpose\nbsp{}[[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]]. This section presents a generalization of the proposed complementary filter synthesis method to address this gap. #+name: fig:detail_control_sensor_fusion_three @@ -7739,7 +7739,7 @@ This section presents a generalization of the proposed complementary filter synt #+end_subfigure #+end_figure -The synthesis objective is to compute a set of $n$ stable transfer functions $[H_1(s),\ H_2(s),\ \dots,\ H_n(s)]$ that satisfy conditions eqref:eq:detail_control_sensor_hinf_cond_compl_gen and eqref:eq:detail_control_sensor_hinf_cond_perf_gen. +The synthesis objective is to compute a set of $n$ stable transfer functions $[H_1(s),\ H_2(s),\ \dots,\ H_n(s)]$ that satisfy conditions\nbsp{}eqref:eq:detail_control_sensor_hinf_cond_compl_gen and eqref:eq:detail_control_sensor_hinf_cond_perf_gen. \begin{subequations}\label{eq:detail_control_sensor_hinf_problem_gen} \begin{align} @@ -7750,8 +7750,8 @@ The synthesis objective is to compute a set of $n$ stable transfer functions $[H The transfer functions $[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weights selected to specify the maximum complementary filters' norm during synthesis. -This synthesis objective is closely related to the one described in Section ref:ssec:detail_control_sensor_hinf_method, and the proposed synthesis method represents a generalization of the approach previously presented. -A set of $n$ complementary filters can be shaped by applying standard $\mathcal{H}_\infty\text{-synthesis}$ to the generalized plant $P_n(s)$ described by eqref:eq:detail_control_sensor_generalized_plant_n_filters. +This synthesis objective is closely related to the one described in Section\nbsp{}ref:ssec:detail_control_sensor_hinf_method, and the proposed synthesis method represents a generalization of the approach previously presented. +A set of $n$ complementary filters can be shaped by applying standard $\mathcal{H}_\infty\text{-synthesis}$ to the generalized plant $P_n(s)$ described by\nbsp{}eqref:eq:detail_control_sensor_generalized_plant_n_filters. \begin{equation}\label{eq:detail_control_sensor_generalized_plant_n_filters} \begin{bmatrix} z_1 \\ \vdots \\ z_n \\ v \end{bmatrix} = P_n(s) \begin{bmatrix} w \\ u_1 \\ \vdots \\ u_{n-1} \end{bmatrix}; \quad @@ -7765,14 +7765,14 @@ A set of $n$ complementary filters can be shaped by applying standard $\mathcal{ \end{bmatrix} \end{equation} -If the synthesis is successful, a set of $n-1$ filters $[H_2(s),\ H_3(s),\ \dots,\ H_n(s)]$ is obtained such that eqref:eq:detail_control_sensor_hinf_syn_obj_gen is satisfied. +If the synthesis is successful, a set of $n-1$ filters $[H_2(s),\ H_3(s),\ \dots,\ H_n(s)]$ is obtained such that\nbsp{}eqref:eq:detail_control_sensor_hinf_syn_obj_gen is satisfied. \begin{equation}\label{eq:detail_control_sensor_hinf_syn_obj_gen} \left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1 \end{equation} -$H_1(s)$ is then defined using eqref:eq:detail_control_sensor_h1_comp_h2_hn, which ensures the complementary property for the set of $n$ filters eqref:eq:detail_control_sensor_hinf_cond_compl_gen. -Condition eqref:eq:detail_control_sensor_hinf_cond_perf_gen is satisfied through eqref:eq:detail_control_sensor_hinf_syn_obj_gen. +$H_1(s)$ is then defined using\nbsp{}eqref:eq:detail_control_sensor_h1_comp_h2_hn, which ensures the complementary property for the set of $n$ filters\nbsp{}eqref:eq:detail_control_sensor_hinf_cond_compl_gen. +Condition\nbsp{}eqref:eq:detail_control_sensor_hinf_cond_perf_gen is satisfied through\nbsp{}eqref:eq:detail_control_sensor_hinf_syn_obj_gen. \begin{equation}\label{eq:detail_control_sensor_h1_comp_h2_hn} H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big] @@ -7780,9 +7780,9 @@ Condition eqref:eq:detail_control_sensor_hinf_cond_perf_gen is satisfied through To validate the proposed method for synthesizing a set of three complementary filters, an example is provided. The sensors to be merged are a displacement sensor (effective from DC up to $\SI{1}{Hz}$), a geophone (effective from $1$ to $\SI{10}{Hz}$), and an accelerometer (effective above $\SI{10}{Hz}$). -Three weighting functions are designed using formula eqref:eq:detail_control_sensor_weight_formula, and their inverse magnitudes are shown in Figure ref:fig:detail_control_sensor_three_complementary_filters_results (dashed curves). +Three weighting functions are designed using formula\nbsp{}eqref:eq:detail_control_sensor_weight_formula, and their inverse magnitudes are shown in Figure\nbsp{}ref:fig:detail_control_sensor_three_complementary_filters_results (dashed curves). -Consider the generalized plant $P_3(s)$ shown in Figure ref:fig:detail_control_sensor_comp_filter_three_hinf_fb, which is also described by eqref:eq:detail_control_sensor_generalized_plant_three_filters. +Consider the generalized plant $P_3(s)$ shown in Figure\nbsp{}ref:fig:detail_control_sensor_comp_filter_three_hinf_fb, which is also described by\nbsp{}eqref:eq:detail_control_sensor_generalized_plant_three_filters. \begin{equation}\label{eq:detail_control_sensor_generalized_plant_three_filters} \begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix} @@ -7807,14 +7807,14 @@ Consider the generalized plant $P_3(s)$ shown in Figure ref:fig:detail_control_s #+end_figure Standard $\mathcal{H}_\infty\text{-synthesis}$ is performed on the generalized plant $P_3(s)$. -Two filters, $H_2(s)$ and $H_3(s)$, are obtained such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer from $w$ to $[z_1,\ z_2,\ z_3]$ of the system in Figure ref:fig:detail_control_sensor_comp_filter_three_hinf_fb is less than one. -Filter $H_1(s)$ is defined using eqref:eq:detail_control_sensor_h1_compl_h2_h3, thus ensuring the complementary property of the obtained set of filters. +Two filters, $H_2(s)$ and $H_3(s)$, are obtained such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer from $w$ to $[z_1,\ z_2,\ z_3]$ of the system in Figure\nbsp{}ref:fig:detail_control_sensor_comp_filter_three_hinf_fb is less than one. +Filter $H_1(s)$ is defined using\nbsp{}eqref:eq:detail_control_sensor_h1_compl_h2_h3, thus ensuring the complementary property of the obtained set of filters. \begin{equation}\label{eq:detail_control_sensor_h1_compl_h2_h3} H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big] \end{equation} -Figure ref:fig:detail_control_sensor_three_complementary_filters_results displays the three synthesized complementary filters (solid lines), confirming the successful synthesis. +Figure\nbsp{}ref:fig:detail_control_sensor_three_complementary_filters_results displays the three synthesized complementary filters (solid lines), confirming the successful synthesis. **** Conclusion @@ -7831,42 +7831,42 @@ Looking forward, it would be interesting to investigate how sensor fusion (parti **** Introduction :ignore: -The control of parallel manipulators (and any MIMO system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of SISO control for the decoupled plant (discussed in section ref:sec:detail_control_cf). +The control of parallel manipulators (and any MIMO system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of SISO control for the decoupled plant (discussed in section\nbsp{}ref:sec:detail_control_cf). When sensors are integrated within the struts, decentralized control may be applied, as the system is already well decoupled at low frequency. -For instance, [[cite:&furutani04_nanom_cuttin_machin_using_stewar]] implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate PI controllers for each strut. -A similar control architecture was proposed in [[cite:&du14_piezo_actuat_high_precis_flexib]] using strain gauge sensors integrated in each strut. +For instance,\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar]] implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate PI controllers for each strut. +A similar control architecture was proposed in\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]] using strain gauge sensors integrated in each strut. An alternative strategy involves decoupling the system in the Cartesian frame using Jacobian matrices. As demonstrated during the study of Stewart platform kinematics, Jacobian matrices can be utilized to map actuator forces to forces and torques applied on the top platform. This approach enables the implementation of controllers in a defined frame. -It has been applied with various sensor types including force sensors [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], relative displacement sensors [[cite:&kim00_robus_track_contr_desig_dof_paral_manip]], and inertial sensors [[cite:&li01_simul_vibrat_isolat_point_contr;&abbas14_vibrat_stewar_platf]]. +It has been applied with various sensor types including force sensors\nbsp{}[[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], relative displacement sensors\nbsp{}[[cite:&kim00_robus_track_contr_desig_dof_paral_manip]], and inertial sensors\nbsp{}[[cite:&li01_simul_vibrat_isolat_point_contr;&abbas14_vibrat_stewar_platf]]. The Cartesian frame in which the system is decoupled is typically chosen at the point of interest (i.e., where the motion is of interest) or at the center of mass. Modal decoupling represents another noteworthy decoupling strategy, wherein the "local" plant inputs and outputs are mapped to the modal space. In this approach, multiple SISO plants, each corresponding to a single mode, can be controlled independently. -This decoupling strategy has been implemented for active damping applications [[cite:&holterman05_activ_dampin_based_decoup_colloc_contr]], which is logical as it is often desirable to dampen specific modes. -The strategy has also been employed in [[cite:&pu11_six_degree_of_freed_activ]] for vibration isolation purposes using geophones, and in [[cite:&yang19_dynam_model_decoup_contr_flexib]] using force sensors. +This decoupling strategy has been implemented for active damping applications\nbsp{}[[cite:&holterman05_activ_dampin_based_decoup_colloc_contr]], which is logical as it is often desirable to dampen specific modes. +The strategy has also been employed in\nbsp{}[[cite:&pu11_six_degree_of_freed_activ]] for vibration isolation purposes using geophones, and in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]] using force sensors. Another completely different strategy would be to implement a multivariable control directly on the coupled system. -$\mathcal{H}_\infty$ and $\mu\text{-synthesis}$ were applied to a Stewart platform model in [[cite:&lei08_multi_objec_robus_activ_vibrat]]. -In [[cite:&xie17_model_contr_hybrid_passiv_activ]], decentralized force feedback was first applied, followed by $\mathcal{H}_2\text{-synthesis}$ for vibration isolation based on accelerometers. -$\mathcal{H}_\infty\text{-synthesis}$ was also employed in [[cite:&jiao18_dynam_model_exper_analy_stewar]] for active damping based on accelerometers. -A comparative study between $\mathcal{H}_\infty\text{-synthesis}$ and decentralized control in the frame of the struts was performed in [[cite:&thayer02_six_axis_vibrat_isolat_system]]. +$\mathcal{H}_\infty$ and $\mu\text{-synthesis}$ were applied to a Stewart platform model in\nbsp{}[[cite:&lei08_multi_objec_robus_activ_vibrat]]. +In\nbsp{}[[cite:&xie17_model_contr_hybrid_passiv_activ]], decentralized force feedback was first applied, followed by $\mathcal{H}_2\text{-synthesis}$ for vibration isolation based on accelerometers. +$\mathcal{H}_\infty\text{-synthesis}$ was also employed in\nbsp{}[[cite:&jiao18_dynam_model_exper_analy_stewar]] for active damping based on accelerometers. +A comparative study between $\mathcal{H}_\infty\text{-synthesis}$ and decentralized control in the frame of the struts was performed in\nbsp{}[[cite:&thayer02_six_axis_vibrat_isolat_system]]. Their experimental closed-loop results indicated that the $\mathcal{H}_\infty$ controller did not outperform the decentralized controller in the frame of the struts. These limitations were attributed to the model's poor ability to predict off-diagonal dynamics, which is crucial for $\mathcal{H}_\infty\text{-synthesis}$. The purpose of this section is to compare several methods for the decoupling of parallel manipulators, an analysis that appears to be lacking in the literature. -A simplified parallel manipulator model is introduced in Section ref:ssec:detail_control_decoupling_model as a test case for evaluating decoupling strategies. -The decentralized plant (transfer functions from actuators to sensors integrated in the struts) is examined in Section ref:ssec:detail_control_decoupling_decentralized. -Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section ref:ssec:detail_control_decoupling_jacobian), modal decoupling (Section ref:ssec:detail_control_decoupling_modal), and Singular Value Decomposition (SVD) decoupling (Section ref:ssec:detail_control_decoupling_svd). -Finally, a comparative analysis with concluding observations is provided in Section ref:ssec:detail_control_decoupling_comp. +A simplified parallel manipulator model is introduced in Section\nbsp{}ref:ssec:detail_control_decoupling_model as a test case for evaluating decoupling strategies. +The decentralized plant (transfer functions from actuators to sensors integrated in the struts) is examined in Section\nbsp{}ref:ssec:detail_control_decoupling_decentralized. +Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_jacobian), modal decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_modal), and Singular Value Decomposition (SVD) decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_svd). +Finally, a comparative analysis with concluding observations is provided in Section\nbsp{}ref:ssec:detail_control_decoupling_comp. **** Test Model <> Instead of utilizing the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis. -The system illustrated in Figure ref:fig:detail_control_decoupling_model_test is used for this purpose. +The system illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test is used for this purpose. It possesses three degrees of freedom (DoF) and incorporates three parallel struts. Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform. @@ -7898,7 +7898,7 @@ Two reference frames are defined within this model: frame $\{M\}$ with origin $O #+end_scriptsize #+end_minipage -The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass eqref:eq:detail_control_decoupling_model_eom, where $\bm{\mathcal{X}}_{\{M\}}$ represents the two translations and one rotation with respect to the center of mass, and $\bm{\mathcal{F}}_{\{M\}}$ denotes the forces and torque applied at the center of mass. +The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass\nbsp{}eqref:eq:detail_control_decoupling_model_eom, where $\bm{\mathcal{X}}_{\{M\}}$ represents the two translations and one rotation with respect to the center of mass, and $\bm{\mathcal{F}}_{\{M\}}$ denotes the forces and torque applied at the center of mass. \begin{equation}\label{eq:detail_control_decoupling_model_eom} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t), \quad @@ -7913,7 +7913,7 @@ The equations of motion are derived by applying Newton's second law to the suspe \end{bmatrix} \end{equation} -The Jacobian matrix $\bm{J}_{\{M\}}$ is employed to map the spring, damping, and actuator forces to XY forces and Z torque expressed at the center of mass eqref:eq:detail_control_decoupling_jacobian_CoM. +The Jacobian matrix $\bm{J}_{\{M\}}$ is employed to map the spring, damping, and actuator forces to XY forces and Z torque expressed at the center of mass\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoM. \begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM} \bm{J}_{\{M\}} = \begin{bmatrix} @@ -7923,13 +7923,13 @@ The Jacobian matrix $\bm{J}_{\{M\}}$ is employed to map the spring, damping, and \end{bmatrix} \end{equation} -Subsequently, the equation of motion relating the actuator forces $\tau$ to the motion of the mass $\bm{\mathcal{X}}_{\{M\}}$ is derived eqref:eq:detail_control_decoupling_plant_cartesian. +Subsequently, the equation of motion relating the actuator forces $\tau$ to the motion of the mass $\bm{\mathcal{X}}_{\{M\}}$ is derived\nbsp{}eqref:eq:detail_control_decoupling_plant_cartesian. \begin{equation}\label{eq:detail_control_decoupling_plant_cartesian} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t) \end{equation} -The matrices representing the payload inertia, actuator stiffness, and damping are shown in eqref:eq:detail_control_decoupling_system_matrices. +The matrices representing the payload inertia, actuator stiffness, and damping are shown in\nbsp{}eqref:eq:detail_control_decoupling_system_matrices. \begin{equation}\label{eq:detail_control_decoupling_system_matrices} \bm{M}_{\{M\}} = \begin{bmatrix} @@ -7949,27 +7949,27 @@ The matrices representing the payload inertia, actuator stiffness, and damping a \end{bmatrix} \end{equation} -The parameters employed for the subsequent analysis are summarized in Table ref:tab:detail_control_decoupling_test_model_params, which includes values for geometric parameters ($l_a$, $h_a$), mechanical properties (actuator stiffness $k$ and damping $c$), and inertial characteristics (payload mass $m$ and rotational inertia $I$). +The parameters employed for the subsequent analysis are summarized in Table\nbsp{}ref:tab:detail_control_decoupling_test_model_params, which includes values for geometric parameters ($l_a$, $h_a$), mechanical properties (actuator stiffness $k$ and damping $c$), and inertial characteristics (payload mass $m$ and rotational inertia $I$). **** Control in the frame of the struts <> The dynamics in the frame of the struts are first examined. -The equation of motion relating actuator forces $\bm{\mathcal{\tau}}$ to strut relative motion $\bm{\mathcal{L}}$ is derived from equation eqref:eq:detail_control_decoupling_plant_cartesian by mapping the Cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix $\bm{J}_{\{M\}}$ defined in eqref:eq:detail_control_decoupling_jacobian_CoM. -The obtained transfer function from $\bm{\mathcal{\tau}}$ to $\bm{\mathcal{L}}$ is shown in eqref:eq:detail_control_decoupling_plant_decentralized. +The equation of motion relating actuator forces $\bm{\mathcal{\tau}}$ to strut relative motion $\bm{\mathcal{L}}$ is derived from equation\nbsp{}eqref:eq:detail_control_decoupling_plant_cartesian by mapping the Cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix $\bm{J}_{\{M\}}$ defined in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoM. +The obtained transfer function from $\bm{\mathcal{\tau}}$ to $\bm{\mathcal{L}}$ is shown in\nbsp{}eqref:eq:detail_control_decoupling_plant_decentralized. \begin{equation}\label{eq:detail_control_decoupling_plant_decentralized} \frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1} \end{equation} -At low frequencies, the plant converges to a diagonal constant matrix whose diagonal elements are equal to the actuator stiffnesses eqref:eq:detail_control_decoupling_plant_decentralized_low_freq. +At low frequencies, the plant converges to a diagonal constant matrix whose diagonal elements are equal to the actuator stiffnesses\nbsp{}eqref:eq:detail_control_decoupling_plant_decentralized_low_freq. At high frequencies, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal. \begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq} \bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}} \end{equation} -The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is illustrated in Figure ref:fig:detail_control_decoupling_coupled_plant_bode. +The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_coupled_plant_bode. This representation confirms that at low frequencies (below the first suspension mode), the plant is well decoupled. Depending on the symmetry present in the system, certain diagonal elements may exhibit identical values, as demonstrated for struts 2 and 3 in this example. @@ -7981,7 +7981,7 @@ Depending on the symmetry present in the system, certain diagonal elements may e <> ***** Jacobian Matrix -The Jacobian matrix $\bm{J}_{\{O\}}$ serves a dual purpose in the decoupling process: it converts strut velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$, and it transforms actuator forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$, as expressed in equation eqref:eq:detail_control_decoupling_jacobian. +The Jacobian matrix $\bm{J}_{\{O\}}$ serves a dual purpose in the decoupling process: it converts strut velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$, and it transforms actuator forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$, as expressed in equation\nbsp{}eqref:eq:detail_control_decoupling_jacobian. \begin{subequations}\label{eq:detail_control_decoupling_jacobian} \begin{align} @@ -7990,7 +7990,7 @@ The Jacobian matrix $\bm{J}_{\{O\}}$ serves a dual purpose in the decoupling pro \end{align} \end{subequations} -The resulting plant (Figure ref:fig:detail_control_jacobian_decoupling_arch) have inputs and outputs with clear physical interpretations: +The resulting plant (Figure\nbsp{}ref:fig:detail_control_jacobian_decoupling_arch) have inputs and outputs with clear physical interpretations: - $\bm{\mathcal{F}}_{\{O\}}$ represents forces/torques applied on the payload at the origin of frame $\{O\}$ - $\bm{\mathcal{X}}_{\{O\}}$ represents translations/rotation of the payload expressed in frame $\{O\}$ @@ -7998,7 +7998,7 @@ The resulting plant (Figure ref:fig:detail_control_jacobian_decoupling_arch) hav #+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 eqref:eq:detail_control_decoupling_plant_jacobian. +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. \begin{equation}\label{eq:detail_control_decoupling_plant_jacobian} \frac{\bm{\mathcal{X}}_{\{O\}}}{\bm{\mathcal{F}}_{\{O\}}}(s) = \bm{G}_{\{O\}}(s) = \left( \bm{J}_{\{O\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{O\}} s^2 + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{O\}} s + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{O\}} \right)^{-1} @@ -8009,7 +8009,7 @@ Two natural reference frames are particularly relevant: the center of mass and t ***** Center Of Mass -When the decoupling frame is located at the center of mass (frame $\{M\}$ in Figure ref:fig:detail_control_decoupling_model_test), the Jacobian matrix and its inverse are expressed as in eqref:eq:detail_control_decoupling_jacobian_CoM_inverse. +When the decoupling frame is located at the center of mass (frame $\{M\}$ in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test), the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoM_inverse. \begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse} \bm{J}_{\{M\}} = \begin{bmatrix} @@ -8023,13 +8023,13 @@ When the decoupling frame is located at the center of mass (frame $\{M\}$ in Fig \end{bmatrix} \end{equation} -Analytical formula of the plant $\bm{G}_{\{M\}}(s)$ is derived eqref:eq:detail_control_decoupling_plant_CoM. +Analytical formula of the plant $\bm{G}_{\{M\}}(s)$ is derived\nbsp{}eqref:eq:detail_control_decoupling_plant_CoM. \begin{equation}\label{eq:detail_control_decoupling_plant_CoM} \frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1} \end{equation} -At high frequencies, the plant converges to the inverse of the mass matrix, which is a diagonal matrix eqref:eq:detail_control_decoupling_plant_CoM_high_freq. +At high frequencies, the plant converges to the inverse of the mass matrix, which is a diagonal matrix\nbsp{}eqref:eq:detail_control_decoupling_plant_CoM_high_freq. \begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq} \bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix} @@ -8039,13 +8039,13 @@ At high frequencies, the plant converges to the inverse of the mass matrix, whic \end{bmatrix} \end{equation} -Consequently, the plant exhibits effective decoupling at frequencies above the highest suspension mode as shown in Figure ref:fig:detail_control_decoupling_jacobian_plant_CoM. -This strategy is typically employed in systems with low-frequency suspension modes [[cite:&butler11_posit_contr_lithog_equip]], where the plant approximates decoupled mass lines. +Consequently, the plant exhibits effective decoupling at frequencies above the highest suspension mode as shown in Figure\nbsp{}ref:fig:detail_control_decoupling_jacobian_plant_CoM. +This strategy is typically employed in systems with low-frequency suspension modes\nbsp{}[[cite:&butler11_posit_contr_lithog_equip]], where the plant approximates decoupled mass lines. The low-frequency coupling observed in this configuration has a clear physical interpretation. When a static force is applied at the center of mass, the suspended mass rotates around the center of stiffness. This rotation is due to torque induced by the stiffness of the first actuator (i.e. the one on the left side), which is not aligned with the force application point. -This phenomenon is illustrated in Figure ref:fig:detail_control_decoupling_model_test_CoM. +This phenomenon is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test_CoM. #+name: fig:detail_control_jacobian_decoupling_plant_CoM_results #+caption: Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_decoupling_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoM}). @@ -8067,7 +8067,7 @@ This phenomenon is illustrated in Figure ref:fig:detail_control_decoupling_model ***** Center Of Stiffness -When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in eqref:eq:detail_control_decoupling_jacobian_CoK_inverse. +When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoK_inverse. \begin{equation}\label{eq:detail_control_decoupling_jacobian_CoK_inverse} \bm{J}_{\{K\}} = \begin{bmatrix} @@ -8086,20 +8086,20 @@ However, it could alternatively be determined through analytical methods to ensu It should be noted that the existence of such a center of stiffness (i.e. a frame $\{K\}$ for which $\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}$ is diagonal) is not guaranteed for arbitrary systems. This property is typically achievable only in systems exhibiting specific symmetrical characteristics, as is the case in the present example. -The analytical expression for the plant in this configuration was then computed eqref:eq:detail_control_decoupling_plant_CoK. +The analytical expression for the plant in this configuration was then computed\nbsp{}eqref:eq:detail_control_decoupling_plant_CoK. \begin{equation}\label{eq:detail_control_decoupling_plant_CoK} \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1} \end{equation} -Figure ref:fig:detail_control_decoupling_jacobian_plant_CoK_results presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the center of stiffness. -The plant is well decoupled below the suspension mode with the lowest frequency eqref:eq:detail_control_decoupling_plant_CoK_low_freq, making it particularly suitable for systems with high stiffness. +Figure\nbsp{}ref:fig:detail_control_decoupling_jacobian_plant_CoK_results presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the center of stiffness. +The plant is well decoupled below the suspension mode with the lowest frequency\nbsp{}eqref:eq:detail_control_decoupling_plant_CoK_low_freq, making it particularly suitable for systems with high stiffness. \begin{equation}\label{eq:detail_control_decoupling_plant_CoK_low_freq} \bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-\intercal} \end{equation} -The physical reason for high-frequency coupling is illustrated in Figure ref:fig:detail_control_decoupling_model_test_CoK. +The physical reason for high-frequency coupling is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test_CoK. When a high-frequency force is applied at a point not aligned with the center of mass, it induces rotation around the center of mass. #+name: fig:detail_control_decoupling_jacobian_plant_CoK_results @@ -8124,28 +8124,28 @@ When a high-frequency force is applied at a point not aligned with the center of <> ***** Theory :ignore: -Modal decoupling represents an approach based on the principle that a mechanical system's behavior can be understood as a combination of contributions from various modes [[cite:&rankers98_machin]]. -To convert the dynamics in the modal space, the equation of motion are first written with respect to the center of mass eqref:eq:detail_control_decoupling_equation_motion_CoM. +Modal decoupling represents an approach based on the principle that a mechanical system's behavior can be understood as a combination of contributions from various modes\nbsp{}[[cite:&rankers98_machin]]. +To convert the dynamics in the modal space, the equation of motion are first written with respect to the center of mass\nbsp{}eqref:eq:detail_control_decoupling_equation_motion_CoM. \begin{equation}\label{eq:detail_control_decoupling_equation_motion_CoM} \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t) \end{equation} -For modal decoupling, a change of variables is introduced eqref:eq:detail_control_decoupling_modal_coordinates where $\bm{\mathcal{X}}_{m}$ represents the modal amplitudes and $\bm{\Phi}$ is a $n \times n$[fn:detail_control_2] matrix whose columns correspond to the mode shapes of the system, computed from $\bm{M}_{\{M\}}$ and $\bm{K}_{\{M\}}$. +For modal decoupling, a change of variables is introduced\nbsp{}eqref:eq:detail_control_decoupling_modal_coordinates where $\bm{\mathcal{X}}_{m}$ represents the modal amplitudes and $\bm{\Phi}$ is a $n \times n$[fn:detail_control_2] matrix whose columns correspond to the mode shapes of the system, computed from $\bm{M}_{\{M\}}$ and $\bm{K}_{\{M\}}$. \begin{equation}\label{eq:detail_control_decoupling_modal_coordinates} \bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m} \end{equation} -By pre-multiplying equation eqref:eq:detail_control_decoupling_equation_motion_CoM by $\bm{\Phi}^{\intercal}$ and applying the change of variable eqref:eq:detail_control_decoupling_modal_coordinates, a new set of equations of motion is obtained eqref:eq:detail_control_decoupling_equation_modal_coordinates where $\bm{\tau}_m$ represents the modal input, while $\bm{M}_m$, $\bm{C}_m$, and $\bm{K}_m$ denote the modal mass, damping, and stiffness matrices respectively. +By pre-multiplying equation\nbsp{}eqref:eq:detail_control_decoupling_equation_motion_CoM by $\bm{\Phi}^{\intercal}$ and applying the change of variable\nbsp{}eqref:eq:detail_control_decoupling_modal_coordinates, a new set of equations of motion is obtained\nbsp{}eqref:eq:detail_control_decoupling_equation_modal_coordinates where $\bm{\tau}_m$ represents the modal input, while $\bm{M}_m$, $\bm{C}_m$, and $\bm{K}_m$ denote the modal mass, damping, and stiffness matrices respectively. \begin{equation}\label{eq:detail_control_decoupling_equation_modal_coordinates} \underbrace{\bm{\Phi}^{\intercal} \bm{M} \bm{\Phi}}_{\bm{M}_m} \bm{\ddot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{C} \bm{\Phi}}_{\bm{C}_m} \bm{\dot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{K} \bm{\Phi}}_{\bm{K}_m} \bm{\mathcal{X}}_m(t) = \underbrace{\bm{\Phi}^{\intercal} \bm{J}^{\intercal} \bm{\tau}(t)}_{\bm{\tau}_m(t)} \end{equation} -The inherent mathematical structure of the mass, damping, and stiffness matrices [[cite:&lang17_under, chapt. 8]] ensures that modal matrices are diagonal [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt. 2.3]]. -This diagonalization transforms equation eqref:eq:detail_control_decoupling_equation_modal_coordinates into a set of $n$ decoupled equations, enabling independent control of each mode without cross-interaction. +The inherent mathematical structure of the mass, damping, and stiffness matrices\nbsp{}[[cite:&lang17_under, chapt. 8]] ensures that modal matrices are diagonal\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt. 2.3]]. +This diagonalization transforms equation\nbsp{}eqref:eq:detail_control_decoupling_equation_modal_coordinates into a set of $n$ decoupled equations, enabling independent control of each mode without cross-interaction. -To implement this approach from a decentralized plant, the architecture shown in Figure ref:fig:detail_control_decoupling_modal is employed. +To implement this approach from a decentralized plant, the architecture shown in Figure\nbsp{}ref:fig:detail_control_decoupling_modal is employed. Inputs of the decoupling plant are the modal modal inputs $\bm{\tau}_m$ and the outputs are the modal amplitudes $\bm{\mathcal{X}}_m$. This implementation requires knowledge of the system's equations of motion, from which the mode shapes matrix $\bm{\Phi}$ is derived. The resulting decoupled system features diagonal elements each representing second-order resonant systems that are straightforward to control individually. @@ -8157,7 +8157,7 @@ The resulting decoupled system features diagonal elements each representing seco ***** Example :ignore: Modal decoupling was then applied to the test model. -First, the eigenvectors $\bm{\Phi}$ of $\bm{M}_{\{M\}}^{-1}\bm{K}_{\{M\}}$ were computed eqref:eq:detail_control_decoupling_modal_eigenvectors. +First, the eigenvectors $\bm{\Phi}$ of $\bm{M}_{\{M\}}^{-1}\bm{K}_{\{M\}}$ were computed\nbsp{}eqref:eq:detail_control_decoupling_modal_eigenvectors. While analytical derivation of eigenvectors could be obtained for such a simple system, they are typically computed numerically for practical applications. \begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors} @@ -8168,7 +8168,7 @@ While analytical derivation of eigenvectors could be obtained for such a simple \end{bmatrix},\ \alpha = \sqrt{\left( I + m (h_a^2 - 2 l_a^2) \right)^2 + 8 m^2 h_a^2 l_a^2} \end{equation} -The numerical values for the eigenvector matrix and its inverse are shown in eqref:eq:detail_control_decoupling_modal_eigenvectors_matrices. +The numerical values for the eigenvector matrix and its inverse are shown in\nbsp{}eqref:eq:detail_control_decoupling_modal_eigenvectors_matrices. \begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors_matrices} \bm{\Phi} = \begin{bmatrix} @@ -8183,8 +8183,8 @@ The numerical values for the eigenvector matrix and its inverse are shown in eqr \end{bmatrix} \end{equation} -The two computed matrices were implemented in the control architecture of Figure ref:fig:detail_control_decoupling_modal, resulting in three distinct second order plants as depicted in Figure ref:fig:detail_control_decoupling_modal_plant. -Each of these diagonal elements corresponds to a specific mode, as shown in Figure ref:fig:detail_control_decoupling_model_test_modal, resulting in a perfectly decoupled system. +The two computed matrices were implemented in the control architecture of Figure\nbsp{}ref:fig:detail_control_decoupling_modal, resulting in three distinct second order plants as depicted in Figure\nbsp{}ref:fig:detail_control_decoupling_modal_plant. +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}) @@ -8208,7 +8208,7 @@ Each of these diagonal elements corresponds to a specific mode, as shown in Figu <> ***** Singular Value Decomposition -Singular Value Decomposition (SVD) represents a powerful mathematical tool with extensive applications in data analysis [[cite:&brunton22_data, chapt. 1]] and multivariable control systems where it is particularly valuable for analyzing directional properties in multivariable systems [[cite:&skogestad07_multiv_feedb_contr]]. +Singular Value Decomposition (SVD) represents a powerful mathematical tool with extensive applications in data analysis\nbsp{}[[cite:&brunton22_data, chapt. 1]] and multivariable control systems where it is particularly valuable for analyzing directional properties in multivariable systems\nbsp{}[[cite:&skogestad07_multiv_feedb_contr]]. The SVD constitutes a unique matrix decomposition applicable to any complex matrix $\bm{X} \in \mathbb{C}^{n \times m}$, expressed as: @@ -8225,11 +8225,11 @@ The procedure for SVD-based decoupling begins with identifying the system dynami A specific frequency is then selected for optimal decoupling, with the targeted crossover frequency $\omega_c$ often serving as an appropriate choice. Since real matrices are required for the decoupling transformation, a real approximation of the complex measured response at the selected frequency must be computed. -In this work, the method proposed in [[cite:&kouvaritakis79_theor_pract_charac_locus_desig_method]] was used as it preserves maximal orthogonality in the directional properties of the input complex matrix. +In this work, the method proposed in\nbsp{}[[cite:&kouvaritakis79_theor_pract_charac_locus_desig_method]] was used as it preserves maximal orthogonality in the directional properties of the input complex matrix. Following this approximation, a real matrix $\tilde{\bm{G}}(\omega_c)$ is obtained, and SVD is performed on this matrix. The resulting (real) unitary matrices $\bm{U}$ and $\bm{V}$ are structured such that $\bm{V}^{-\intercal} \tilde{\bm{G}}(\omega_c) \bm{U}^{-1}$ forms a diagonal matrix. -These singular input and output matrices are then applied to decouple the system as illustrated in Figure ref:fig:detail_control_decoupling_svd, and the decoupled plant is described by eqref:eq:detail_control_decoupling_plant_svd. +These singular input and output matrices are then applied to decouple the system as illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling_svd, and the decoupled plant is described by\nbsp{}eqref:eq:detail_control_decoupling_plant_svd. \begin{equation}\label{eq:detail_control_decoupling_plant_svd} \bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-\intercal} @@ -8248,7 +8248,7 @@ Furthermore, the quality of decoupling depends significantly on the accuracy of Plant decoupling using the Singular Value Decomposition was then applied on the test model. A decoupling frequency of $\SI{100}{Hz}$ was used. -The plant response at that frequency, as well as its real approximation and the obtained $\bm{U}$ and $\bm{V}$ matrices are shown in eqref:eq:detail_control_decoupling_svd_example. +The plant response at that frequency, as well as its real approximation and the obtained $\bm{U}$ and $\bm{V}$ matrices are shown in\nbsp{}eqref:eq:detail_control_decoupling_svd_example. \begin{equation}\label{eq:detail_control_decoupling_svd_example} \begin{align} @@ -8274,8 +8274,8 @@ The plant response at that frequency, as well as its real approximation and the \end{align} \end{equation} -Using these $\bm{U}$ and $\bm{V}$ matrices, the decoupled plant is computed according to equation eqref:eq:detail_control_decoupling_plant_svd. -The resulting plant, depicted in Figure ref:fig:detail_control_decoupling_svd_plant, exhibits remarkable decoupling across a broad frequency range, extending well beyond the vicinity of $\omega_c$. +Using these $\bm{U}$ and $\bm{V}$ matrices, the decoupled plant is computed according to equation\nbsp{}eqref:eq:detail_control_decoupling_plant_svd. +The resulting plant, depicted in Figure\nbsp{}ref:fig:detail_control_decoupling_svd_plant, exhibits remarkable decoupling across a broad frequency range, extending well beyond the vicinity of $\omega_c$. Additionally, the diagonal terms manifest as second-order dynamic systems, facilitating straightforward controller design. #+name: fig:detail_control_decoupling_svd_plant @@ -8283,8 +8283,8 @@ Additionally, the diagonal terms manifest as second-order dynamic systems, facil [[file:figs/detail_control_decoupling_svd_plant.png]] As it was surprising to obtain such a good decoupling at all frequencies, a variant system with identical dynamics but different sensor configurations was examined. -Instead of using relative motion sensors collocated with the struts, three relative motion sensors were positioned as shown in Figure ref:fig:detail_control_decoupling_model_test_alt. -Although Jacobian matrices could theoretically be used to map these sensors to the frame of the struts, application of the same SVD decoupling procedure yielded the plant response shown in Figure ref:fig:detail_control_decoupling_svd_alt_plant, which exhibits significantly greater coupling. +Instead of using relative motion sensors collocated with the struts, three relative motion sensors were positioned as shown in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test_alt. +Although Jacobian matrices could theoretically be used to map these sensors to the frame of the struts, application of the same SVD decoupling procedure yielded the plant response shown in Figure\nbsp{}ref:fig:detail_control_decoupling_svd_alt_plant, which exhibits significantly greater coupling. Notably, the coupling demonstrates local minima near the decoupling frequency, consistent with the fact that the decoupling matrices were derived specifically for that frequency point. #+name: fig:detail_control_svd_decoupling_not_symmetrical @@ -8306,13 +8306,13 @@ Notably, the coupling demonstrates local minima near the decoupling frequency, c #+end_figure The exceptional performance of SVD decoupling on the plant with collocated sensors warrants further investigation. -This effectiveness may be attributed to the symmetrical properties of the plant, as evidenced in the Bode plots of the decentralized plant shown in Figure ref:fig:detail_control_decoupling_coupled_plant_bode. -The phenomenon potentially relates to previous research on SVD controllers applied to systems with specific symmetrical characteristics [[cite:&hovd97_svd_contr_contr]]. +This effectiveness may be attributed to the symmetrical properties of the plant, as evidenced in the Bode plots of the decentralized plant shown in Figure\nbsp{}ref:fig:detail_control_decoupling_coupled_plant_bode. +The phenomenon potentially relates to previous research on SVD controllers applied to systems with specific symmetrical characteristics\nbsp{}[[cite:&hovd97_svd_contr_contr]]. **** Comparison of decoupling strategies <> -While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table ref:tab:detail_control_decoupling_strategies_comp. +While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table\nbsp{}ref:tab:detail_control_decoupling_strategies_comp. Each method employs a distinct conceptual framework: Jacobian decoupling is "topology-driven", relying on the geometric configuration of the system; modal decoupling is "physics-driven", based on the system's dynamical equations; and SVD decoupling is "data-driven", utilizing measured frequency response functions. @@ -8362,37 +8362,37 @@ SVD decoupling can be implemented using measured data without requiring a model, **** Introduction :ignore: -Once the system is properly decoupled using one of the approaches described in Section ref:sec:detail_control_decoupling, SISO controllers can be individually tuned for each decoupled "directions". +Once the system is properly decoupled using one of the approaches described in Section\nbsp{}ref:sec:detail_control_decoupling, SISO controllers can be individually tuned for each decoupled "directions". Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented. -In some cases "fixed" controller structures are utilized, such as PI and PID controllers, whose parameters are manually tuned [[cite:&furutani04_nanom_cuttin_machin_using_stewar;&du14_piezo_actuat_high_precis_flexib;&yang19_dynam_model_decoup_contr_flexib]]. +In some cases "fixed" controller structures are utilized, such as PI and PID controllers, whose parameters are manually tuned\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar;&du14_piezo_actuat_high_precis_flexib;&yang19_dynam_model_decoup_contr_flexib]]. Another popular method is Open-Loop shaping, which was used during the conceptual phase. -Open-loop shaping involves tuning the controller through a series of "standard" filters (leads, lags, notches, low-pass filters, ...) to shape the open-loop transfer function $G(s)K(s)$ according to desired specifications, including bandwidth, gain and phase margins [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 4.4.7]]. +Open-loop shaping involves tuning the controller through a series of "standard" filters (leads, lags, notches, low-pass filters, ...) to shape the open-loop transfer function $G(s)K(s)$ according to desired specifications, including bandwidth, gain and phase margins\nbsp{}[[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 4.4.7]]. Open-Loop shaping is very popular because the open-loop transfer function is a linear function of the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics. Another key advantage is that controllers can be tuned directly from measured frequency response functions of the plant without requiring an explicit model. However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions. -Specifications can therefore be expressed in terms of the magnitude of closed-loop transfer functions, such as the sensitivity, plant sensitivity, and complementary sensitivity transfer functions [[cite:&skogestad07_multiv_feedb_contr, chapt. 3]]. +Specifications can therefore be expressed in terms of the magnitude of closed-loop transfer functions, such as the sensitivity, plant sensitivity, and complementary sensitivity transfer functions\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 3]]. With open-loop shaping, closed-loop transfer functions are changed only indirectly, which may make it difficult to directly address the specifications that are in terms of the closed-loop transfer functions. -In order to synthesize a controller that directly shapes the closed-loop transfer functions (and therefore the performance metric), $\mathcal{H}_\infty\text{-synthesis}$ may be used [[cite:&skogestad07_multiv_feedb_contr]]. -This approach requires a good model of the plant and expertise in selecting weighting functions that will define the wanted shape of different closed-loop transfer functions [[cite:&bibel92_guidel_h]]. -$\mathcal{H}_{\infty}\text{-synthesis}$ has been applied for the Stewart platform [[cite:&jiao18_dynam_model_exper_analy_stewar]], yet when benchmarked against more basic decentralized controllers, the performance gains proved small [[cite:&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. +In order to synthesize a controller that directly shapes the closed-loop transfer functions (and therefore the performance metric), $\mathcal{H}_\infty\text{-synthesis}$ may be used\nbsp{}[[cite:&skogestad07_multiv_feedb_contr]]. +This approach requires a good model of the plant and expertise in selecting weighting functions that will define the wanted shape of different closed-loop transfer functions\nbsp{}[[cite:&bibel92_guidel_h]]. +$\mathcal{H}_{\infty}\text{-synthesis}$ has been applied for the Stewart platform\nbsp{}[[cite:&jiao18_dynam_model_exper_analy_stewar]], yet when benchmarked against more basic decentralized controllers, the performance gains proved small\nbsp{}[[cite:&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. In this section, an alternative controller synthesis scheme is proposed in which complementary filters are used for directly shaping the closed-loop transfer functions (i.e., directly addressing the closed-loop performances). -In Section ref:ssec:detail_control_cf_control_arch, the proposed control architecture is presented. -In Section ref:ssec:detail_control_cf_trans_perf, typical performance requirements are translated into the shape of the complementary filters. -The design of the complementary filters is briefly discussed in Section ref:ssec:detail_control_cf_analytical_complementary_filters, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real time. -Finally, in Section ref:ssec:detail_control_cf_simulations, a numerical example is used to show how the proposed control architecture can be implemented in practice. +In Section\nbsp{}ref:ssec:detail_control_cf_control_arch, the proposed control architecture is presented. +In Section\nbsp{}ref:ssec:detail_control_cf_trans_perf, typical performance requirements are translated into the shape of the complementary filters. +The design of the complementary filters is briefly discussed in Section\nbsp{}ref:ssec:detail_control_cf_analytical_complementary_filters, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real time. +Finally, in Section\nbsp{}ref:ssec:detail_control_cf_simulations, a numerical example is used to show how the proposed control architecture can be implemented in practice. **** Control Architecture <> ***** Virtual Sensor Fusion -The idea of using complementary filters in the control architecture originates from sensor fusion techniques [[cite:&collette15_sensor_fusion_method_high_perfor]], where two sensors are combined using complementary filters. -Building upon this concept, "virtual sensor fusion" [[cite:&verma20_virtual_sensor_fusion_high_precis_contr]] replaces one physical sensor with a model $G$ of the plant. -The corresponding control architecture is illustrated in Figure ref:fig:detail_control_cf_arch, where $G^\prime$ represents the physical plant to be controlled, $G$ is a model of the plant, $k$ is the controller, and $H_L$ and $H_H$ are complementary filters satisfying $H_L(s) + H_H(s) = 1$. +The idea of using complementary filters in the control architecture originates from sensor fusion techniques\nbsp{}[[cite:&collette15_sensor_fusion_method_high_perfor]], where two sensors are combined using complementary filters. +Building upon this concept, "virtual sensor fusion"\nbsp{}[[cite:&verma20_virtual_sensor_fusion_high_precis_contr]] replaces one physical sensor with a model $G$ of the plant. +The corresponding control architecture is illustrated in Figure\nbsp{}ref:fig:detail_control_cf_arch, where $G^\prime$ represents the physical plant to be controlled, $G$ is a model of the plant, $k$ is the controller, and $H_L$ and $H_H$ are complementary filters satisfying $H_L(s) + H_H(s) = 1$. In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness. #+name: fig:detail_control_cf_arch_and_eq @@ -8413,12 +8413,12 @@ In this arrangement, the physical plant is controlled at low frequencies, while #+end_subfigure #+end_figure -Although the control architecture shown in Figure ref:fig:detail_control_cf_arch appears to be a multi-loop system, it should be noted that no non-linear saturation-type elements are present in the inner loop (containing $k$, $G$, and $H_H$, all numerically implemented). -Consequently, this structure is mathematically equivalent to the single-loop architecture illustrated in Figure ref:fig:detail_control_cf_arch_eq. +Although the control architecture shown in Figure\nbsp{}ref:fig:detail_control_cf_arch appears to be a multi-loop system, it should be noted that no non-linear saturation-type elements are present in the inner loop (containing $k$, $G$, and $H_H$, all numerically implemented). +Consequently, this structure is mathematically equivalent to the single-loop architecture illustrated in Figure\nbsp{}ref:fig:detail_control_cf_arch_eq. ***** Asymptotic behavior -When considering the extreme case of very high values for $k$, the effective controller $K(s)$ converges to the inverse of the plant model multiplied by the inverse of the high-pass filter, as expressed in eqref:eq:detail_control_cf_high_k. +When considering the extreme case of very high values for $k$, the effective controller $K(s)$ converges to the inverse of the plant model multiplied by the inverse of the high-pass filter, as expressed in\nbsp{}eqref:eq:detail_control_cf_high_k. \begin{equation}\label{eq:detail_control_cf_high_k} \lim_{k\to\infty} K(s) = \lim_{k\to\infty} \frac{k}{1+H_H(s) G(s) k} = \big( H_H(s) G(s) \big)^{-1} @@ -8427,8 +8427,8 @@ When considering the extreme case of very high values for $k$, the effective con If the resulting $K$ is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization. Furthermore, for $K$ to be stable, both $G$ and $H_H$ must be minimum phase transfer functions. -With these assumptions, the resulting control architecture is illustrated in Figure ref:fig:detail_control_cf_arch_class, where the complementary filters $H_L$ and $H_H$ remain the only tuning parameters. -The dynamics of this closed-loop system are described by equations eqref:eq:detail_control_cf_cl_system_y and eqref:eq:detail_control_cf_cl_system_y. +With these assumptions, the resulting control architecture is illustrated in Figure\nbsp{}ref:fig:detail_control_cf_arch_class, where the complementary filters $H_L$ and $H_H$ remain the only tuning parameters. +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 @@ -8441,7 +8441,7 @@ The dynamics of this closed-loop system are described by equations eqref:eq:deta \end{align} \end{subequations} -At frequencies where the model accurately represents the physical plant ($G^{-1} G^{\prime} \approx 1$), the denominator simplifies to $H_H + G^\prime G^{-1} H_L \approx H_H + H_L = 1$, and the closed-loop transfer functions are then described by equations eqref:eq:detail_control_cf_cl_performance_y and eqref:eq:detail_control_cf_cl_performance_u. +At frequencies where the model accurately represents the physical plant ($G^{-1} G^{\prime} \approx 1$), the denominator simplifies to $H_H + G^\prime G^{-1} H_L \approx H_H + H_L = 1$, and the closed-loop transfer functions are then described by equations\nbsp{}eqref:eq:detail_control_cf_cl_performance_y and eqref:eq:detail_control_cf_cl_performance_u. \begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect} \begin{alignat}{5} @@ -8456,10 +8456,10 @@ Hence, when the plant model closely approximates the actual dynamics, the closed **** Translating the performance requirements into the shape of the complementary filters <> ***** Introduction :ignore: -Performance specifications in a feedback system can usually be expressed as upper bounds on the magnitudes of closed-loop transfer functions such as the sensitivity and complementary sensitivity transfer functions [[cite:&bibel92_guidel_h]]. +Performance specifications in a feedback system can usually be expressed as upper bounds on the magnitudes of closed-loop transfer functions such as the sensitivity and complementary sensitivity transfer functions\nbsp{}[[cite:&bibel92_guidel_h]]. The design of a controller $K(s)$ to obtain the desired shape of these closed-loop transfer functions is known as closed-loop shaping. -In the proposed control architecture, the closed-loop transfer functions eqref:eq:detail_control_cf_sf_cl_tf_K_inf are expressed in terms of the complementary filters $H_L(s)$ and $H_H(s)$ rather than directly through the controller $K(s)$. +In the proposed control architecture, the closed-loop transfer functions\nbsp{}eqref:eq:detail_control_cf_sf_cl_tf_K_inf are expressed in terms of the complementary filters $H_L(s)$ and $H_H(s)$ rather than directly through the controller $K(s)$. Therefore, performance requirements must be translated into constraints on the shape of these complementary filters. ***** Nominal Stability (NS) @@ -8471,7 +8471,7 @@ Consequently, stable and minimum phase complementary filters must be employed. ***** Nominal Performance (NP) -Performance specifications can be formalized using weighting functions $w_H$ and $w_L$, where performance is achieved when eqref:eq:detail_control_cf_weights is satisfied. +Performance specifications can be formalized using weighting functions $w_H$ and $w_L$, where performance is achieved when\nbsp{}eqref:eq:detail_control_cf_weights is satisfied. The weighting functions define the maximum magnitude of the closed-loop transfer functions as a function of frequency, effectively determining their "shape". \begin{subequations}\label{eq:detail_control_cf_weights} @@ -8481,7 +8481,7 @@ The weighting functions define the maximum magnitude of the closed-loop transfer \end{align} \end{subequations} -For the nominal system, $S = H_H$ and $T = H_L$, hence the performance specifications can be converted on the shape of the complementary filters eqref:eq:detail_control_cf_nominal_performance. +For the nominal system, $S = H_H$ and $T = H_L$, hence the performance specifications can be converted on the shape of the complementary filters\nbsp{}eqref:eq:detail_control_cf_nominal_performance. \begin{equation}\label{eq:detail_control_cf_nominal_performance} \Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*} @@ -8502,8 +8502,8 @@ Therefore, by carefully selecting the shape of the complementary filters, nomina Robust stability refers to a control system's ability to maintain stability despite discrepancies between the actual system $G^\prime$ and the model $G$ used for controller design. These discrepancies may arise from unmodeled dynamics or nonlinearities. -To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure ref:fig:detail_control_cf_input_uncertainty is employed. -The set of possible plants $\Pi_i$ is described by eqref:eq:detail_control_cf_multiplicative_uncertainty, with the weighting function $w_I$ selected such that all possible plants $G^\prime$ are contained within the set $\Pi_i$. +To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure\nbsp{}ref:fig:detail_control_cf_input_uncertainty is employed. +The set of possible plants $\Pi_i$ is described by\nbsp{}eqref:eq:detail_control_cf_multiplicative_uncertainty, with the weighting function $w_I$ selected such that all possible plants $G^\prime$ are contained within the set $\Pi_i$. \begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty} \Pi_i: \quad G^\prime(s) = G(s)\big(1 + w_I(s)\Delta_I(s)\big); \quad |\Delta_I(j\omega)| \le 1 \ \forall\omega @@ -8527,13 +8527,13 @@ The set of possible plants $\Pi_i$ is described by eqref:eq:detail_control_cf_mu #+end_subfigure #+end_figure -When considering input multiplicative uncertainty, robust stability can be derived graphically from the Nyquist plot (illustrated in Figure ref:fig:detail_control_cf_nyquist_uncertainty), yielding to eqref:eq:detail_control_cf_robust_stability_graphically, as demonstrated in [[cite:&skogestad07_multiv_feedb_contr, chapt. 7.5.1]]. +When considering input multiplicative uncertainty, robust stability can be derived graphically from the Nyquist plot (illustrated in Figure\nbsp{}ref:fig:detail_control_cf_nyquist_uncertainty), yielding to\nbsp{}eqref:eq:detail_control_cf_robust_stability_graphically, as demonstrated in\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 7.5.1]]. \begin{equation}\label{eq:detail_control_cf_robust_stability_graphically} \text{RS} \Longleftrightarrow \left|w_I(j\omega) L(j\omega) \right| \le \left| 1 + L(j\omega) \right| \quad \forall\omega \end{equation} -After algebraic manipulation, robust stability is guaranteed when the low-pass complementary filter $H_L$ satisfies eqref:eq:detail_control_cf_condition_robust_stability. +After algebraic manipulation, robust stability is guaranteed when the low-pass complementary filter $H_L$ satisfies\nbsp{}eqref:eq:detail_control_cf_condition_robust_stability. \begin{equation}\label{eq:detail_control_cf_condition_robust_stability} \boxed{\text{RS} \Longleftrightarrow |w_I(j\omega) H_L(j\omega)| \le 1 \quad \forall \omega} @@ -8541,7 +8541,7 @@ After algebraic manipulation, robust stability is guaranteed when the low-pass c ***** Robust Performance (RP) -Robust performance ensures that performance specifications eqref:eq:detail_control_cf_weights are met even when the plant dynamics fluctuates within specified bounds eqref:eq:detail_control_cf_robust_perf_S. +Robust performance ensures that performance specifications\nbsp{}eqref:eq:detail_control_cf_weights are met even when the plant dynamics fluctuates within specified bounds\nbsp{}eqref:eq:detail_control_cf_robust_perf_S. \begin{equation}\label{eq:detail_control_cf_robust_perf_S} \text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega @@ -8553,18 +8553,18 @@ Transforming this condition into constraints on the complementary filters yields \boxed{\text{RP} \Longleftrightarrow | w_H(j\omega) H_H(j\omega) | + | w_I(j\omega) H_L(j\omega) | \le 1, \ \forall\omega} \end{equation} -The robust performance condition effectively combines both nominal performance eqref:eq:detail_control_cf_nominal_performance and robust stability conditions eqref:eq:detail_control_cf_condition_robust_stability. -If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2 [[cite:&skogestad07_multiv_feedb_contr, chapt. 7.6]]. +The robust performance condition effectively combines both nominal performance\nbsp{}eqref:eq:detail_control_cf_nominal_performance and robust stability conditions\nbsp{}eqref:eq:detail_control_cf_condition_robust_stability. +If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 7.6]]. Therefore, for SISO systems, ensuring robust stability and nominal performance is typically sufficient. **** Complementary filter design <> -As proposed in Section ref:sec:detail_control_sensor, complementary filters can be shaped using standard $\mathcal{H}_{\infty}\text{-synthesis}$ techniques. +As proposed in Section\nbsp{}ref:sec:detail_control_sensor, complementary filters can be shaped using standard $\mathcal{H}_{\infty}\text{-synthesis}$ techniques. This approach is particularly well-suited since performance requirements were expressed as upper bounds on the magnitude of the complementary filters. Alternatively, analytical formulas for complementary filters may be employed. -For some applications, first-order complementary filters as shown in Equation eqref:eq:detail_control_cf_1st_order are sufficient. +For some applications, first-order complementary filters as shown in Equation\nbsp{}eqref:eq:detail_control_cf_1st_order are sufficient. \begin{subequations}\label{eq:detail_control_cf_1st_order} \begin{align} @@ -8573,7 +8573,7 @@ For some applications, first-order complementary filters as shown in Equation eq \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 eqref:eq:detail_control_cf_1st_order_z. +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} @@ -8582,7 +8582,7 @@ These filters can be transformed into the digital domain using the Bilinear tran \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 ref:fig:detail_control_cf_arch_tunable_params. +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 @@ -8591,7 +8591,7 @@ This real-time tunability allows rapid testing of different control bandwidths t For many practical applications, first order complementary filters are not sufficient. Specifically, a slope of $+2$ at low frequencies for the sensitivity transfer function (enabling accurate tracking of ramp inputs) and a slope of $-2$ for the complementary sensitivity transfer function are often desired. -For these cases, the complementary filters analytical formula in Equation eqref:eq:detail_control_cf_2nd_order is proposed. +For these cases, the complementary filters analytical formula in Equation\nbsp{}eqref:eq:detail_control_cf_2nd_order is proposed. \begin{subequations}\label{eq:detail_control_cf_2nd_order} \begin{align} @@ -8600,7 +8600,7 @@ For these cases, the complementary filters analytical formula in Equation eqref: \end{align} \end{subequations} -The influence of parameters $\alpha$ and $\omega_0$ on the frequency response of these complementary filters is illustrated in Figure ref:fig:detail_control_cf_analytical_effect. +The influence of parameters $\alpha$ and $\omega_0$ on the frequency response of these complementary filters is illustrated in Figure\nbsp{}ref:fig:detail_control_cf_analytical_effect. The parameter $\alpha$ primarily affects the damping characteristics near the crossover frequency as well as high and low frequency magnitudes, while $\omega_0$ determines the frequency at which the transition between high-pass and low-pass behavior occurs. These filters can also be implemented in the digital domain with analytical formulas, preserving the ability to adjust $\alpha$ and $\omega_0$ in real-time. @@ -8630,15 +8630,15 @@ To implement the proposed control architecture in practice, the following proced 1. Identify the plant to be controlled to obtain the plant model $G$. 2. Design the weighting function $w_I$ such that all possible plants $G^\prime$ are contained within the uncertainty set $\Pi_i$. -3. Translate performance requirements into upper bounds on the complementary filters as explained in Section ref:ssec:detail_control_cf_trans_perf. -4. Design the weighting functions $w_H$ and $w_L$ and generate the complementary filters using $\mathcal{H}_{\infty}\text{-synthesis}$ as described in Section ref:ssec:detail_control_sensor_hinf_method. +3. Translate performance requirements into upper bounds on the complementary filters as explained in Section\nbsp{}ref:ssec:detail_control_cf_trans_perf. +4. Design the weighting functions $w_H$ and $w_L$ and generate the complementary filters using $\mathcal{H}_{\infty}\text{-synthesis}$ as described in Section\nbsp{}ref:ssec:detail_control_sensor_hinf_method. If the synthesis fails to produce filters satisfying the defined upper bounds, either revise the requirements or develop a more accurate model $G$ that will allow for a smaller $w_I$. - For simpler cases, the analytical formulas for complementary filters presented in Section ref:ssec:detail_control_cf_analytical_complementary_filters can be employed. + For simpler cases, the analytical formulas for complementary filters presented in Section\nbsp{}ref:ssec:detail_control_cf_analytical_complementary_filters can be employed. 5. If $K(s) = H_H^{-1}(s) G^{-1}(s)$ is not proper, add low-pass filters with sufficiently high corner frequencies to ensure realizability. ***** Plant :ignore: -To evaluate this control architecture, a simple test model representative of many synchrotron positioning stages is utilized (Figure ref:fig:detail_control_cf_test_model). +To evaluate this control architecture, a simple test model representative of many synchrotron positioning stages is utilized (Figure\nbsp{}ref:fig:detail_control_cf_test_model). In this model, a payload with mass $m$ is positioned on top of a stage. The objective is to accurately position the sample relative to the X-ray beam. @@ -8646,20 +8646,20 @@ The relative position $y$ between the payload and the X-ray is measured, which t Various disturbance forces affect positioning stability, including stage vibrations $d_w$ and direct forces applied to the sample $d_F$ (such as cable forces). The positioning stage itself is characterized by stiffness $k$, internal damping $c$, and a controllable force $F$. -The model of the plant $G(s)$ from actuator force $F$ to displacement $y$ is described by Equation eqref:eq:detail_control_cf_test_plant_tf. +The model of the plant $G(s)$ from actuator force $F$ to displacement $y$ is described by Equation\nbsp{}eqref:eq:detail_control_cf_test_plant_tf. \begin{equation}\label{eq:detail_control_cf_test_plant_tf} G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = 1\si{\N/\mu\m},\ c = 10^2\si{\N\per(\m\per\s)} \end{equation} The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics and payload dynamics. -These uncertainties are represented using a multiplicative input uncertainty weight eqref:eq:detail_control_cf_test_plant_uncertainty, which specifies the magnitude of uncertainty as a function of frequency. +These uncertainties are represented using a multiplicative input uncertainty weight\nbsp{}eqref:eq:detail_control_cf_test_plant_uncertainty, which specifies the magnitude of uncertainty as a function of frequency. \begin{equation}\label{eq:detail_control_cf_test_plant_uncertainty} w_I(s) = 10 \cdot \frac{(s+100)^2}{(s+1000)^2} \end{equation} -Figure ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the nominal plant dynamics and the complete set of possible plants $\Pi_i$ encompassed by the uncertainty model. +Figure\nbsp{}ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the nominal plant dynamics and the complete set of possible plants $\Pi_i$ encompassed by the uncertainty model. #+name: fig:detail_control_cf_test_model_plant #+caption: Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of the transfer function $G(s)$ from $F$ to $y$ and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}). @@ -8681,21 +8681,21 @@ Figure ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the nominal ***** Requirements and choice of complementary filters -As discussed in Section ref:ssec:detail_control_cf_trans_perf, nominal performance requirements can be expressed as upper bounds on the shape of the complementary filters. +As discussed in Section\nbsp{}ref:ssec:detail_control_cf_trans_perf, nominal performance requirements can be expressed as upper bounds on the shape of the complementary filters. For this example, the requirements are: - track ramp inputs (i.e. constant velocity scans) with zero steady-state error: a $+2$ slope at low frequencies for the magnitude of the sensitivity function $|S(j\omega)|$ is required - filtering of measurement noise above $\SI{300}{Hz}$, where sensor noise is significant (requiring a filtering factor of approximately 100 above this frequency) - maximizing disturbance rejection Additionally, robust stability must be ensured, requiring the closed-loop system to remain stable despite the dynamic uncertainties modeled by $w_I$. -This condition is satisfied when the magnitude of the low-pass complementary filter $|H_L(j\omega)|$ remains below the inverse of the uncertainty weight magnitude $|w_I(j\omega)|$, as expressed in Equation eqref:eq:detail_control_cf_condition_robust_stability. +This condition is satisfied when the magnitude of the low-pass complementary filter $|H_L(j\omega)|$ remains below the inverse of the uncertainty weight magnitude $|w_I(j\omega)|$, as expressed in Equation\nbsp{}eqref:eq:detail_control_cf_condition_robust_stability. Robust performance is achieved when both nominal performance and robust stability conditions are simultaneously satisfied. -All requirements imposed on $H_L$ and $H_H$ are visualized in Figure ref:fig:detail_control_cf_specs_S_T. +All requirements imposed on $H_L$ and $H_H$ are visualized in Figure\nbsp{}ref:fig:detail_control_cf_specs_S_T. While $\mathcal{H}_\infty\text{-synthesis}$ could be employed to design the complementary filters, analytical formulas were used for this relatively simple example. -The second-order complementary filters from Equation eqref:eq:detail_control_cf_2nd_order were selected with parameters $\alpha = 1$ and $\omega_0 = 2\pi \cdot 20\,\text{Hz}$. -There magnitudes are displayed in Figure ref:fig:detail_control_cf_specs_S_T, confirming that these complementary filters are fulfilling the specifications. +The second-order complementary filters from Equation\nbsp{}eqref:eq:detail_control_cf_2nd_order were selected with parameters $\alpha = 1$ and $\omega_0 = 2\pi \cdot 20\,\text{Hz}$. +There magnitudes are displayed in Figure\nbsp{}ref:fig:detail_control_cf_specs_S_T, confirming that these complementary filters are fulfilling the specifications. #+name: fig:detail_control_cf_specs_S_T_obtained_filters #+caption: Performance requirement and complementary filters used (\subref{fig:detail_control_cf_specs_S_T}). Obtained controller from the complementary filters and the plant inverse is shown in (\subref{fig:detail_control_cf_bode_Kfb}). @@ -8718,13 +8718,13 @@ There magnitudes are displayed in Figure ref:fig:detail_control_cf_specs_S_T, co ***** Controller analysis The controller to be implemented takes the form $K(s) = \tilde{G}^{-1}(s) H_H^{-1}(s)$, where $\tilde{G}^{-1}(s)$ represents the plant inverse, which must be both stable and proper. -To ensure properness, low-pass filters with high corner frequencies are added as shown in Equation eqref:eq:detail_control_cf_test_plant_inverse. +To ensure properness, low-pass filters with high corner frequencies are added as shown in Equation\nbsp{}eqref:eq:detail_control_cf_test_plant_inverse. \begin{equation}\label{eq:detail_control_cf_test_plant_inverse} \tilde{G}^{-1}(s) = \frac{m s^2 + c s + k}{1 + \frac{s}{2\pi \cdot 1000} + \left( \frac{s}{2\pi \cdot 1000} \right)^2} \end{equation} -The Bode plot of the controller multiplied by the complementary low-pass filter, $K(s) \cdot H_L(s)$, is presented in Figure ref:fig:detail_control_cf_bode_Kfb. +The Bode plot of the controller multiplied by the complementary low-pass filter, $K(s) \cdot H_L(s)$, is presented in Figure\nbsp{}ref:fig:detail_control_cf_bode_Kfb. The frequency response reveals several important characteristics: - The presence of two integrators at low frequencies, enabling accurate tracking of ramp inputs - A notch at the plant resonance frequency (arising from the plant inverse) @@ -8732,10 +8732,10 @@ The frequency response reveals several important characteristics: ***** Robustness and Performance analysis -Robust stability is assessed using the Nyquist plot shown in Figure ref:fig:detail_control_cf_nyquist_robustness. +Robust stability is assessed using the Nyquist plot shown in Figure\nbsp{}ref:fig:detail_control_cf_nyquist_robustness. Even when considering all possible plants within the uncertainty set, the Nyquist plot remains sufficiently distant from the critical point $(-1,0)$, indicating robust stability with adequate margins. -Performance is evaluated by examining the closed-loop sensitivity and complementary sensitivity transfer functions, as illustrated in Figure ref:fig:detail_control_cf_robust_perf. +Performance is evaluated by examining the closed-loop sensitivity and complementary sensitivity transfer functions, as illustrated in Figure\nbsp{}ref:fig:detail_control_cf_robust_perf. 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 @@ -8765,7 +8765,7 @@ The method shares conceptual similarities with mixed-sensitivity $\mathcal{H}_{\ While $\mathcal{H}_{\infty}\text{-synthesis}$ offers greater flexibility and can be readily generalized to MIMO plants, the presented approach provides a simpler alternative that requires minimal design effort. Implementation only necessitates extracting a model of the plant and selecting appropriate analytical complementary filters, making it particularly interesting for applications where simplicity and intuitive parameter tuning are valued. -Due to time constraints, an extensive literature review comparing this approach with similar existing architectures, such as Internal Model Control [[cite:&saxena12_advan_inter_model_contr_techn]], was not conducted. +Due to time constraints, an extensive literature review comparing this approach with similar existing architectures, such as Internal Model Control\nbsp{}[[cite:&saxena12_advan_inter_model_contr_techn]], was not conducted. Consequently, it remains unclear whether the proposed architecture offers significant advantages over existing methods in the literature. The control architecture has been presented for SISO systems, but can be applied to MIMO systems when sufficient decoupling is achieved. @@ -8775,13 +8775,13 @@ It will be experimentally validated with the NASS during the experimental phase. <> In order to optimize the control of the Nano Active Stabilization System, several aspects of control theory were studied. -Different approaches to combine sensors were compared in Section ref:sec:detail_control_sensor. +Different approaches to combine sensors were compared in Section\nbsp{}ref:sec:detail_control_sensor. While High Authority Control-Low Authority Control (HAC-LAC) was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters. It was demonstrated that the performance of such fusion depends significantly on the magnitude of the complementary filters. To address this challenge, a synthesis method based on $\mathcal{H}_\infty\text{-synthesis}$ was proposed, allowing for intuitive shaping of the complementary filters through weighting functions. For the NASS, while HAC-LAC remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation. -Various decoupling strategies for parallel manipulators were examined in Section ref:sec:detail_control_decoupling, including decentralized control, Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. +Various decoupling strategies for parallel manipulators were examined in Section\nbsp{}ref:sec:detail_control_decoupling, including decentralized control, Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling. The main characteristics of each approach were highlighted, providing valuable insights into their respective strengths and limitations. Among the examined methods, Jacobian decoupling was determined to be most appropriate for the NASS, as it provides straightforward implementation while preserving the physical meaning of inputs and outputs. @@ -8796,16 +8796,16 @@ Experimental validation of this method on the NASS will be conducted during the *** Introduction :ignore: This chapter presents an approach to select and validate appropriate instrumentation for the Nano Active Stabilization System (NASS), ensuring each component meets specific performance requirements. -Figure ref:fig:detail_instrumentation_plant illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis. +Figure\nbsp{}ref:fig:detail_instrumentation_plant illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis. The selection process follows a three-stage methodology. -First, dynamic error budgeting is performed in Section ref:sec:detail_instrumentation_dynamic_error_budgeting to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier). +First, dynamic error budgeting is performed in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier). This analysis utilizes the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements. From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived. -Section ref:sec:detail_instrumentation_choice then presents the selection of appropriate components based on these noise specifications and additional requirements. +Section\nbsp{}ref:sec:detail_instrumentation_choice then presents the selection of appropriate components based on these noise specifications and additional requirements. -Finally, Section ref:sec:detail_instrumentation_characterization validates the selected components through experimental testing. +Finally, Section\nbsp{}ref:sec:detail_instrumentation_characterization validates the selected components through experimental testing. Each instrument is characterized individually, measuring actual noise levels and performance characteristics. The measured noise characteristics are then incorporated into the multi-body model to confirm that the combined effect of all instrumentation noise sources remains within acceptable limits. @@ -8819,22 +8819,22 @@ The measured noise characteristics are then incorporated into the multi-body mod The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system. -The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section ref:ssec:detail_instrumentation_cl_sensitivity). +The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section\nbsp{}ref:ssec:detail_instrumentation_cl_sensitivity). This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs. Only the vertical direction is considered in this analysis as it presents the most stringent requirements; the horizontal directions are subject to less demanding constraints. -From these transfer functions, the maximum acceptable Amplitude Spectral Density (ASD) of the noise sources is derived (Section ref:ssec:detail_instrumentation_max_noise_specs). +From these transfer functions, the maximum acceptable Amplitude Spectral Density (ASD) of the noise sources is derived (Section\nbsp{}ref:ssec:detail_instrumentation_max_noise_specs). Since the voltage amplifier gain affects the amplification of DAC noise, an assumption of an amplifier gain of 20 was made. **** Closed-Loop Sensitivity to Instrumentation Disturbances <> -Several key noise sources are considered in the analysis (Figure ref:fig:detail_instrumentation_plant). +Several key noise sources are considered in the analysis (Figure\nbsp{}ref:fig:detail_instrumentation_plant). These include the output voltage noise of the DAC ($n_{da}$), the output voltage noise of the voltage amplifier ($n_{amp}$), and the voltage noise of the ADC measuring the force sensor stacks ($n_{ad}$). Encoder noise, which is only used to estimate $R_z$, has been found to have minimal impact on the vertical sample error and is therefore omitted from this analysis for clarity. -The transfer functions from these three noise sources (for one strut) to the vertical error of the sample are estimated from the multi-body model, which includes the APA300ML and the designed flexible joints (Figure ref:fig:detail_instrumentation_noise_sensitivities). +The transfer functions from these three noise sources (for one strut) to the vertical error of the sample are estimated from the multi-body model, which includes the APA300ML and the designed flexible joints (Figure\nbsp{}ref:fig:detail_instrumentation_noise_sensitivities). #+name: fig:detail_instrumentation_noise_sensitivities #+caption: Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy. @@ -8875,10 +8875,10 @@ The amplifier should accept an analog input voltage, preferably in the range of Small signal bandwidth is particularly important for feedback applications as it can limit the overall bandwidth of the complete feedback system. -A simplified electrical model of a voltage amplifier connected to a piezoelectric stack is shown in Figure ref:fig:detail_instrumentation_amp_output_impedance. -This model is valid for small signals and provides insight into the small signal bandwidth limitation [[cite:&fleming14_desig_model_contr_nanop_system, chap. 14]]. +A simplified electrical model of a voltage amplifier connected to a piezoelectric stack is shown in Figure\nbsp{}ref:fig:detail_instrumentation_amp_output_impedance. +This model is valid for small signals and provides insight into the small signal bandwidth limitation\nbsp{}[[cite:&fleming14_desig_model_contr_nanop_system, chap. 14]]. In this model, $R_o$ represents the output impedance of the amplifier. -When combined with the piezoelectric load (represented as a capacitance $C_p$), it forms a first order low pass filter described by eqref:eq:detail_instrumentation_amp_output_impedance. +When combined with the piezoelectric load (represented as a capacitance $C_p$), it forms a first order low pass filter described by\nbsp{}eqref:eq:detail_instrumentation_amp_output_impedance. \begin{equation}\label{eq:detail_instrumentation_amp_output_impedance} \frac{V_a}{V_i}(s) = \frac{1}{1 + \frac{s}{\omega_0}}, \quad \omega_0 = \frac{1}{R_o C_p} @@ -8910,19 +8910,19 @@ Therefore, ideally, a voltage amplifier capable of providing $0.3\,A$ of current ***** Output voltage noise -As established in Section ref:sec:detail_instrumentation_dynamic_error_budgeting, the output noise of the voltage amplifier should be below $20\,\text{mV RMS}$. +As established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the output noise of the voltage amplifier should be below $20\,\text{mV RMS}$. -It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure ref:fig:detail_instrumentation_amp_output_impedance. -Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement [[cite:&spengen20_high_voltag_amplif]]. +It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure\nbsp{}ref:fig:detail_instrumentation_amp_output_impedance. +Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement\nbsp{}[[cite:&spengen20_high_voltag_amplif]]. For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\mu F$ and a bandwidth exceeding $5\,\text{kHz}$. ***** Choice of voltage amplifier -The specifications are summarized in Table ref:tab:detail_instrumentation_amp_choice. +The specifications are summarized in Table\nbsp{}ref:tab:detail_instrumentation_amp_choice. The most critical characteristics are the small signal bandwidth ($>5\,\text{kHz}$) and the output voltage noise ($<20\,\text{mV RMS}$). -Several voltage amplifiers were considered, with their datasheet information presented in Table ref:tab:detail_instrumentation_amp_choice. +Several voltage amplifiers were considered, with their datasheet information presented in Table\nbsp{}ref:tab:detail_instrumentation_amp_choice. One challenge encountered during the selection process was that manufacturers typically do not specify output noise as a function of frequency (i.e., the ASD of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies. This approach does not account for the frequency dependency of the noise, which is crucial for accurate error budgeting. @@ -8956,7 +8956,7 @@ The proper selection of these components is critical for system performance. ***** Synchronicity and Jitter -For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements [[cite:&abramovitch22_pract_method_real_world_contr_system;&abramovitch23_tutor_real_time_comput_issues_contr_system]]. +For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements\nbsp{}[[cite:&abramovitch22_pract_method_real_world_contr_system;&abramovitch23_tutor_real_time_comput_issues_contr_system]]. Therefore, the ADC and DAC must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals. Based on this requirement, priority was given to ADC and DAC components specifically marketed by Speedgoat to ensure optimal integration. @@ -8970,14 +8970,14 @@ Then, the /bandwidth/ specifies the maximum frequency of a measured signal (typi Finally, /delay/ (or /latency/) refers to the time interval between the analog signal at the input of the ADC and the digital information transferred to the control system. Sigma-Delta ADCs can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency. -Typically, the latency can reach 20 times the sampling period [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 8.4]]. +Typically, the latency can reach 20 times the sampling period\nbsp{}[[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 8.4]]. Consequently, while Sigma-Delta ADCs are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor. For real-time control applications, SAR-ADCs (Successive Approximation ADCs) remain the predominant choice due to their single-sample latency characteristics. ***** ADC Noise -Based on the dynamic error budget established in Section ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise ASD should not exceed $11\,\mu V/\sqrt{\text{Hz}}$. +Based on the dynamic error budget established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise ASD should not exceed $11\,\mu V/\sqrt{\text{Hz}}$. ADCs are subject to various noise sources. Quantization noise, which results from the discrete nature of digital representation, is one of these sources. @@ -8986,13 +8986,13 @@ To determine the minimum bit depth $n$ required to meet the noise specifications The quantization step size, denoted as $q = \Delta V/2^n$, represents the voltage equivalent of the least significant bit, with $\Delta V$ the full range of the ADC in volts, and $F_s$ the sampling frequency in Hertz. The quantization noise ranges between $\pm q/2$, and its probability density function is constant across this range (uniform distribution). -Since the integral of this probability density function $p(e)$ equals one, its value is $1/q$ for $-q/2 < e < q/2$, as illustrated in Figure ref:fig:detail_instrumentation_adc_quantization. +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 $e$ [[file:figs/detail_instrumentation_adc_quantization.png]] -The variance (or time-average power) of the quantization noise is expressed by eqref:eq:detail_instrumentation_quant_power. +The variance (or time-average power) of the quantization noise is expressed by\nbsp{}eqref:eq:detail_instrumentation_quant_power. \begin{equation}\label{eq:detail_instrumentation_quant_power} P_q = \int_{-q/2}^{q/2} e^2 p(e) de = \frac{q^2}{12} @@ -9002,13 +9002,13 @@ To compute the power spectral density of the quantization noise, which is define Under this assumption, the autocorrelation function approximates a delta function in the time domain. Since the Fourier transform of a delta function equals one, the power spectral density becomes frequency-independent (white noise). -By Parseval's theorem, the power spectral density of the quantization noise $\Phi_q$ can be linked to the ADC sampling frequency and quantization step size eqref:eq:detail_instrumentation_psd_quant_noise. +By Parseval's theorem, the power spectral density of the quantization noise $\Phi_q$ can be linked to the ADC sampling frequency and quantization step size\nbsp{}eqref:eq:detail_instrumentation_psd_quant_noise. \begin{equation}\label{eq:detail_instrumentation_psd_quant_noise} \int_{-F_s/2}^{F_s/2} \Phi_q(f) d f = \int_{-q/2}^{q/2} e^2 p(e) de \quad \Longrightarrow \quad \Phi_q = \frac{q^2}{12 F_s} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 F_s} \quad \text{in } \left[ \frac{V^2}{\text{Hz}} \right] \end{equation} -From a specified noise amplitude spectral density $\Gamma_{\text{max}}$, the minimum number of bits required to keep quantization noise below $\Gamma_{\text{max}}$ is calculated using eqref:eq:detail_instrumentation_min_n. +From a specified noise amplitude spectral density $\Gamma_{\text{max}}$, the minimum number of bits required to keep quantization noise below $\Gamma_{\text{max}}$ is calculated using\nbsp{}eqref:eq:detail_instrumentation_min_n. \begin{equation}\label{eq:detail_instrumentation_min_n} n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right) @@ -9030,14 +9030,14 @@ The selected model is the IO131, which features 16 analog inputs based on the AD The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, $\pm 10\,V$ range, conversion time of $10\,\mu s$, and simultaneous update capability. Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements. -This will be experimentally verified in Section ref:sec:detail_instrumentation_characterization. +This will be experimentally verified in Section\nbsp{}ref:sec:detail_instrumentation_characterization. **** Relative Displacement Sensors The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below $6\,\text{nm RMS}$ (derived from the $15\,\text{nm RMS}$ vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding $100\,\mu m$. -Several sensor technologies are capable of meeting these requirements [[cite:&fleming13_review_nanom_resol_posit_sensor]]. -These include optical encoders (Figure ref:fig:detail_instrumentation_sensor_encoder), capacitive sensors (Figure ref:fig:detail_instrumentation_sensor_capacitive), and eddy current sensors (Figure ref:fig:detail_instrumentation_sensor_eddy_current), each with their own advantages and implementation considerations. +Several sensor technologies are capable of meeting these requirements\nbsp{}[[cite:&fleming13_review_nanom_resol_posit_sensor]]. +These include optical encoders (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_encoder), capacitive sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_capacitive), and eddy current sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_eddy_current), each with their own advantages and implementation considerations. #+name: fig:detail_instrumentation_sensor_examples #+caption: Relative motion sensors considered for measuring the nano-hexapod strut motion @@ -9063,8 +9063,8 @@ These include optical encoders (Figure ref:fig:detail_instrumentation_sensor_enc #+end_subfigure #+end_figure -From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the APA, as illustrated in Figure ref:fig:detail_instrumentation_capacitive_implementation. -In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure ref:fig:detail_instrumentation_encoder_implementation. +From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the APA, as illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_capacitive_implementation. +In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure\nbsp{}ref:fig:detail_instrumentation_encoder_implementation. #+name: fig:detail_instrumentation_sensor_implementation #+caption: Implementation of relative displacement sensor to measure the motion of the APA @@ -9089,7 +9089,7 @@ Measurements conducted on the slip-ring integrated in the micro-station revealed To mitigate this issue, preference was given to sensors that transmit displacement measurements digitally, as these are inherently less susceptible to noise and cross-talk. Based on this criterion, an optical encoder with digital output was selected, where signal interpolation is performed directly in the sensor head. -The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table ref:tab:detail_instrumentation_sensor_specs, alongside alternative options that were considered. +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 @@ -9110,14 +9110,14 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic ***** Measured Noise The measurement of ADC noise was performed by short-circuiting its input with a $50\,\Omega$ resistor and recording the digital values at a sampling rate of $10\,\text{kHz}$. -The amplitude spectral density of the recorded values was computed and is presented in Figure ref:fig:detail_instrumentation_adc_noise_measured. +The amplitude spectral density of the recorded values was computed and is presented in Figure\nbsp{}ref:fig:detail_instrumentation_adc_noise_measured. The ADC noise exhibits characteristics of white noise with an amplitude spectral density of $5.6\,\mu V/\sqrt{\text{Hz}}$ (equivalent to $0.4\,\text{mV RMS}$), which satisfies the established specifications. All ADC channels demonstrated similar performance, so only one channel's noise profile is shown. -If necessary, oversampling can be applied to further reduce the noise [[cite:&lab13_improv_adc]]. +If necessary, oversampling can be applied to further reduce the noise\nbsp{}[[cite:&lab13_improv_adc]]. To gain $w$ additional bits of resolution, the oversampling frequency $f_{os}$ should be set to $f_{os} = 4^w \cdot F_s$. Given that the ADC can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4). -This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV) [[cite:&hauser91_princ_overs_d_conver]]. +This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV)\nbsp{}[[cite:&hauser91_princ_overs_d_conver]]. #+name: fig:detail_instrumentation_adc_noise_measured #+caption: Measured ADC noise (IO318) @@ -9126,7 +9126,7 @@ This approach is effective because the noise approximates white noise and its am ***** Reading of piezoelectric force sensor To further validate the ADC's capability to effectively measure voltage generated by a piezoelectric stack, a test was conducted using the APA95ML. -The setup is illustrated in Figure ref:fig:detail_instrumentation_force_sensor_adc_setup, where two stacks are used as actuators (connected in parallel) and one stack serves as a sensor. +The setup is illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_setup, where two stacks are used as actuators (connected in parallel) and one stack serves as a sensor. The voltage amplifier employed in this setup has a gain of 20. #+name: fig:detail_instrumentation_force_sensor_adc_setup @@ -9134,10 +9134,10 @@ The voltage amplifier employed in this setup has a gain of 20. [[file:figs/detail_instrumentation_force_sensor_adc_setup.png]] Step signals with an amplitude of $1\,V$ were generated using the DAC, and the ADC signal was recorded. -The excitation signal (steps) and the measured voltage across the sensor stack are displayed in Figure ref:fig:detail_instrumentation_step_response_force_sensor. +The excitation signal (steps) and the measured voltage across the sensor stack are displayed in Figure\nbsp{}ref:fig:detail_instrumentation_step_response_force_sensor. Two notable observations were made: an offset voltage of $2.26\,V$ was present, and the measured voltage exhibited an exponential decay response to the step input. -These phenomena can be explained by examining the electrical schematic shown in Figure ref:fig:detail_instrumentation_force_sensor_adc, where the ADC has an input impedance $R_i$ and an input bias current $i_n$. +These phenomena can be explained by examining the electrical schematic shown in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc, where the ADC has an input impedance $R_i$ and an input bias current $i_n$. The input impedance $R_i$ of the ADC, in combination with the capacitance $C_p$ of the piezoelectric stack sensor, forms an RC circuit with a time constant $\tau = R_i C_p$. The charge generated by the piezoelectric effect across the stack's capacitance gradually discharges into the input resistor of the ADC. @@ -9147,7 +9147,7 @@ An exponential curve was fitted to the experimental data, yielding a time consta With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu F$, the internal impedance of the Speedgoat ADC was calculated as $R_i = \tau/C_p = 1.5\,M\Omega$, which closely aligns with the specified value of $1\,M\Omega$ found in the datasheet. #+name: fig:detail_instrumentation_force_sensor -#+caption: Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from [[cite:&reza06_piezoel_trans_vibrat_contr_dampin]]. Measured voltage $V_s$ while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}). +#+caption: Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from\nbsp{}[[cite:&reza06_piezoel_trans_vibrat_contr_dampin]]. Measured voltage $V_s$ while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_force_sensor_adc}Electrical Schematic} @@ -9164,15 +9164,15 @@ With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu F$, #+end_subfigure #+end_figure -The constant voltage offset can be explained by the input bias current $i_n$ of the ADC, represented in Figure ref:fig:detail_instrumentation_force_sensor_adc. +The constant voltage offset can be explained by the input bias current $i_n$ of the ADC, represented in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc. At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the ADC, and therefore the input bias current $i_n$ passing through the internal resistance $R_i$ produces a constant voltage offset $V_{\text{off}} = R_i \cdot i_n$. The input bias current $i_n$ is estimated from $i_n = V_{\text{off}}/R_i = 1.5\mu A$. -In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor $R_p$ can be added in parallel to the force sensor, as illustrated in Figure ref:fig:detail_instrumentation_force_sensor_adc_R. +In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor $R_p$ can be added in parallel to the force sensor, as illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_R. This modification produces two beneficial effects: a reduction of input voltage offset through the relationship $V_{\text{off}} = (R_p R_i)/(R_p + R_i) i_n$, and an increase in the high pass corner frequency $f_c$ according to the equations $\tau = 1/(2\pi f_c) = (R_i R_p)/(R_i + R_p) C_p$. -To validate this approach, a resistor $R_p \approx 82\,k\Omega$ was added in parallel with the force sensor as shown in Figure ref:fig:detail_instrumentation_force_sensor_adc_R. -After incorporating this resistor, the same step response tests were performed, with results displayed in Figure ref:fig:detail_instrumentation_step_response_force_sensor_R. +To validate this approach, a resistor $R_p \approx 82\,k\Omega$ was added in parallel with the force sensor as shown in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_R. +After incorporating this resistor, the same step response tests were performed, with results displayed in Figure\nbsp{}ref:fig:detail_instrumentation_step_response_force_sensor_R. The measurements confirmed the expected improvements, with a substantially reduced offset voltage ($V_{\text{off}} = 0.15\,V$) and a much faster time constant ($\tau = 0.45\,s$). These results validate both the model of the ADC and the effectiveness of the added parallel resistor as a solution. @@ -9200,12 +9200,12 @@ Because the ADC noise may be too low to measure the noise of other instruments ( A Femto DLPVA-101-B-S amplifier with adjustable gains from 20dB up to 80dB was selected for this purpose. The first step was to characterize the input[fn:detail_instrumentation_2] noise of the amplifier. -This was accomplished by short-circuiting its input with a $50\,\Omega$ resistor and measuring the output voltage with the ADC (Figure ref:fig:detail_instrumentation_femto_meas_setup). +This was accomplished by short-circuiting its input with a $50\,\Omega$ resistor and measuring the output voltage with the ADC (Figure\nbsp{}ref:fig:detail_instrumentation_femto_meas_setup). The maximum amplifier gain of 80dB (equivalent to 10000) was used for this measurement. The measured voltage $n$ was then divided by 10000 to determine the equivalent noise at the input of the voltage amplifier $n_a$. In this configuration, the noise contribution from the ADC $q_{ad}$ is rendered negligible due to the high gain employed. -The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the (negligible) contribution of the ADC noise are presented in Figure ref:fig:detail_instrumentation_femto_input_noise. +The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the (negligible) contribution of the ADC noise are presented in Figure\nbsp{}ref:fig:detail_instrumentation_femto_input_noise. #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage @@ -9225,12 +9225,12 @@ The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the **** Digital to Analog Converters ***** Output Voltage Noise -To measure the output noise of the DAC, the setup schematically represented in Figure ref:fig:detail_instrumentation_dac_setup was utilized. +To measure the output noise of the DAC, the setup schematically represented in Figure\nbsp{}ref:fig:detail_instrumentation_dac_setup was utilized. The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the ADC. The Amplitude Spectral Density $\Gamma_{n_{da}}(\omega)$ of the measured signal was computed, and verification was performed to confirm that the contributions of ADC noise and amplifier noise were negligible in the measurement. -The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure ref:fig:detail_instrumentation_dac_output_noise. +The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure\nbsp{}ref:fig:detail_instrumentation_dac_output_noise. The noise profile is predominantly white with an ASD of $0.6\,\mu V/\sqrt{\text{Hz}}$. Minor $50\,\text{Hz}$ noise is present, along with some low frequency $1/f$ noise, but these are not expected to pose issues as they are well within specifications. It should be noted that all DAC channels demonstrated similar performance, so only one channel measurement is presented. @@ -9243,7 +9243,7 @@ It should be noted that all DAC channels demonstrated similar performance, so on To measure the transfer function from DAC to ADC and verify that the bandwidth and latency of both instruments is sufficient, a direct connection was established between the DAC output and the ADC input. A white noise signal was generated by the DAC, and the ADC response was recorded. -The resulting frequency response function from the digital DAC signal to the digital ADC signal is presented in Figure ref:fig:detail_instrumentation_dac_adc_tf. +The resulting frequency response function from the digital DAC signal to the digital ADC signal is presented in Figure\nbsp{}ref:fig:detail_instrumentation_dac_adc_tf. The observed frequency response function corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer. #+name: fig:detail_instrumentation_dac @@ -9266,7 +9266,7 @@ The observed frequency response function corresponds to exactly one sample delay **** Piezoelectric Voltage Amplifier ***** Output Voltage Noise -The measurement setup for evaluating the PD200 amplifier noise is illustrated in Figure ref:fig:detail_instrumentation_pd200_setup. +The measurement setup for evaluating the PD200 amplifier noise is illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_pd200_setup. The input of the PD200 amplifier was shunted with a $50\,\Ohm$ resistor to ensure that only the inherent noise of the amplifier itself was measured. The pre-amplifier gain was increased to produce a signal substantially larger than the noise floor of the ADC. Two piezoelectric stacks from the APA95ML were connected to the PD200 output to provide an appropriate load for the amplifier. @@ -9276,13 +9276,13 @@ Two piezoelectric stacks from the APA95ML were connected to the PD200 output to [[file:figs/detail_instrumentation_pd200_setup.png]] The Amplitude Spectral Density $\Gamma_{n}(\omega)$ of the signal measured by the ADC was computed. -From this, the Amplitude Spectral Density of the output voltage noise of the PD200 amplifier $n_p$ was derived, accounting for the gain of the pre-amplifier according to eqref:eq:detail_instrumentation_amp_asd. +From this, the Amplitude Spectral Density of the output voltage noise of the PD200 amplifier $n_p$ was derived, accounting for the gain of the pre-amplifier according to\nbsp{}eqref:eq:detail_instrumentation_amp_asd. \begin{equation}\label{eq:detail_instrumentation_amp_asd} \Gamma_{n_p}(\omega) = \frac{\Gamma_n(\omega)}{|G_p(j\omega) G_a(j\omega)|} \end{equation} -The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure ref:fig:detail_instrumentation_pd200_noise. +The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure\nbsp{}ref:fig:detail_instrumentation_pd200_noise. Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or ADC noise. The measurements from all six amplifiers are displayed in this figure. @@ -9303,7 +9303,7 @@ This measurement approach eliminates the influence of ADC-DAC-related time delay All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to 500Hz, well within the specified requirements. -The identified dynamics shown in Figure ref:fig:detail_instrumentation_pd200_tf can be accurately modeled as either a first-order low-pass filter or as a simple constant gain. +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 @@ -9314,10 +9314,10 @@ The identified dynamics shown in Figure ref:fig:detail_instrumentation_pd200_tf To measure the noise of the encoder, the head and ruler were rigidly fixed together to ensure that no relative motion would be detected. Under these conditions, any measured signal would correspond solely to the encoder noise. -The measurement setup is shown in Figure ref:fig:detail_instrumentation_vionic_bench. +The measurement setup is shown in Figure\nbsp{}ref:fig:detail_instrumentation_vionic_bench. To minimize environmental disturbances, the entire bench was covered with a plastic bubble sheet during measurements. -The amplitude spectral density of the measured displacement (which represents the measurement noise) is presented in Figure ref:fig:detail_instrumentation_vionic_asd. +The amplitude spectral density of the measured displacement (which represents the measurement noise) is presented in Figure\nbsp{}ref:fig:detail_instrumentation_vionic_asd. The noise profile exhibits characteristics of white noise with an amplitude of approximately $1\,\text{nm RMS}$, which complies with the system requirements. #+attr_latex: :options [b]{0.48\linewidth} @@ -9340,7 +9340,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier. -The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure ref:fig:detail_instrumentation_cl_noise_budget. +The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure\nbsp{}ref:fig:detail_instrumentation_cl_noise_budget. The total motion induced by all noise sources combined is approximately $1.5\,\text{nm RMS}$, which remains well within the specified limit of $15\,\text{nm RMS}$. This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application. @@ -9419,24 +9419,24 @@ Following the completion of this design phase and the subsequent procurement of :UNNUMBERED: t :END: -The experimental validation follows a systematic approach, beginning with the characterization of individual components before advancing to evaluate the assembled system's performance (illustrated in Figure ref:fig:chapter3_overview). -Section ref:sec:test_apa focuses on the Amplified Piezoelectric Actuator (APA300ML), examining its electrical properties, and dynamical behavior. +The experimental validation follows a systematic approach, beginning with the characterization of individual components before advancing to evaluate the assembled system's performance (illustrated in Figure\nbsp{}ref:fig:chapter3_overview). +Section\nbsp{}ref:sec:test_apa focuses on the Amplified Piezoelectric Actuator (APA300ML), examining its electrical properties, and dynamical behavior. Two models are developed and validated: a simplified two degrees-of-freedom model and a more complex super-element extracted from finite element analysis. The implementation of Integral Force Feedback is also experimentally evaluated to assess its effectiveness in adding damping to the system. -In Section ref:sec:test_joints, the flexible joints are characterized to ensure they meet the required specifications for stiffness and stroke. +In Section\nbsp{}ref:sec:test_joints, the flexible joints are characterized to ensure they meet the required specifications for stiffness and stroke. A dedicated test bench is developed to measure the bending stiffness, with error analysis performed to validate the measurement accuracy. -Section ref:sec:test_struts examines the assembly and testing of the struts, which integrate the APAs and flexible joints. +Section\nbsp{}ref:sec:test_struts examines the assembly and testing of the struts, which integrate the APAs and flexible joints. The mounting procedure is detailed, with particular attention to ensure consistent performance across multiple struts. Dynamical measurements are performed to verify whether the dynamics of the struts are corresponding to the multi-body model. -The assembly and testing of the complete nano-hexapod is presented in Section ref:sec:test_nhexa. +The assembly and testing of the complete nano-hexapod is presented in Section\nbsp{}ref:sec:test_nhexa. A suspended table is developed to isolate the hexapod's dynamics from support dynamics, enabling accurate identification of its dynamical properties. The experimental frequency response functions are compared with the multi-body model predictions to validate the modeling approach. The effects of various payload masses are also investigated. -Finally, Section ref:sec:test_id31 presents the validation of the NASS on the ID31 beamline. +Finally, Section\nbsp{}ref:sec:test_id31 presents the validation of the NASS on the ID31 beamline. A short-stroke metrology system is developed to measure the sample position relative to the granite base. The HAC-LAC control architecture is implemented and tested under various experimental conditions, including payload masses up to $39\,\text{kg}$ and for typical experiments, including tomography scans, reflectivity measurements, and diffraction tomography. @@ -9450,20 +9450,20 @@ The HAC-LAC control architecture is implemented and tested under various experim <> *** Introduction :ignore: -In this chapter, the goal is to ensure that the received APA300ML (shown in Figure ref:fig:test_apa_received) are complying with the requirements and that the dynamical models of the actuator accurately represent its dynamics. +In this chapter, the goal is to ensure that the received APA300ML (shown in Figure\nbsp{}ref:fig:test_apa_received) are complying with the requirements and that the dynamical models of the actuator accurately represent its dynamics. -In section ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks and the achievable stroke. +In section\nbsp{}ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks and the achievable stroke. The flexible modes of the APA300ML, which were estimated using a finite element model, are compared with measurements. -Using a dedicated test bench, dynamical measurements are performed (Section ref:sec:test_apa_dynamics). +Using a dedicated test bench, dynamical measurements are performed (Section\nbsp{}ref:sec:test_apa_dynamics). The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated. Integral Force Feedback is experimentally applied, and the damped plants are estimated for several feedback gains. Two different models of the APA300ML are presented. -First, in Section ref:sec:test_apa_model_2dof, a two degrees-of-freedom model is presented, tuned, and compared with the measured dynamics. +First, in Section\nbsp{}ref:sec:test_apa_model_2dof, a two degrees-of-freedom model is presented, tuned, and compared with the measured dynamics. This model is proven to accurately represent the APA300ML's axial dynamics while having low complexity. -Then, in Section ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a finite element model and imported into the multi-body model. +Then, in Section\nbsp{}ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a finite element model and imported into the multi-body model. This more complex model also captures well capture the axial dynamics of the APA300ML. #+name: fig:test_apa_received @@ -9477,18 +9477,18 @@ This more complex model also captures well capture the axial dynamics of the APA **** Introduction :ignore: Before measuring the dynamical characteristics of the APA300ML, simple measurements are performed. -First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section ref:ssec:test_apa_geometrical_measurements. -Then, the capacitance of the piezoelectric stacks is measured in Section ref:ssec:test_apa_electrical_measurements. -The achievable stroke of the APA300ML is measured using a displacement probe in Section ref:ssec:test_apa_stroke_measurements. -Finally, in Section ref:ssec:test_apa_spurious_resonances, the flexible modes of the APA are measured and compared with a finite element model. +First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section\nbsp{}ref:ssec:test_apa_geometrical_measurements. +Then, the capacitance of the piezoelectric stacks is measured in Section\nbsp{}ref:ssec:test_apa_electrical_measurements. +The achievable stroke of the APA300ML is measured using a displacement probe in Section\nbsp{}ref:ssec:test_apa_stroke_measurements. +Finally, in Section\nbsp{}ref:ssec:test_apa_spurious_resonances, the flexible modes of the APA are measured and compared with a finite element model. **** Geometrical Measurements <> To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness. -As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:test_apa_3] is used to measure the height of four points on each of the APA300ML interfaces. +As shown in Figure\nbsp{}ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:test_apa_3] is used to measure the height of four points on each of the APA300ML interfaces. From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting[fn:test_apa_4] a plane through all the points. -The measured flatness values, summarized in Table ref:tab:test_apa_flatness_meas, are within the specifications. +The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatness_meas, are within the specifications. #+attr_latex: :options [b]{0.48\textwidth} #+begin_minipage @@ -9520,14 +9520,14 @@ The measured flatness values, summarized in Table ref:tab:test_apa_flatness_meas From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu F$ and $26\,\mu F$ with a nominal capacitance of $20\,\mu F$. -The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter[fn:test_apa_1] shown in Figure ref:fig:test_apa_lcr_meter. +The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter[fn:test_apa_1] shown in Figure\nbsp{}ref:fig:test_apa_lcr_meter. The two stacks used as the actuator and the stack used as the force sensor were measured separately. -The measured capacitance values are summarized in Table ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu F$. +The measured capacitance values are summarized in Table\nbsp{}ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu F$. However, the measured capacitance of the stacks of "APA 3" is only half of the expected capacitance. This may indicate a manufacturing defect. The measured capacitance is found to be lower than the specified value. -This may be because the manufacturer measures the capacitance with large signals ($-20\,V$ to $150\,V$), whereas it was here measured with small signals [[cite:&wehrsdorfer95_large_signal_measur_piezoel_stack]]. +This may be because the manufacturer measures the capacitance with large signals ($-20\,V$ to $150\,V$), whereas it was here measured with small signals\nbsp{}[[cite:&wehrsdorfer95_large_signal_measur_piezoel_stack]]. #+attr_latex: :options [b]{0.48\textwidth} #+begin_minipage @@ -9557,24 +9557,24 @@ This may be because the manufacturer measures the capacitance with large signals **** Stroke and Hysteresis Measurement <> -To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe[fn:test_apa_2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench. +To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe[fn:test_apa_2] is located on the other side as shown in Figure\nbsp{}ref:fig:test_apa_stroke_bench. The voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:test_apa_12] and a voltage amplifier[fn:test_apa_13]. -Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_voltage). +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 #+attr_latex: :width 0.7\linewidth [[file:figs/test_apa_stroke_bench.jpg]] -The measured APA displacement is shown as a function of the applied voltage in Figure ref:fig:test_apa_stroke_hysteresis. +The measured APA displacement is shown as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis. Typical hysteresis curves for piezoelectric stack actuators can be observed. The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks. This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu m$, therefore $\approx 200\,\mu m$ if only two stacks are used). For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu m$. -It is clear from Figure ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared with the other units. -This confirms the abnormal electrical measurements made in Section ref:ssec:test_apa_electrical_measurements. +It is clear from Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared with the other units. +This confirms the abnormal electrical measurements made in Section\nbsp{}ref:ssec:test_apa_electrical_measurements. 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. @@ -9600,11 +9600,11 @@ From now on, only the six remaining amplified piezoelectric actuators that behav <> In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. -To experimentally estimate these modes, the APA is fixed at one end (see Figure ref:fig:test_apa_meas_setup_modes). +To experimentally estimate these modes, the APA is fixed at one end (see Figure\nbsp{}ref:fig:test_apa_meas_setup_modes). A Laser Doppler Vibrometer[fn:test_apa_6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:test_apa_7] is used to excite the flexible modes. Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated. -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 ref:fig:test_apa_mode_shapes. +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 @@ -9648,9 +9648,9 @@ The flexible modes for the same condition (i.e. one mechanical interface of the #+end_subfigure #+end_figure -The measured frequency response functions computed from the experimental setups of figures ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure ref:fig:test_apa_meas_freq_compare. +The measured frequency response functions computed from the experimental setups of figures\nbsp{}ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure\nbsp{}ref:fig:test_apa_meas_freq_compare. The $y$ bending mode is observed at $280\,\text{Hz}$ and the $x$ bending mode is at $412\,\text{Hz}$. -These modes are measured at higher frequencies than the frequencies estimated from the Finite Element Model (see frequencies in Figure ref:fig:test_apa_mode_shapes). +These modes are measured at higher frequencies than the frequencies estimated from the Finite Element Model (see frequencies in Figure\nbsp{}ref:fig:test_apa_mode_shapes). This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model). This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used). Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades. @@ -9662,8 +9662,8 @@ Another explanation is the shape difference between the manufactured APA300ML an *** Dynamical measurements <> **** Introduction :ignore: -After the measurements on the APA were performed in Section ref:sec:test_apa_basic_meas, a new test bench was used to better characterize the dynamics of the APA300ML. -This test bench, depicted in Figure ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a 5kg granite block that is vertically guided by an air bearing. +After the measurements on the APA were performed in Section\nbsp{}ref:sec:test_apa_basic_meas, a new test bench was used to better characterize the dynamics of the APA300ML. +This test bench, depicted in Figure\nbsp{}ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a 5kg granite block that is vertically guided by an air bearing. Thus, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors. An encoder[fn:test_apa_8] is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA. @@ -9685,12 +9685,12 @@ An encoder[fn:test_apa_8] is used to measure the relative movement between the t #+end_subfigure #+end_figure -The bench is schematically shown in Figure ref:fig:test_apa_schematic with the associated signals. -It will be first used to estimate the hysteresis from the piezoelectric stack (Section ref:ssec:test_apa_hysteresis) as well as the axial stiffness of the APA300ML (Section ref:ssec:test_apa_stiffness). -The frequency response functions from the DAC voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section ref:ssec:test_apa_meas_dynamics. -The presence of a non-minimum phase zero found on the transfer function from $u$ to $V_s$ is investigated in Section ref:ssec:test_apa_non_minimum_phase. -To limit the low-frequency gain of the transfer function from $u$ to $V_s$, a resistor is added across the force sensor stack (Section ref:ssec:test_apa_resistance_sensor_stack). -Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section ref:ssec:test_apa_iff_locus. +The bench is schematically shown in Figure\nbsp{}ref:fig:test_apa_schematic with the associated signals. +It will be first used to estimate the hysteresis from the piezoelectric stack (Section\nbsp{}ref:ssec:test_apa_hysteresis) as well as the axial stiffness of the APA300ML (Section\nbsp{}ref:ssec:test_apa_stiffness). +The frequency response functions from the DAC voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section\nbsp{}ref:ssec:test_apa_meas_dynamics. +The presence of a non-minimum phase zero found on the transfer function from $u$ to $V_s$ is investigated in Section\nbsp{}ref:ssec:test_apa_non_minimum_phase. +To limit the low-frequency gain of the transfer function from $u$ to $V_s$, a resistor is added across the force sensor stack (Section\nbsp{}ref:ssec:test_apa_resistance_sensor_stack). +Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section\nbsp{}ref:ssec:test_apa_iff_locus. #+name: fig:test_apa_schematic #+caption: Schematic of the Test Bench used to measure the dynamics of the APA300ML. $u$ is the output DAC voltage, $V_a$ the output amplifier voltage (i.e. voltage applied across the actuator stacks), $d_e$ the measured displacement by the encoder and $V_s$ the measured voltage across the sensor stack. @@ -9702,8 +9702,8 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a Because the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload. A quasi static[fn:test_apa_9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the DAC. -For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure ref:fig:test_apa_meas_hysteresis. -This is the typical behavior expected from a PZT stack actuator, where the hysteresis increases as a function of the applied voltage amplitude [[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]]. +For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_meas_hysteresis. +This is the typical behavior expected from a PZT stack actuator, where the hysteresis increases as a function of the applied voltage amplitude\nbsp{}[[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]]. #+name: fig:test_apa_meas_hysteresis #+caption: Displacement as a function of applied voltage for multiple excitation amplitudes @@ -9713,18 +9713,18 @@ This is the typical behavior expected from a PZT stack actuator, where the hyste <> To estimate the stiffness of the APA, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $\Delta d_e$ is measured using the encoder. -The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness, with $g \approx 9.8\,m/s^2$ the acceleration of gravity. +The APA stiffness can then be estimated from equation\nbsp{}eqref:eq:test_apa_stiffness, with $g \approx 9.8\,m/s^2$ the acceleration of gravity. \begin{equation} \label{eq:test_apa_stiffness} k_{\text{apa}} = \frac{m_a g}{\Delta d_e} \end{equation} -The measured displacement $d_e$ as a function of time is shown in Figure ref:fig:test_apa_meas_stiffness_time. +The measured displacement $d_e$ as a function of time is shown in Figure\nbsp{}ref:fig:test_apa_meas_stiffness_time. It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep), and that the displacement does not return to the initial position after the mass is removed (probably due to piezoelectric hysteresis). These two effects induce some uncertainties in the measured stiffness. -The stiffnesses are computed for all APAs from the two displacements $d_1$ and $d_2$ (see Figure ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$. -These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,N/\mu m$. +The stiffnesses are computed for all APAs from the two displacements $d_1$ and $d_2$ (see Figure\nbsp{}ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$. +These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,N/\mu m$. #+attr_latex: :options [b]{0.57\textwidth} #+begin_minipage @@ -9750,7 +9750,7 @@ These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_st | 8 | 1.73 | 1.98 | #+end_minipage -The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. +The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section\nbsp{}ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. \begin{equation} \label{eq:test_apa_res_freq} \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} @@ -9758,7 +9758,7 @@ The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by The obtained stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and using the "static deflection" method. -It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]]. +It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack\nbsp{}[[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]]. To estimate this effect for the APA300ML, its stiffness is estimated using the "static deflection" method in two cases: - $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance ADC) @@ -9771,28 +9771,28 @@ The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ In this section, the dynamics from the excitation voltage $u$ to the encoder measured displacement $d_e$ and to the force sensor voltage $V_s$ is identified. -First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_encoder. +First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure\nbsp{}ref:fig:test_apa_frf_encoder. The obtained frequency response functions are similar to those of a (second order) mass-spring-damper system with: - A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$. The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA - A lightly damped resonance at $95\,\text{Hz}$ - A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the APA support. - The flexible modes studied in section ref:ssec:test_apa_spurious_resonances seem not to impact the measured axial motion of the actuator. + The flexible modes studied in section\nbsp{}ref:ssec:test_apa_spurious_resonances seem not to impact the measured axial motion of the actuator. -The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_force. +The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure\nbsp{}ref:fig:test_apa_frf_force. A lightly damped resonance (pole) is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance (zero) at $41\,\text{Hz}$. No additional resonances are present up to at least $2\,\text{kHz}$ indicating that Integral Force Feedback can be applied without stability issues from high-frequency flexible modes. The zero at $41\,\text{Hz}$ seems to be non-minimum phase (the phase /decreases/ by 180 degrees whereas it should have /increased/ by 180 degrees for a minimum phase zero). -This is investigated in Section ref:ssec:test_apa_non_minimum_phase. +This is investigated in Section\nbsp{}ref:ssec:test_apa_non_minimum_phase. As illustrated by the Root Locus plot, the poles of the /closed-loop/ system converges to the zeros of the /open-loop/ plant as the feedback gain increases. -The significance of this behavior varies with the type of sensor used, as explained in [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]]. +The significance of this behavior varies with the type of sensor used, as explained in\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]]. Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain. Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell. From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$ (which is close to what is found using a finite element model). -All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure ref:fig:test_apa_frf_encoder and at the force sensor in Figure ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell. +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 @@ -9815,15 +9815,15 @@ All the identified dynamics of the six APA300ML (both when looking at the encode **** Non Minimum Phase Zero? <> -It was surprising to observe a non-minimum phase zero on the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force). +It was surprising to observe a non-minimum phase zero on the transfer function from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_frf_force). It was initially thought that this non-minimum phase behavior was an artifact arising from the measurement. -A longer measurement was performed using different excitation signals (noise, slow sine sweep, etc.) to determine if the phase behavior of the zero changes (Figure ref:fig:test_apa_non_minimum_phase). -The coherence (Figure ref:fig:test_apa_non_minimum_phase_coherence) is good even in the vicinity of the lightly damped zero, and the phase (Figure ref:fig:test_apa_non_minimum_phase_zoom) clearly indicates non-minimum phase behavior. +A longer measurement was performed using different excitation signals (noise, slow sine sweep, etc.) to determine if the phase behavior of the zero changes (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase). +The coherence (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase_coherence) is good even in the vicinity of the lightly damped zero, and the phase (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase_zoom) clearly indicates non-minimum phase behavior. -Such non-minimum phase zero when using load cells has also been observed on other mechanical systems [[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. +Such non-minimum phase zero when using load cells has also been observed on other mechanical systems\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. It could be induced to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear. -However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice. +However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure\nbsp{}ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice. #+name: fig:test_apa_non_minimum_phase #+caption: Measurement of the anti-resonance found in the transfer function from $u$ to $V_s$. The coherence \subref{fig:test_apa_non_minimum_phase_coherence} is quite good around the anti-resonance frequency. The phase \subref{fig:test_apa_non_minimum_phase_zoom} shoes a non-minimum phase behavior. @@ -9850,7 +9850,7 @@ A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, As explained before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain. -The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor were measured and compared in Figure ref:fig:test_apa_effect_resistance. +The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor were measured and compared in Figure\nbsp{}ref:fig:test_apa_effect_resistance. It is confirmed that the added resistor has the effect of adding a high-pass filter with a cut-off frequency of $\approx 0.39\,\text{Hz}$. #+name: fig:test_apa_effect_resistance @@ -9860,27 +9860,27 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil **** Integral Force Feedback <> -To implement the Integral Force Feedback strategy, the measured frequency response function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation eqref:eq:test_apa_iff_manual_fit. +To implement the Integral Force Feedback strategy, the measured frequency response function from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation\nbsp{}eqref:eq:test_apa_iff_manual_fit. The parameters were manually tuned, and the obtained values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$. \begin{equation} \label{eq:test_apa_iff_manual_fit} G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s} \end{equation} -A comparison between the identified plant and the manually tuned transfer function is shown in Figure ref:fig:test_apa_iff_plant_comp_manual_fit. +A comparison between the identified plant and the manually tuned transfer function is shown in Figure\nbsp{}ref:fig:test_apa_iff_plant_comp_manual_fit. #+name: fig:test_apa_iff_plant_comp_manual_fit #+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\mu s$ is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero. [[file:figs/test_apa_iff_plant_comp_manual_fit.png]] -The implemented Integral Force Feedback Controller transfer function is shown in equation eqref:eq:test_apa_Kiff_formula. +The implemented Integral Force Feedback Controller transfer function is shown in equation\nbsp{}eqref:eq:test_apa_Kiff_formula. It contains a high-pass filter (cut-off frequency of $2\,\text{Hz}$) to limit the low-frequency gain, a low-pass filter to add integral action above $20\,\text{Hz}$, a second low-pass filter to add robustness to high-frequency resonances, and a tunable gain $g$. \begin{equation} \label{eq:test_apa_Kiff_formula} K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000} \end{equation} -To estimate how the dynamics of the APA changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure ref:fig:test_apa_iff_schematic is used. +To estimate how the dynamics of the APA changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure\nbsp{}ref:fig:test_apa_iff_schematic is used. The transfer function from the "damped" plant input $u\prime$ to the encoder displacement $d_e$ is identified for several IFF controller gains $g$. #+name: fig:test_apa_iff_schematic @@ -9888,13 +9888,13 @@ The transfer function from the "damped" plant input $u\prime$ to the encoder dis [[file:figs/test_apa_iff_schematic.png]] The identified dynamics were then fitted by second order transfer functions[fn:test_apa_10]. -A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown in Figure ref:fig:test_apa_identified_damped_plants for different gains $g$. +A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown in Figure\nbsp{}ref:fig:test_apa_identified_damped_plants for different gains $g$. It is clear that a large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies. The evolution of the pole in the complex plane as a function of the controller gain $g$ (i.e. the "root locus") is computed in two cases. -First using the IFF plant model eqref:eq:test_apa_iff_manual_fit and the implemented controller eqref:eq:test_apa_Kiff_formula. +First using the IFF plant model\nbsp{}eqref:eq:test_apa_iff_manual_fit and the implemented controller\nbsp{}eqref:eq:test_apa_Kiff_formula. Second using the fitted transfer functions of the damped plants experimentally identified for several controller gains. -The two obtained root loci are compared in Figure 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. +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} @@ -9918,7 +9918,7 @@ The two obtained root loci are compared in Figure ref:fig:test_apa_iff_root_locu <> ***** Introduction :ignore: -In this section, a multi-body model (Figure ref:fig:test_apa_bench_model) of the measurement bench is used to tune the two degrees-of-freedom model of the APA using the measured frequency response functions. +In this section, a multi-body model (Figure\nbsp{}ref:fig:test_apa_bench_model) of the measurement bench is used to tune the two degrees-of-freedom model of the APA using the measured frequency response functions. This two degrees-of-freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and a low number of associated states. After the model is presented, the procedure for tuning the model is described, and the obtained model dynamics is compared with the measurements. @@ -9930,7 +9930,7 @@ After the model is presented, the procedure for tuning the model is described, a ***** Two degrees-of-freedom APA Model -The model of the amplified piezoelectric actuator is shown in Figure ref:fig:test_apa_2dof_model. +The model of the amplified piezoelectric actuator is shown in Figure\nbsp{}ref:fig:test_apa_2dof_model. It can be decomposed into three components: - the shell whose axial properties are represented by $k_1$ and $c_1$ - the actuator stacks whose contribution to the axial stiffness is represented by $k_a$ and $c_a$. @@ -9950,7 +9950,7 @@ Such a simple model has some limitations: ***** Tuning of the APA model :ignore: -9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics. +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 $V_s$ @@ -9959,18 +9959,18 @@ Such a simple model has some limitations: First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. Both methods lead to an estimated mass of $m = 5.7\,\text{kg}$. -Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure ref:fig:test_apa_frf_force. +Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section\nbsp{}ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure\nbsp{}ref:fig:test_apa_frf_force. Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $5\,Ns/m$. Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:test_apa_5]. Therefore, we have $k_e = 2 k_a$ and $c_e = 2 c_a$ as the actuator stack is composed of two stacks in series. -In this case, the total stiffness of the APA model is described by eqref:eq:test_apa_2dof_stiffness. +In this case, the total stiffness of the APA model is described by\nbsp{}eqref:eq:test_apa_2dof_stiffness. \begin{equation}\label{eq:test_apa_2dof_stiffness} k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a \end{equation} -Knowing from eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\text{tot}} = 2\,N/\mu m$, we get from eqref:eq:test_apa_2dof_stiffness that $k_a = 2.5\,N/\mu m$ and $k_e = 5\,N/\mu m$. +Knowing from\nbsp{}eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\text{tot}} = 2\,N/\mu m$, we get from\nbsp{}eqref:eq:test_apa_2dof_stiffness that $k_a = 2.5\,N/\mu m$ and $k_e = 5\,N/\mu m$. \begin{equation}\label{eq:test_apa_tot_stiffness} \omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,N/\mu m \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s @@ -9981,7 +9981,7 @@ $c_a = 50\,Ns/m$ and $c_e = 100\,Ns/m$ are obtained. In the last step, $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions. -The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_simscape are summarized in Table ref:tab:test_apa_2dof_parameters. +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 @@ -10001,9 +10001,9 @@ The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model ***** Obtained Dynamics :ignore: -The dynamics of the two degrees-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the multi-body model. -This is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf. -A good match can be observed between the model and the experimental data, both for the encoder (Figure ref:fig:test_apa_2dof_comp_frf_enc) and for the force sensor (Figure ref:fig:test_apa_2dof_comp_frf_force). +The dynamics of the two degrees-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table\nbsp{}ref:tab:test_apa_2dof_parameters) from the multi-body model. +This is compared with the experimental data in Figure\nbsp{}ref:fig:test_apa_2dof_comp_frf. +A good match can be observed between the model and the experimental data, both for the encoder (Figure\nbsp{}ref:fig:test_apa_2dof_comp_frf_enc) and for the force sensor (Figure\nbsp{}ref:fig:test_apa_2dof_comp_frf_force). This indicates that this model represents well the axial dynamics of the APA300ML. #+name: fig:test_apa_2dof_comp_frf @@ -10029,8 +10029,8 @@ This indicates that this model represents well the axial dynamics of the APA300M ***** Introduction :ignore: In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:test_apa_11]. -It is then imported into multi-body (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in ref:sec:test_apa_model_2dof. -This procedure is illustrated in Figure ref:fig:test_apa_super_element_simscape. +It is then imported into multi-body (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in\nbsp{}ref:sec:test_apa_model_2dof. +This procedure is illustrated in Figure\nbsp{}ref:fig:test_apa_super_element_simscape. Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then made accessible in Simscape as shown at the right by the "frames" =F1= to =F5=. For the APA300ML /super element/, 5 /remote points/ are defined. @@ -10051,7 +10051,7 @@ By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained. To ensure that the sensitivities $g_a$ and $g_s$ are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material. -From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force. +From\nbsp{}[[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by\nbsp{}eqref:eq:test_apa_piezo_strain_to_voltage and from\nbsp{}[[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by\nbsp{}eqref:eq:test_apa_piezo_voltage_to_force. \begin{subequations} \begin{align} @@ -10062,7 +10062,7 @@ From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation bet Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML. However, based on the available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. -The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties. +The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table\nbsp{}ref:tab:test_apa_piezo_properties. From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained, which are close to the constants identified using the experimentally identified transfer functions. @@ -10082,9 +10082,9 @@ From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained, w ***** Comparison of the obtained dynamics -The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf. +The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" are compared with the experimentally identified frequency response functions in Figure\nbsp{}ref:fig:test_apa_super_element_comp_frf. A good match between the model and the experimental results was observed. -It is however surprising that the model is "softer" than the measured system, as finite element models usually overestimate the stiffness (see Section ref:ssec:test_apa_spurious_resonances for possible explanations). +It is however surprising that the model is "softer" than the measured system, as finite element models usually overestimate the stiffness (see Section\nbsp{}ref:ssec:test_apa_spurious_resonances for possible explanations). 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). @@ -10114,19 +10114,19 @@ Using this simple test bench, it can be concluded that the /super element/ model In this study, the amplified piezoelectric actuators "APA300ML" have been characterized to ensure that they fulfill all the requirements determined during the detailed design phase. -Geometrical features such as the flatness of its interfaces, electrical capacitance, and achievable strokes were measured in Section ref:sec:test_apa_basic_meas. +Geometrical features such as the flatness of its interfaces, electrical capacitance, and achievable strokes were measured in Section\nbsp{}ref:sec:test_apa_basic_meas. These simple measurements allowed for the early detection of a manufacturing defect in one of the APA300ML. -Then in Section ref:sec:test_apa_dynamics, using a dedicated test bench, the dynamics of all the APA300ML were measured and were found to all match very well (Figure ref:fig:test_apa_frf_dynamics). +Then in Section\nbsp{}ref:sec:test_apa_dynamics, using a dedicated test bench, the dynamics of all the APA300ML were measured and were found to all match very well (Figure\nbsp{}ref:fig:test_apa_frf_dynamics). This consistency indicates good manufacturing tolerances, facilitating the modeling and control of the nano-hexapod. -Although a non-minimum zero was identified in the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_non_minimum_phase), it was found not to be problematic because a large amount of damping could be added using the integral force feedback strategy (Figure ref:fig:test_apa_iff). +Although a non-minimum zero was identified in the transfer function from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase), it was found not to be problematic because a large amount of damping could be added using the integral force feedback strategy (Figure\nbsp{}ref:fig:test_apa_iff). Then, two different models were used to represent the APA300ML dynamics. -In Section ref:sec:test_apa_model_2dof, a simple two degrees-of-freedom mass-spring-damper model was presented and tuned based on the measured dynamics. +In Section\nbsp{}ref:sec:test_apa_model_2dof, a simple two degrees-of-freedom mass-spring-damper model was presented and tuned based on the measured dynamics. After following a tuning procedure, the model dynamics was shown to match very well with the experiment. However, this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions. -In Section ref:sec:test_apa_model_flexible, a /super element/ extracted from a finite element model was used to model the APA300ML. +In Section\nbsp{}ref:sec:test_apa_model_flexible, a /super element/ extracted from a finite element model was used to model the APA300ML. Here, the /super element/ represents the dynamics of the APA300ML in all directions. However, only the axial dynamics could be compared with the experimental results, yielding a good match. The benefit of employing this model over the two degrees-of-freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved. @@ -10140,7 +10140,7 @@ At both ends of the nano-hexapod struts, a flexible joint is used. Ideally, these flexible joints would behave as perfect spherical joints, that is to say no bending and torsional stiffness, infinite shear and axial stiffness, unlimited bending and torsional stroke, no friction, and no backlash. Deviations from these ideal properties will impact the dynamics of the Nano-Hexapod and could limit the attainable performance. -During the detailed design phase, specifications in terms of stiffness and stroke were determined and are summarized in Table ref:tab:test_joints_specs. +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 @@ -10154,9 +10154,9 @@ During the detailed design phase, specifications in terms of stiffness and strok | Torsion Stiffness | $< 500\,Nm/\text{rad}$ | 260 | | Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | -After optimization using a finite element model, the geometry shown in Figure ref:fig:test_joints_schematic has been obtained and the corresponding flexible joint characteristics are summarized in Table ref:tab:test_joints_specs. +After optimization using a finite element model, the geometry shown in Figure\nbsp{}ref:fig:test_joints_schematic has been obtained and the corresponding flexible joint characteristics are summarized in Table\nbsp{}ref:tab:test_joints_specs. This flexible joint is a monolithic piece of stainless steel[fn:test_joints_1] manufactured using wire electrical discharge machining. -It serves several functions, as shown in Figure ref:fig:test_joints_iso, such as: +It serves several functions, as shown in Figure\nbsp{}ref:fig:test_joints_iso, such as: - Rigid interfacing with the nano-hexapod plates (yellow surfaces) - Rigid interfacing with the amplified piezoelectric actuator (blue surface) - Allow two rotations between the "yellow" and the "blue" interfaces. @@ -10186,7 +10186,7 @@ It serves several functions, as shown in Figure ref:fig:test_joints_iso, such as #+end_subfigure #+end_figure -Sixteen flexible joints have been ordered (shown in Figure ref:fig:test_joints_received) such that some selection can be made for the twelve that will be used on the nano-hexapod. +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 @@ -10208,11 +10208,11 @@ Sixteen flexible joints have been ordered (shown in Figure ref:fig:test_joints_r In this document, the received flexible joints are characterized to ensure that they fulfill the requirements and such that they can well be modeled. -First, the flexible joints are visually inspected, and the minimum gaps (responsible for most of the joint compliance) are measured (Section ref:sec:test_joints_flex_dim_meas). +First, the flexible joints are visually inspected, and the minimum gaps (responsible for most of the joint compliance) are measured (Section\nbsp{}ref:sec:test_joints_flex_dim_meas). Then, a test bench was developed to measure the bending stiffness of the flexible joints. -The development of this test bench is presented in Section ref:sec:test_joints_test_bench_desc, including a noise budget and some requirements in terms of instrumentation. +The development of this test bench is presented in Section\nbsp{}ref:sec:test_joints_test_bench_desc, including a noise budget and some requirements in terms of instrumentation. The test bench is then used to measure the bending stiffnesses of all the flexible joints. -Results are shown in Section ref:sec:test_joints_bending_stiffness_meas +Results are shown in Section\nbsp{}ref:sec:test_joints_bending_stiffness_meas *** Dimensional Measurements <> @@ -10222,8 +10222,8 @@ Two dimensions are critical for the bending stiffness of the flexible joints. These dimensions can be measured using a profilometer. The dimensions of the flexible joint in the Y-Z plane will contribute to the X-bending stiffness, whereas the dimensions in the X-Z plane will contribute to the Y-bending stiffness. -The setup used to measure the dimensions of the "X" flexible beam is shown in Figure ref:fig:test_joints_profilometer_setup. -What is typically observed is shown in Figure ref:fig:test_joints_profilometer_image. +The setup used to measure the dimensions of the "X" flexible beam is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_setup. +What is typically observed is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_image. It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\mu m$, #+name: fig:test_joints_profilometer @@ -10249,7 +10249,7 @@ The specified flexible beam thickness (gap) is $250\,\mu m$. Four gaps are measured for each flexible joint (2 in the $x$ direction and 2 in the $y$ direction). The "beam thickness" is then estimated as the mean between the gaps measured on opposite sides. -A histogram of the measured beam thicknesses is shown in Figure ref:fig:test_joints_size_hist. +A histogram of the measured beam thicknesses is shown in Figure\nbsp{}ref:fig:test_joints_size_hist. The measured thickness is less than the specified value of $250\,\mu m$, but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. However, what is more important than the true value of the thickness is the consistency between all flexible joints. @@ -10260,7 +10260,7 @@ However, what is more important than the true value of the thickness is the cons **** Bad flexible joints -Using this profilometer allowed to detect flexible joints with manufacturing defects such as non-symmetrical shapes (see Figure ref:fig:test_joints_bad_shape) or flexible joints with machining chips stuck in the gap (see Figure ref:fig:test_joints_bad_chips). +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 @@ -10286,7 +10286,7 @@ Using this profilometer allowed to detect flexible joints with manufacturing def The most important characteristic of the flexible joint to be measured is its bending stiffness $k_{R_x} \approx k_{R_y}$. To estimate the bending stiffness, the basic idea is to apply a torque $T_{x}$ to the flexible joints and to measure its angular deflection $\theta_{x}$. -The bending stiffness can then be computed from equation eqref:eq:test_joints_bending_stiffness. +The bending stiffness can then be computed from equation\nbsp{}eqref:eq:test_joints_bending_stiffness. \begin{equation}\label{eq:test_joints_bending_stiffness} \boxed{k_{R_x} = \frac{T_x}{\theta_x}, \quad k_{R_y} = \frac{T_y}{\theta_y}} @@ -10296,22 +10296,22 @@ The bending stiffness can then be computed from equation eqref:eq:test_joints_be <> ***** Torque and Rotation measurement To apply torque $T_{y}$ between the two mobile parts of the flexible joint, a known "linear" force $F_{x}$ can be applied instead at a certain distance $h$ with respect to the rotation point. -In this case, the equivalent applied torque can be estimated from equation eqref:eq:test_joints_force_torque_distance. +In this case, the equivalent applied torque can be estimated from equation\nbsp{}eqref:eq:test_joints_force_torque_distance. Note that the application point of the force should be sufficiently far from the rotation axis such that the resulting bending motion is much larger than the displacement due to shear. -Such effects are studied in Section ref:ssec:test_joints_error_budget. +Such effects are studied in Section\nbsp{}ref:ssec:test_joints_error_budget. \begin{equation}\label{eq:test_joints_force_torque_distance} T_y = h F_x, \quad T_x = h F_y \end{equation} Similarly, instead of directly measuring the bending motion $\theta_y$ of the flexible joint, its linear motion $d_x$ at a certain distance $h$ from the rotation points is measured. -The equivalent rotation is estimated from eqref:eq:test_joints_rot_displ. +The equivalent rotation is estimated from\nbsp{}eqref:eq:test_joints_rot_displ. \begin{equation}\label{eq:test_joints_rot_displ} \theta_y = \tan^{-1}\left(\frac{d_x}{h}\right) \approx \frac{d_x}{h}, \quad \theta_x = \tan^{-1} \left( \frac{d_y}{h} \right) \approx \frac{d_y}{h} \end{equation} -Then, the bending stiffness can be estimated from eqref:eq:test_joints_stiff_displ_force. +Then, the bending stiffness can be estimated from\nbsp{}eqref:eq:test_joints_stiff_displ_force. \begin{subequations}\label{eq:test_joints_stiff_displ_force} \begin{align} @@ -10320,7 +10320,7 @@ k_{R_y} &= \frac{T_y}{\theta_y} = \frac{h F_x}{\tan^{-1}\left( \frac{d_x}{h} \ri \end{align} \end{subequations} -The working principle of the measurement bench is schematically shown in Figure ref:fig:test_joints_bench_working_principle. +The working principle of the measurement bench is schematically shown in Figure\nbsp{}ref:fig:test_joints_bench_working_principle. One part of the flexible joint is fixed to a rigid frame while a (known) force $F_x$ is applied to the other side of the flexible joint. The deflection of the joint $d_x$ is measured using a displacement sensor. @@ -10330,15 +10330,15 @@ The deflection of the joint $d_x$ is measured using a displacement sensor. ***** Required external applied force The bending stiffness is foreseen to be $k_{R_y} \approx k_{R_x} \approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx \theta_{x,\text{max}}\approx 25\,mrad$. -The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 22.5\,mm$ (see Figure ref:fig:test_joints_bench_working_principle). +The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 22.5\,mm$ (see Figure\nbsp{}ref:fig:test_joints_bench_working_principle). -The bending $\theta_y$ of the flexible joint due to the force $F_x$ is given by equation eqref:eq:test_joints_deflection_force. +The bending $\theta_y$ of the flexible joint due to the force $F_x$ is given by equation\nbsp{}eqref:eq:test_joints_deflection_force. \begin{equation}\label{eq:test_joints_deflection_force} \theta_y = \frac{T_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}} \end{equation} -Therefore, the force that must be applied to test the full range of the flexible joints is given by equation eqref:eq:test_joints_max_force. +Therefore, the force that must be applied to test the full range of the flexible joints is given by equation\nbsp{}eqref:eq:test_joints_max_force. The measurement range of the force sensor should then be higher than $5.5\,N$. \begin{equation}\label{eq:test_joints_max_force} @@ -10347,7 +10347,7 @@ The measurement range of the force sensor should then be higher than $5.5\,N$. ***** Required actuator stroke and sensors range The flexible joint is designed to allow a bending motion of $\pm 25\,mrad$. -The corresponding stroke at the location of the force sensor is given by eqref:eq:test_joints_max_stroke. +The corresponding stroke at the location of the force sensor is given by\nbsp{}eqref:eq:test_joints_max_stroke. To test the full range of the flexible joint, the means of applying a force (explained in the next section) should allow a motion of at least $0.5\,mm$. Similarly, the measurement range of the displacement sensor should also be higher than $0.5\,mm$. @@ -10357,39 +10357,39 @@ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \approx 0.5\,mm ***** Force and Displacement measurements To determine the applied force, a load cell will be used in series with the mechanism that applied the force. -The measured deflection of the flexible joint will be indirectly estimated from the displacement of the force sensor itself (see Section ref:ssec:test_joints_test_bench). +The measured deflection of the flexible joint will be indirectly estimated from the displacement of the force sensor itself (see Section\nbsp{}ref:ssec:test_joints_test_bench). Indirectly measuring the deflection of the flexible joint induces some errors because of the limited stiffness between the force sensor and the displacement sensor. -Such an effect will be estimated in the error budget (Section ref:ssec:test_joints_error_budget) +Such an effect will be estimated in the error budget (Section\nbsp{}ref:ssec:test_joints_error_budget) **** Error budget <> ***** Introduction :ignore: To estimate the accuracy of the measured bending stiffness that can be obtained using this measurement principle, an error budget is performed. -Based on equation eqref:eq:test_joints_stiff_displ_force, several errors can affect the accuracy of the measured bending stiffness: +Based on equation\nbsp{}eqref:eq:test_joints_stiff_displ_force, several errors can affect the accuracy of the measured bending stiffness: - Errors in the measured torque $M_x, M_y$: this is mainly due to inaccuracies in the load cell and of the height estimation $h$ - Errors in the measured bending motion of the flexible joints $\theta_x, \theta_y$: errors from limited shear stiffness, from the deflection of the load cell itself, and inaccuracy of the height estimation $h$ -If only the bending stiffness is considered, the induced displacement is described by eqref:eq:test_joints_dbx. +If only the bending stiffness is considered, the induced displacement is described by\nbsp{}eqref:eq:test_joints_dbx. \begin{equation}\label{eq:test_joints_dbx} d_{x,b} = h \tan(\theta_y) = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) \end{equation} ***** Effect of Shear -The applied force $F_x$ will induce some shear $d_{x,s}$ which is described by eqref:eq:test_joints_shear_displ with $k_s$ the shear stiffness of the flexible joint. +The applied force $F_x$ will induce some shear $d_{x,s}$ which is described by\nbsp{}eqref:eq:test_joints_shear_displ with $k_s$ the shear stiffness of the flexible joint. \begin{equation}\label{eq:test_joints_shear_displ} d_{x,s} = \frac{F_x}{k_s} \end{equation} -The measured displacement $d_x$ is affected shear, as shown in equation eqref:eq:test_joints_displ_shear. +The measured displacement $d_x$ is affected shear, as shown in equation\nbsp{}eqref:eq:test_joints_displ_shear. \begin{equation}\label{eq:test_joints_displ_shear} d_x = d_{x,b} + d_{x,s} = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_s} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_s} \right) \end{equation} -The estimated bending stiffness $k_{\text{est}}$ then depends on the shear stiffness eqref:eq:test_joints_error_shear. +The estimated bending stiffness $k_{\text{est}}$ then depends on the shear stiffness\nbsp{}eqref:eq:test_joints_error_shear. \begin{equation}\label{eq:test_joints_error_shear} k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_s h^2}}_{\epsilon_{s}} \Bigl) @@ -10401,7 +10401,7 @@ With an estimated shear stiffness $k_s = 13\,N/\mu m$ from the finite element mo As explained in the previous section, because the measurement of the flexible joint deflection is indirectly performed with the encoder, errors will be made if the load cell experiences some compression. Suppose the load cell has an internal stiffness $k_f$, the same reasoning that was made for the effect of shear can be applied here. -The estimation error of the bending stiffness due to the limited stiffness of the load cell is then described by eqref:eq:test_joints_error_load_cell_stiffness. +The estimation error of the bending stiffness due to the limited stiffness of the load cell is then described by\nbsp{}eqref:eq:test_joints_error_load_cell_stiffness. \begin{equation}\label{eq:test_joints_error_load_cell_stiffness} k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_F h^2}}_{\epsilon_f} \Bigl) @@ -10410,13 +10410,13 @@ The estimation error of the bending stiffness due to the limited stiffness of th With an estimated load cell stiffness of $k_f \approx 1\,N/\mu m$ (from the documentation), the errors due to the load cell limited stiffness is around $\epsilon_f = 1\,\%$. ***** Estimation error due to height estimation error -Now consider an error $\delta h$ in the estimation of the height $h$ as described by eqref:eq:test_joints_est_h_error. +Now consider an error $\delta h$ in the estimation of the height $h$ as described by\nbsp{}eqref:eq:test_joints_est_h_error. \begin{equation}\label{eq:test_joints_est_h_error} h_{\text{est}} = h + \delta h \end{equation} -The computed bending stiffness will be eqref:eq:test_joints_stiffness_height_error. +The computed bending stiffness will be\nbsp{}eqref:eq:test_joints_stiffness_height_error. \begin{equation}\label{eq:test_joints_stiffness_height_error} k_{R_y, \text{est}} \approx h_{\text{est}}^2 \frac{F_x}{d_x} \approx k_{R_y} \Bigl( 1 + \underbrace{2 \frac{\delta h}{h} + \frac{\delta h ^2}{h^2}}_{\epsilon_h} \Bigl) @@ -10425,14 +10425,14 @@ The computed bending stiffness will be eqref:eq:test_joints_stiffness_height_err The height estimation is foreseen to be accurate to within $|\delta h| < 0.4\,mm$ which corresponds to a stiffness error $\epsilon_h < 3.5\,\%$. ***** Estimation error due to force and displacement sensors accuracy -An optical encoder is used to measure the displacement (see Section ref:ssec:test_joints_test_bench) whose maximum non-linearity is $40\,nm$. +An optical encoder is used to measure the displacement (see Section\nbsp{}ref:ssec:test_joints_test_bench) whose maximum non-linearity is $40\,nm$. As the measured displacement is foreseen to be $0.5\,mm$, the error $\epsilon_d$ due to the encoder non-linearity is negligible $\epsilon_d < 0.01\,\%$. The accuracy of the load cell is specified at $1\,\%$ and therefore, estimation errors of the bending stiffness due to the limited load cell accuracy should be $\epsilon_F < 1\,\%$ ***** Conclusion -The different sources of errors are summarized in Table ref:tab:test_joints_error_budget. +The different sources of errors are summarized in Table\nbsp{}ref:tab:test_joints_error_budget. The most important source of error is the estimation error of the distance between the flexible joint rotation axis and its contact with the force sensor. 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. @@ -10451,19 +10451,19 @@ An overall accuracy of $\approx 5\,\%$ can be expected with this measurement ben **** Mechanical Design <> -As explained in Section ref:ssec:test_joints_meas_principle, the flexible joint's bending stiffness is estimated by applying a known force to the flexible joint's tip and by measuring its deflection at the same point. +As explained in Section\nbsp{}ref:ssec:test_joints_meas_principle, the flexible joint's bending stiffness is estimated by applying a known force to the flexible joint's tip and by measuring its deflection at the same point. The force is applied using a load cell[fn:test_joints_2] such that the applied force to the flexible joint's tip is directly measured. -To control the height and direction of the applied force, a cylinder cut in half is fixed at the tip of the force sensor (pink element in Figure ref:fig:test_joints_bench_side) that initially had a flat surface. +To control the height and direction of the applied force, a cylinder cut in half is fixed at the tip of the force sensor (pink element in Figure\nbsp{}ref:fig:test_joints_bench_side) that initially had a flat surface. Doing so, the contact between the flexible joint cylindrical tip and the force sensor is a point (intersection of two cylinders) at a precise height, and the force is applied in a known direction. To translate the load cell at a constant height, it is fixed to a translation stage[fn:test_joints_3] which is moved by hand. Instead of measuring the displacement directly at the tip of the flexible joint (with a probe or an interferometer for instance), the displacement of the load cell itself is measured. To do so, an encoder[fn:test_joints_4] is used, which measures the motion of a ruler. -This ruler is fixed to the translation stage in line (i.e. at the same height) with the application point to reduce Abbe errors (see Figure ref:fig:test_joints_bench_overview). +This ruler is fixed to the translation stage in line (i.e. at the same height) with the application point to reduce Abbe errors (see Figure\nbsp{}ref:fig:test_joints_bench_overview). The flexible joint can be rotated by $90^o$ in order to measure the bending stiffness in the two directions. -The obtained CAD design of the measurement bench is shown in Figure ref:fig:test_joints_bench_overview while a zoom on the flexible joint with the associated important quantities is shown in Figure ref:fig:test_joints_bench_side. +The obtained CAD design of the measurement bench is shown in Figure\nbsp{}ref:fig:test_joints_bench_overview while a zoom on the flexible joint with the associated important quantities is shown in Figure\nbsp{}ref:fig:test_joints_bench_side. #+name: fig:test_joints_bench #+caption: CAD view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in \subref{fig:test_joints_bench_overview} while a zoom on the flexible joint is shown in \subref{fig:test_joints_bench_side} @@ -10487,8 +10487,8 @@ The obtained CAD design of the measurement bench is shown in Figure ref:fig:test <> **** Introduction :ignore: -A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure ref:fig:test_joints_picture_bench_overview. -A closer view of the force sensor tip is shown in Figure ref:fig:test_joints_picture_bench_zoom. +A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure\nbsp{}ref:fig:test_joints_picture_bench_overview. +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 @@ -10510,10 +10510,10 @@ A closer view of the force sensor tip is shown in Figure ref:fig:test_joints_pic **** Load Cell Calibration In order to estimate the measured errors of the load cell "FC2231", it is compared against another load cell[fn:test_joints_5]. -The two load cells are measured simultaneously while they are pushed against each other (see Figure ref:fig:test_joints_force_sensor_calib_picture). +The two load cells are measured simultaneously while they are pushed against each other (see Figure\nbsp{}ref:fig:test_joints_force_sensor_calib_picture). The contact between the two load cells is well defined as one has a spherical interface and the other has a flat surface. -The measured forces are compared in Figure ref:fig:test_joints_force_sensor_calib_fit. +The measured forces are compared in Figure\nbsp{}ref:fig:test_joints_force_sensor_calib_fit. The gain mismatch between the two load cells is approximately $4\,\%$ which is higher than that specified in the data sheets. However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\,N$ and $5\,N$. @@ -10537,9 +10537,9 @@ However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\, **** Load Cell Stiffness The objective of this measurement is to estimate the stiffness $k_F$ of the force sensor. -To do so, a stiff element (much stiffer than the estimated $k_F \approx 1\,N/\mu m$) is mounted in front of the force sensor, as shown in Figure ref:fig:test_joints_meas_force_sensor_stiffness_picture. +To do so, a stiff element (much stiffer than the estimated $k_F \approx 1\,N/\mu m$) is mounted in front of the force sensor, as shown in Figure\nbsp{}ref:fig:test_joints_meas_force_sensor_stiffness_picture. Then, the force sensor is pushed against this stiff element while the force sensor and the encoder displacement are measured. -The measured displacement as a function of the measured force is shown in Figure ref:fig:test_joints_force_sensor_stiffness_fit. +The measured displacement as a function of the measured force is shown in Figure\nbsp{}ref:fig:test_joints_force_sensor_stiffness_fit. The load cell stiffness can then be estimated by computing a linear fit and is found to be $k_F \approx 0.68\,N/\mu m$. #+name: fig:test_joints_meas_force_sensor_stiffness @@ -10563,12 +10563,12 @@ The load cell stiffness can then be estimated by computing a linear fit and is f **** Bending Stiffness estimation The actual stiffness is now estimated by manually moving the translation stage from a start position where the force sensor is not yet in contact with the flexible joint to a position where the flexible joint is on its mechanical stop. -The measured force and displacement as a function of time are shown in Figure ref:fig:test_joints_meas_bend_time. +The measured force and displacement as a function of time are shown in Figure\nbsp{}ref:fig:test_joints_meas_bend_time. Three regions can be observed: first, the force sensor tip is not in contact with the flexible joint and the measured force is zero; then, the flexible joint deforms linearly; and finally, the flexible joint comes in contact with the mechanical stop. -The angular motion $\theta_{y}$ computed from the displacement $d_x$ is displayed as function of the measured torque $T_{y}$ in Figure ref:fig:test_joints_meas_F_d_lin_fit. +The angular motion $\theta_{y}$ computed from the displacement $d_x$ is displayed as function of the measured torque $T_{y}$ in Figure\nbsp{}ref:fig:test_joints_meas_F_d_lin_fit. The bending stiffness of the flexible joint can be estimated by computing the slope of the curve in the linear regime (red dashed line) and is found to be $k_{R_y} = 4.4\,Nm/\text{rad}$. -The bending stroke can also be estimated as shown in Figure ref:fig:test_joints_meas_F_d_lin_fit and is found to be $\theta_{y,\text{max}} = 20.9\,\text{mrad}$. +The bending stroke can also be estimated as shown in Figure\nbsp{}ref:fig:test_joints_meas_F_d_lin_fit and is found to be $\theta_{y,\text{max}} = 20.9\,\text{mrad}$. #+name: fig:test_joints_meas_example #+caption: Results obtained on the first flexible joint. The measured force and displacement are shown in \subref{fig:test_joints_meas_bend_time}. The estimated angular displacement $\theta_x$ as a function of the estimated applied torque $T_{x}$ is shown in \subref{fig:test_joints_meas_F_d_lin_fit}. The bending stiffness $k_{R_x}$ of the flexible joint can be estimated by computing a best linear fit (red dashed line). @@ -10591,10 +10591,10 @@ The bending stroke can also be estimated as shown in Figure ref:fig:test_joints_ **** Measured flexible joint stiffness The same measurement was performed for all the 16 flexible joints, both in the $x$ and $y$ directions. -The measured angular motion as a function of the applied torque is shown in Figure ref:fig:test_joints_meas_bending_all_raw_data for the 16 flexible joints. +The measured angular motion as a function of the applied torque is shown in Figure\nbsp{}ref:fig:test_joints_meas_bending_all_raw_data for the 16 flexible joints. This gives a first idea of the dispersion of the measured bending stiffnesses (i.e. slope of the linear region) and of the angular stroke. -A histogram of the measured bending stiffnesses is shown in Figure ref:fig:test_joints_bend_stiff_hist. +A histogram of the measured bending stiffnesses is shown in Figure\nbsp{}ref:fig:test_joints_bend_stiff_hist. Most of the bending stiffnesses are between $4.6\,Nm/rad$ and $5.0\,Nm/rad$. #+name: fig:test_joints_meas_bending_results @@ -10622,7 +10622,7 @@ Most of the bending stiffnesses are between $4.6\,Nm/rad$ and $5.0\,Nm/rad$. <> The flexible joints are a key element of the nano-hexapod. -Careful dimensional measurements (Section ref:sec:test_joints_flex_dim_meas) allowed for the early identification of faulty flexible joints. +Careful dimensional measurements (Section\nbsp{}ref:sec:test_joints_flex_dim_meas) allowed for the early identification of faulty flexible joints. This was crucial in preventing potential complications that could have arisen from the installation of faulty joints on the nano-hexapod. A dedicated test bench was developed to asses the bending stiffness of the flexible joints. @@ -10635,7 +10635,7 @@ Furthermore, the data obtained from these measurements have provided the necessa <> *** Introduction :ignore: -The Nano-Hexapod struts (shown in Figure 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]. +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 @@ -10643,16 +10643,16 @@ The Nano-Hexapod struts (shown in Figure ref:fig:test_struts_picture_strut) are [[file:figs/test_struts_picture_strut.jpg]] After the strut elements have been individually characterized (see previous sections), the struts are assembled. -The mounting procedure of the struts is explained in Section ref:sec:test_struts_mounting. +The mounting procedure of the struts is explained in Section\nbsp{}ref:sec:test_struts_mounting. A mounting bench was used to ensure coaxiality between the two ends of the struts. In this way, no angular stroke is lost when mounted to the nano-hexapod. -The flexible modes of the struts were then experimentally measured and compared with a finite element model (Section ref:sec:test_struts_flexible_modes). +The flexible modes of the struts were then experimentally measured and compared with a finite element model (Section\nbsp{}ref:sec:test_struts_flexible_modes). -Dynamic measurements of the strut are performed with the same test bench used to characterize the APA300ML dynamics (Section ref:sec:test_struts_dynamical_meas). -It was found that the dynamics from the acrshort:dac voltage to the displacement measured by the encoder is complex due to the flexible modes of the struts (Section ref:sec:test_struts_flexible_modes). +Dynamic measurements of the strut are performed with the same test bench used to characterize the APA300ML dynamics (Section\nbsp{}ref:sec:test_struts_dynamical_meas). +It was found that the dynamics from the acrshort:dac voltage to the displacement measured by the encoder is complex due to the flexible modes of the struts (Section\nbsp{}ref:sec:test_struts_flexible_modes). -The strut models were then compared with the measured dynamics (Section ref:sec:test_struts_simscape). +The strut models were then compared with the measured dynamics (Section\nbsp{}ref:sec:test_struts_simscape). The model dynamics from the acrshort:dac voltage to the axial motion of the strut (measured by an interferometer) and to the force sensor voltage well match the experimental results. However, this is not the case for the dynamics from acrshort:dac voltage to the encoder displacement. It is found that the complex dynamics is due to a misalignment between the flexible joints and the acrshort:apa. @@ -10667,10 +10667,10 @@ A mounting bench was developed to ensure: - Precise alignment of the APA with the two flexible joints - Reproducible and consistent assembly between all struts -A CAD view of the mounting bench is shown in Figure ref:fig:test_struts_mounting_bench_first_concept. -It consists of a "main frame" (Figure ref:fig:test_struts_mounting_step_0) precisely machined to ensure both correct strut length and strut coaxiality. +A CAD view of the mounting bench is shown in Figure\nbsp{}ref:fig:test_struts_mounting_bench_first_concept. +It consists of a "main frame" (Figure\nbsp{}ref:fig:test_struts_mounting_step_0) precisely machined to ensure both correct strut length and strut coaxiality. The coaxiality is ensured by good flatness (specified at $20\,\mu m$) between surfaces A and B and between surfaces C and D. -Such flatness was checked using a FARO arm[fn:test_struts_1] (see Figure ref:fig:test_struts_check_dimensions_bench) and was found to comply with the requirements. +Such flatness was checked using a FARO arm[fn:test_struts_1] (see Figure\nbsp{}ref:fig:test_struts_check_dimensions_bench) and was found to comply with the requirements. The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes. #+name: fig:test_struts_mounting @@ -10709,9 +10709,9 @@ The strut length (defined by the distance between the rotation points of the two #+end_subfigure #+end_figure -The flexible joints were not directly fixed to the mounting bench but were fixed to a cylindrical "sleeve" shown in Figures ref:fig:test_struts_cylindrical_mounting_part_top and ref:fig:test_struts_cylindrical_mounting_part_bot. +The flexible joints were not directly fixed to the mounting bench but were fixed to a cylindrical "sleeve" shown in Figures\nbsp{}ref:fig:test_struts_cylindrical_mounting_part_top and ref:fig:test_struts_cylindrical_mounting_part_bot. The goal of these "sleeves" is to avoid mechanical stress that could damage the flexible joints during the mounting process. -These "sleeves" have one dowel groove (that are fitted to the dowel holes shown in Figure ref:fig:test_struts_mounting_step_0) that will determine the length of the mounted strut. +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" @@ -10737,16 +10737,16 @@ These "sleeves" have one dowel groove (that are fitted to the dowel holes shown #+end_subfigure #+end_figure -The "sleeves" were mounted to the main element as shown in Figure ref:fig:test_struts_mounting_step_0. +The "sleeves" were mounted to the main element as shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_0. The left sleeve has a thigh fit such that its orientation is fixed (it is roughly aligned horizontally), while the right sleeve has a loose fit such that it can rotate (it will get the same orientation as the fixed one when tightening the screws). -The cylindrical washers and the APA300ML are stacked on top of the flexible joints, as shown in Figure ref:fig:test_struts_mounting_step_2 and screwed together using a torque screwdriver. -A dowel pin is used to laterally align the APA300ML with the flexible joints (see the dowel slot on the flexible joints in Figure ref:fig:test_struts_mounting_joints). +The cylindrical washers and the APA300ML are stacked on top of the flexible joints, as shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_2 and screwed together using a torque screwdriver. +A dowel pin is used to laterally align the APA300ML with the flexible joints (see the dowel slot on the flexible joints in Figure\nbsp{}ref:fig:test_struts_mounting_joints). Two cylindrical washers are used to allow proper mounting even when the two APA interfaces are not parallel. -The encoder and ruler are then fixed to the strut and properly aligned, as shown in Figure ref:fig:test_struts_mounting_step_3. +The encoder and ruler are then fixed to the strut and properly aligned, as shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_3. -Finally, the strut can be disassembled from the mounting bench (Figure ref:fig:test_struts_mounting_step_4). +Finally, the strut can be disassembled from the mounting bench (Figure\nbsp{}ref:fig:test_struts_mounting_step_4). Thanks to this mounting procedure, the coaxiality and length between the two flexible joint's interfaces can be obtained within the desired tolerances. #+name: fig:test_struts_mounting_steps @@ -10787,7 +10787,7 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle A Finite Element Model[fn:test_struts_3] of the struts is developed and is used to estimate the flexible modes. The inertia of the encoder (estimated at $15\,g$) is considered. The two cylindrical interfaces were fixed (boundary conditions), and the first three flexible modes were computed. -The mode shapes are displayed in Figure ref:fig:test_struts_mode_shapes: an "X-bending" mode at 189Hz, a "Y-bending" mode at 285Hz and a "Z-torsion" mode at 400Hz. +The mode shapes are displayed in Figure\nbsp{}ref:fig:test_struts_mode_shapes: an "X-bending" mode at 189Hz, a "Y-bending" mode at 285Hz and a "Z-torsion" mode at 400Hz. #+name: fig:test_struts_mode_shapes #+caption: Spurious resonances of the struts estimated from a Finite Element Model @@ -10814,11 +10814,11 @@ The mode shapes are displayed in Figure ref:fig:test_struts_mode_shapes: an "X-b #+end_figure To experimentally measure these mode shapes, a Laser vibrometer[fn:test_struts_7] was used. -It measures the difference of motion between two beam path (red points in Figure ref:fig:test_struts_meas_modes). +It measures the difference of motion between two beam path (red points in Figure\nbsp{}ref:fig:test_struts_meas_modes). The strut is then excited by an instrumented hammer, and the transfer function from the hammer to the measured rotation is computed. -The setup used to measure the "X-bending" mode is shown in Figure ref:fig:test_struts_meas_x_bending. -The "Y-bending" mode is measured as shown in Figure ref:fig:test_struts_meas_y_bending and the "Z-torsion" measurement setup is shown in Figure ref:fig:test_struts_meas_z_torsion. +The setup used to measure the "X-bending" mode is shown in Figure\nbsp{}ref:fig:test_struts_meas_x_bending. +The "Y-bending" mode is measured as shown in Figure\nbsp{}ref:fig:test_struts_meas_y_bending and the "Z-torsion" measurement setup is shown in Figure\nbsp{}ref:fig:test_struts_meas_z_torsion. These tests were performed with and without the encoder being fixed to the strut. #+name: fig:test_struts_meas_modes @@ -10845,10 +10845,10 @@ These tests were performed with and without the encoder being fixed to the strut #+end_subfigure #+end_figure -The obtained frequency response functions for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure ref:fig:test_struts_spur_res_frf_no_enc when the encoder is not fixed to the strut and in Figure ref:fig:test_struts_spur_res_frf_enc when the encoder is fixed to the strut. +The obtained frequency response functions for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_no_enc when the encoder is not fixed to the strut and in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_enc when the encoder is fixed to the strut. #+name: fig:test_struts_spur_res_frf -#+caption: Measured frequency response functions without the encoder ref:fig:test_struts_spur_res_frf and with the encoder 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} @@ -10865,7 +10865,7 @@ The obtained frequency response functions for the three configurations (X-bendin #+end_subfigure #+end_figure -Table ref:tab:test_struts_spur_mode_freqs summarizes the measured resonance frequencies and the computed ones using the acrfull:fem. +Table\nbsp{}ref:tab:test_struts_spur_mode_freqs summarizes the measured resonance frequencies and the computed ones using the acrfull:fem. The resonance frequencies of the 3 modes are only slightly decreased when the encoder is fixed to the strut. In addition, the computed resonance frequencies from the acrshort:fem are very close to the measured frequencies when the encoder is fixed to the strut. This validates the quality of the acrshort:fem. @@ -10886,8 +10886,8 @@ This validates the quality of the acrshort:fem. In order to measure the dynamics of the strut, the test bench used to measure the APA300ML dynamics is being used again. -The strut mounted on the bench is shown in Figure ref:fig:test_struts_bench_leg_overview -A schematic of the bench and the associated signals are shown in Figure ref:fig:test_struts_bench_schematic. +The strut mounted on the bench is shown in Figure\nbsp{}ref:fig:test_struts_bench_leg_overview +A schematic of the bench and the associated signals are shown in Figure\nbsp{}ref:fig:test_struts_bench_schematic. A fiber interferometer[fn:test_struts_4] is used to measure the motion of the granite (i.e. the axial motion of the strut). #+name: fig:test_struts_bench_leg @@ -10908,14 +10908,14 @@ A fiber interferometer[fn:test_struts_4] is used to measure the motion of the gr #+end_subfigure #+end_figure -First, the effect of the encoder on the measured dynamics is investigated in Section ref:ssec:test_struts_effect_encoder. -The dynamics observed by the encoder and interferometers are compared in Section ref:ssec:test_struts_comp_enc_int. -Finally, all measured struts are compared in terms of dynamics in Section ref:ssec:test_struts_comp_all_struts. +First, the effect of the encoder on the measured dynamics is investigated in Section\nbsp{}ref:ssec:test_struts_effect_encoder. +The dynamics observed by the encoder and interferometers are compared in Section\nbsp{}ref:ssec:test_struts_comp_enc_int. +Finally, all measured struts are compared in terms of dynamics in Section\nbsp{}ref:ssec:test_struts_comp_all_struts. **** Effect of the Encoder on the measured dynamics <> -System identification was performed without the encoder being fixed to the strut (Figure ref:fig:test_struts_bench_leg_front) and with one encoder being fixed to the strut (Figure ref:fig:test_struts_bench_leg_coder). +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} @@ -10935,10 +10935,10 @@ System identification was performed without the encoder being fixed to the strut #+end_subfigure #+end_figure -The obtained frequency response functions are compared in Figure ref:fig:test_struts_effect_encoder. -It was found that the encoder had very little effect on the transfer function from excitation voltage $u$ to the axial motion of the strut $d_a$ as measured by the interferometer (Figure ref:fig:test_struts_effect_encoder_int). +The obtained frequency response functions are compared in Figure\nbsp{}ref:fig:test_struts_effect_encoder. +It was found that the encoder had very little effect on the transfer function from excitation voltage $u$ to the axial motion of the strut $d_a$ as measured by the interferometer (Figure\nbsp{}ref:fig:test_struts_effect_encoder_int). This means that the axial motion of the strut is unaffected by the presence of the encoder. -Similarly, it has little effect on the transfer function from $u$ to the sensor stack voltage $V_s$ (Figure ref:fig:test_struts_effect_encoder_iff). +Similarly, it has little effect on the transfer function from $u$ to the sensor stack voltage $V_s$ (Figure\nbsp{}ref:fig:test_struts_effect_encoder_iff). 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 @@ -10968,10 +10968,10 @@ This means that the encoder should have little effect on the effectiveness of th **** Comparison of the encoder and interferometer <> -The dynamics measured by the encoder (i.e. $d_e/u$) and interferometers (i.e. $d_a/u$) are compared in Figure ref:fig:test_struts_comp_enc_int. -The dynamics from the excitation voltage $u$ to the displacement measured by the encoder $d_e$ presents a behavior that is much more complex than the dynamics of the displacement measured by the interferometer (comparison made in Figure ref:fig:test_struts_comp_enc_int). +The dynamics measured by the encoder (i.e. $d_e/u$) and interferometers (i.e. $d_a/u$) are compared in Figure\nbsp{}ref:fig:test_struts_comp_enc_int. +The dynamics from the excitation voltage $u$ to the displacement measured by the encoder $d_e$ presents a behavior that is much more complex than the dynamics of the displacement measured by the interferometer (comparison made in Figure\nbsp{}ref:fig:test_struts_comp_enc_int). Three additional resonance frequencies can be observed at 197Hz, 290Hz and 376Hz. -These resonance frequencies match the frequencies of the flexible modes studied in Section ref:sec:test_struts_flexible_modes. +These resonance frequencies match the frequencies of the flexible modes studied in Section\nbsp{}ref:sec:test_struts_flexible_modes. The good news is that these resonances are not impacting the axial motion of the strut (which is what is important for the hexapod positioning). However, these resonances make the use of an encoder fixed to the strut difficult from a control perspective. @@ -10980,7 +10980,7 @@ However, these resonances make the use of an encoder fixed to the strut difficul <> The dynamics of all the mounted struts (only 5 at the time of the experiment) were then measured on the same test bench. -The obtained dynamics from $u$ to $d_a$ are compared in Figure ref:fig:test_struts_comp_interf_plants while is dynamics from $u$ to $V_s$ are compared in Figure ref:fig:test_struts_comp_iff_plants. +The obtained dynamics from $u$ to $d_a$ are compared in Figure\nbsp{}ref:fig:test_struts_comp_interf_plants while is dynamics from $u$ to $V_s$ are compared in Figure\nbsp{}ref:fig:test_struts_comp_iff_plants. A very good match can be observed between the struts. #+name: fig:test_struts_comp_plants @@ -11007,7 +11007,7 @@ A very good match can be observed between the struts. #+end_subfigure #+end_figure -The same comparison is made for the transfer function from $u$ to $d_e$ (encoder output) in Figure ref:fig:test_struts_comp_enc_plants. +The same comparison is made for the transfer function from $u$ to $d_e$ (encoder output) in Figure\nbsp{}ref:fig:test_struts_comp_enc_plants. In this study, large dynamics differences were observed between the 5 struts. Although the same resonance frequencies were seen for all of the struts (95Hz, 200Hz, 300Hz and 400Hz), the amplitude of the peaks were not the same. In addition, the location or even presence of complex conjugate zeros changes from one strut to another. @@ -11017,13 +11017,13 @@ The reason for this variability will be studied in the next section thanks to th <> **** Introduction :ignore: -The multi-body model of the strut was included in the multi-body model of the test bench (see Figure ref:fig:test_struts_simscape_model). -The obtained model was first used to compare the measured FRF with the existing model (Section ref:ssec:test_struts_comp_model). +The multi-body model of the strut was included in the multi-body model of the test bench (see Figure\nbsp{}ref:fig:test_struts_simscape_model). +The obtained model was first used to compare the measured FRF with the existing model (Section\nbsp{}ref:ssec:test_struts_comp_model). -Using a flexible APA model (extracted from a acrshort:fem), the effect of a misalignment of the APA with respect to flexible joints is studied (Section ref:ssec:test_struts_effect_misalignment). +Using a flexible APA model (extracted from a acrshort:fem), the effect of a misalignment of the APA with respect to flexible joints is studied (Section\nbsp{}ref:ssec:test_struts_effect_misalignment). It was found that misalignment has a large impact on the dynamics from $u$ to $d_e$. -This misalignment is estimated and measured in Section ref:ssec:test_struts_meas_misalignment. -The struts were then disassembled and reassemble a second time to optimize alignment (Section ref:sec:test_struts_meas_all_aligned_struts). +This misalignment is estimated and measured in Section\nbsp{}ref:ssec:test_struts_meas_misalignment. +The struts were then disassembled and reassemble a second time to optimize alignment (Section\nbsp{}ref:sec:test_struts_meas_all_aligned_struts). #+name: fig:test_struts_simscape_model #+caption: Screenshot of the multi-body model of the strut fixed to the bench @@ -11036,9 +11036,9 @@ The struts were then disassembled and reassemble a second time to optimize align Two models of the APA300ML are used here: a simple two-degrees-of-freedom model and a model using a super-element extracted from a acrlong:fem. These two models of the APA300ML were tuned to best match the measured frequency response functions of the APA alone. The flexible joints were modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness). -These two models are compared with the measured frequency responses in Figure ref:fig:test_struts_comp_frf_flexible_model. +These two models are compared with the measured frequency responses in Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model. -The model dynamics from DAC voltage $u$ to the axial motion of the strut $d_a$ (Figure ref:fig:test_struts_comp_frf_flexible_model_int) and from DAC voltage $u$ to the force sensor voltage $V_s$ (Figure ref:fig:test_struts_comp_frf_flexible_model_iff) are well matching the experimental identification. +The model dynamics from DAC voltage $u$ to the axial motion of the strut $d_a$ (Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model_int) and from DAC voltage $u$ to the force sensor voltage $V_s$ (Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model_iff) are well matching the experimental identification. However, the transfer function from $u$ to encoder displacement $d_e$ are not well matching for both models. For the 2DoF model, this is normal because the resonances affecting the dynamics are not modelled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis). @@ -11071,10 +11071,10 @@ For the flexible model, it will be shown in the next section that by adding some **** Effect of strut misalignment <> -As shown in Figure ref:fig:test_struts_comp_enc_plants, the identified dynamics from DAC voltage $u$ to encoder measured displacement $d_e$ are very different from one strut to the other. +As shown in Figure\nbsp{}ref:fig:test_struts_comp_enc_plants, the identified dynamics from DAC voltage $u$ to encoder measured displacement $d_e$ are very different from one strut to the other. In this section, it is investigated whether poor alignment of the strut (flexible joints with respect to the APA) can explain such dynamics. -For instance, consider Figure ref:fig:test_struts_misalign_schematic where there is a misalignment in the $y$ direction between the two flexible joints (well aligned thanks to the mounting procedure in Section ref:sec:test_struts_mounting) and the APA300ML. -In this case, the "x-bending" mode at 200Hz (see Figure ref:fig:test_struts_meas_x_bending) can be expected to have greater impact on the dynamics from the actuator to the encoder. +For instance, consider Figure\nbsp{}ref:fig:test_struts_misalign_schematic where there is a misalignment in the $y$ direction between the two flexible joints (well aligned thanks to the mounting procedure in Section\nbsp{}ref:sec:test_struts_mounting) and the APA300ML. +In this case, the "x-bending" mode at 200Hz (see Figure\nbsp{}ref:fig:test_struts_meas_x_bending) can be expected to have greater impact on the dynamics from the actuator to the encoder. #+name: fig:test_struts_misalign_schematic #+caption: Mis-alignement between the joints and the APA @@ -11082,19 +11082,19 @@ In this case, the "x-bending" mode at 200Hz (see Figure ref:fig:test_struts_meas [[file:figs/test_struts_misalign_schematic.png]] To verify this assumption, the dynamics from the output DAC voltage $u$ to the measured displacement by the encoder $d_e$ is computed using the flexible APA model for several misalignments in the $y$ direction. -The obtained dynamics are shown in Figure ref:fig:test_struts_effect_misalignment_y. +The obtained dynamics are shown in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y. The alignment of the APA with the flexible joints has a large influence on the dynamics from actuator voltage to the measured displacement by the encoder. The misalignment in the $y$ direction mostly influences: -- the presence of the flexible mode at 200Hz (see mode shape in Figure ref:fig:test_struts_mode_shapes_1) +- the presence of the flexible mode at 200Hz (see mode shape in Figure\nbsp{}ref:fig:test_struts_mode_shapes_1) - the location of the complex conjugate zero between the first two resonances: - if $d_{y} < 0$: there is no zero between the two resonances and possibly not even between the second and third resonances - if $d_{y} > 0$: there is a complex conjugate zero between the first two resonances - the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero) The same can be done for misalignments in the $x$ direction. -The obtained dynamics (Figure ref:fig:test_struts_effect_misalignment_x) are showing that misalignment in the $x$ direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure ref:fig:test_struts_mode_shapes_2). +The obtained dynamics (Figure\nbsp{}ref:fig:test_struts_effect_misalignment_x) are showing that misalignment in the $x$ direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure\nbsp{}ref:fig:test_struts_mode_shapes_2). -A comparison of the experimental frequency response functions in Figure ref:fig:test_struts_comp_enc_plants with the model dynamics for several $y$ misalignments in Figure ref:fig:test_struts_effect_misalignment_y indicates a clear similarity. +A comparison of the experimental frequency response functions in Figure\nbsp{}ref:fig:test_struts_comp_enc_plants with the model dynamics for several $y$ misalignments in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y indicates a clear similarity. This similarity suggests that the identified differences in dynamics are caused by misalignment. #+name: fig:test_struts_effect_misalignment @@ -11118,18 +11118,18 @@ This similarity suggests that the identified differences in dynamics are caused **** Measured strut misalignment <> -During the initial mounting of the struts, as presented in Section ref:sec:test_struts_mounting, the positioning pins that were used to position the APA with respect to the flexible joints in the $y$ directions were not used (not received at the time). +During the initial mounting of the struts, as presented in Section\nbsp{}ref:sec:test_struts_mounting, the positioning pins that were used to position the APA with respect to the flexible joints in the $y$ directions were not used (not received at the time). Therefore, large $y$ misalignments are expected. To estimate the misalignments between the two flexible joints and the APA: -- the struts were fixed horizontally on the mounting bench, as shown in Figure ref:fig:test_struts_mounting_step_3 but without the encoder +- the struts were fixed horizontally on the mounting bench, as shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_3 but without the encoder - using a length gauge[fn:test_struts_2], the height difference between the flexible joints surface and the APA shell surface was measured for both the top and bottom joints and for both sides - as the thickness of the flexible joint is $21\,mm$ and the thickness of the APA shell is $20\,mm$, $0.5\,mm$ of height difference should be measured if the two are perfectly aligned -Large variations in the $y$ misalignment are found from one strut to the other (results are summarized in Table ref:tab:test_struts_meas_y_misalignment). +Large variations in the $y$ misalignment are found from one strut to the other (results are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). To check the validity of the measurement, it can be verified that the sum of the measured thickness difference on each side is $1\,mm$ (equal to the thickness difference between the flexible joint and the APA). -Thickness differences for all the struts were found to be between $0.94\,mm$ and $1.00\,mm$ which indicate low errors compared to the misalignments found in Table ref:tab:test_struts_meas_y_misalignment. +Thickness differences for all the struts were found to be between $0.94\,mm$ and $1.00\,mm$ which indicate low errors compared to the misalignments found in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment. #+name: tab:test_struts_meas_y_misalignment #+caption: Measured $y$ misalignment at the top and bottom of the APA. Measurements are in $mm$ @@ -11143,14 +11143,14 @@ Thickness differences for all the struts were found to be between $0.94\,mm$ and | 4 | -0.01 | 0.54 | | 5 | 0.15 | 0.02 | -By using the measured $y$ misalignment in the model with the flexible APA model, the model dynamics from $u$ to $d_e$ is closer to the measured dynamics, as shown in Figure ref:fig:test_struts_comp_dy_tuned_model_frf_enc. -A better match in the dynamics can be obtained by fine-tuning both the $x$ and $y$ misalignments (yellow curves in Figure ref:fig:test_struts_comp_dy_tuned_model_frf_enc). +By using the measured $y$ misalignment in the model with the flexible APA model, the model dynamics from $u$ to $d_e$ is closer to the measured dynamics, as shown in Figure\nbsp{}ref:fig:test_struts_comp_dy_tuned_model_frf_enc. +A better match in the dynamics can be obtained by fine-tuning both the $x$ and $y$ misalignments (yellow curves in Figure\nbsp{}ref:fig:test_struts_comp_dy_tuned_model_frf_enc). This confirms that misalignment between the APA and the strut axis (determined by the two flexible joints) is critical and inducing large variations in the dynamics from DAC voltage $u$ to encoder measured displacement $d_e$. If encoders are fixed to the struts, the APA and flexible joints must be precisely aligned when mounting the struts. In the next section, the struts are re-assembled with a "positioning pin" to better align the APA with the flexible joints. -With a better alignment, the amplitude of the spurious resonances is expected to decrease, as shown in Figure ref:fig:test_struts_effect_misalignment_y. +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 @@ -11163,7 +11163,7 @@ After receiving the positioning pins, the struts were mounted again with the pos This should improve the alignment of the APA with the two flexible joints. The alignment is then estimated using a length gauge, as described in the previous sections. -Measured $y$ alignments are summarized in Table ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\mu m$ for all the struts, which is much better than before (see Table ref:tab:test_struts_meas_y_misalignment). +Measured $y$ alignments are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\mu m$ for all the struts, which is much better than before (see Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). #+name: tab:test_struts_meas_y_misalignment_with_pin #+caption: Measured $y$ misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in $mm$. @@ -11178,8 +11178,8 @@ Measured $y$ alignments are summarized in Table ref:tab:test_struts_meas_y_misal | 5 | 0.0 | 0.0 | | 6 | -0.005 | 0.055 | -The dynamics of the re-aligned struts were then measured on the same test bench (Figure ref:fig:test_struts_bench_leg). -A comparison of the initial strut dynamics and the dynamics of the re-aligned struts (i.e. with the positioning pin) is presented in Figure ref:fig:test_struts_comp_enc_frf_realign. +The dynamics of the re-aligned struts were then measured on the same test bench (Figure\nbsp{}ref:fig:test_struts_bench_leg). +A comparison of the initial strut dynamics and the dynamics of the re-aligned struts (i.e. with the positioning pin) is presented in Figure\nbsp{}ref:fig:test_struts_comp_enc_frf_realign. Even though the struts are now much better aligned, not much improvement can be observed. The dynamics of the six aligned struts were also quite different from one another. @@ -11210,18 +11210,18 @@ Therefore, the encoders will be fixed directly to the nano-hexapod plates rather *** Introduction :ignore: Prior to the nano-hexapod assembly, all the struts were mounted and individually characterized. -In Section ref:sec:test_nhexa_mounting, the assembly procedure of the nano-hexapod is presented. +In Section\nbsp{}ref:sec:test_nhexa_mounting, the assembly procedure of the nano-hexapod is presented. To identify the dynamics of the nano-hexapod, a special suspended table was developed, which consisted of a stiff "optical breadboard" suspended on top of four soft springs. -The Nano-Hexapod was then mounted on top of the suspended table such that its dynamics is not affected by complex dynamics except from the suspension modes of the table that can be well characterized and modeled (Section ref:sec:test_nhexa_table). +The Nano-Hexapod was then mounted on top of the suspended table such that its dynamics is not affected by complex dynamics except from the suspension modes of the table that can be well characterized and modeled (Section\nbsp{}ref:sec:test_nhexa_table). -The obtained nano-hexapod dynamics is analyzed in Section ref:sec:test_nhexa_dynamics, and compared with the multi-body model in Section ref:sec:test_nhexa_model. +The obtained nano-hexapod dynamics is analyzed in Section\nbsp{}ref:sec:test_nhexa_dynamics, and compared with the multi-body model in Section\nbsp{}ref:sec:test_nhexa_model. *** Nano-Hexapod Assembly Procedure <> The assembly of the nano-hexapod is critical for both avoiding additional stress in the flexible joints (that would result in a loss of stroke) and for precisely determining the Jacobian matrix. -The goal was to fix the six struts to the two nano-hexapod plates (shown in Figure ref:fig:test_nhexa_nano_hexapod_plates) while the two plates were parallel and aligned vertically so that all the flexible joints did not experience any stress. -To do so, a precisely machined mounting tool (Figure ref:fig:test_nhexa_center_part_hexapod_mounting) is used to position the two nano-hexapod plates during the assembly procedure. +The goal was to fix the six struts to the two nano-hexapod plates (shown in Figure\nbsp{}ref:fig:test_nhexa_nano_hexapod_plates) while the two plates were parallel and aligned vertically so that all the flexible joints did not experience any stress. +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} @@ -11243,9 +11243,9 @@ To do so, a precisely machined mounting tool (Figure ref:fig:test_nhexa_center_p #+end_subfigure #+end_figure -The mechanical tolerances of the received plates were checked using a FARO arm[fn:test_nhexa_1] (Figure ref:fig:test_nhexa_plates_tolerances) and were found to comply with the requirements[fn:test_nhexa_2]. +The mechanical tolerances of the received plates were checked using a FARO arm[fn:test_nhexa_1] (Figure\nbsp{}ref:fig:test_nhexa_plates_tolerances) and were found to comply with the requirements[fn:test_nhexa_2]. The same was done for the mounting tool[fn:test_nhexa_3]. -The two plates were then fixed to the mounting tool, as shown in Figure ref:fig:test_nhexa_mounting_tool_hexapod_top_view. +The two plates were then fixed to the mounting tool, as shown in Figure\nbsp{}ref:fig:test_nhexa_mounting_tool_hexapod_top_view. The main goal of this "mounting tool" is to position the flexible joint interfaces (the "V" shapes) of both plates so that a cylinder can rest on the 4 flat interfaces at the same time. #+name: fig:test_nhexa_dimensional_check @@ -11270,7 +11270,7 @@ The main goal of this "mounting tool" is to position the flexible joint interfac The quality of the positioning can be estimated by measuring the "straightness" of the top and bottom "V" interfaces. This corresponds to the diameter of the smallest cylinder which contains all points along the measured axis. -This was again done using the FARO arm, and the results for all six struts are summarized in Table ref:tab:measured_straightness. +This was again done using the FARO arm, and the results for all six struts are summarized in Table\nbsp{}ref:tab:measured_straightness. The straightness was found to be better than $15\,\mu m$ for all struts[fn:test_nhexa_4], which is sufficiently good to not induce significant stress of the flexible joint during assembly. #+name: tab:measured_straightness @@ -11286,7 +11286,7 @@ The straightness was found to be better than $15\,\mu m$ for all struts[fn:test_ | 5 | $7\, \mu m$ | $5\, \mu m$ | | 6 | $6\, \mu m$ | $7\, \mu m$ | -The encoder rulers and heads were then fixed to the top and bottom plates, respectively (Figure ref:fig:test_nhexa_mount_encoder), and the encoder heads were aligned to maximize the received contrast. +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} @@ -11312,7 +11312,7 @@ The six struts were then fixed to the bottom and top plates one by one. First, the top flexible joint is fixed so that its flat reference surface is in contact with the top plate. This step precisely determines the position of the flexible joint with respect to the top plate. The bottom flexible joint is then fixed. -After mounting all six struts, the mounting tool (Figure ref:fig:test_nhexa_center_part_hexapod_mounting) can be disassembled, and the nano-hexapod as shown in Figure ref:fig:test_nhexa_nano_hexapod_mounted is fully assembled. +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 @@ -11333,9 +11333,9 @@ Another key advantage is that the suspension modes of the table can be easily re Therefore, the measured dynamics of the nano-hexapod on top of the suspended table can be compared to a multi-body model representing the same experimental conditions. The model of the Nano-Hexapod can thus be precisely tuned to match the measured dynamics. -The developed suspended table is described in Section ref:ssec:test_nhexa_table_setup. -The modal analysis of the table is done in ref:ssec:test_nhexa_table_identification. -Finally, the multi-body model representing the suspended table was tuned to match the measured modes (Section ref:ssec:test_nhexa_table_model). +The developed suspended table is described in Section\nbsp{}ref:ssec:test_nhexa_table_setup. +The modal analysis of the table is done in\nbsp{}ref:ssec:test_nhexa_table_identification. +Finally, the multi-body model representing the suspended table was tuned to match the measured modes (Section\nbsp{}ref:ssec:test_nhexa_table_model). **** Experimental Setup <> @@ -11343,7 +11343,7 @@ Finally, the multi-body model representing the suspended table was tuned to matc The design of the suspended table is quite straightforward. First, an optical table with high frequency flexible mode was selected[fn:test_nhexa_5]. Then, four springs[fn:test_nhexa_6] were selected with low spring rate such that the suspension modes are below 10Hz. -Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure ref:fig:test_nhexa_suspended_table_cad). +Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure\nbsp{}ref:fig:test_nhexa_suspended_table_cad). #+name: fig:test_nhexa_suspended_table_cad #+caption: CAD View of the vibration table. The purple cylinders are representing the soft springs. @@ -11354,9 +11354,9 @@ Finally, some interface elements were designed, and mechanical lateral mechanica <> In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers[fn:test_nhexa_7] were fixed to the breadboard. -Using an instrumented hammer, the first 9 modes could be identified and are summarized in Table ref:tab:test_nhexa_suspended_table_modes. +Using an instrumented hammer, the first 9 modes could be identified and are summarized in Table\nbsp{}ref:tab:test_nhexa_suspended_table_modes. The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below 10Hz. -The next modes are the flexible modes of the breadboard as shown in Figure ref:fig:test_nhexa_table_flexible_modes, and are located above 700Hz. +The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{}ref:fig:test_nhexa_table_flexible_modes, and are located above 700Hz. #+attr_latex: :options [t]{0.45\textwidth} #+begin_minipage @@ -11420,7 +11420,7 @@ The model order is 12, which corresponds to the 6 suspension modes. The inertia properties of the parts were determined from the geometry and material densities. The stiffness of the springs was initially set from the datasheet nominal value of $17.8\,N/mm$ and then reduced down to $14\,N/mm$ to better match the measured suspension modes. The stiffness of the springs in the horizontal plane is set at $0.5\,N/mm$. -The obtained suspension modes of the multi-body model are compared with the measured modes in Table ref:tab:test_nhexa_suspended_table_simscape_modes. +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 @@ -11436,7 +11436,7 @@ The obtained suspension modes of the multi-body model are compared with the meas **** Introduction :ignore: -The Nano-Hexapod was then mounted on top of the suspended table, as shown in Figure ref:fig:test_nhexa_hexa_suspended_table. +The Nano-Hexapod was then mounted on top of the suspended table, as shown in Figure\nbsp{}ref:fig:test_nhexa_hexa_suspended_table. All instrumentation (Speedgoat with ADC, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) were configured and connected to the nano-hexapod using many cables. #+name: fig:test_nhexa_hexa_suspended_table @@ -11444,12 +11444,12 @@ All instrumentation (Speedgoat with ADC, DAC, piezoelectric voltage amplifiers a #+attr_latex: :width 0.7\linewidth [[file:figs/test_nhexa_hexa_suspended_table.jpg]] -A modal analysis of the nano-hexapod is first performed in Section ref:ssec:test_nhexa_enc_struts_modal_analysis. +A modal analysis of the nano-hexapod is first performed in Section\nbsp{}ref:ssec:test_nhexa_enc_struts_modal_analysis. The results of the modal analysis will be useful to better understand the measured dynamics from actuators to sensors. -A block diagram of the (open-loop) system is shown in Figure ref:fig:test_nhexa_nano_hexapod_signals. -The frequency response functions from controlled signals $\mathbf{u}$ to the force sensors voltages $\mathbf{V}_s$ and to the encoders measured displacements $\mathbf{d}_e$ are experimentally identified in Section ref:ssec:test_nhexa_identification. -The effect of the payload mass on the dynamics is discussed in Section ref:ssec:test_nhexa_added_mass. +A block diagram of the (open-loop) system is shown in Figure\nbsp{}ref:fig:test_nhexa_nano_hexapod_signals. +The frequency response functions from controlled signals $\mathbf{u}$ to the force sensors voltages $\mathbf{V}_s$ and to the encoders measured displacements $\mathbf{d}_e$ are experimentally identified in Section\nbsp{}ref:ssec:test_nhexa_identification. +The effect of the payload mass on the dynamics is discussed in Section\nbsp{}ref:ssec:test_nhexa_added_mass. #+name: fig:test_nhexa_nano_hexapod_signals #+caption: Block diagram of the studied system. The command signal generated by the speedgoat is $\mathbf{u}$, and the measured dignals are $\mathbf{d}_{e}$ and $\mathbf{V}_s$. Units are indicated in square brackets. @@ -11460,7 +11460,7 @@ The effect of the payload mass on the dynamics is discussed in Section ref:ssec: <> To facilitate the future analysis of the measured plant dynamics, a basic modal analysis of the nano-hexapod is performed. -Five 3-axis accelerometers were fixed on the top platform of the nano-hexapod (Figure ref:fig:test_nhexa_modal_analysis) and the top platform was excited using an instrumented hammer. +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 @@ -11468,8 +11468,8 @@ Five 3-axis accelerometers were fixed on the top platform of the nano-hexapod (F [[file:figs/test_nhexa_modal_analysis.jpg]] Between 100Hz and 200Hz, 6 suspension modes (i.e. rigid body modes of the top platform) were identified. -At around 700Hz, two flexible modes of the top plate were observed (see Figure ref:fig:test_nhexa_hexa_flexible_modes). -These modes are summarized in Table ref:tab:test_nhexa_hexa_modal_modes_list. +At around 700Hz, two flexible modes of the top plate were observed (see Figure\nbsp{}ref:fig:test_nhexa_hexa_flexible_modes). +These modes are summarized in Table\nbsp{}ref:tab:test_nhexa_hexa_modal_modes_list. #+name: tab:test_nhexa_hexa_modal_modes_list #+caption: Description of the identified modes of the Nano-Hexapod @@ -11509,15 +11509,15 @@ These modes are summarized in Table ref:tab:test_nhexa_hexa_modal_modes_list. The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) to the six measured displacement by the encoders ($d_{e1}$ to $d_{e6}$) and to the six force sensors ($V_{s1}$ to $V_{s6}$) were identified by generating low-pass filtered white noise for each command signal, one by one. -The $6 \times 6$ FRF matrix from $\mathbf{u}$ ot $\mathbf{d}_e$ is shown in Figure ref:fig:test_nhexa_identified_frf_de. +The $6 \times 6$ FRF matrix from $\mathbf{u}$ ot $\mathbf{d}_e$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de. The diagonal terms are displayed using colored lines, and all the 30 off-diagonal terms are displayed by gray lines. All six diagonal terms are well superimposed up to at least $1\,kHz$, indicating good manufacturing and mounting uniformity. Below the first suspension mode, good decoupling can be observed (the amplitude of all off-diagonal terms are $\approx 20$ times smaller than the diagonal terms), indicating the correct assembly of all parts. From 10Hz up to 1kHz, around 10 resonance frequencies can be observed. -The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section ref:ssec:test_nhexa_enc_struts_modal_analysis. -Three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section ref:ssec:test_nhexa_comp_model_coupling). +The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section\nbsp{}ref:ssec:test_nhexa_enc_struts_modal_analysis. +Three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_coupling). Except for the mode at 237Hz, their impact on the dynamics is small. The two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform. Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. @@ -11530,7 +11530,7 @@ This would not have occurred if the encoders were fixed to the struts. #+attr_latex: :width \linewidth [[file:figs/test_nhexa_identified_frf_de.png]] -Similarly, the $6 \times 6$ FRF matrix from $\mathbf{u}$ to $\mathbf{V}_s$ is shown in Figure ref:fig:test_nhexa_identified_frf_Vs. +Similarly, the $6 \times 6$ FRF matrix from $\mathbf{u}$ to $\mathbf{V}_s$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs. Alternating poles and zeros can be observed up to at least 2kHz, which is a necessary characteristics for applying decentralized IFF. Similar to what was observed for the encoder outputs, all the "diagonal" terms are well superimposed, indicating that the same controller can be applied to all the struts. The first flexible mode of the struts as 235Hz has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF. @@ -11546,7 +11546,7 @@ The first flexible mode of the struts as 235Hz has large amplitude, and therefor One major challenge for controlling the NASS is the wanted robustness to a variation of payload mass; therefore, it is necessary to understand how the dynamics of the nano-hexapod changes with a change in payload mass. To study how the dynamics changes with the payload mass, up to three "cylindrical masses" of $13\,kg$ each can be added for a total of $\approx 40\,kg$. -These three cylindrical masses on top of the nano-hexapod are shown in Figure ref:fig:test_nhexa_table_mass_3. +These three cylindrical masses on top of the nano-hexapod are shown in Figure\nbsp{}ref:fig:test_nhexa_table_mass_3. #+name: fig:test_nhexa_table_mass_3 #+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,kg$ @@ -11554,19 +11554,19 @@ These three cylindrical masses on top of the nano-hexapod are shown in Figure re #+attr_latex: :width 0.8\linewidth [[file:figs/test_nhexa_table_mass_3.jpg]] -The obtained frequency response functions from actuator signal $u_i$ to the associated encoder $d_{ei}$ for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure ref:fig:test_nhexa_identified_frf_de_masses. +The obtained frequency response functions from actuator signal $u_i$ to the associated encoder $d_{ei}$ for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de_masses. As expected, the frequency of the suspension modes decreased with increasing payload mass. The low frequency gain does not change because it is linked to the stiffness property of the nano-hexapod and not to its mass property. The frequencies of the two flexible modes of the top plate first decreased significantly when the first mass was added (from $\approx 700\,Hz$ to $\approx 400\,Hz$). This is because the added mass is composed of two half cylinders that are not fixed together. Therefore, it adds a lot of mass to the top plate without increasing stiffness in one direction. -When more than one "mass layer" is added, the half cylinders are added at some angles such that rigidity is added in all directions (see how the three mass "layers" are positioned in Figure ref:fig:test_nhexa_table_mass_3). +When more than one "mass layer" is added, the half cylinders are added at some angles such that rigidity is added in all directions (see how the three mass "layers" are positioned in Figure\nbsp{}ref:fig:test_nhexa_table_mass_3). In this case, the frequency of these flexible modes is increased. In practice, the payload should be one solid body, and no decrease in the frequency of this flexible mode should be observed. The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass increased. -The measured FRFs from $u_i$ to $V_{si}$ are shown in Figure ref:fig:test_nhexa_identified_frf_Vs_masses. +The measured FRFs from $u_i$ to $V_{si}$ are shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs_masses. For all tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust manner. #+name: fig:test_nhexa_identified_frf_masses @@ -11592,8 +11592,8 @@ For all tested payloads, the measured FRF always have alternating poles and zero **** Introduction :ignore: -In this section, the dynamics measured in Section ref:sec:test_nhexa_dynamics is compared with those estimated from the multi-body model. -The nano-hexapod multi-body model was therefore added on top of the vibration table multi-body model, as shown in Figure ref:fig:test_nhexa_hexa_simscape. +In this section, the dynamics measured in Section\nbsp{}ref:sec:test_nhexa_dynamics is compared with those estimated from the multi-body model. +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 @@ -11601,10 +11601,10 @@ The nano-hexapod multi-body model was therefore added on top of the vibration ta [[file:figs/test_nhexa_hexa_simscape.png]] The model should exhibit certain characteristics that are verified in this section. -First, it should match the measured system dynamics from actuators to sensors presented in Section ref:sec:test_nhexa_dynamics. -Both the "direct" terms (Section ref:ssec:test_nhexa_comp_model) and "coupling" terms (Section ref:ssec:test_nhexa_comp_model_coupling) of the multi-body model are compared with the measured dynamics. +First, it should match the measured system dynamics from actuators to sensors presented in Section\nbsp{}ref:sec:test_nhexa_dynamics. +Both the "direct" terms (Section\nbsp{}ref:ssec:test_nhexa_comp_model) and "coupling" terms (Section\nbsp{}ref:ssec:test_nhexa_comp_model_coupling) of the multi-body model are compared with the measured dynamics. Second, it should also represents how the system dynamics changes when a payload is fixed to the top platform. -This is checked in Section ref:ssec:test_nhexa_comp_model_masses. +This is checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_masses. **** Nano-Hexapod model dynamics <> @@ -11615,7 +11615,7 @@ The parameters of the APA model were determined from the test bench of the APA. The $6 \times 6$ transfer function matrices from $\mathbf{u}$ to $\mathbf{d}_e$ and from $\mathbf{u}$ to $\mathbf{V}_s$ are then extracted from the multi-body model. First, is it evaluated how well the models matches the "direct" terms of the measured FRF matrix. -To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_diag. +To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag. It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modeled. The three resonances that were attributed to "internal" flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable because the APAs are here modeled as a simple uniaxial 2-DoF system. At higher frequencies, no resonances can be observed in the model, as the top plate and the encoder supports are modeled as rigid bodies. @@ -11642,7 +11642,7 @@ At higher frequencies, no resonances can be observed in the model, as the top pl <> Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of MIMO systems. -Instead of comparing the full 36 elements of the $6 \times 6$ FFR matrix from $\mathbf{u}$ to $\mathbf{d}_e$, only the first "column" is compared (Figure ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$. +Instead of comparing the full 36 elements of the $6 \times 6$ FFR matrix from $\mathbf{u}$ to $\mathbf{d}_e$, only the first "column" is compared (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$. It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at 237Hz. Similar results are observed for all other coupling terms and for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$. @@ -11651,7 +11651,7 @@ Similar results are observed for all other coupling terms and for the transfer f [[file:figs/test_nhexa_comp_simscape_de_all.png]] The APA300ML was then modeled with a /super-element/ extracted from a FE-software. -The obtained transfer functions from $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$ are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_de_all_flex. +The obtained transfer functions from $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$ are compared with the measured FRF in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all_flex. While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modeled. Even the mode 395Hz can be observed in the model. Therefore, if the modes of the struts are to be modeled, the /super-element/ of the APA300ML can be used at the cost of obtaining a much higher order model. @@ -11664,9 +11664,9 @@ Therefore, if the modes of the struts are to be modeled, the /super-element/ of <> Another important characteristic of the model is that it should represents the dynamics of the system well for all considered payloads. -The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure ref:fig:test_nhexa_comp_simscape_diag_masses. +The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag_masses. The observed shift of the suspension modes to lower frequencies with increased payload mass is well represented by the multi-body model. -The complex conjugate zeros also well match the experiments both for the encoder outputs (Figure ref:fig:test_nhexa_comp_simscape_de_diag_masses) and the force sensor outputs (Figure ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). +The complex conjugate zeros also well match the experiments both for the encoder outputs (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_diag_masses) and the force sensor outputs (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). Note that the model displays smaller damping than that observed experimentally for high values of the payload mass. One option could be to tune the damping as a function of the mass (similar to what is done with the Rayleigh damping). @@ -11690,7 +11690,7 @@ However, as decentralized IFF will be applied, the damping is actively brought, #+end_subfigure #+end_figure -In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from $u_1$ to $d_{e1}$ to $d_{e6}$ are compared in the case of the 39kg payload in Figure ref:fig:test_nhexa_comp_simscape_de_all_high_mass. +In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from $u_1$ to $d_{e1}$ to $d_{e6}$ are compared in the case of the 39kg payload in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all_high_mass. Excellent match between experimental and model coupling is observed. Therefore, the model effectively represents the system coupling for different payloads. @@ -11706,18 +11706,18 @@ Therefore, the model effectively represents the system coupling for different pa The goal of this test bench was to obtain an accurate model of the nano-hexapod that could then be included on top of the micro-station model. The adopted strategy was to identify the nano-hexapod dynamics under conditions in which all factors that could have affected the nano-hexapod dynamics were considered. -This was achieved by developing a suspended table with low frequency suspension modes that can be accurately modeled (Section ref:sec:test_nhexa_table). +This was achieved by developing a suspended table with low frequency suspension modes that can be accurately modeled (Section\nbsp{}ref:sec:test_nhexa_table). Although the dynamics of the nano-hexapod was indeed impacted by the dynamics of the suspended platform, this impact was also considered in the multi-body model. -The dynamics of the nano-hexapod was then identified in Section ref:sec:test_nhexa_dynamics. +The dynamics of the nano-hexapod was then identified in Section\nbsp{}ref:sec:test_nhexa_dynamics. Below the first suspension mode, good decoupling could be observed for the transfer function from $\bm{u}$ to $\bm{d}_e$, which enables the design of a decentralized positioning controller based on the encoders for relative positioning purposes. Many other modes were present above 700Hz, which will inevitably limit the achievable bandwidth. The observed effect of the payload's mass on the dynamics was quite large, which also represents a complex control challenge. -The frequency response functions from the six DAC voltages $\bm{u}$ to the six force sensors voltages $\bm{V}_s$ all have alternating complex conjugate poles and complex conjugate zeros for all the tested payloads (Figure ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). +The frequency response functions from the six DAC voltages $\bm{u}$ to the six force sensors voltages $\bm{V}_s$ all have alternating complex conjugate poles and complex conjugate zeros for all the tested payloads (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). This indicates that it is possible to implement decentralized Integral Force Feedback in a robust manner. -The developed multi-body model of the nano-hexapod was found to accurately represents the suspension modes of the Nano-Hexapod (Section ref:sec:test_nhexa_model). +The developed multi-body model of the nano-hexapod was found to accurately represents the suspension modes of the Nano-Hexapod (Section\nbsp{}ref:sec:test_nhexa_model). Both FRF matrices from $\mathbf{u}$ to $\mathbf{V}_s$ and from $\mathbf{u}$ to $\mathbf{d}_e$ are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective. At frequencies above the suspension modes, the Nano-Hexapod model became inaccurate because the flexible modes were not modeled. It was found that modeling the APA300ML using a /super-element/ allows to model the internal resonances of the struts. @@ -11732,22 +11732,22 @@ If a model of the nano-hexapod was developed in one time, it would be difficult <> *** Introduction :ignore: -To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure ref:fig:test_id31_micro_station_nano_hexapod. +To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure\nbsp{}ref:fig:test_id31_micro_station_nano_hexapod. This section presents a comprehensive experimental evaluation of the complete system's performance on the ID31 beamline, focusing on its ability to maintain precise sample positioning under various experimental conditions. Initially, the project planned to develop a long-stroke ($\approx 1 \, cm^3$) 5-DoF metrology system to measure the sample position relative to the granite base. However, the complexity of this development prevented its completion before the experimental testing phase on ID31. -To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\mu m^3$) metrology system was developed, which is presented in Section ref:sec:test_id31_metrology. +To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\mu m^3$) metrology system was developed, which is presented in Section\nbsp{}ref:sec:test_id31_metrology. Then, several key aspects of the system validation are examined. -Section ref:sec:test_id31_open_loop_plant analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities. +Section\nbsp{}ref:sec:test_id31_open_loop_plant analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities. These measurements were compared with predictions from the multi-body model to verify its accuracy and applicability to control design. -Sections ref:sec:test_id31_iff and ref:sec:test_id31_hac focus on the implementation and validation of the HAC-LAC control architecture. -First, Section ref:sec:test_id31_iff demonstrates the application of decentralized Integral Force Feedback for robust active damping of the nano-hexapod suspension modes. -This is followed in Section ref:sec:test_id31_hac by the implementation of the high authority controller, which addresses low-frequency disturbances and completes the control system design. +Sections\nbsp{}ref:sec:test_id31_iff and ref:sec:test_id31_hac focus on the implementation and validation of the HAC-LAC control architecture. +First, Section\nbsp{}ref:sec:test_id31_iff demonstrates the application of decentralized Integral Force Feedback for robust active damping of the nano-hexapod suspension modes. +This is followed in Section\nbsp{}ref:sec:test_id31_hac by the implementation of the high authority controller, which addresses low-frequency disturbances and completes the control system design. -Finally, Section ref:sec:test_id31_experiments evaluates the NASS's positioning performances through a comprehensive series of experiments that mirror typical scientific applications. +Finally, Section\nbsp{}ref:sec:test_id31_experiments evaluates the NASS's positioning performances through a comprehensive series of experiments that mirror typical scientific applications. 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 @@ -11775,7 +11775,7 @@ These include tomography scans at various speeds and with different payload mass The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite. As a long-stroke ($\approx 1 \,cm^3$) metrology system was not yet developed, a stroke stroke ($\approx 100\,\mu m^3$) was used instead to validate the nano-hexapod control. -The first considered option was to use the "Spindle error analyzer" shown in Figure ref:fig:test_id31_lion. +The first considered option was to use the "Spindle error analyzer" shown in Figure\nbsp{}ref:fig:test_id31_lion. This system comprises 5 capacitive sensors facing two reference spheres. 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. @@ -11803,8 +11803,8 @@ However, as the gap between the capacitive sensors and the spheres is very small #+end_subfigure #+end_figure -Instead of using capacitive sensors, 5 fibered interferometers were used in a similar manner (Figure ref:fig:test_id31_interf). -At the end of each fiber, a sensor head[fn:test_id31_2] (Figure ref:fig:test_id31_interf_head) is used, which consists of a lens precisely positioned with respect to the fiber's end. +Instead of using capacitive sensors, 5 fibered interferometers were used in a similar manner (Figure\nbsp{}ref:fig:test_id31_interf). +At the end of each fiber, a sensor head[fn:test_id31_2] (Figure\nbsp{}ref:fig:test_id31_interf_head) is used, which consists of a lens precisely positioned with respect to the fiber's end. The lens focuses the light on the surface of the sphere, such that the reflected light comes back into the fiber and produces an interference. In this way, the gap between the head and the reference sphere is much larger (here around $40\,mm$), thereby removing the risk of collision. @@ -11814,10 +11814,10 @@ Indeed, when the spheres are moving perpendicularly to the beam axis, the reflec **** Metrology Kinematics <> -The proposed short-stroke metrology system is schematized in Figure ref:fig:test_id31_metrology_kinematics. +The proposed short-stroke metrology system is schematized in Figure\nbsp{}ref:fig:test_id31_metrology_kinematics. The point of interest is indicated by the blue frame $\{B\}$, which is located $H = 150\,mm$ above the nano-hexapod's top platform. The spheres have a diameter $d = 25.4\,mm$, and the indicated dimensions are $l_1 = 60\,mm$ and $l_2 = 16.2\,mm$. -To compute the pose of $\{B\}$ with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured (small) displacements $[d_1,\ d_2,\ d_3,\ d_4,\ d_5]$ by the interferometers are first written as a function of the (small) linear and angular motion of the $\{B\}$ frame $[D_x,\ D_y,\ D_z,\ R_x,\ R_y]$ eqref:eq:test_id31_metrology_kinematics. +To compute the pose of $\{B\}$ with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured (small) displacements $[d_1,\ d_2,\ d_3,\ d_4,\ d_5]$ by the interferometers are first written as a function of the (small) linear and angular motion of the $\{B\}$ frame $[D_x,\ D_y,\ D_z,\ R_x,\ R_y]$\nbsp{}eqref:eq:test_id31_metrology_kinematics. \begin{equation}\label{eq:test_id31_metrology_kinematics} d_1 = D_y - l_2 R_x, \quad d_2 = D_y + l_1 R_x, \quad d_3 = -D_x - l_2 R_y, \quad d_4 = -D_x + l_1 R_y, \quad d_5 = -D_z @@ -11839,7 +11839,7 @@ d_1 = D_y - l_2 R_x, \quad d_2 = D_y + l_1 R_x, \quad d_3 = -D_x - l_2 R_y, \qua [[file:figs/test_id31_align_top_sphere_comparators.jpg]] #+end_minipage -The five equations eqref:eq:test_id31_metrology_kinematics can be written in matrix form, and then inverted to have the pose of the $\{B\}$ frame as a linear combination of the measured five distances by the interferometers eqref:eq:test_id31_metrology_kinematics_inverse. +The five equations\nbsp{}eqref:eq:test_id31_metrology_kinematics can be written in matrix form, and then inverted to have the pose of the $\{B\}$ frame as a linear combination of the measured five distances by the interferometers\nbsp{}eqref:eq:test_id31_metrology_kinematics_inverse. \begin{equation}\label{eq:test_id31_metrology_kinematics_inverse} \begin{bmatrix} @@ -11859,7 +11859,7 @@ The five equations eqref:eq:test_id31_metrology_kinematics can be written in mat <> The two reference spheres must be well aligned with the rotation axis of the spindle. -To achieve this, two measuring probes were used as shown in Figure ref:fig:test_id31_align_top_sphere_comparators. +To achieve this, two measuring probes were used as shown in Figure\nbsp{}ref:fig:test_id31_align_top_sphere_comparators. To not damage the sensitive sphere surface, the probes are instead positioned on the cylinder on which the sphere is mounted. The probes are first fixed to the bottom (fixed) cylinder to align the first sphere with the spindle axis. @@ -11872,7 +11872,7 @@ However, this first alignment should be sufficient to position the two sphere wi **** Tip-Tilt adjustment of the interferometers <> -The short-stroke metrology system was placed on top of the main granite using granite blocs (Figure ref:fig:test_id31_short_stroke_metrology_overview). +The short-stroke metrology system was placed on top of the main granite using granite blocs (Figure\nbsp{}ref:fig:test_id31_short_stroke_metrology_overview). Granite is used for its good mechanical and thermal stability. #+name: fig:test_id31_short_stroke_metrology_overview @@ -11880,7 +11880,7 @@ Granite is used for its good mechanical and thermal stability. #+attr_latex: :width 0.8\linewidth [[file:figs/test_id31_short_stroke_metrology_overview.jpg]] -The interferometer beams must be placed with respect to the two reference spheres as close as possible to the ideal case shown in Figure ref:fig:test_id31_metrology_kinematics. +The interferometer beams must be placed with respect to the two reference spheres as close as possible to the ideal case shown in Figure\nbsp{}ref:fig:test_id31_metrology_kinematics. Therefore, their positions and angles must be well adjusted with respect to the two spheres. First, the vertical positions of the spheres is adjusted using the micro-hexapod to match the heights of the interferometers. Then, the horizontal position of the gantry is adjusted such that the intensity of the light reflected back in the fiber of the top interferometer is maximized. @@ -11895,7 +11895,7 @@ After the alignment procedure, the top interferometer should coincide with the s **** Fine Alignment of reference spheres using interferometers <> -Thanks to the first alignment of the two reference spheres with the spindle axis (Section ref:ssec:test_id31_metrology_sphere_rought_alignment) and to the fine adjustment of the interferometer orientations (Section ref:ssec:test_id31_metrology_alignment), the spindle can perform complete rotations while still having interference for all five interferometers. +Thanks to the first alignment of the two reference spheres with the spindle axis (Section\nbsp{}ref:ssec:test_id31_metrology_sphere_rought_alignment) and to the fine adjustment of the interferometer orientations (Section\nbsp{}ref:ssec:test_id31_metrology_alignment), the spindle can perform complete rotations while still having interference for all five interferometers. Therefore, this metrology can be used to better align the axis defined by the centers of the two spheres with the spindle axis. The alignment process requires few iterations. @@ -11903,7 +11903,7 @@ First, the spindle is scanned, and alignment errors are recorded. From the errors, the motion of the micro-hexapod to better align the spheres with the spindle axis is computed and the micro-hexapod is positioned accordingly. Then, the spindle is scanned again, and new alignment errors are recorded. -This iterative process is first performed for angular errors (Figure ref:fig:test_id31_metrology_align_rx_ry) and then for lateral errors (Figure ref:fig:test_id31_metrology_align_dx_dy). +This iterative process is first performed for angular errors (Figure\nbsp{}ref:fig:test_id31_metrology_align_rx_ry) and then for lateral errors (Figure\nbsp{}ref:fig:test_id31_metrology_align_dx_dy). The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ in $R_x$ and $R_y$ orientations, $\pm 1\,\mu m$ in $D_x$ and $D_y$ directions, and less than $0.1\,\mu m$ vertically. #+name: fig:test_id31_metrology_align @@ -11930,7 +11930,7 @@ The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ i Because the interferometers point to spheres and not flat surfaces, the lateral acceptance is limited. To estimate the metrology acceptance, the micro-hexapod was used to perform three accurate scans of $\pm 1\,mm$, respectively along the $x$, $y$ and $z$ axes. During these scans, the 5 interferometers are recorded individually, and the ranges in which each interferometer had enough coupling efficiency to be able to measure the displacement were estimated. -Results are summarized in Table ref:tab:test_id31_metrology_acceptance. +Results are summarized in Table\nbsp{}ref:tab:test_id31_metrology_acceptance. The obtained lateral acceptance for pure displacements in any direction is estimated to be around $+/-0.5\,mm$, which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS. #+name: tab:test_id31_metrology_acceptance @@ -11953,8 +11953,8 @@ However, the validation of the nano-hexapod, the associated instrumentation, and Only the bandwidth and noise characteristics of the external metrology are important. However, some elements that affect the accuracy of the metrology system are discussed here. -First, the "metrology kinematics" (discussed in Section ref:ssec:test_id31_metrology_kinematics) is only approximate (i.e. valid for small displacements). -This can be easily seen when performing lateral $[D_x,\,D_y]$ scans using the micro-hexapod while recording the vertical interferometer (Figure ref:fig:test_id31_xy_map_sphere). +First, the "metrology kinematics" (discussed in Section\nbsp{}ref:ssec:test_id31_metrology_kinematics) is only approximate (i.e. valid for small displacements). +This can be easily seen when performing lateral $[D_x,\,D_y]$ scans using the micro-hexapod while recording the vertical interferometer (Figure\nbsp{}ref:fig:test_id31_xy_map_sphere). As the top interferometer points to a sphere and not to a plane, lateral motion of the sphere is seen as a vertical motion by the top interferometer. Then, the reference spheres have some deviations relative to an ideal sphere [fn:test_id31_6]. @@ -11965,8 +11965,8 @@ As the light from the interferometer travels through air (as opposed to being in Therefore, any variation in air temperature, pressure or humidity will induce measurement errors. For instance, for a measurement length of $40\,mm$, a temperature variation of $0.1\,{}^oC$ (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of $\approx 4\,nm$. -Interferometers are also affected by noise [[cite:&watchi18_review_compac_inter]]. -The effect of noise on the translation and rotation measurements is estimated in Figure ref:fig:test_id31_interf_noise. +Interferometers are also affected by noise\nbsp{}[[cite:&watchi18_review_compac_inter]]. +The effect of noise on the translation and rotation measurements is estimated in Figure\nbsp{}ref:fig:test_id31_interf_noise. #+name: fig:test_id31_metrology_errors #+caption: Estimated measurement errors of the metrology. Cross-coupling between lateral motion and vertical measurement is shown in (\subref{fig:test_id31_xy_map_sphere}). The effect of interferometer noise on the measured translations and rotations is shown in (\subref{fig:test_id31_interf_noise}). @@ -11990,7 +11990,7 @@ The effect of noise on the translation and rotation measurements is estimated in <> **** Introduction :ignore: -The NASS plant is schematically illustrated in Figure ref:fig:test_id31_block_schematic_plant. +The NASS plant is schematically illustrated in Figure\nbsp{}ref:fig:test_id31_block_schematic_plant. The input $\bm{u} = [u_1,\ u_2,\ u_3,\ u_4,\ u_5,\ u_6]$ is the command signal, which corresponds to the voltages generated for each piezoelectric actuator. After amplification, the voltages across the piezoelectric stack actuators are $\bm{V}_a = [V_{a1},\ V_{a2},\ V_{a3},\ V_{a4},\ V_{a5},\ V_{a6}]$. @@ -12014,11 +12014,11 @@ Voltages generated by the force sensor piezoelectric stacks $\bm{V}_s = [V_{s1}, The dynamics of the plant is first identified for a fixed spindle angle (at $0\,\text{deg}$) and without any payload. The model dynamics is also identified under the same conditions. -A comparison between the model and the measured dynamics is presented in Figure ref:fig:test_id31_first_id. +A comparison between the model and the measured dynamics is presented in Figure\nbsp{}ref:fig:test_id31_first_id. A good match can be observed for the diagonal dynamics (except the high frequency modes which are not modeled). -However, the coupling of the transfer function from command signals $\bm{u}$ to the estimated strut motion from the external metrology $\bm{\epsilon\mathcal{L}}$ is larger than expected (Figure ref:fig:test_id31_first_id_int). +However, the coupling of the transfer function from command signals $\bm{u}$ to the estimated strut motion from the external metrology $\bm{\epsilon\mathcal{L}}$ is larger than expected (Figure\nbsp{}ref:fig:test_id31_first_id_int). -The experimental time delay estimated from the FRF (Figure ref:fig:test_id31_first_id_int) is larger than expected. +The experimental time delay estimated from the FRF (Figure\nbsp{}ref:fig:test_id31_first_id_int) is larger than expected. After investigation, it was found that the additional delay was due to a digital processing unit[fn:test_id31_3] that was used to get the interferometers' signals in the Speedgoat. This issue was later solved. @@ -12043,14 +12043,14 @@ This issue was later solved. **** Better Angular Alignment <> -One possible explanation of the increased coupling observed in Figure ref:fig:test_id31_first_id_int is the poor alignment between the external metrology axes (i.e. the interferometer supports) and the nano-hexapod axes. +One possible explanation of the increased coupling observed in Figure\nbsp{}ref:fig:test_id31_first_id_int is the poor alignment between the external metrology axes (i.e. the interferometer supports) and the nano-hexapod axes. To estimate this alignment, a decentralized low-bandwidth feedback controller based on the nano-hexapod encoders was implemented. This allowed to perform two straight motions of the nano-hexapod along its $x$ and $y$ axes. -During these two motions, external metrology measurements were recorded and the results are shown in Figure ref:fig:test_id31_Rz_align_error_and_correct. +During these two motions, external metrology measurements were recorded and the results are shown in Figure\nbsp{}ref:fig:test_id31_Rz_align_error_and_correct. It was found that there was a misalignment of 2.7 degrees (rotation along the vertical axis) between the interferometer axes and nano-hexapod axes. This was corrected by adding an offset to the spindle angle. After alignment, the same motion was performed using the nano-hexapod while recording the signal of the external metrology. -Results shown in Figure ref:fig:test_id31_Rz_align_correct are indeed indicating much better alignment. +Results shown in Figure\nbsp{}ref:fig:test_id31_Rz_align_correct are indeed indicating much better alignment. #+name: fig:test_id31_Rz_align_error_and_correct #+caption: Measurement of the Nano-Hexapod axes in the frame of the external metrology. Before alignment (\subref{fig:test_id31_Rz_align_error}) and after alignment (\subref{fig:test_id31_Rz_align_correct}). @@ -12070,8 +12070,8 @@ Results shown in Figure ref:fig:test_id31_Rz_align_correct are indeed indicating #+end_subfigure #+end_figure -The dynamics of the plant was identified again after fine alignment and compared with the model dynamics in Figure ref:fig:test_id31_first_id_int_better_rz_align. -Compared to the initial identification shown in Figure ref:fig:test_id31_first_id_int, the obtained coupling was decreased and was close to the coupling obtained with the multi-body model. +The dynamics of the plant was identified again after fine alignment and compared with the model dynamics in Figure\nbsp{}ref:fig:test_id31_first_id_int_better_rz_align. +Compared to the initial identification shown in Figure\nbsp{}ref:fig:test_id31_first_id_int, the obtained coupling was decreased and was close to the coupling obtained with the multi-body model. At low frequency (below $10\,\text{Hz}$), all off-diagonal elements have an amplitude $\approx 100$ times lower than the diagonal elements, indicating that a low bandwidth feedback controller can be implemented in a decentralized manner (i.e. $6$ SISO controllers). Between $650\,\text{Hz}$ and $1000\,\text{Hz}$, several modes can be observed, which are due to flexible modes of the top platform and the modes of the two spheres adjustment mechanism. The flexible modes of the top platform can be passively damped, whereas the modes of the two reference spheres should not be present in the final application. @@ -12083,12 +12083,12 @@ The flexible modes of the top platform can be passively damped, whereas the mode **** Effect of Payload Mass <> -To determine how the system dynamics changes with the payload, open-loop identification was performed for four payload conditions shown in Figure ref:fig:test_id31_picture_masses. -The obtained direct terms are compared with the model dynamics in Figure ref:fig:test_id31_comp_simscape_diag_masses. +To determine how the system dynamics changes with the payload, open-loop identification was performed for four payload conditions shown in Figure\nbsp{}ref:fig:test_id31_picture_masses. +The obtained direct terms are compared with the model dynamics in Figure\nbsp{}ref:fig:test_id31_comp_simscape_diag_masses. It was found that the model well predicts the measured dynamics under all payload conditions. Therefore, the model can be used for model-based control if necessary. -It is interesting to note that the anti-resonances in the force sensor plant now appear as minimum-phase, as the model predicts (Figure ref:fig:test_id31_comp_simscape_iff_diag_masses). +It is interesting to note that the anti-resonances in the force sensor plant now appear as minimum-phase, as the model predicts (Figure\nbsp{}ref:fig:test_id31_comp_simscape_iff_diag_masses). #+name: fig:test_id31_picture_masses #+caption: The four tested payload conditions. (\subref{fig:test_id31_picture_mass_m0}) without payload. (\subref{fig:test_id31_picture_mass_m1}) with $13\,\text{kg}$ payload. (\subref{fig:test_id31_picture_mass_m2}) with $26\,\text{kg}$ payload. (\subref{fig:test_id31_picture_mass_m3}) with $39\,\text{kg}$ payload. @@ -12141,10 +12141,10 @@ It is interesting to note that the anti-resonances in the force sensor plant now **** Effect of Spindle Rotation <> -To verify that all the kinematics in Figure ref:fig:test_id31_block_schematic_plant are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at $36\,\text{deg}/s$ and at $180\,\text{deg}/s$. +To verify that all the kinematics in Figure\nbsp{}ref:fig:test_id31_block_schematic_plant are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at $36\,\text{deg}/s$ and at $180\,\text{deg}/s$. -The obtained dynamics from command signal $u$ to estimated strut error $\epsilon\mathcal{L}$ are displayed in Figure ref:fig:test_id31_effect_rotation. -Both direct terms (Figure ref:fig:test_id31_effect_rotation_direct) and coupling terms (Figure ref:fig:test_id31_effect_rotation_coupling) are unaffected by the rotation. +The obtained dynamics from command signal $u$ to estimated strut error $\epsilon\mathcal{L}$ are displayed in Figure\nbsp{}ref:fig:test_id31_effect_rotation. +Both direct terms (Figure\nbsp{}ref:fig:test_id31_effect_rotation_direct) and coupling terms (Figure\nbsp{}ref:fig:test_id31_effect_rotation_coupling) are unaffected by the rotation. The same can be observed for the dynamics from command signal to encoders and to force sensors. This confirms that spindle's rotation has no significant effect on plant dynamics. This also indicates that the metrology kinematics is correct and is working in real time. @@ -12181,7 +12181,7 @@ The spindle rotation had no visible effect on the measured dynamics, indicating **** Introduction :ignore: In this section, the low authority control part is first validated. -It consists of a decentralized Integral Force Feedback controller $\bm{K}_{\text{IFF}}$, with all the diagonal terms being equal eqref:eq:test_id31_Kiff. +It consists of a decentralized Integral Force Feedback controller $\bm{K}_{\text{IFF}}$, with all the diagonal terms being equal\nbsp{}eqref:eq:test_id31_Kiff. \begin{equation}\label{eq:test_id31_iff_diagonal} \bm{K}_{\text{IFF}} = K_{\text{IFF}} \cdot \bm{I}_6 = \begin{bmatrix} @@ -12191,7 +12191,7 @@ K_{\text{IFF}} & & 0 \\ \end{bmatrix} \end{equation} -The decentralized Integral Force Feedback is implemented as shown in the block diagram of Figure ref:fig:test_id31_iff_block_diagram. +The decentralized Integral Force Feedback is implemented as shown in the block diagram of Figure\nbsp{}ref:fig:test_id31_iff_block_diagram. #+name: fig:test_id31_iff_block_diagram #+caption: Block diagram of the implemented decentralized IFF controller. The controller $\bm{K}_{\text{IFF}}$ is a diagonal controller with the same elements for every diagonal term $K_{\text{IFF}}$. @@ -12202,9 +12202,9 @@ The decentralized Integral Force Feedback is implemented as shown in the block d As the multi-body model is used to evaluate the stability of the IFF controller and to optimize the achievable damping, it is first checked whether this model accurately represents the system dynamics. -In the previous section (Figure ref:fig:test_id31_comp_simscape_iff_diag_masses), it was shown that the model well captures the dynamics from each actuator to its collocated force sensor, and that for all considered payloads. +In the previous section (Figure\nbsp{}ref:fig:test_id31_comp_simscape_iff_diag_masses), it was shown that the model well captures the dynamics from each actuator to its collocated force sensor, and that for all considered payloads. Nevertheless, it is also important to model accurately the coupling in the system. -To verify that, instead of comparing the 36 elements of the $6 \times 6$ frequency response matrix from $\bm{u}$ to $\bm{V_s}$, only 6 elements are compared in Figure ref:fig:test_id31_comp_simscape_Vs. +To verify that, instead of comparing the 36 elements of the $6 \times 6$ frequency response matrix from $\bm{u}$ to $\bm{V_s}$, only 6 elements are compared in Figure\nbsp{}ref:fig:test_id31_comp_simscape_Vs. Similar results were obtained for all other 30 elements and for the different payload conditions. This confirms that the multi-body model can be used to tune the IFF controller. @@ -12216,14 +12216,14 @@ This confirms that the multi-body model can be used to tune the IFF controller. <> A decentralized IFF controller was designed to add damping to the suspension modes of the nano-hexapod for all considered payloads. -The frequency of the suspension modes are ranging from $\approx 30\,\text{Hz}$ to $\approx 250\,\text{Hz}$ (Figure ref:fig:test_id31_comp_simscape_iff_diag_masses), and therefore, the IFF controller should provide integral action in this frequency range. -A second-order high-pass filter (cut-off frequency of $10\,\text{Hz}$) was added to limit the low frequency gain eqref:eq:test_id31_Kiff. +The frequency of the suspension modes are ranging from $\approx 30\,\text{Hz}$ to $\approx 250\,\text{Hz}$ (Figure\nbsp{}ref:fig:test_id31_comp_simscape_iff_diag_masses), and therefore, the IFF controller should provide integral action in this frequency range. +A second-order high-pass filter (cut-off frequency of $10\,\text{Hz}$) was added to limit the low frequency gain\nbsp{}eqref:eq:test_id31_Kiff. \begin{equation}\label{eq:test_id31_Kiff} K_{\text{IFF}} = g_0 \cdot \underbrace{\frac{1}{s}}_{\text{int}} \cdot \underbrace{\frac{s^2/\omega_z^2}{s^2/\omega_z^2 + 2\xi_z s /\omega_z + 1}}_{\text{2nd order LPF}},\quad \left(g_0 = -100,\ \omega_z = 2\pi10\,\text{rad/s},\ \xi_z = 0.7\right) \end{equation} -The bode plot of the decentralized IFF controller is shown in Figure ref:fig:test_id31_Kiff_bode_plot and the "decentralized loop-gains" for all considered payload masses are shown in Figure ref:fig:test_id31_Kiff_loop_gain. +The bode plot of the decentralized IFF controller is shown in Figure\nbsp{}ref:fig:test_id31_Kiff_bode_plot and the "decentralized loop-gains" for all considered payload masses are shown in Figure\nbsp{}ref:fig:test_id31_Kiff_loop_gain. 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 @@ -12244,7 +12244,7 @@ It can be seen that the loop-gain is larger than $1$ around the suspension modes #+end_subfigure #+end_figure -To estimate the added damping, a root-locus plot was computed using the multi-body model (Figure ref:fig:test_id31_iff_root_locus). +To estimate the added damping, a root-locus plot was computed using the multi-body model (Figure\nbsp{}ref:fig:test_id31_iff_root_locus). It can be seen that for all considered payloads, the poles are bounded to the "left-half plane" indicating that the decentralized IFF is robust. The closed-loop poles for the chosen gain value are represented by black crosses. It can be seen that while damping can be added for all payloads (as compared to the open-loop case), the optimal value of the gain is different for each payload. @@ -12285,11 +12285,11 @@ However, in this study, it was chosen to implement a "fixed" (i.e. non-adaptive) <> As the model accurately represents the system dynamics, it can be used to estimate the damped plant, i.e. the transfer functions from $\bm{u}^\prime$ to $\bm{\mathcal{L}}$. -The obtained damped plants are compared to the open-loop plants in Figure ref:fig:test_id31_comp_ol_iff_plant_model. +The obtained damped plants are compared to the open-loop plants in Figure\nbsp{}ref:fig:test_id31_comp_ol_iff_plant_model. The peak amplitudes corresponding to the suspension modes were approximately reduced by a factor $10$ for all considered payloads, indicating the effectiveness of the decentralized IFF control strategy. To experimentally validate the Decentralized IFF controller, it was implemented and the damped plants (i.e. the transfer function from $\bm{u}^\prime$ to $\bm{\epsilon\mathcal{L}}$) were identified for all payload conditions. -The obtained frequency response functions are compared with the model in Figure ref:fig:test_id31_hac_plant_effect_mass verifying the good correlation between the predicted damped plant using the multi-body model and the experimental results. +The obtained frequency response functions are compared with the model in Figure\nbsp{}ref:fig:test_id31_hac_plant_effect_mass verifying the good correlation between the predicted damped plant using the multi-body model and the experimental results. #+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 @@ -12325,9 +12325,9 @@ The good correlation between the modeled and measured damped plants confirms the **** Introduction :ignore: In this section, a High-Authority-Controller is developed to actively stabilize the sample position. -The corresponding control architecture is shown in Figure ref:fig:test_id31_iff_hac_schematic. +The corresponding control architecture is shown in Figure\nbsp{}ref:fig:test_id31_iff_hac_schematic. -As the diagonal terms of the damped plants were found to be all equal (thanks to the system's symmetry and manufacturing and mounting uniformity, see Figure ref:fig:test_id31_hac_plant_effect_mass), a diagonal high authority controller $\bm{K}_{\text{HAC}}$ is implemented with all diagonal terms being equal eqref:eq:eq:test_id31_hac_diagonal. +As the diagonal terms of the damped plants were found to be all equal (thanks to the system's symmetry and manufacturing and mounting uniformity, see Figure\nbsp{}ref:fig:test_id31_hac_plant_effect_mass), a diagonal high authority controller $\bm{K}_{\text{HAC}}$ is implemented with all diagonal terms being equal\nbsp{}eqref:eq:eq:test_id31_hac_diagonal. \begin{equation}\label{eq:eq:test_id31_hac_diagonal} \bm{K}_{\text{HAC}} = K_{\text{HAC}} \cdot \bm{I}_6 = \begin{bmatrix} @@ -12344,7 +12344,7 @@ K_{\text{HAC}} & & 0 \\ **** Damped Plant <> -To verify whether the multi-body model accurately represents the measured damped dynamics, both the direct terms and coupling terms corresponding to the first actuator are compared in Figure ref:fig:test_id31_comp_simscape_hac. +To verify whether the multi-body model accurately represents the measured damped dynamics, both the direct terms and coupling terms corresponding to the first actuator are compared in Figure\nbsp{}ref:fig:test_id31_comp_simscape_hac. 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 @@ -12352,8 +12352,8 @@ Considering the complexity of the system's dynamics, the model can be considered [[file:figs/test_id31_comp_simscape_hac.png]] The challenge here is to tune a high authority controller such that it is robust to the change in dynamics due to different payloads being used. -Without using the HAC-LAC strategy, it would be necessary to design a controller that provides good performance for all undamped dynamics (blue curves in Figure ref:fig:test_id31_comp_all_undamped_damped_plants), which is a very complex control problem. -With the HAC-LAC strategy, the designed controller must be robust to all the damped dynamics (red curves in Figure ref:fig:test_id31_comp_all_undamped_damped_plants), which is easier from a control perspective. +Without using the HAC-LAC strategy, it would be necessary to design a controller that provides good performance for all undamped dynamics (blue curves in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants), which is a very complex control problem. +With the HAC-LAC strategy, the designed controller must be robust to all the damped dynamics (red curves in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants), which is easier from a control perspective. This is one of the key benefits of using the HAC-LAC strategy. #+name: fig:test_id31_comp_all_undamped_damped_plants @@ -12364,19 +12364,19 @@ This is one of the key benefits of using the HAC-LAC strategy. <> The control strategy here is to apply a diagonal control in the frame of the struts; thus, it is important to determine the frequency at which the multivariable effects become significant, as this represents a critical limitation of the control approach. -To conduct this interaction analysis, the acrfull:rga $\bm{\Lambda_G}$ is first computed using eqref:eq:test_id31_rga for the plant dynamics identified with the multiple payload masses. +To conduct this interaction analysis, the acrfull:rga $\bm{\Lambda_G}$ is first computed using\nbsp{}eqref:eq:test_id31_rga for the plant dynamics identified with the multiple payload masses. \begin{equation}\label{eq:test_id31_rga} \bm{\Lambda_G}(\omega) = \bm{G}(j\omega) \star \left(\bm{G}(j\omega)^{-1}\right)^{\intercal}, \quad (\star \text{ means element wise multiplication}) \end{equation} -Then, acrshort:rga numbers are computed using eqref:eq:test_id31_rga_number and are use as a metric for interaction [[cite:&skogestad07_multiv_feedb_contr chapt. 3.4]]. +Then, acrshort:rga numbers are computed using\nbsp{}eqref:eq:test_id31_rga_number and are use as a metric for interaction\nbsp{}[[cite:&skogestad07_multiv_feedb_contr chapt. 3.4]]. \begin{equation}\label{eq:test_id31_rga_number} \text{RGA number}(\omega) = \|\bm{\Lambda_G}(\omega) - \bm{I}\|_{\text{sum}} \end{equation} -The obtained acrshort:rga numbers are compared in Figure ref:fig:test_id31_hac_rga_number. +The obtained acrshort:rga numbers are compared in Figure\nbsp{}ref:fig:test_id31_hac_rga_number. The results indicate that higher payload masses increase the coupling when implementing control in the strut reference frame (i.e., decentralized approach). This indicates that achieving high bandwidth feedback control is increasingly challenging as the payload mass increases. This behavior can be attributed to the fundamental approach of implementing control in the frame of the struts. @@ -12390,17 +12390,17 @@ This design choice, while beneficial for system simplicity, introduces inherent **** Robust Controller Design <> -A diagonal controller was designed to be robust against changes in payload mass, which means that every damped plant shown in Figure ref:fig:test_id31_comp_all_undamped_damped_plants must be considered during the controller design. -For this controller design, a crossover frequency of $5\,\text{Hz}$ was chosen to limit the multivariable effects, as explain in Section ref:sec:test_id31_hac_interaction_analysis. +A diagonal controller was designed to be robust against changes in payload mass, which means that every damped plant shown in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants must be considered during the controller design. +For this controller design, a crossover frequency of $5\,\text{Hz}$ was chosen to limit the multivariable effects, as explain in Section\nbsp{}ref:sec:test_id31_hac_interaction_analysis. One integrator is added to increase the low-frequency gain, a lead is added around $5\,\text{Hz}$ to increase the stability margins and a first-order low-pass filter with a cut-off frequency of $30\,\text{Hz}$ is added to improve the robustness to dynamical uncertainty at high frequency. -The controller transfer function is shown in eqref:eq:test_id31_robust_hac. +The controller transfer function is shown in\nbsp{}eqref:eq:test_id31_robust_hac. \begin{equation}\label{eq:test_id31_robust_hac} K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi5\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi30\,\text{rad/s} \right) \end{equation} -The obtained "decentralized" loop-gains (i.e. the diagonal element of the controller times the diagonal terms of the plant) are shown in Figure ref:fig:test_id31_hac_loop_gain. -The closed-loop stability was verified by computing the characteristic Loci (Figure ref:fig:test_id31_hac_characteristic_loci). +The obtained "decentralized" loop-gains (i.e. the diagonal element of the controller times the diagonal terms of the plant) are shown in Figure\nbsp{}ref:fig:test_id31_hac_loop_gain. +The closed-loop stability was verified by computing the characteristic Loci (Figure\nbsp{}ref:fig:test_id31_hac_characteristic_loci). 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 @@ -12426,8 +12426,8 @@ However, small stability margins were observed for the highest mass, indicating To estimate the performances that can be expected with this HAC-LAC architecture and the designed controller, simulations of tomography experiments were performed[fn:test_id31_4]. The rotational velocity was set to $180\,\text{deg/s}$, and no payload was added on top of the nano-hexapod. -An open-loop simulation and a closed-loop simulation were performed and compared in Figure ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim. -The obtained closed-loop positioning accuracy was found to comply with the requirements as it succeeded to keep the point of interest on the beam (Figure ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz). +An open-loop simulation and a closed-loop simulation were performed and compared in Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim. +The obtained closed-loop positioning accuracy was found to comply with the requirements as it succeeded to keep the point of interest on the beam (Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz). #+name: fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim #+caption: Position error of the sample in the XY (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy}) and YZ (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}) planes during a simulation of a tomography experiment at $180\,\text{deg/s}$. No payload is placed on top of the nano-hexapod. @@ -12450,11 +12450,11 @@ The obtained closed-loop positioning accuracy was found to comply with the requi **** Robustness estimation with simulation of Tomography scans <> -To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure ref:fig:test_id31_picture_masses (i.e. $0\,kg$, $13\,kg$, $26\,kg$ and $39\,kg$). +To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure\nbsp{}ref:fig:test_id31_picture_masses (i.e. $0\,kg$, $13\,kg$, $26\,kg$ and $39\,kg$). The rotational velocity was set at $6\,\text{deg/s}$, which is the typical rotational velocity for heavy samples. The closed-loop systems were stable under all payload conditions, indicating good control robustness. -However, the positioning errors worsen as the payload mass increases, especially in the lateral $D_y$ direction, as shown in Figure ref:fig:test_id31_hac_tomography_Wz36_simulation. +However, the positioning errors worsen as the payload mass increases, especially in the lateral $D_y$ direction, as shown in Figure\nbsp{}ref:fig:test_id31_hac_tomography_Wz36_simulation. However, it was decided that this controller should be tested experimentally and improved only if necessary. #+name: fig:test_id31_hac_tomography_Wz36_simulation @@ -12484,24 +12484,24 @@ These results demonstrate both the effectiveness and limitations of implementing **** Introduction :ignore: In this section, the goal is to evaluate the performance of the NASS and validate its use to perform typical scientific experiments. -However, the online metrology prototype (presented in Section ref:sec:test_id31_metrology) does not allow samples to be placed on top of the nano-hexapod while being illuminated by the x-ray beam. +However, the online metrology prototype (presented in Section\nbsp{}ref:sec:test_id31_metrology) does not allow samples to be placed on top of the nano-hexapod while being illuminated by the x-ray beam. Nevertheless, to fully validate the NASS, typical motions performed during scientific experiments can be mimicked, and the positioning performances can be evaluated. Several scientific experiments were replicated, such as: -- Tomography scans: continuous rotation of the Spindle along the vertical axis (Section ref:ssec:test_id31_scans_tomography) -- Reflectivity scans: $R_y$ rotations using the tilt-stage (Section ref:ssec:test_id31_scans_reflectivity) -- Vertical layer scans: $D_z$ step motion or ramp scans using the nano-hexapod (Section ref:ssec:test_id31_scans_dz) -- Lateral scans: $D_y$ scans using the $T_y$ translation stage (Section ref:ssec:test_id31_scans_dy) -- Diffraction Tomography:continuous $R_z$ rotation using the Spindle and lateral $D_y$ scans performed at the same time using the translation stage (Section ref:ssec:test_id31_scans_diffraction_tomo) +- Tomography scans: continuous rotation of the Spindle along the vertical axis (Section\nbsp{}ref:ssec:test_id31_scans_tomography) +- Reflectivity scans: $R_y$ rotations using the tilt-stage (Section\nbsp{}ref:ssec:test_id31_scans_reflectivity) +- Vertical layer scans: $D_z$ step motion or ramp scans using the nano-hexapod (Section\nbsp{}ref:ssec:test_id31_scans_dz) +- Lateral scans: $D_y$ scans using the $T_y$ translation stage (Section\nbsp{}ref:ssec:test_id31_scans_dy) +- Diffraction Tomography:continuous $R_z$ rotation using the Spindle and lateral $D_y$ scans performed at the same time using the translation stage (Section\nbsp{}ref:ssec:test_id31_scans_diffraction_tomo) -Unless explicitly stated, all closed-loop experiments were performed using the robust (i.e. conservative) high authority controller designed in Section ref:ssec:test_id31_iff_hac_controller. +Unless explicitly stated, all closed-loop experiments were performed using the robust (i.e. conservative) high authority controller designed in Section\nbsp{}ref:ssec:test_id31_iff_hac_controller. For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to $200\,nm\times 100\,nm$. In this case, the goal is to keep the sample's point of interest in the beam, and therefore the $D_y$ and $D_z$ positioning errors should be less than $200\,nm$ and $100\,nm$ peak-to-peak, respectively. The $R_y$ error should be less than $1.7\,\mu\text{rad}$ peak-to-peak. -In terms of RMS errors, this corresponds to $30\,nm$ in $D_y$, $15\,nm$ in $D_z$ and $250\,\text{nrad}$ in $R_y$ (a summary of the specifications is given in Table ref:tab:test_id31_experiments_specifications). +In terms of RMS errors, this corresponds to $30\,nm$ in $D_y$, $15\,nm$ in $D_z$ and $250\,\text{nrad}$ in $R_y$ (a summary of the specifications is given in Table\nbsp{}ref:tab:test_id31_experiments_specifications). -Results obtained for all experiments are summarized and compared to the specifications in Section ref:ssec:test_id31_scans_conclusion. +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 @@ -12516,13 +12516,13 @@ Results obtained for all experiments are summarized and compared to the specific <> ***** Slow Tomography scans -First, tomography scans were performed with a rotational velocity of $6\,\text{deg/s}$ for all considered payload masses (shown in Figure ref:fig:test_id31_picture_masses). +First, tomography scans were performed with a rotational velocity of $6\,\text{deg/s}$ for all considered payload masses (shown in Figure\nbsp{}ref:fig:test_id31_picture_masses). Each experimental sequence consisted of two complete spindle rotations: an initial open-loop rotation followed by a closed-loop rotation. -The experimental results for the $26\,\text{kg}$ payload are presented in Figure ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit. +The experimental results for the $26\,\text{kg}$ payload are presented in Figure\nbsp{}ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit. Due to the static deformation of the micro-station stages under payload loading, a significant eccentricity was observed between the point of interest and the spindle rotation axis. To establish a theoretical lower bound for open-loop errors, an ideal scenario was assumed, where the point of interest perfectly aligns with the spindle rotation axis. -This idealized case was simulated by first calculating the eccentricity through circular fitting (represented by the dashed black circle in Figure ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit), and then subtracting it from the measured data, as shown in Figure ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed. +This idealized case was simulated by first calculating the eccentricity through circular fitting (represented by the dashed black circle in Figure\nbsp{}ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit), and then subtracting it from the measured data, as shown in Figure\nbsp{}ref:fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed. While this approach likely underestimates actual open-loop errors, as perfect alignment is practically unattainable, it enables a more balanced comparison with closed-loop performance. #+name: fig:test_id31_tomo_m2_1rpm_robust_hac_iff @@ -12543,9 +12543,9 @@ While this approach likely underestimates actual open-loop errors, as perfect al #+end_subfigure #+end_figure -The residual motion (i.e. after compensating for eccentricity) in the $Y-Z$ is compared against the minimum beam size, as illustrated in Figure ref:fig:test_id31_tomo_Wz36_results. +The residual motion (i.e. after compensating for eccentricity) in the $Y-Z$ is compared against the minimum beam size, as illustrated in Figure\nbsp{}ref:fig:test_id31_tomo_Wz36_results. Results are indicating the NASS succeeds in keeping the sample's point of interest on the beam, except for the highest mass of $39\,\text{kg}$ for which the lateral motion is a bit too high. -These experimental findings are consistent with the predictions from the tomography simulations presented in Section ref:ssec:test_id31_iff_hac_robustness. +These experimental findings are consistent with the predictions from the tomography simulations presented in Section\nbsp{}ref:ssec:test_id31_iff_hac_robustness. #+name: fig:test_id31_tomo_Wz36_results #+caption: Measured errors in the $Y-Z$ plane during tomography experiments at $6\,\text{deg/s}$ for all considered payloads. In the open-loop case, the effect of eccentricity is removed from the data. @@ -12554,8 +12554,8 @@ These experimental findings are consistent with the predictions from the tomogra ***** Fast Tomography scans A tomography experiment was then performed with the highest rotational velocity of the Spindle: $180\,\text{deg/s}$[fn:test_id31_7]. -The trajectory of the point of interest during the fast tomography scan is shown in Figure ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp. -Although the experimental results closely match the simulation results (Figure ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim), the actual performance was slightly lower than predicted. +The trajectory of the point of interest during the fast tomography scan is shown in Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp. +Although the experimental results closely match the simulation results (Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim), the actual performance was slightly lower than predicted. Nevertheless, even with this robust (i.e. conservative) HAC implementation, the system performance was already close to the specified requirements. #+name: fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp @@ -12582,9 +12582,9 @@ A comparative analysis was conducted using three tomography scans at $180\,\text The scans were performed under three conditions: open-loop, with decentralized IFF control, and with the complete HAC-LAC strategy. For these specific measurements, an enhanced high authority controller was optimized for low payload masses to meet the performance requirements. -Figure ref:fig:test_id31_hac_cas_cl presents the cumulative amplitude spectra of the position errors for all three cases. +Figure\nbsp{}ref:fig:test_id31_hac_cas_cl presents the cumulative amplitude spectra of the position errors for all three cases. The results reveal two distinct control contributions: the decentralized IFF effectively attenuates vibrations near the nano-hexapod suspension modes (an achievement not possible with HAC alone), while the high authority controller suppresses low-frequency vibrations primarily arising from Spindle guiding errors. -Notably, the spectral patterns in Figure ref:fig:test_id31_hac_cas_cl closely resemble the cumulative amplitude spectra computed in the project's early stages. +Notably, the spectral patterns in Figure\nbsp{}ref:fig:test_id31_hac_cas_cl closely resemble the cumulative amplitude spectra computed in the project's early stages. This experiment also illustrates that when needed, performance can be enhanced by designing controllers for specific experimental conditions rather than relying solely on robust controllers that can accommodate all payload ranges. @@ -12616,7 +12616,7 @@ This experiment also illustrates that when needed, performance can be enhanced b <> X-ray reflectivity measurements involve scanning thin structures, particularly solid/liquid interfaces, through the beam by varying the $R_y$ angle. -In this experiment, a $R_y$ scan was executed at a rotational velocity of $100\,\mu rad/s$, and the closed-loop positioning errors were monitored (Figure ref:fig:test_id31_reflectivity). +In this experiment, a $R_y$ scan was executed at a rotational velocity of $100\,\mu rad/s$, and the closed-loop positioning errors were monitored (Figure\nbsp{}ref:fig:test_id31_reflectivity). The results confirmed that the NASS successfully maintained the point of interest within the specified beam parameters throughout the scanning process. #+name: fig:test_id31_reflectivity @@ -12654,11 +12654,11 @@ These vertical scans can be executed either continuously or in a step-by-step ma The vertical step motion was performed exclusively with the nano-hexapod. Testing was conducted across step sizes ranging from $10\,nm$ to $1\,\mu m$. -Results are presented in Figure ref:fig:test_id31_dz_mim_steps. -The system successfully resolved 10nm steps (red curve in Figure ref:fig:test_id31_dz_mim_10nm_steps) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements. +Results are presented in Figure\nbsp{}ref:fig:test_id31_dz_mim_steps. +The system successfully resolved 10nm steps (red curve in Figure\nbsp{}ref:fig:test_id31_dz_mim_10nm_steps) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements. In step-by-step scanning procedures, the settling time is a critical parameter as it significantly affects the total experiment duration. -The system achieved a response time of approximately $70\,ms$ to reach the target position (within $\pm 20\,nm$), as demonstrated by the $1\,\mu m$ step response in Figure ref:fig:test_id31_dz_mim_1000nm_steps. +The system achieved a response time of approximately $70\,ms$ to reach the target position (within $\pm 20\,nm$), as demonstrated by the $1\,\mu m$ step response in Figure\nbsp{}ref:fig:test_id31_dz_mim_1000nm_steps. The settling duration typically decreases for smaller step sizes. #+name: fig:test_id31_dz_mim_steps @@ -12690,7 +12690,7 @@ The settling duration typically decreases for smaller step sizes. For these and subsequent experiments, the NASS performs "ramp scans" (constant velocity scans). To eliminate tracking errors, the feedback controller incorporates two integrators, compensating for the plant's lack of integral action at low frequencies. -Initial testing at $10\,\mu m/s$ demonstrated positioning errors well within specifications (indicated by dashed lines in Figure ref:fig:test_id31_dz_scan_10ums). +Initial testing at $10\,\mu m/s$ demonstrated positioning errors well within specifications (indicated by dashed lines in Figure\nbsp{}ref:fig:test_id31_dz_scan_10ums). #+name: fig:test_id31_dz_scan_10ums #+caption: $D_z$ scan at a velocity of $10\,\mu m/s$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry}) @@ -12716,7 +12716,7 @@ Initial testing at $10\,\mu m/s$ demonstrated positioning errors well within spe #+end_subfigure #+end_figure -A subsequent scan at $100\,\mu m/s$ - the maximum velocity for high-precision $D_z$ scans[fn:test_id31_8] - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure ref:fig:test_id31_dz_scan_100ums). +A subsequent scan at $100\,\mu m/s$ - the maximum velocity for high-precision $D_z$ scans[fn:test_id31_8] - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure\nbsp{}ref:fig:test_id31_dz_scan_100ums). Since detectors typically operate only during the constant velocity phase, these transient deviations do not compromise the measurement quality. However, performance during acceleration phases could be enhanced through the implementation of feedforward control. @@ -12756,12 +12756,12 @@ The scanning range is constrained $\pm 100\,\mu m$ due to the limited acceptance ***** Slow scan Initial testing utilized a scanning velocity of $10\,\mu m/s$, which is typical for these experiments. -Figure ref:fig:test_id31_dy_10ums compares the positioning errors between open-loop (without NASS) and closed-loop operation. -In the scanning direction, open-loop measurements reveal periodic errors (Figure ref:fig:test_id31_dy_10ums_dy) attributable to the $T_y$ stage's stepper motor. +Figure\nbsp{}ref:fig:test_id31_dy_10ums compares the positioning errors between open-loop (without NASS) and closed-loop operation. +In the scanning direction, open-loop measurements reveal periodic errors (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy) attributable to the $T_y$ stage's stepper motor. These micro-stepping errors, which are inherent to stepper motor operation, occur 200 times per motor rotation with approximately $1\,\text{mrad}$ angular error amplitude. -Given the $T_y$ stage's lead screw pitch of $2\,mm$, these errors manifest as $10\,\mu m$ periodic oscillations with $\approx 300\,nm$ amplitude, which can indeed be seen in the open-loop measurements (Figure ref:fig:test_id31_dy_10ums_dy). +Given the $T_y$ stage's lead screw pitch of $2\,mm$, these errors manifest as $10\,\mu m$ periodic oscillations with $\approx 300\,nm$ amplitude, which can indeed be seen in the open-loop measurements (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy). -In the vertical direction (Figure ref:fig:test_id31_dy_10ums_dz), open-loop errors likely stem from metrology measurement error because the top interferometer points at a spherical target surface (see Figure ref:fig:test_id31_xy_map_sphere). +In the vertical direction (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dz), open-loop errors likely stem from metrology measurement error because the top interferometer points at a spherical target surface (see Figure\nbsp{}ref:fig:test_id31_xy_map_sphere). Under closed-loop control, positioning errors remain within specifications in all directions. #+name: fig:test_id31_dy_10ums @@ -12790,14 +12790,14 @@ Under closed-loop control, positioning errors remain within specifications in al ***** Fast Scan -The system performance was evaluated at an increased scanning velocity of $100\,\mu m/s$, and the results are presented in Figure ref:fig:test_id31_dy_100ums. +The system performance was evaluated at an increased scanning velocity of $100\,\mu m/s$, and the results are presented in Figure\nbsp{}ref:fig:test_id31_dy_100ums. At this velocity, the micro-stepping errors generate $10\,\text{Hz}$ vibrations, which are further amplified by micro-station resonances. These vibrations exceeded the NASS feedback controller bandwidth, resulting in limited attenuation under closed-loop control. -This limitation exemplifies why stepper motors are suboptimal for "long-stroke/short-stroke" systems requiring precise scanning performance [[cite:&dehaeze22_fastj_uhv]]. +This limitation exemplifies why stepper motors are suboptimal for "long-stroke/short-stroke" systems requiring precise scanning performance\nbsp{}[[cite:&dehaeze22_fastj_uhv]]. Two potential solutions exist for improving high-velocity scanning performance. First, the $T_y$ stage's stepper motor could be replaced by a three-phase torque motor. -Alternatively, since closed-loop errors in $D_z$ and $R_y$ directions remain within specifications (Figures ref:fig:test_id31_dy_100ums_dz and ref:fig:test_id31_dy_100ums_ry), detector triggering could be based on measured $D_y$ position rather than time or $T_y$ setpoint, reducing sensitivity to $D_y$ vibrations. +Alternatively, since closed-loop errors in $D_z$ and $R_y$ directions remain within specifications (Figures\nbsp{}ref:fig:test_id31_dy_100ums_dz and ref:fig:test_id31_dy_100ums_ry), detector triggering could be based on measured $D_y$ position rather than time or $T_y$ setpoint, reducing sensitivity to $D_y$ vibrations. For applications requiring small $D_y$ scans, the nano-hexapod can be used exclusively, although with limited stroke capability. #+name: fig:test_id31_dy_100ums @@ -12830,15 +12830,15 @@ For applications requiring small $D_y$ scans, the nano-hexapod can be used exclu In diffraction tomography experiments, the micro-station performs combined motions: continuous rotation around the $R_z$ axis while performing lateral scans along $D_y$. For this validation, the spindle maintained a constant rotational velocity of $6\,\text{deg/s}$ while the nano-hexapod performs the lateral scanning motion. To avoid high-frequency vibrations typically induced by the stepper motor, the $T_y$ stage was not utilized, which constrained the scanning range to approximately $\pm 100\,\mu m/s$. -The system performance was evaluated at three lateral scanning velocities: $0.1\,mm/s$, $0.5\,mm/s$, and $1\,mm/s$. Figure ref:fig:test_id31_diffraction_tomo_setpoint presents both the $D_y$ position setpoints and the corresponding measured $D_y$ positions for all tested velocities. +The system performance was evaluated at three lateral scanning velocities: $0.1\,mm/s$, $0.5\,mm/s$, and $1\,mm/s$. Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_setpoint presents both the $D_y$ position setpoints and the corresponding measured $D_y$ positions for all tested velocities. #+name: fig:test_id31_diffraction_tomo_setpoint #+caption: Dy motion for several configured velocities [[file:figs/test_id31_diffraction_tomo_setpoint.png]] -The positioning errors measured along $D_y$, $D_z$, and $R_y$ directions are displayed in Figure ref:fig:test_id31_diffraction_tomo. -The system maintained positioning errors within specifications for both $D_z$ and $R_y$ (Figures ref:fig:test_id31_diffraction_tomo_dz and ref:fig:test_id31_diffraction_tomo_ry). -However, the lateral positioning errors exceeded specifications during the acceleration and deceleration phases (Figure ref:fig:test_id31_diffraction_tomo_dy). +The positioning errors measured along $D_y$, $D_z$, and $R_y$ directions are displayed in Figure\nbsp{}ref:fig:test_id31_diffraction_tomo. +The system maintained positioning errors within specifications for both $D_z$ and $R_y$ (Figures\nbsp{}ref:fig:test_id31_diffraction_tomo_dz and ref:fig:test_id31_diffraction_tomo_ry). +However, the lateral positioning errors exceeded specifications during the acceleration and deceleration phases (Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_dy). These large errors occurred only during $\approx 20\,ms$ intervals; thus, a delay of $20\,ms$ could be implemented in the detector the avoid integrating the beam when these large errors are occurring. Alternatively, a feedforward controller could improve the lateral positioning accuracy during these transient phases. @@ -12886,7 +12886,7 @@ For lateral scanning, the system performed well at moderate speeds ($10\,\mu m/s The most challenging test case - diffraction tomography combining rotation and lateral scanning - demonstrated the system's ability to maintain vertical and angular stability while highlighting some limitations in lateral positioning during rapid accelerations. These limitations could be addressed through feedforward control or alternative detector triggering strategies. -Overall, the experimental results validate the effectiveness of the developed control architecture and demonstrate that the NASS meets most design specifications across a wide range of operating conditions (summarized in Table ref:tab:test_id31_experiments_results_summary). +Overall, the experimental results validate the effectiveness of the developed control architecture and demonstrate that the NASS meets most design specifications across a wide range of operating conditions (summarized in Table\nbsp{}ref:tab:test_id31_experiments_results_summary). The identified limitations, primarily related to high-speed lateral scanning and heavy payload handling, provide clear directions for future improvements. #+name: tab:test_id31_experiments_results_summary @@ -13027,7 +13027,7 @@ With the implementation of an accurate online metrology system, the NASS will be [fn:ustation_9]The special optics (straightness interferometer and reflector) are manufactured by Agilent (10774A). [fn:ustation_8]C8 capacitive sensors and CPL290 capacitive driver electronics from Lion Precision. [fn:ustation_7]The Spindle Error Analyzer is made by Lion Precision. -[fn:ustation_6]The tools presented here are largely taken from [[cite:&taghirad13_paral]]. +[fn:ustation_6]The tools presented here are largely taken from\nbsp{}[[cite:&taghirad13_paral]]. [fn:ustation_5]Rotations are non commutative in 3D. [fn:ustation_4]Ball cage (N501) and guide bush (N550) from Mahr are used. [fn:ustation_3]Modified Zonda Hexapod by Symetrie. @@ -13050,12 +13050,12 @@ With the implementation of an accurate online metrology system, the NASS will be [fn:test_apa_13]PD200 from PiezoDrive. The gain is $20\,V/V$ [fn:test_apa_12]The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of $\pm 10\,V$ and 16-bits resolution [fn:test_apa_11]Ansys\textsuperscript{\textregistered} was used -[fn:test_apa_10]The transfer function fitting was computed using the =vectfit3= routine, see [[cite:&gustavsen99_ration_approx_frequen_domain_respon]] +[fn:test_apa_10]The transfer function fitting was computed using the =vectfit3= routine, see \nbsp{}[[cite:&gustavsen99_ration_approx_frequen_domain_respon]] [fn:test_apa_9]Frequency of the sinusoidal wave is $1\,\text{Hz}$ [fn:test_apa_8]Renishaw Vionic, resolution of $2.5\,nm$ [fn:test_apa_7]Kistler 9722A [fn:test_apa_6]Polytec controller 3001 with sensor heads OFV512 -[fn:test_apa_5]Note that this is not completely correct as it was shown in Section ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. +[fn:test_apa_5]Note that this is not completely correct as it was shown in Section\nbsp{}ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. [fn:test_apa_4]The Matlab =fminsearch= command is used to fit the plane [fn:test_apa_3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu m$ [fn:test_apa_2]Millimar 1318 probe, specified linearity better than $1\,\mu m$ @@ -13086,8 +13086,8 @@ With the implementation of an accurate online metrology system, the NASS will be [fn:test_id31_8]Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and 100nm "resolution" (equal to the vertical beam size). [fn:test_id31_7]The highest rotational velocity of $360\,\text{deg/s}$ could not be tested due to an issue in the Spindle's controller. [fn:test_id31_6]The roundness of the spheres is specified at $50\,nm$. -[fn:test_id31_5]The "IcePAP" [[cite:&janvier13_icepap]] which is developed at the ESRF. +[fn:test_id31_5]The "IcePAP"\nbsp{}[[cite:&janvier13_icepap]] which is developed at the ESRF. [fn:test_id31_4]Note that the eccentricity of the "point of interest" with respect to the Spindle rotation axis has been tuned based on measurements. -[fn:test_id31_3]The "PEPU" [[cite:&hino18_posit_encod_proces_unit]] was used for digital protocol conversion between the interferometers and the Speedgoat. +[fn:test_id31_3]The "PEPU"\nbsp{}[[cite:&hino18_posit_encod_proces_unit]] was used for digital protocol conversion between the interferometers and the Speedgoat. [fn:test_id31_2]M12/F40 model from Attocube. [fn:test_id31_1]Depending on the measuring range, gap can range from $\approx 1\,\mu m$ to $\approx 100\,\mu m$.