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Removed third level headings for first two sections

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Thomas Dehaeze 2025-04-23 09:13:23 +02:00
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@ -986,11 +986,11 @@ Two key effects that may limit that positioning performances are then considered
*** Micro Station Model *** Micro Station Model
<<sec:uniaxial_micro_station_model>> <<sec:uniaxial_micro_station_model>>
**** Introduction :ignore: ***** Introduction :ignore:
In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station. In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station.
**** Measured dynamics ***** Measured dynamics
The measurement setup is schematically shown in Figure\nbsp{}ref:fig:uniaxial_ustation_meas_dynamics_schematic where two vertical hammer hits are performed, one on the Granite (force $F_{g}$) and the other on the positioning 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 positioning hexapod's top platform (force $F_{h}$).
The vertical inertial motion of the granite $x_{g}$ and the top platform of the positioning hexapod $x_{h}$ are measured using geophones[fn:uniaxial_1]. The vertical inertial motion of the granite $x_{g}$ and the top platform of the positioning hexapod $x_{h}$ are measured using geophones[fn:uniaxial_1].
@ -1016,7 +1016,7 @@ Due to the poor coherence at low frequencies, these acrlongpl:frf will only be s
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Uniaxial Model ***** Uniaxial Model
The uniaxial model of the micro-station is shown in Figure\nbsp{}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 acrshortpl:dof. It consists of a mass spring damper system with three acrshortpl:dof.
A mass-spring-damper system represents the granite (with mass $m_g$, stiffness $k_g$ and damping $c_g$). A mass-spring-damper system represents the granite (with mass $m_g$, stiffness $k_g$ and damping $c_g$).
@ -1040,7 +1040,7 @@ The parameters obtained are summarized in Table\nbsp{}ref:tab:uniaxial_ustation_
Two disturbances are considered which are shown in red: the floor motion $x_f$ and the stage vibrations represented by $f_t$. Two disturbances are considered which are shown in red: the floor motion $x_f$ and the stage vibrations represented by $f_t$.
The hammer impacts $F_{h}, F_{g}$ are shown in blue, whereas the measured inertial motions $x_{h}, x_{g}$ are shown in black. 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 ***** Comparison of model and measurements
The transfer functions from the forces injected by the hammers to the measured inertial motion of the positioning hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model. The transfer functions from the forces injected by the hammers to the measured inertial motion of the positioning hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model.
Because the uniaxial model has three acrshortpl:dof, only three modes with frequencies at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$ are modeled. Because the uniaxial model has three acrshortpl:dof, only three modes with frequencies at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$ are modeled.
@ -1055,7 +1055,7 @@ More accurate models will be used later on.
*** Active Platform Model *** Active Platform Model
<<sec:uniaxial_nano_station_model>> <<sec:uniaxial_nano_station_model>>
**** Introduction :ignore: ***** Introduction :ignore:
A model of the active platform and sample is now added on top of the uniaxial model of the micro-station (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). A model of the active platform and sample is now added on top of the uniaxial model of the micro-station (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass).
Disturbances (shown in red) are gls:fs the direct forces applied to the sample (for example cable forces), gls:ft representing the vibrations induced when scanning the different stages and gls:xf the floor motion. 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.
@ -1081,14 +1081,14 @@ The effect of resonances between the sample's acrshort:poi and the active platfo
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Active Platform Parameters ***** Active Platform Parameters
The active platform is represented by a mass spring damper system (shown in blue in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). The active platform is represented by a mass spring damper system (shown in blue in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass).
Its mass gls:mn is set to $15\,\text{kg}$ while its stiffness $k_n$ can vary depending on the chosen architecture/technology. 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 $50\,\text{kg}$. 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 active platform stiffness of is set at $k_n = 10\,\text{N}/\mu\text{m}$ and the sample mass is chosen at $m_s = 10\,\text{kg}$. As a first example, the active platform stiffness of is set at $k_n = 10\,\text{N}/\mu\text{m}$ and the sample mass is chosen at $m_s = 10\,\text{kg}$.
**** Obtained Dynamic Response ***** Obtained Dynamic Response
The sensitivity to disturbances (i.e., the transfer functions from $x_f,f_t,f_s$ to $d$) can be extracted from the uniaxial model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass and are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_first_params. The 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. 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.
@ -1120,7 +1120,7 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nb
*** Disturbance Identification *** Disturbance Identification
<<sec:uniaxial_disturbances>> <<sec:uniaxial_disturbances>>
**** Introduction :ignore: ***** Introduction :ignore:
To quantify disturbances (red signals in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), three geophones[fn:uniaxial_2] are used. 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 positioning hexapod's top platform (see Figure\nbsp{}ref:fig:uniaxial_ustation_meas_disturbances). One is located on the floor, another one on the granite, and the last one on the positioning hexapod's top platform (see Figure\nbsp{}ref:fig:uniaxial_ustation_meas_disturbances).
The geophone located on the floor was used to measure the floor motion $x_f$ while the other two geophones were used to measure vibrations introduced by scanning of the $T_y$ stage and $R_z$ stage (see Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup). 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).
@ -1143,7 +1143,7 @@ The geophone located on the floor was used to measure the floor motion $x_f$ whi
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Ground Motion ***** Ground Motion
To acquire the geophone signals, the measurement setup shown in Figure\nbsp{}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 $60\,\text{dB}$ before going to the acrfull:adc. The voltage generated by the geophone is amplified using a low noise voltage amplifier[fn:uniaxial_3] with a gain of $60\,\text{dB}$ before going to the acrfull:adc.
This is done to improve the signal-to-noise ratio. This is done to improve the signal-to-noise ratio.
@ -1186,7 +1186,7 @@ The estimated acrshort:asd $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Stage Vibration ***** Stage Vibration
To estimate the vibrations induced by scanning the micro-station stages, two geophones are used, as shown in Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup. 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 positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. The vertical relative velocity between the top platform of the positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm.
The vibrations induced by the $T_y$ stage are not considered here because they have less amplitude than the vibrations induced by the $R_z$ stage and because the $T_y$ stage can be scanned at lower velocities if the induced vibrations are found to be an issue. The 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.
@ -1209,16 +1209,15 @@ The amplitude spectral density $\Gamma_{f_{t}}$ of the disturbance force is them
*** Open-Loop Dynamic Noise Budgeting *** Open-Loop Dynamic Noise Budgeting
<<sec:uniaxial_noise_budgeting>> <<sec:uniaxial_noise_budgeting>>
**** Introduction :ignore: ***** Introduction :ignore:
Now that a model of the acrshort:nass has been obtained (see section\nbsp{}ref:sec:uniaxial_nano_station_model) and that the disturbances have been estimated (see section\nbsp{}ref:sec:uniaxial_disturbances), it is possible to perform an /open-loop dynamic noise budgeting/. Now that a model of the acrshort:nass has been obtained (see section\nbsp{}ref:sec:uniaxial_nano_station_model) and that the disturbances have been estimated (see section\nbsp{}ref:sec:uniaxial_disturbances), it is possible to perform an /open-loop dynamic noise budgeting/.
To perform such noise budgeting, the disturbances need to be modeled by their spectral densities (done in section\nbsp{}ref:sec:uniaxial_disturbances). To perform such 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). Then, the transfer functions from disturbances to the performance metric (here the distance $d$) are computed.
Finally, these two types of information are combined to estimate the corresponding spectral density of the performance metric. 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\nbsp{}ref:ssec:uniaxial_noise_budget_result). This is very useful to identify what is limiting the performance of the system, and to compare the achievable performance with different system parameters.
**** Sensitivity to disturbances ***** Sensitivity to disturbances
<<ssec:uniaxial_noise_budget_sensitivity>>
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. 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 active platform stiffnesses: This is done for two extreme sample masses $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$ and three active platform stiffnesses:
@ -1255,8 +1254,7 @@ The obtained sensitivity to disturbances for the three active platform stiffness
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Open-Loop Dynamic Noise Budgeting ***** Open-Loop Dynamic Noise Budgeting
<<ssec:uniaxial_noise_budget_result>>
Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion $d$ for each active platform and sample configuration. Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion $d$ for each active platform and sample configuration.
The acrfull:cas of the relative motion $d$ due to both floor motion $x_f$ and stage vibrations $f_t$ are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three active platform stiffnesses. The acrfull:cas of the relative motion $d$ due to both floor motion $x_f$ and stage vibrations $f_t$ are shown in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_stiffnesses for the three active platform stiffnesses.
It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft active platform below $5\,\text{Hz}$. It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft active platform below $5\,\text{Hz}$.
@ -1282,7 +1280,7 @@ The conclusion is that the sample mass has little effect on the cumulative ampli
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Conclusion ***** Conclusion
The open-loop residual vibrations of $d$ can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. The open-loop residual vibrations of $d$ can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses.
This residual vibration of $d$ is found to be in the order of $100\,\text{nm RMS}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$), $200\,\text{nm RMS}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$) and $1\,\mu\text{m}\,\text{RMS}$ for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$). This residual vibration of $d$ is found to be in the order of $100\,\text{nm RMS}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$), $200\,\text{nm RMS}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$) and $1\,\mu\text{m}\,\text{RMS}$ for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$).
@ -1299,10 +1297,10 @@ The advantage of the soft active platform can be explained by its natural isolat
*** Active Damping *** Active Damping
<<sec:uniaxial_active_damping>> <<sec:uniaxial_active_damping>>
**** Introduction :ignore: ***** Introduction :ignore:
In this section, three active damping techniques are applied to the active platform (see Figure\nbsp{}ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]], Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] and Direct Velocity Feedback (DVF)\nbsp{}[[cite:&karnopp74_vibrat_contr_using_semi_activ_force_gener;&serrand00_multic_feedb_contr_isolat_base_excit_vibrat;&preumont02_force_feedb_versus_accel_feedb]]. In this section, three active damping techniques are applied to the active platform (see Figure\nbsp{}ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]], Relative Damping Control (RDC)\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] and Direct Velocity Feedback (DVF)\nbsp{}[[cite:&karnopp74_vibrat_contr_using_semi_activ_force_gener;&serrand00_multic_feedb_contr_isolat_base_excit_vibrat;&preumont02_force_feedb_versus_accel_feedb]].
These damping strategies are first described (Section\nbsp{}ref:ssec:uniaxial_active_damping_strategies) and are then compared in terms of achievable damping of the active platform mode (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping), reduction of the effect of disturbances (i.e., $x_f$, $f_t$ and $f_s$) on the displacement $d$ (Sections\nbsp{}ref:ssec:uniaxial_active_damping_sensitivity_disturbances). These damping strategies are first described and are then compared in terms of achievable damping of the active platform mode, reduction of the effect of disturbances (i.e., $x_f$, $f_t$ and $f_s$) on the displacement $d$.
#+name: fig:uniaxial_active_damping_strategies #+name: fig:uniaxial_active_damping_strategies
#+caption: Three active damping strategies. Integral Force Feedback (\subref{fig:uniaxial_active_damping_strategies_iff}) using a force sensor, Relative Damping Control (\subref{fig:uniaxial_active_damping_strategies_rdc}) using a relative displacement sensor, and Direct Velocity Feedback (\subref{fig:uniaxial_active_damping_strategies_dvf}) using a geophone #+caption: Three active damping strategies. Integral Force Feedback (\subref{fig:uniaxial_active_damping_strategies_iff}) using a force sensor, Relative Damping Control (\subref{fig:uniaxial_active_damping_strategies_rdc}) using a relative displacement sensor, and Direct Velocity Feedback (\subref{fig:uniaxial_active_damping_strategies_dvf}) using a geophone
@ -1328,8 +1326,6 @@ These damping strategies are first described (Section\nbsp{}ref:ssec:uniaxial_ac
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Active Damping Strategies
<<ssec:uniaxial_active_damping_strategies>>
***** Integral Force Feedback (IFF) ***** Integral Force Feedback (IFF)
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. 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.
@ -1413,8 +1409,7 @@ This is usually referred to as "/sky hook damper/".
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Plant Dynamics for Active Damping ***** Plant Dynamics for Active Damping
<<ssec:uniaxial_active_damping_plants>>
The plant dynamics for all three active damping techniques are shown in Figure\nbsp{}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. 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.
@ -1449,9 +1444,7 @@ Therefore, it is expected that the micro-station dynamics might impact the achie
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Achievable Damping and Damped Plants ***** Achievable Damping and Damped Plants
<<ssec:uniaxial_active_damping_achievable_damping>>
To compare the added damping using the three considered active damping strategies, the root locus plot is used. 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\nbsp{}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. Increasing the angle with the imaginary axis therefore means that more damping is added to the considered resonance.
@ -1529,9 +1522,7 @@ All three active damping techniques yielded similar damped plants.
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Sensitivity to disturbances and Noise Budgeting ***** Sensitivity to disturbances and Noise Budgeting
<<ssec:uniaxial_active_damping_sensitivity_disturbances>>
Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped. Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped.
The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping. The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping.
The comparison is done with the active platform having a stiffness $k_n = 1\,\text{N}/\mu\text{m}$. The comparison is done with the active platform having a stiffness $k_n = 1\,\text{N}/\mu\text{m}$.
@ -1595,16 +1586,15 @@ All three active damping methods give similar results.
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Conclusion ***** Conclusion
Three active damping strategies have been studied for the acrfull:nass. Three active damping strategies have been studied for the acrfull:nass.
Equivalent mechanical representations were derived in Section\nbsp{}ref:ssec:uniaxial_active_damping_strategies which are helpful for understanding the specific effects of each strategy. Equivalent mechanical representations were derived 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. The plant dynamics were then compared and were found to all have alternating poles and zeros, which helps in the design of the active damping controller.
However, this property is not guaranteed for acrshort:dvf. However, this property is not guaranteed for acrshort:dvf.
The achievable damping of the active platform suspension mode can be made as large as possible for all three active damping techniques (Section\nbsp{}ref:ssec:uniaxial_active_damping_achievable_damping). The achievable damping of the active platform suspension mode can be made as large as possible for all three active damping techniques.
Even some damping can be applied to some micro-station modes in specific cases. Even some damping can be applied to some micro-station modes in specific cases.
The obtained damped plants were found to be similar. The obtained damped plants were found to be similar.
The damping strategies were then compared in terms of disturbance reduction in Section\nbsp{}ref:ssec:uniaxial_active_damping_sensitivity_disturbances. The damping strategies were then compared in terms of disturbance reduction.
The comparison between the three active damping strategies is summarized in Table\nbsp{}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. It is difficult to conclude on the best active damping strategy for the acrfull:nass yet.
@ -1628,7 +1618,7 @@ Which one will be used will be determined by the use of more accurate models and
*** Position Feedback Controller *** Position Feedback Controller
<<sec:uniaxial_position_control>> <<sec:uniaxial_position_control>>
**** Introduction :ignore: ***** Introduction :ignore:
The gls:haclac architecture is shown in Figure\nbsp{}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: This corresponds to a /two step/ control strategy:
- First, an active damping controller $\bm{K}_{\textsc{LAC}}$ is implemented (see Section\nbsp{}ref:sec:uniaxial_active_damping). - First, an active damping controller $\bm{K}_{\textsc{LAC}}$ is implemented (see Section\nbsp{}ref:sec:uniaxial_active_damping).
@ -1658,8 +1648,7 @@ This control architecture applied to the uniaxial model is shown in Figure\nbsp{
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Damped Plant Dynamics ***** Damped Plant Dynamics
<<ssec:uniaxial_position_control_damped_dynamics>>
The damped plants obtained for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses. The damped plants obtained for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses.
For $k_n = 0.01\,\text{N}/\mu\text{m}$ and $k_n = 1\,\text{N}/\mu\text{m}$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). For $k_n = 0.01\,\text{N}/\mu\text{m}$ and $k_n = 1\,\text{N}/\mu\text{m}$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid).
However, this is not the case for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). However, this is not the case for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff).
@ -1690,8 +1679,7 @@ This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Position Feedback Controller ***** Position Feedback Controller
<<ssec:uniaxial_position_control_design>>
The objective is to design high-authority feedback controllers for the three active platforms. The objective is to design high-authority feedback controllers for the three active platforms.
This controller must be robust to the change of sample's mass (from $1\,\text{kg}$ up to $50\,\text{kg}$). This controller must be robust to the change of sample's mass (from $1\,\text{kg}$ up to $50\,\text{kg}$).
@ -1800,8 +1788,7 @@ The goal is to have a first estimation of the attainable performance.
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Closed-Loop Noise Budgeting ***** Closed-Loop Noise Budgeting
<<ssec:uniaxial_position_control_cl_noise_budget>>
The acrlong:hac are then implemented and the closed-loop sensitivities to disturbances are computed. The acrlong:hac are then implemented and the closed-loop sensitivities to disturbances are computed.
These are compared with the open-loop and damped plants cases in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff active platform with $25\,\text{kg}$ sample's mass). These are compared with the open-loop and damped plants cases in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff active platform with $25\,\text{kg}$ sample's mass).
@ -1859,7 +1846,7 @@ Obtained root mean square values of the distance $d$ are better for the soft act
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Conclusion ***** Conclusion
On the basis of the open-loop noise budgeting made in Section\nbsp{}ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,\text{nm RMS}$ was estimated. On the basis of the open-loop noise budgeting made in Section\nbsp{}ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain a vibration level of $\approx 20\,\text{nm RMS}$ was estimated.
To achieve such bandwidth, the acrshort:haclac strategy was followed, which consists of first using an active damping controller (studied in Section\nbsp{}ref:sec:uniaxial_active_damping) and then adding a high authority position feedback controller. 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.
@ -1871,7 +1858,7 @@ A slight advantage can be given to the soft active platform as it requires less
*** Effect of limited micro-station compliance *** Effect of limited micro-station compliance
<<sec:uniaxial_support_compliance>> <<sec:uniaxial_support_compliance>>
**** Introduction :ignore: ***** 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. 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\nbsp{}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.
@ -1900,7 +1887,7 @@ The second one consists of the active platform fixed on top of the micro-station
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Neglected support compliance ***** Neglected support compliance
The limited compliance of the micro-station is first neglected and the uniaxial model shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_nano_hexapod_only is used. The 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 active platform mass (including the payload) is set at $20\,\text{kg}$ and three active platform stiffnesses are considered, such that their resonance frequencies are at $\omega_{n} = 10\,\text{Hz}$, $\omega_{n} = 70\,\text{Hz}$ and $\omega_{n} = 400\,\text{Hz}$. The active platform mass (including the payload) is set at $20\,\text{kg}$ and three active platform stiffnesses are considered, such that their resonance frequencies are at $\omega_{n} = 10\,\text{Hz}$, $\omega_{n} = 70\,\text{Hz}$ and $\omega_{n} = 400\,\text{Hz}$.
@ -1931,7 +1918,7 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Effect of support compliance on $L/F$ ***** Effect of support compliance on $L/F$
Some support compliance is now added and the model shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system is used. 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 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).
@ -1968,7 +1955,7 @@ If a soft active platform is used, the support dynamics appears in the dynamics
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Effect of support compliance on $d/F$ ***** Effect of support compliance on $d/F$
When the motion to be controlled is the relative displacement $d$ between the granite and the active platform's top platform (which is the case for the acrshort:nass), the effect of the support compliance on the plant dynamics is opposite to that previously observed. When the motion to be controlled is the relative displacement $d$ between the granite and the active platform's top platform (which is the case for the acrshort:nass), the effect of the support compliance on the plant dynamics is opposite to that previously observed.
Indeed, using a "soft" active platform (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_soft). Indeed, using a "soft" active platform (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_d_soft).
@ -1998,7 +1985,7 @@ Conversely, if a "stiff" active platform is used, the support dynamics appears i
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Conclusion ***** Conclusion
To study the impact of support compliance on plant dynamics, simple models shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_models were used. 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 active platform) with respect to $\omega_\mu$ (modes of the support), the interaction between the support and the active platform dynamics can drastically change (observations made are summarized in Table\nbsp{}ref:tab:uniaxial_effect_compliance). Depending on the quantity to be controlled ($L$ or $d$ in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system) and on the relative location of $\omega_\nu$ (suspension mode of the active platform) with respect to $\omega_\mu$ (modes of the support), the interaction between the support and the active platform dynamics can drastically change (observations made are summarized in Table\nbsp{}ref:tab:uniaxial_effect_compliance).
@ -2018,7 +2005,7 @@ Note that the observations made in this section are also affected by the ratio b
*** Effect of Payload Dynamics *** Effect of Payload Dynamics
<<sec:uniaxial_payload_dynamics>> <<sec:uniaxial_payload_dynamics>>
**** Introduction :ignore: ***** Introduction :ignore:
Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic). Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic).
However, such a sample may present internal dynamics, and its fixation to the active platform may have limited stiffness. However, such a sample may present internal dynamics, and its fixation to the active platform may have limited stiffness.
@ -2042,8 +2029,7 @@ To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Impact on plant dynamics ***** Impact on plant dynamics
<<ssec:uniaxial_payload_dynamics_effect_dynamics>>
To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used. To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used.
Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are considered. Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are considered.
@ -2095,8 +2081,7 @@ Even though the added sample's flexibility still shifts the high frequency mass
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Impact on close loop performances ***** Impact on close loop performances
<<ssec:uniaxial_payload_dynamics_effect_stability>>
Having a flexibility between the measured position (i.e., the top platform of the active platform) and the acrshort:poi to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). Having a flexibility between the measured position (i.e., the top platform of the active platform) and the acrshort:poi to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration).
@ -2137,7 +2122,7 @@ What happens is that above $\omega_s$, even though the motion $d$ can be control
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Conclusion ***** Conclusion
Payload dynamics is usually a major concern when designing a positioning system. Payload dynamics is usually a major concern when designing a positioning system.
In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the active platform $\omega_n$. In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the active platform $\omega_n$.
@ -2147,7 +2132,7 @@ Such additional dynamics can induce stability issues depending on their position
The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload. The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload.
This is why high-bandwidth soft positioning stages are usually restricted to constant and calibrated payloads (CD-player, lithography machines, isolation system for gravitational wave detectors, ...), whereas stiff positioning systems are usually used when the control must be robust to a change of payload's mass (stiff piezo nano-positioning stages for instance). This is why high-bandwidth soft positioning stages are usually restricted to constant and calibrated payloads (CD-player, lithography machines, isolation system for gravitational wave detectors, ...), whereas stiff positioning systems are usually used when the control must be robust to a change of payload's mass (stiff piezo nano-positioning stages for instance).
Having some flexibility between the measurement point and the acrshort:poi (i.e., the sample point to be position on the x-ray) also degrades the position stability as shown in Section\nbsp{}ref:ssec:uniaxial_payload_dynamics_effect_stability. Having some flexibility between the measurement point and the acrshort:poi (i.e., the sample point to be position on the x-ray) also degrades the position stability.
Therefore, it is important to take special care when designing sampling environments, especially if a soft active platform is used. Therefore, it is important to take special care when designing sampling environments, especially if a soft active platform is used.
*** Conclusion *** Conclusion
@ -2199,7 +2184,7 @@ The goal is to determine whether the rotation imposes performance limitation on
*** System Description and Analysis *** System Description and Analysis
<<sec:rotating_system_description>> <<sec:rotating_system_description>>
**** Introduction :ignore: ***** Introduction :ignore:
The system used to study gyroscopic effects consists of a 2-acrshortpl:dof translation stage on top of a rotating stage (Figure\nbsp{}ref:fig:rotating_3dof_model_schematic). The system used to study gyroscopic effects consists of a 2-acrshortpl:dof translation stage on top of a rotating stage (Figure\nbsp{}ref:fig:rotating_3dof_model_schematic).
The rotating stage is supposed to be ideal, meaning it induces a perfect rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\s}$. The 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 $F_u, F_v$. 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 $F_u, F_v$.
@ -2213,7 +2198,7 @@ After the dynamics of this system is studied, the objective will be to dampen th
#+attr_latex: :scale 0.8 #+attr_latex: :scale 0.8
[[file:figs/rotating_3dof_model_schematic.png]] [[file:figs/rotating_3dof_model_schematic.png]]
**** Equations of motion and transfer functions ***** Equations of motion and transfer functions
To obtain the equations of motion for the system represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic, the Lagrangian equation\nbsp{}eqref:eq:rotating_lagrangian_equations is used. To obtain the equations of motion for the system represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic, the Lagrangian equation\nbsp{}eqref:eq:rotating_lagrangian_equations is used.
$L = T - V$ is the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. $L = T - V$ is the Lagrangian, $T$ the kinetic 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\nbsp{}eqref:eq:rotating_energy_functions_lagrange. These terms are derived in\nbsp{}eqref:eq:rotating_energy_functions_lagrange.
@ -2271,7 +2256,7 @@ The elements of the transfer function matrix $\bm{G}_d$ are described by equatio
\end{align} \end{align}
\end{subequations} \end{subequations}
**** System Poles: Campbell Diagram ***** System Poles: Campbell Diagram
The poles of $\bm{G}_d$ are the complex solutions $p$ of equation\nbsp{}eqref:eq:rotating_poles (i.e. the roots of its denominator). The poles of $\bm{G}_d$ 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} \begin{equation}\label{eq:rotating_poles}
@ -2310,7 +2295,7 @@ Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forc
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** System Dynamics: Effect of rotation ***** 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. The system dynamics from actuator forces $[F_u, F_v]$ to the relative motion $[d_u, d_v]$ is identified for several rotating velocities.
Looking at the transfer function matrix $\bm{G}_d$ in equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. Looking at the transfer function matrix $\bm{G}_d$ in equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite.
The bode plots of these two terms are shown in Figure\nbsp{}ref:fig:rotating_bode_plot for several rotational speeds $\Omega$. The bode plots of these two terms are shown in Figure\nbsp{}ref:fig:rotating_bode_plot for several rotational speeds $\Omega$.
@ -2338,7 +2323,7 @@ For $\Omega > \omega_0$, the low-frequency pair of complex conjugate poles $p_{-
*** Integral Force Feedback *** Integral Force Feedback
<<sec:rotating_iff_pure_int>> <<sec:rotating_iff_pure_int>>
**** Introduction :ignore: ***** Introduction :ignore:
The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation. The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation.
As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances\nbsp{}[[cite:&collette11_review_activ_vibrat_isolat_strat]] and to make the plant easier to control for the high authority controller. As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances\nbsp{}[[cite:&collette11_review_activ_vibrat_isolat_strat]] and to make the plant easier to control for the high authority controller.
@ -2355,7 +2340,7 @@ Recently, an $\mathcal{H}_\infty$ optimization criterion has been used to derive
However, none of these studies have been applied to rotating systems. 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. 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 ***** System and Equations of motion
To apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_iff). 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. 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.
@ -2429,7 +2414,7 @@ This small displacement then increases the centrifugal force $m\Omega^2d_u = \fr
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
**** Effect of rotation speed on IFF plant dynamics ***** 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\nbsp{}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: 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. - when $\Omega < \omega_0$: the low-frequency gain is no longer zero and two (non-minimum phase) real zeros appear at low-frequencies.
@ -2455,7 +2440,7 @@ As expected from the derived equations of motion:
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Decentralized Integral Force Feedback ***** Decentralized Integral Force Feedback
The control diagram for decentralized acrshort:iff is shown in Figure\nbsp{}ref:fig:rotating_iff_diagram. 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. The decentralized acrshort:iff controller $\bm{K}_F$ corresponds to a diagonal controller with integrators\nbsp{}eqref:eq:rotating_Kf_pure_int.
@ -2477,7 +2462,7 @@ The control system is thus cancels the spring forces, which makes the suspended
*** Integral Force Feedback with a High-Pass Filter *** Integral Force Feedback with a High-Pass Filter
<<sec:rotating_iff_pseudo_int>> <<sec:rotating_iff_pseudo_int>>
**** Introduction :ignore: ***** 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. As explained in the previous section, the instability of the IFF controller applied to the rotating system is due to the high gain of the integrator at low-frequency.
To limit the low-frequency controller gain, a acrfull:hpf can be added to the controller, as shown in equation\nbsp{}eqref:eq:rotating_iff_lhf. 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 is equivalent to slightly shifting the controller pole to the left along the real axis.
@ -2488,7 +2473,7 @@ This is however not the reason why this acrlong:hpf is added here.
\boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} \boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}}
\end{equation} \end{equation}
**** Modified Integral Force Feedback Controller ***** 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\nbsp{}eqref:eq:rotating_iff_lhf where $\omega_i$ characterize the frequency down to which the signal is integrated. 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 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 effect of the added HPF limits the low-frequency gain to finite values as expected.
@ -2519,7 +2504,7 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the controll
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Optimal IFF with HPF parameters $\omega_i$ and $g$ ***** Optimal IFF with HPF parameters $\omega_i$ and $g$
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 $\omega_i$. 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. 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.
@ -2549,7 +2534,7 @@ For larger values of $\omega_i$, the attainable damping ratio decreases as a fun
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Obtained Damped Plant ***** 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\nbsp{}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. 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 $\omega_i$. Therefore, there is a trade-off between achievable damping and added coupling when tuning $\omega_i$.
@ -2576,7 +2561,7 @@ The same trade-off can be seen between achievable damping and loss of compliance
*** IFF with a stiffness in parallel with the force sensor *** IFF with a stiffness in parallel with the force sensor
<<sec:rotating_iff_parallel_stiffness>> <<sec:rotating_iff_parallel_stiffness>>
**** Introduction :ignore: ***** 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. 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\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. 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.
@ -2585,7 +2570,7 @@ Such springs are schematically shown in Figure\nbsp{}ref:fig:rotating_3dof_model
#+attr_latex: :scale 0.8 #+attr_latex: :scale 0.8
[[file:figs/rotating_3dof_model_schematic_iff_parallel_springs.png]] [[file:figs/rotating_3dof_model_schematic_iff_parallel_springs.png]]
**** Equations ***** Equations
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. 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{equation}\label{eq:rotating_measured_force_kp}
@ -2624,7 +2609,7 @@ Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiff
\boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2} \boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2}
\end{equation} \end{equation}
**** Effect of parallel stiffness on the IFF plant ***** 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 $k_p > m \Omega^2$. 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\nbsp{}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 $k_p > m \Omega^2$. The two real zeros for $k_p < m \Omega^2$ are transformed into two complex conjugate zeros for $k_p > m \Omega^2$.
@ -2651,7 +2636,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Effect of $k_p$ on the attainable damping ***** 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. 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\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition]], the parallel stiffness $k_p$ is expected to have some impact on the attainable damping. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition]], the parallel stiffness $k_p$ is expected to have some impact on the attainable damping.
To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus_effect_kp. 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.
@ -2677,7 +2662,7 @@ This is confirmed by the Figure\nbsp{}ref:fig:rotating_iff_kp_optimal_gain where
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Damped plant ***** Damped plant
The parallel stiffness are chosen to be $k_p = 2 m \Omega^2$ and the damped plant is computed. 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\nbsp{}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. Even though the two resonances are well damped, the IFF changes the low-frequency behavior of the plant, which is usually not desired.
@ -2718,7 +2703,7 @@ The added acrshort:hpf gives almost the same damping properties to the suspensio
*** Relative Damping Control *** Relative Damping Control
<<sec:rotating_relative_damp_control>> <<sec:rotating_relative_damp_control>>
**** Introduction :ignore: ***** Introduction :ignore:
To apply a "Relative Damping Control" strategy, relative motion sensors are added in parallel with the actuators as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic_rdc. To apply a "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. Two controllers $K_d$ are used to feed back the relative motion to the actuator.
These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter\nbsp{}eqref:eq:rotating_rdc_controller. These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter\nbsp{}eqref:eq:rotating_rdc_controller.
@ -2732,7 +2717,7 @@ K_d(s) = g \cdot \frac{s}{s + \omega_d}
#+attr_latex: :scale 0.8 #+attr_latex: :scale 0.8
[[file:figs/rotating_3dof_model_schematic_rdc.png]] [[file:figs/rotating_3dof_model_schematic_rdc.png]]
**** Equations of motion ***** Equations of motion
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. 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. 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.
@ -2754,7 +2739,7 @@ Therefore, for $\Omega < \sqrt{k/m}$ (i.e. stable system), the transfer function
z = \pm j \sqrt{\omega_0^2 - \omega^2}, \quad p_1 = \pm j (\omega_0 - \omega), \quad p_2 = \pm j (\omega_0 + \omega) z = \pm j \sqrt{\omega_0^2 - \omega^2}, \quad p_1 = \pm j (\omega_0 - \omega), \quad p_2 = \pm j (\omega_0 + \omega)
\end{equation} \end{equation}
**** Decentralized Relative Damping Control ***** 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\nbsp{}ref:sec:rotating_system_description and are shown in Figure\nbsp{}ref:fig:rotating_bode_plot (page\nbsp{}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\nbsp{}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).
@ -2786,12 +2771,12 @@ It does not increase the low-frequency coupling as compared to the Integral Forc
*** Comparison of Active Damping Techniques *** Comparison of Active Damping Techniques
<<sec:rotating_comp_act_damp>> <<sec:rotating_comp_act_damp>>
**** Introduction :ignore: ***** Introduction :ignore:
These two proposed IFF modifications and relative damping control are compared in terms of added damping and closed-loop behavior. These two proposed IFF modifications and relative damping control are compared in terms of added damping and closed-loop behavior.
For the following comparisons, the cut-off frequency for the added HPF is set to $\omega_i = 0.1 \omega_0$ and the stiffness of the parallel springs is set to $k_p = 5 m \Omega^2$ (corresponding to $\alpha = 0.05$). For the following comparisons, the cut-off frequency for the added HPF is set to $\omega_i = 0.1 \omega_0$ and the stiffness of the parallel springs is set to $k_p = 5 m \Omega^2$ (corresponding to $\alpha = 0.05$).
These values are chosen one the basis of previous discussions about optimal parameters. These values are chosen one the basis of previous discussions about optimal parameters.
**** Root Locus ***** Root Locus
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. 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. 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. This means that the closed-loop behavior of both systems will differ when large control gains are used.
@ -2818,12 +2803,12 @@ It is interesting to note that the maximum added damping is very similar for bot
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Obtained Damped Plant ***** Obtained Damped Plant
The actively damped plants are computed for the three techniques and compared in Figure\nbsp{}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. 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. The acrshort:iff strategy is adding some coupling at low-frequency, which may negatively impact the positioning performance.
**** Transmissibility And Compliance ***** Transmissibility And Compliance
The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance. The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance.
The transmissibility is defined as the transfer function from the displacement of the rotating stage along $\vec{i}_x$ to the displacement of the payload along the same direction. The transmissibility is defined as the transfer function from the displacement of the rotating stage along $\vec{i}_x$ to the displacement of the payload along the same direction.
It is used to characterize the amount of vibration is transmitted through the suspended platform to the payload. It is used to characterize the amount of vibration is transmitted through the suspended platform to the payload.
@ -2855,12 +2840,12 @@ This is very well known characteristics of these common active damping technique
*** Rotating Active Platform *** Rotating Active Platform
<<sec:rotating_nano_hexapod>> <<sec:rotating_nano_hexapod>>
**** Introduction :ignore: ***** Introduction :ignore:
The previous analysis is now applied to a model representing a rotating active platform. The previous analysis is now applied to a model representing a rotating active platform.
Three active platform stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. Three active platform stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$.
Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) because this is the worst identified case scenario in terms of gyroscopic effects. Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) because this is the worst identified case scenario in terms of gyroscopic effects.
**** Nano-Active-Stabilization-System - Plant Dynamics ***** Nano-Active-Stabilization-System - Plant Dynamics
For the NASS, the maximum rotating velocity is $\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}$ for a suspended mass on top of the active platform's actuators equal to $m_n + m_s = \SI{16}{\kilo\gram}$. For the NASS, the maximum rotating velocity is $\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}$ for a suspended mass on top of the active platform's actuators equal to $m_n + m_s = \SI{16}{\kilo\gram}$.
The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \approx \SI{0.6}{\newton\per\mm}$. The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \approx \SI{0.6}{\newton\per\mm}$.
@ -2893,7 +2878,7 @@ The coupling (or interaction) in a acrshort:mimo $2 \times 2$ system can be visu
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Optimal IFF with a High-Pass Filter ***** Optimal IFF with a High-Pass Filter
Integral Force Feedback with an added acrlong:hpf is applied to the three active platforms. Integral Force Feedback with an added acrlong:hpf is applied to the three active platforms.
First, the parameters ($\omega_i$ and $g$) of the IFF controller that yield the best simultaneous damping are determined from Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain. 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: The IFF parameters are chosen as follows:
@ -2937,7 +2922,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
| $1\,\text{N}/\mu\text{m}$ | 39 | 427 | 0.93 | | $1\,\text{N}/\mu\text{m}$ | 39 | 427 | 0.93 |
| $100\,\text{N}/\mu\text{m}$ | 500 | 3775 | 0.94 | | $100\,\text{N}/\mu\text{m}$ | 500 | 3775 | 0.94 |
**** Optimal IFF with Parallel Stiffness ***** Optimal IFF with Parallel Stiffness
For each considered active platform stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total active platform stiffness). For each considered active platform stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total active platform stiffness).
To keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased ($k_a = k_n - k_p$, with $k_n$ the total active platform stiffness). To keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased ($k_a = k_n - k_p$, with $k_n$ the total active platform stiffness).
A high-pass filter is also added to limit the low-frequency gain with a cut-off frequency $\omega_i$ equal to one tenth of the system resonance ($\omega_i = \omega_0/10$). 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 ($\omega_i = \omega_0/10$).
@ -2972,7 +2957,7 @@ The corresponding optimal controller gains and achievable damping are summarized
#+latex: \captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses} #+latex: \captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses}
#+end_minipage #+end_minipage
**** Optimal Relative Motion Control ***** Optimal Relative Motion Control
For each considered active platform stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure\nbsp{}ref:fig:rotating_rdc_optimal_gain). For each considered active platform stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure\nbsp{}ref:fig:rotating_rdc_optimal_gain).
The gain is chosen such that 99% of modal damping is obtained (obtained gains are summarized in Table\nbsp{}ref:tab:rotating_rdc_opt_params_nass). 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).
@ -2997,7 +2982,7 @@ The gain is chosen such that 99% of modal damping is obtained (obtained gains ar
#+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC} #+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC}
#+end_minipage #+end_minipage
**** Comparison of the obtained damped plants ***** 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\nbsp{}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: Similar to what was concluded in the previous analysis:
@ -3031,12 +3016,12 @@ Similar to what was concluded in the previous analysis:
*** Nano-Active-Stabilization-System with rotation *** Nano-Active-Stabilization-System with rotation
<<sec:rotating_nass>> <<sec:rotating_nass>>
**** Introduction :ignore: ***** Introduction :ignore:
Until now, the model used to study gyroscopic effects consisted of an infinitely stiff rotating stage with a X-Y suspended stage on top. Until now, the model used to study gyroscopic effects consisted of an infinitely stiff rotating stage with a X-Y suspended stage on top.
While quite simplistic, this allowed us to study the effects of rotation and the associated limitations when active damping is to be applied. While quite simplistic, this allowed us to study the effects of rotation and the associated limitations when active damping is to be applied.
In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. 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 ***** Nano Active Stabilization System model
To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view):
- the floor whose motion is imposed - 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 granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$)
@ -3051,7 +3036,7 @@ A payload is rigidly fixed to the active platform and the $x,y$ motion of the pa
#+attr_latex: :scale 0.7 #+attr_latex: :scale 0.7
[[file:figs/rotating_nass_model.png]] [[file:figs/rotating_nass_model.png]]
**** System dynamics ***** System dynamics
The dynamics of the undamped and damped plants are identified using the optimal parameters found in Section\nbsp{}ref:sec:rotating_nano_hexapod. 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. 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.
@ -3085,7 +3070,7 @@ It can be observed that:
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
**** Effect of disturbances ***** Effect of disturbances
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). 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. 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.