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@ -114,31 +114,33 @@ Prefix is =rotating=
| iff | IFF | Integral Force Feedback | | iff | IFF | Integral Force Feedback |
| rdc | RDC | Relative Damping Control | | rdc | RDC | Relative Damping Control |
| drga | DRGA | Dynamical Relative Gain Array | | drga | DRGA | Dynamical Relative Gain Array |
| hpf | HPF | high-pass filter |
| lpf | LPF | low-pass filter |
* Introduction :ignore: * Introduction :ignore:
An important aspect of the acrfull:nass is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. 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. 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 ref:sec:rotating_system_description)
This model is simple enough to be able to derive its dynamics analytically and to well understand its behavior, while still allowing to capture the important physical effects in play. 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 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 ref:sec:rotating_iff_pure_int).
Two modifications of the Integral Force Feedback are then proposed. Two modifications of the Integral Force Feedback are then proposed.
The first one consists of adding an 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 ref:sec:rotating_iff_pseudo_int).
It is shown that the acrshort:iff controller is stable for some values of the gain, and that damping can be added to the suspension modes. It is shown that the acrshort:iff controller is stable for some gain values, and that damping can be added to the suspension modes.
Optimal high pass filter cut-off frequency is computed. 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 ref:sec:rotating_iff_parallel_stiffness).
Under a certain condition, the unconditional stability of the the IFF controller is regained. Under certain conditions, the unconditional stability of the IFF controller is regained.
Optimal parallel stiffness is then computed. The optimal parallel stiffness is then computed.
This study of adapting acrshort:iff for the damping of rotating platforms was 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 [[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). 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, obtained damped plant and closed-loop compliance and transmissibility (Section ref:sec:rotating_comp_act_damp). 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).
The previous analysis is applied on 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 optimal active damping controller are 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 ref:sec:rotating_nano_hexapod).
Up until this section, the study was performed on a very simplistic model that just captures the rotation aspect and the model parameters were not tuned to corresponds to the NASS. 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 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 if the rotation imposes performance limitation for the NASS. The goal is to determine whether the rotation imposes performance limitation on the NASS.
#+name: fig:rotating_overview #+name: fig:rotating_overview
#+caption: Overview of this chapter's organization. Sections are indicated by the red circles. #+caption: Overview of this chapter's organization. Sections are indicated by the red circles.
@ -152,13 +154,13 @@ The goal is to determine if the rotation imposes performance limitation for the
<<sec:rotating_system_description>> <<sec:rotating_system_description>>
** Introduction :ignore: ** Introduction :ignore:
The studied system 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 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 $F_u, F_v$.
A payload with a mass $m$ in $\si{\kilo\gram}$, is mounted on the (rotating) suspended platform. A payload with a mass $m$ in $\si{\kilo\gram}$, is mounted on the (rotating) suspended platform.
Two reference frames are used: an /inertial/ frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ and a /uniform rotating/ frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ rigidly fixed on top of the rotating stage with $\vec{i}_w$ aligned with the rotation axis. Two reference frames are used: an /inertial/ frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ and a /uniform rotating/ frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ rigidly fixed on top of the rotating stage with $\vec{i}_w$ aligned with the rotation axis.
The position of the payload is represented by $(d_u, d_v, 0)$ expressed in the rotating frame. The position of the payload is represented by $(d_u, d_v, 0)$ expressed in the rotating frame.
After the dynamics of this system is studied, the objective will be to damp the two suspension modes of the payload while the rotating stage performs a constant rotation. After the dynamics of this system is studied, the objective will be to dampen the two suspension modes of the payload while the rotating stage performs a constant rotation.
#+begin_src latex :file rotating_3dof_model_schematic.pdf #+begin_src latex :file rotating_3dof_model_schematic.pdf
\begin{tikzpicture} \begin{tikzpicture}
@ -255,7 +257,7 @@ mdl = 'rotating_model';
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 ref:fig:rotating_3dof_model_schematic, the Lagrangian equation 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 eqref:eq:rotating_energy_functions_lagrange. These terms are derived in eqref:eq:rotating_energy_functions_lagrange.
Note that the equation of motion corresponding to the constant rotation along $\vec{i}_w$ is disregarded as this motion is considered to be imposed by the rotation stage. 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} \begin{equation}\label{eq:rotating_lagrangian_equations}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
@ -278,9 +280,9 @@ Substituting equations eqref:eq:rotating_energy_functions_lagrange into equation
\end{subequations} \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 eqref:eq:rotating_eom_coupled:
- /Centrifugal forces/: that can been seen as an added /negative stiffness/ $- m \Omega^2$ along $\vec{i}_u$ and $\vec{i}_v$ - /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. - /Coriolis forces/: that adds /coupling/ between the two orthogonal directions.
One can verify that without rotation ($\Omega = 0$) the system becomes equivalent to two /uncoupled/ one degree of freedom mass-spring-damper systems. One can verify that without rotation ($\Omega = 0$), the system becomes equivalent to two /uncoupled/ one 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. 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. The four transfer functions in $\mathbf{G}_d$ are shown in equation eqref:eq:rotating_Gd_indiv_el.
@ -297,7 +299,7 @@ The four transfer functions in $\mathbf{G}_d$ are shown in equation eqref:eq:rot
\end{subequations} \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. 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 transfer function matrix $\mathbf{G}_d$ are now described by equation eqref:eq:rotating_Gd_w0_xi_k. The elements of the transfer function matrix $\mathbf{G}_d$ are described by equation eqref:eq:rotating_Gd_w0_xi_k.
\begin{equation} \label{eq:rotating_xi_and_omega} \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}} \omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}}
\end{equation} \end{equation}
@ -477,9 +479,9 @@ save('./mat/rotating_generic_plants.mat', 'Gs', 'Wzs');
** 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 $\mathbf{G}_d$ 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. 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 plot of these two terms are shown in Figure ref:fig:rotating_bode_plot for several rotational speeds $\Omega$. The bode plots of these two terms are shown in Figure ref:fig:rotating_bode_plot for several rotational speeds $\Omega$.
These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as $\Omega$ increases. 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). 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).
#+begin_src matlab :results none #+begin_src matlab :results none
%% Bode plot of the direct and coupling terms for several rotating velocities %% Bode plot of the direct and coupling terms for several rotating velocities
@ -589,20 +591,20 @@ exportFig('figs/rotating_bode_plot_coupling.pdf', 'width', 'half', 'height', 600
** 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 active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances cite:collette11_review_activ_vibrat_isolat_strat and to make the plant easier to control for the high authority controller. As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances cite:collette11_review_activ_vibrat_isolat_strat and to make the plant easier to control for the high authority controller.
Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) cite:lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc, Integral Force Feedback (IFF) cite:preumont91_activ and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb. Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF) cite:lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc, Integral Force Feedback (IFF) cite:preumont91_activ and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb.
In [[cite:&preumont91_activ]], the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system. In [[cite:&preumont91_activ]], the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system.
When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitates to guarantee the stability of the closed loop system cite:preumont02_force_feedb_versus_accel_feedb. 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 latter shown that this property holds for multiple collated actuator/sensor pairs cite:preumont08_trans_zeros_struc_contr_with. It was later shown that this property holds for multiple collated actuator/sensor pairs cite:preumont08_trans_zeros_struc_contr_with.
The main advantages of IFF over other active damping techniques are the guaranteed stability even in presence of flexible dynamics, good performance and robustness properties cite:preumont02_force_feedb_versus_accel_feedb. The main advantages of IFF over other active damping techniques are the guaranteed stability even in the presence of flexible dynamics, good performance, and robustness properties cite:preumont02_force_feedb_versus_accel_feedb.
Several improvements of the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping cite:teo15_optim_integ_force_feedb_activ_vibrat_contr or adding an high pass filter to recover the loss of compliance at low frequency cite:chesne16_enhan_dampin_flexib_struc_using_force_feedb. Several improvements to the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping cite:teo15_optim_integ_force_feedb_activ_vibrat_contr or adding a high-pass filter to recover the loss of compliance at low-frequency cite:chesne16_enhan_dampin_flexib_struc_using_force_feedb.
Recently, an $\mathcal{H}_\infty$ optimization criterion has been used to derive optimal gains for the IFF controller cite:zhao19_optim_integ_force_feedb_contr. \par Recently, an $\mathcal{H}_\infty$ optimization criterion has been used to derive optimal gains for the IFF controller cite:zhao19_optim_integ_force_feedb_contr. \par
However, none of these study have been applied to a rotating system. However, none of these studies have been applied to rotating systems.
In this section, Integral Force Feedback strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alters 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.
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -636,7 +638,7 @@ load('rotating_generic_plants.mat', 'Gs', 'Wzs');
#+end_src #+end_src
** System and Equations of motion ** System and Equations of motion
In order to apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure ref:fig:rotating_3dof_model_schematic_iff). 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. 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.
\begin{equation}\label{eq:rotating_iff_controller} \begin{equation}\label{eq:rotating_iff_controller}
@ -803,10 +805,10 @@ It is interesting to see that the frequency of the pair of complex conjugate zer
This is what usually gives the unconditional stability of IFF when collocated force sensors are used. 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/. 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 $\SI{180}{\degree}$. 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 $\SI{180}{\degree}$.
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 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 taking into account the negative stiffness induced by the rotation). 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. 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.
\begin{equation}\label{eq:rotating_low_freq_gain_iff_plan} \begin{equation}\label{eq:rotating_low_freq_gain_iff_plan}
@ -816,13 +818,13 @@ This small displacement then increases the centrifugal force $m\Omega^2d_u = \fr
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
** Effect of the rotation speed on the 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 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 ref:fig:rotating_iff_bode_plot_effect_rot.
As was 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 zero appears at low frequency. - 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 $\Omega$. The low-frequency gain increases with $\Omega$.
A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles that are split further apart as $\Omega$ increases. A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles, which are split further apart as $\Omega$ increases.
- when $\omega_0 < \Omega$: the low frequency pole becomes unstable. - when $\omega_0 < \Omega$: the low-frequency pole becomes unstable.
#+begin_src matlab :results none #+begin_src matlab :results none
%% Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation %% Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation
@ -886,9 +888,9 @@ exportFig('figs/rotating_iff_bode_plot_effect_rot_direct.pdf', 'width', 'half',
#+end_figure #+end_figure
** Decentralized Integral Force Feedback ** Decentralized Integral Force Feedback
The control diagram for decentralized Integral Force Feedback is shown in Figure ref:fig:rotating_iff_diagram. 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 decentralized IFF controller $\bm{K}_F$ corresponds to a diagonal controller with integrators:
\begin{equation} \label{eq:rotating_Kf_pure_int} \begin{equation} \label{eq:rotating_Kf_pure_int}
\begin{aligned} \begin{aligned}
\mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\ \mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\
@ -896,13 +898,13 @@ The decentralized IFF controller $\bm{K}_F$ corresponds to a diagonal controller
\end{aligned} \end{aligned}
\end{equation} \end{equation}
In order to see how the 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 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 $\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};$) for $g = 0$ and coincide with the transmission zeros (shown by $\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];$) as $g \to \infty$. As explained in cite:preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr, the closed-loop poles start at the open-loop poles (shown by $\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};$) for $g = 0$ and coincide with the transmission zeros (shown by $\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];$) 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 in non-null. Whereas collocated IFF is usually associated with unconditional stability cite:preumont91_activ, this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null.
This can be seen in the Root Locus plot (Figure ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. 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 like so: at low frequency, the loop gain is very large due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure ref:fig:rotating_iff_bode_plot_effect_rot). Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure ref:fig:rotating_iff_bode_plot_effect_rot).
The control system is thus canceling the spring forces which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. 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.
#+begin_src matlab #+begin_src matlab
%% Root Locus for the Decentralized Integral Force Feedback controller %% Root Locus for the Decentralized Integral Force Feedback controller
@ -941,18 +943,18 @@ leg.ItemTokenSize(1) = 8;
exportFig('figs/rotating_root_locus_iff_pure_int.pdf', 'width', 600, 'height', 600); exportFig('figs/rotating_root_locus_iff_pure_int.pdf', 'width', 600, 'height', 600);
#+end_src #+end_src
* Integral Force Feedback with an High Pass Filter * Integral Force Feedback with a High-Pass Filter
:PROPERTIES: :PROPERTIES:
:header-args:matlab+: :tangle matlab/rotating_3_iff_hpf.m :header-args:matlab+: :tangle matlab/rotating_3_iff_hpf.m
:END: :END:
<<sec:rotating_iff_pseudo_int>> <<sec:rotating_iff_pseudo_int>>
** Introduction :ignore: ** Introduction :ignore:
As was explained in the previous section, the instability of the IFF controller applied on 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.
In order to limit the low frequency controller gain, an High Pass Filter (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 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.
This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator cite:preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans. This modification of the IFF controller is typically performed to avoid saturation associated with the pure integrator cite:preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans.
This is however not the reason why this high pass filter is added here. This is however not the reason why this high-pass filter is added here.
\begin{equation}\label{eq:rotating_iff_lhf} \begin{equation}\label{eq:rotating_iff_lhf}
\boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} \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}}
@ -992,11 +994,11 @@ load('rotating_generic_plants.mat', 'Gs', 'Wzs');
** 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 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 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 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 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.
The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure ref:fig:rotating_iff_root_locus_hpf_large. 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 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. 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). 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).
\begin{equation}\label{eq:rotating_gmax_iff_hpf} \begin{equation}\label{eq:rotating_gmax_iff_hpf}
\boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)} \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)}
@ -1012,7 +1014,7 @@ Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF
#+end_src #+end_src
#+begin_src matlab :results none #+begin_src matlab :results none
%% Loop gain for the IFF with pure integrator and modified IFF with added high pass filter %% Loop gain for the IFF with pure integrator and modified IFF with added high-pass filter
freqs = logspace(-2, 1, 1000); freqs = logspace(-2, 1, 1000);
Wz_i = 2; Wz_i = 2;
@ -1048,7 +1050,7 @@ xlim([freqs(1), freqs(end)]);
#+end_src #+end_src
#+name: fig:rotating_iff_modified_loop_gain_root_locus #+name: fig:rotating_iff_modified_loop_gain_root_locus
#+caption: Comparison of the IFF with pure integrator and modified IFF with added high pass filter ($\Omega = 0.1\omega_0$). Loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with $\omega_i = 0.1 \omega_0$ and $g = 2$. Root Locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large}) #+caption: Comparison of the IFF with pure integrator and modified IFF with added high-pass filter ($\Omega = 0.1\omega_0$). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with $\omega_i = 0.1 \omega_0$ and $g = 2$. The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large})
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_modified_loop_gain}Loop gain} #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_modified_loop_gain}Loop gain}
@ -1067,18 +1069,18 @@ xlim([freqs(1), freqs(end)]);
** 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 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 eqref:eq:rotating_iff_lhf: the gain $g$ and the pole's location $\omega_i$.
The optimal values of $\omega_i$ and $g$ are here considered as the values for which the damping of all the closed-loop poles are 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.
In order 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. 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. 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.
In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$. 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 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 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 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.
#+begin_src matlab #+begin_src matlab
%% High Pass Filter Cut-Off Frequency %% High-Pass Filter Cut-Off Frequency
wis = [0.01, 0.05, 0.1]; % [rad/s] wis = [0.01, 0.05, 0.1]; % [rad/s]
#+end_src #+end_src
@ -1167,7 +1169,7 @@ exportFig('figs/rotating_iff_hpf_optimal_gain.pdf', 'width', 'half', 'height', 4
#+end_src #+end_src
#+name: fig:rotating_iff_modified_effect_wi #+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 which is confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain}) #+caption: Root Locus for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as $\omega_i$ increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus} #+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus}
@ -1185,10 +1187,10 @@ exportFig('figs/rotating_iff_hpf_optimal_gain.pdf', 'width', 'half', 'height', 4
#+end_figure #+end_figure
** Obtained Damped Plant ** Obtained Damped Plant
In order 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 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.
There is therefore 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 $\omega_i$.
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 ref:fig:rotating_iff_hpf_effect_wi_compliance).
#+begin_src matlab #+begin_src matlab
%% Compute damped plant %% Compute damped plant
@ -1418,14 +1420,14 @@ The forces measured by the two force sensors represented in Figure ref:fig:rotat
\begin{bmatrix} d_u \\ d_v \end{bmatrix} \begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation} \end{equation}
In order 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 eqref:eq:rotating_kp_alpha.
\begin{equation}\label{eq:rotating_kp_alpha} \begin{equation}\label{eq:rotating_kp_alpha}
k_p = \alpha k, \quad k_a = (1 - \alpha) k k_p = \alpha k, \quad k_a = (1 - \alpha) k
\end{equation} \end{equation}
After the equations of motion derived and transformed in the Laplace domain, the transfer function matrix $\mathbf{G}_k$ in Eq. eqref:eq:rotating_Gk_mimo_tf is computed. 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 Eq. eqref:eq:rotating_Gk_diag and eqref:eq:rotating_Gk_off_diag. Its elements are shown in Eqs. eqref:eq:rotating_Gk_diag and eqref:eq:rotating_Gk_off_diag.
\begin{equation}\label{eq:rotating_Gk_mimo_tf} \begin{equation}\label{eq:rotating_Gk_mimo_tf}
\begin{bmatrix} f_u \\ f_v \end{bmatrix} = \begin{bmatrix} f_u \\ f_v \end{bmatrix} =
@ -1440,9 +1442,9 @@ Its elements are shown in Eq. eqref:eq:rotating_Gk_diag and eqref:eq:rotating_Gk
\end{align} \end{align}
\end{subequations} \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 are kept the same, the zeros of the diagonal terms have changed. 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. 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.
Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiffness/ induced by centrifugal forces $m \Omega^2$, the dynamics from actuator to its collocated force sensor will show /minimum phase behavior/. 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} \begin{equation}\label{eq:rotating_kp_cond_cc_zeros}
\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}
@ -1472,14 +1474,14 @@ io(io_i) = linio([mdl, '/translation_stage'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy] io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy]
#+end_src #+end_src
** Effect of the 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 $k_p > m \Omega^2$.
The 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 ref:fig:rotating_iff_effect_kp.
One can see that the 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 $k_p > m \Omega^2$.
In that case, the systems shows alternating complex conjugate poles and zeros as what is the case in the non-rotating case. 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 in Eq. eqref:eq:rotating_Kf_pure_int. 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.
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 IFF is recovered. 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.
#+begin_src matlab #+begin_src matlab
Wz = 0.1; % The rotation speed [rad/s] Wz = 0.1; % The rotation speed [rad/s]
@ -1608,7 +1610,7 @@ leg.ItemTokenSize(1) = 8;
#+end_src #+end_src
#+name: fig:rotating_iff_plant_effect_kp #+name: fig:rotating_iff_plant_effect_kp
#+caption: Effect of the parallel stiffness on the IFF plant #+caption: Effect of parallel stiffness on the IFF plant
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$} #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$}
@ -1627,9 +1629,9 @@ leg.ItemTokenSize(1) = 8;
** 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 cite:preumont18_vibrat_contr_activ_struc_fourt_edition, the parallel stiffness $k_p$ is foreseen to have some impact on the attainable damping. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros cite:preumont18_vibrat_contr_activ_struc_fourt_edition, the parallel stiffness $k_p$ is expected to have some impact on the attainable damping.
To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure ref:fig:rotating_iff_kp_root_locus_effect_kp. To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure ref:fig:rotating_iff_kp_root_locus_effect_kp.
The frequencies of the transmission zeros of the system are increasing with an increase of the parallel stiffness $k_p$ (thus getting closer to the poles) and the associated attainable damping is reduced. 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. 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 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.
@ -1738,16 +1740,16 @@ exportFig('figs/rotating_iff_kp_optimal_gain.pdf', 'width', 'half', 'height', 45
#+end_src #+end_src
#+name: fig:rotating_iff_optimal_kp #+name: fig:rotating_iff_optimal_kp
#+caption: Effect of the parallel stiffness on the IFF plant #+caption: Effect of parallel stiffness on the IFF plant
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of the parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$} #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$}
#+attr_latex: :options {0.49\linewidth} #+attr_latex: :options {0.49\linewidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :scale 1 #+attr_latex: :scale 1
[[file:figs/rotating_iff_kp_root_locus_effect_kp.png]] [[file:figs/rotating_iff_kp_root_locus_effect_kp.png]]
#+end_subfigure #+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. Corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown as the system is unstable.} #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown because the system is unstable.}
#+attr_latex: :options {0.49\linewidth} #+attr_latex: :options {0.49\linewidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :scale 0.9 #+attr_latex: :scale 0.9
@ -1756,12 +1758,12 @@ exportFig('figs/rotating_iff_kp_optimal_gain.pdf', 'width', 'half', 'height', 45
#+end_figure #+end_figure
** Damped plant ** Damped plant
Let's choose a parallel stiffness equal to $k_p = 2 m \Omega^2$ and compute the 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 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 wanted. 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 due to the fact that "pure" integrators are used, and that the low frequency loop gains becomes large below some frequency. This is because "pure" integrators are used which are inducing large low-frequency loop gains.
In order to lower the low frequency gain, a high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): To lower the low-frequency gain, a high-pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane):
\begin{equation} \begin{equation}
K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix}
1 & 0 \\ 1 & 0 \\
@ -1769,11 +1771,11 @@ In order to lower the low frequency gain, a high pass filter is added to the IFF
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
In order to see how the high pass filter impacts the attainable damping, the controller gain $g$ is kept constant while $\omega_i$ is changed, and the minimum damping ratio of the damped plant is computed. To determine how the 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. 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). 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. Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compare the obtained damped plant again with the undamped and with the "pure" IFF in Figure ref:fig:rotating_iff_kp_added_hpf_damped_plant.
The added high pass filter gives almost the same damping properties to the suspension while giving good low frequency behavior. The added high-pass filter gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior.
#+begin_src matlab #+begin_src matlab
%% Identify dynamics with parallel stiffness = 2mW^2 %% Identify dynamics with parallel stiffness = 2mW^2
@ -1817,7 +1819,7 @@ end
#+end_src #+end_src
#+begin_src matlab :results none #+begin_src matlab :results none
%% Effect of the high pass filter cut-off frequency on the obtained damping %% Effect of the high-pass filter cut-off frequency on the obtained damping
figure; figure;
plot(wis, opt_xi, '-'); plot(wis, opt_xi, '-');
set(gca, 'XScale', 'log'); set(gca, 'XScale', 'log');
@ -1832,7 +1834,7 @@ exportFig('figs/rotating_iff_kp_added_hpf_effect_damping.pdf', 'width', 'third',
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
%% Compute the damped plant with added High Pass Filter %% Compute the damped plant with added High-Pass Filter
Kiff_kp_hpf = (2.2/(s + 0.1*w0))*eye(2); Kiff_kp_hpf = (2.2/(s + 0.1*w0))*eye(2);
Kiff_kp_hpf.InputName = {'fu', 'fv'}; Kiff_kp_hpf.InputName = {'fu', 'fv'};
Kiff_kp_hpf.OutputName = {'Fu', 'Fv'}; Kiff_kp_hpf.OutputName = {'Fu', 'Fv'};
@ -1890,7 +1892,7 @@ exportFig('figs/rotating_iff_kp_added_hpf_damped_plant.pdf', 'width', 700, 'heig
#+end_src #+end_src
#+name: fig:rotating_iff_optimal_hpf #+name: fig:rotating_iff_optimal_hpf
#+caption:Effect of the high pass filter cut-off frequency on the obtained damping #+caption:Effect of high-pass filter cut-off frequency on the obtained damping
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$} #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$}
@ -1914,9 +1916,9 @@ exportFig('figs/rotating_iff_kp_added_hpf_damped_plant.pdf', 'width', 700, 'heig
<<sec:rotating_relative_damp_control>> <<sec:rotating_relative_damp_control>>
** Introduction :ignore: ** Introduction :ignore:
In order 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 ref:fig:rotating_3dof_model_schematic_rdc.
Two controllers $K_d$ are used to fed 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 eqref:eq:rotating_rdc_controller. These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high-pass filter eqref:eq:rotating_rdc_controller.
\begin{equation}\label{eq:rotating_rdc_controller} \begin{equation}\label{eq:rotating_rdc_controller}
K_d(s) = g \cdot \frac{s}{s + \omega_d} K_d(s) = g \cdot \frac{s}{s + \omega_d}
@ -2041,7 +2043,7 @@ The elements of $\bm{G}_d$ were derived in Section ref:sec:rotating_system_descr
\end{align} \end{align}
\end{subequations} \end{subequations}
Neglecting the damping for simplicity ($\xi \ll 1$), the direct terms have two complex conjugate zeros which are 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 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. 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} \begin{equation}\label{eq:rotating_rdc_zeros_poles}
@ -2051,13 +2053,13 @@ Therefore, for $\Omega < \sqrt{k/m}$ (i.e. stable system), the transfer function
** 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 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 ref:sec:rotating_system_description and are shown in Figure ref:fig:rotating_bode_plot (page pageref:fig:rotating_bode_plot).
In order 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 ref:fig:rotating_rdc_root_locus).
The closed-loop system is unconditionally stable as expected and the poles can be damped as much as wanted. The closed-loop system is unconditionally stable as expected and the poles can be damped as much as desired.
Let's select a reasonable "Relative Damping Control" gain, and compute the closed-loop damped system. 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 ref:fig:rotating_rdc_damped_plant.
The rotating aspect does not add any complexity for the use of Relative Damping Control. 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 Integral Force Feedback. It does not increase the low-frequency coupling as compared to the Integral Force Feedback.
#+begin_src matlab :results none #+begin_src matlab :results none
%% Root Locus for Relative Damping Control %% Root Locus for Relative Damping Control
@ -2210,9 +2212,9 @@ exportFig('figs/rotating_rdc_damped_plant.pdf', 'width', 'half', 'height', 500);
<<sec:rotating_comp_act_damp>> <<sec:rotating_comp_act_damp>>
** Introduction :ignore: ** Introduction :ignore:
These two proposed IFF modifications as well as relative damping control are now 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 based on previous discussion about optimal parameters. These values are chosen one the basis of previous discussions about optimal parameters.
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -2319,12 +2321,12 @@ Krdc.OutputName = {'Fu', 'Fv'};
#+end_src #+end_src
** Root Locus ** Root Locus
Figure ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control. Figure 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.
One can observe that the closed loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability. The closed-loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability.
This is not the case for the system where the controller is augmented with an HPF (in blue). This is not the case for the system in which the controller is augmented with an HPF (in blue).
It is interesting to note that the maximum added damping is very similar for both modified IFF techniques. It is interesting to note that the maximum added damping is very similar for both modified IFF techniques.
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
@ -2449,7 +2451,7 @@ exportFig('figs/rotating_comp_techniques_root_locus_zoom.pdf', 'width', 600, 'he
** Obtained Damped Plant ** 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 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.
Integral Force Feedback 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.
#+begin_src matlab #+begin_src matlab
%% Compute Damped plants %% Compute Damped plants
@ -2515,18 +2517,18 @@ exportFig('figs/rotating_comp_techniques_dampled_plants.pdf', 'width', 'half', '
** 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 here defined as the transfer function from a 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 how much 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.
The compliance describes the displacement response of the payload to external forces applied to it. The compliance describes the displacement response of the payload to the external forces applied to it.
This is a useful metric when disturbances are directly applied to the payload. This is a useful metric when disturbances are directly applied to the payload.
It is here 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. 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 are 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 ref:fig:rotating_comp_techniques_trans_compliance).
Using IFF degrades the compliance at low frequency while using relative damping control degrades the transmissibility at high frequency. 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 holds when applied to rotating platforms. This is very well known characteristics of these common active damping techniques that hold when applied to rotating platforms.
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Comparison of the obtained transmissibilty and compliance for the three tested active damping techniques %% Comparison of the obtained transmissibility and compliance for the three tested active damping techniques
freqs = logspace(-2, 2, 1000); freqs = logspace(-2, 2, 1000);
% transmissibility % transmissibility
@ -2576,7 +2578,7 @@ exportFig('figs/rotating_comp_techniques_compliance.pdf', 'width', 'half', 'heig
#+end_src #+end_src
#+name: fig:rotating_comp_techniques_trans_compliance #+name: fig:rotating_comp_techniques_trans_compliance
#+caption: Comparison of the obtained transmissibilty (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques #+caption: Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_transmissibility}Transmissibility} #+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_transmissibility}Transmissibility}
@ -2599,9 +2601,9 @@ exportFig('figs/rotating_comp_techniques_compliance.pdf', 'width', 'half', 'heig
:END: :END:
<<sec:rotating_nano_hexapod>> <<sec:rotating_nano_hexapod>>
** Introduction :ignore: ** Introduction :ignore:
The previous analysis is now applied on a model representing the rotating nano-hexapod. The previous analysis is now applied to a model representing a rotating nano-hexapod.
Three nano-hexapod stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. Three nano-hexapod stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$.
Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) as 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.
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -2707,9 +2709,9 @@ The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \a
Kneg_light = (15+1)*(2*pi)^2; Kneg_light = (15+1)*(2*pi)^2;
#+end_src #+end_src
The transfer functions from 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 ref:fig:rotating_nano_hexapod_dynamics for all three considered nano-hexapod stiffnesses.
The soft nano-hexapod is the most affected by the rotation. 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 for the stiffer nano-hexapods. This can be seen by the large shift of the resonance frequencies, and by the induced coupling, which is larger than that for the stiffer nano-hexapods.
The coupling (or interaction) in a MIMO $2 \times 2$ system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). The coupling (or interaction) in a MIMO $2 \times 2$ system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix).
#+begin_src matlab :results none #+begin_src matlab :results none
@ -2828,7 +2830,7 @@ exportFig('figs/rotating_nano_hexapod_dynamics_pz.pdf', 'width', 'third', 'heigh
#+end_src #+end_src
#+name: fig:rotating_nano_hexapod_dynamics #+name: fig:rotating_nano_hexapod_dynamics
#+caption: Effect of rotation on the nano-hexapod dynamics. Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity #+caption: Effect of rotation on the nano-hexapod dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$} #+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$}
@ -2851,14 +2853,14 @@ exportFig('figs/rotating_nano_hexapod_dynamics_pz.pdf', 'width', 'third', 'heigh
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
** Optimal IFF with High Pass Filter ** Optimal IFF with a High-Pass Filter
Integral Force Feedback with an added High Pass Filter is applied to the three nano-hexapods. 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 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 ref:fig:rotating_iff_hpf_nass_optimal_gain.
The IFF parameters are chosen as follow: 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 the 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 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. 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 obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. - 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 was shown in Section ref:sec:rotating_iff_pseudo_int. 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. 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.
#+begin_src matlab #+begin_src matlab
@ -3000,7 +3002,7 @@ exportFig('figs/rotating_iff_hpf_nass_optimal_gain_pz.pdf', 'width', 'third', 'h
#+end_src #+end_src
#+name: fig:rotating_iff_hpf_nass_optimal_gain #+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 choosen controller parameters used for further analysis are shown by the large dots. #+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.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$} #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$}
@ -3028,7 +3030,7 @@ data2orgtable([wis(i_iff_hpf_vc), opt_iff_hpf_gain_vc(i_iff_hpf_vc), opt_iff_hpf
#+end_src #+end_src
#+name: tab:rotating_iff_hpf_opt_iff_hpf_params_nass #+name: tab:rotating_iff_hpf_opt_iff_hpf_params_nass
#+caption: Obtained optimal parameters ($\omega_i$ and $g$) for the modified IFF controller including a high pass filter. The corresponding achievable simultaneous damping of the two modes $\xi$ is also shown. #+caption: Obtained optimal parameters ($\omega_i$ and $g$) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping of the two modes $\xi$ is also shown.
#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc
#+attr_latex: :center t :booktabs t #+attr_latex: :center t :booktabs t
#+RESULTS: #+RESULTS:
@ -3040,14 +3042,14 @@ data2orgtable([wis(i_iff_hpf_vc), opt_iff_hpf_gain_vc(i_iff_hpf_vc), opt_iff_hpf
** Optimal IFF with Parallel Stiffness ** Optimal IFF with Parallel Stiffness
For each considered nano-hexapod stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total nano-hexapod stiffness). For each considered nano-hexapod stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total nano-hexapod stiffness).
In order 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). To keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased ($k_a = k_n - k_p$, with $k_n$ the total nano-hexapod stiffness).
A high pass filter is also added to limit the low frequency gain with a cut-off frequency $\omega_i$ equal to one tenth of the system resonance ($\omega_i = \omega_0/10$). A high-pass filter is also added to limit the low-frequency gain with a cut-off frequency $\omega_i$ equal to one tenth of the system resonance ($\omega_i = \omega_0/10$).
The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure ref:fig:rotating_iff_kp_nass_optimal_gain). The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure ref:fig:rotating_iff_kp_nass_optimal_gain).
It is shown that the soft nano-hexapod cannot yield good damping as the parallel stiffness cannot be made large enough compared to the negative stiffness induced by the rotation. 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 chosen too high as explained in Section ref:sec:rotating_iff_parallel_stiffness. 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 explain 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]]. 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]].
This distance is larger for stiff nano-hexapod as the open-loop pole will be at higher frequencies while the open-loop zero, which depends on the value of the parallel stiffness, can only be made large for stiff nano-hexapods. 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. 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 ref:tab:rotating_iff_kp_opt_iff_kp_params_nass.
@ -3274,7 +3276,7 @@ data2orgtable([opt_iff_kp_gain_vc(i_kp_vc), opt_iff_kp_xi_vc(i_kp_vc); opt_iff_k
** Optimal Relative Motion Control ** 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). 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 is chosen such that 99% of modal damping is obtained (obtained gains are summarized in Table 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 ref:tab:rotating_rdc_opt_params_nass).
#+begin_src matlab #+begin_src matlab
%% Computes the optimal parameters and attainable simultaneous damping - Piezo nano-hexapod %% Computes the optimal parameters and attainable simultaneous damping - Piezo nano-hexapod
@ -3365,11 +3367,11 @@ exportFig('figs/rotating_rdc_optimal_gain.pdf', 'width', 'half', 'height', 350);
#+end_minipage #+end_minipage
** Comparison of the obtained damped plants ** Comparison of the obtained damped plants
Now that 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 ref:fig:rotating_nass_damped_plant_comp.
Similarly to what was concluded in previous analysis: Similar to what was concluded in the previous analysis:
- acrshort:iff adds coupling below the resonance frequency as compared to the open-loop and acrshort:rdc cases - acrshort:iff adds more coupling below the resonance frequency as compared to the open-loop and acrshort:rdc cases
- All three methods are yielding good damping, except for acrshort:iff applied on the soft nano-hexapod - All three methods yield good damping, except for acrshort:iff applied on the soft nano-hexapod
- Coupling is smaller for stiff nano-hexapods - Coupling is smaller for stiff nano-hexapods
#+begin_src matlab :tangle no #+begin_src matlab :tangle no
@ -3386,7 +3388,7 @@ load('nass_controllers.mat');
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
%% Closed Loop Plants - IFF with HPF %% Closed-Loop Plants - IFF with HPF
G_vc_norot_iff_hpf = feedback(G_vc_norot, Kiff_hpf_vc, 'name'); G_vc_norot_iff_hpf = feedback(G_vc_norot, Kiff_hpf_vc, 'name');
G_vc_fast_iff_hpf = feedback(G_vc_fast, Kiff_hpf_vc, 'name'); G_vc_fast_iff_hpf = feedback(G_vc_fast, Kiff_hpf_vc, 'name');
@ -3396,7 +3398,7 @@ G_md_fast_iff_hpf = feedback(G_md_fast, Kiff_hpf_md, 'name');
G_pz_norot_iff_hpf = feedback(G_pz_norot, Kiff_hpf_pz, 'name'); G_pz_norot_iff_hpf = feedback(G_pz_norot, Kiff_hpf_pz, 'name');
G_pz_fast_iff_hpf = feedback(G_pz_fast, Kiff_hpf_pz, 'name'); G_pz_fast_iff_hpf = feedback(G_pz_fast, Kiff_hpf_pz, 'name');
%% Closed Loop Plants - IFF with Parallel Stiffness %% Closed-Loop Plants - IFF with Parallel Stiffness
G_vc_norot_iff_kp = feedback(G_vc_kp_norot, Kiff_kp_vc, 'name'); G_vc_norot_iff_kp = feedback(G_vc_kp_norot, Kiff_kp_vc, 'name');
G_vc_fast_iff_kp = feedback(G_vc_kp_fast, Kiff_kp_vc, 'name'); G_vc_fast_iff_kp = feedback(G_vc_kp_fast, Kiff_kp_vc, 'name');
@ -3406,7 +3408,7 @@ G_md_fast_iff_kp = feedback(G_md_kp_fast, Kiff_kp_md, 'name');
G_pz_norot_iff_kp = feedback(G_pz_kp_norot, Kiff_kp_pz, 'name'); G_pz_norot_iff_kp = feedback(G_pz_kp_norot, Kiff_kp_pz, 'name');
G_pz_fast_iff_kp = feedback(G_pz_kp_fast, Kiff_kp_pz, 'name'); G_pz_fast_iff_kp = feedback(G_pz_kp_fast, Kiff_kp_pz, 'name');
%% Closed Loop Plants - RDC %% Closed-Loop Plants - RDC
G_vc_norot_rdc = feedback(G_vc_norot, Krdc_vc, 'name'); G_vc_norot_rdc = feedback(G_vc_norot, Krdc_vc, 'name');
G_vc_fast_rdc = feedback(G_vc_fast, Krdc_vc, 'name'); G_vc_fast_rdc = feedback(G_vc_fast, Krdc_vc, 'name');
@ -3555,7 +3557,7 @@ exportFig('figs/rotating_nass_damped_plant_comp_pz.pdf', 'width', 'third', 'heig
#+end_src #+end_src
#+name: fig:rotating_nass_damped_plant_comp #+name: fig:rotating_nass_damped_plant_comp
#+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by the solid lines and coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$. #+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$} #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$}
@ -3584,9 +3586,9 @@ exportFig('figs/rotating_nass_damped_plant_comp_pz.pdf', 'width', 'third', 'heig
:END: :END:
<<sec:rotating_nass>> <<sec:rotating_nass>>
** Introduction :ignore: ** Introduction :ignore:
Up 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 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 taken into account 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.
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -3621,7 +3623,7 @@ load('nass_controllers.mat');
#+end_src #+end_src
** Nano Active Stabilization System model ** Nano Active Stabilization System model
In order to have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modelled 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 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):
- 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 ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$)
- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_t = \SI{600}{\kg}$) - a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_t = \SI{600}{\kg}$)
@ -3637,13 +3639,13 @@ A payload is rigidly fixed to the nano-hexapod and the $x,y$ motion of the paylo
** System dynamics ** System dynamics
The dynamics of the un-damped and damped plants are identified using the optimal parameters found in Section ref:sec:rotating_nano_hexapod. 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 while the coupling terms are shown by the shaded ones. 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.
It can be observed that: 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 - The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod
- Damping added by the three proposed techniques is quite high and the obtained plant is rather easy to control - Damping added using the three proposed techniques is quite high, and the obtained plant is rather easy to control
- There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz) - There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz)
- The two proposed IFF modification yields similar results - The two proposed IFF modifications yield similar results
#+begin_src matlab #+begin_src matlab
%% System parameters %% System parameters
@ -3763,7 +3765,7 @@ G_pz_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'
G_pz_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; G_pz_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'};
%% Compute dampepd plants %% Compute dampepd plants
% Closed Loop Plants - IFF with HPF % Closed-Loop Plants - IFF with HPF
G_vc_norot_iff_hpf = feedback(G_vc_norot, Kiff_hpf_vc, 'name'); G_vc_norot_iff_hpf = feedback(G_vc_norot, Kiff_hpf_vc, 'name');
G_vc_fast_iff_hpf = feedback(G_vc_fast, Kiff_hpf_vc, 'name'); G_vc_fast_iff_hpf = feedback(G_vc_fast, Kiff_hpf_vc, 'name');
@ -3773,7 +3775,7 @@ G_md_fast_iff_hpf = feedback(G_md_fast, Kiff_hpf_md, 'name');
G_pz_norot_iff_hpf = feedback(G_pz_norot, Kiff_hpf_pz, 'name'); G_pz_norot_iff_hpf = feedback(G_pz_norot, Kiff_hpf_pz, 'name');
G_pz_fast_iff_hpf = feedback(G_pz_fast, Kiff_hpf_pz, 'name'); G_pz_fast_iff_hpf = feedback(G_pz_fast, Kiff_hpf_pz, 'name');
% Closed Loop Plants - IFF with Parallel Stiffness % Closed-Loop Plants - IFF with Parallel Stiffness
G_vc_norot_iff_kp = feedback(G_vc_kp_norot, Kiff_kp_vc, 'name'); G_vc_norot_iff_kp = feedback(G_vc_kp_norot, Kiff_kp_vc, 'name');
G_vc_fast_iff_kp = feedback(G_vc_kp_fast, Kiff_kp_vc, 'name'); G_vc_fast_iff_kp = feedback(G_vc_kp_fast, Kiff_kp_vc, 'name');
@ -3783,7 +3785,7 @@ G_md_fast_iff_kp = feedback(G_md_kp_fast, Kiff_kp_md, 'name');
G_pz_norot_iff_kp = feedback(G_pz_kp_norot, Kiff_kp_pz, 'name'); G_pz_norot_iff_kp = feedback(G_pz_kp_norot, Kiff_kp_pz, 'name');
G_pz_fast_iff_kp = feedback(G_pz_kp_fast, Kiff_kp_pz, 'name'); G_pz_fast_iff_kp = feedback(G_pz_kp_fast, Kiff_kp_pz, 'name');
% Closed Loop Plants - RDC % Closed-Loop Plants - RDC
G_vc_norot_rdc = feedback(G_vc_norot, Krdc_vc, 'name'); G_vc_norot_rdc = feedback(G_vc_norot, Krdc_vc, 'name');
G_vc_fast_rdc = feedback(G_vc_fast, Krdc_vc, 'name'); G_vc_fast_rdc = feedback(G_vc_fast, Krdc_vc, 'name');
@ -3975,14 +3977,14 @@ exportFig('figs/rotating_nass_plant_comp_stiffness_pz.pdf', 'width', 'third', 'h
** 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 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 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).
Note that only the transfer function 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 as 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.
Conclusions are similar than with the uniaxial (non-rotating) model: Conclusions are similar than those of the uniaxial (non-rotating) model:
- Regarding the effect of floor motion and forces applied on the payload: - 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 for the soft one - 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
- acrshort:iff degrades the performance at low frequency compared to acrshort:rdc - acrshort:iff degrades the performance at low-frequency compared to acrshort:rdc
- Regarding the effect of micro-station vibrations: - Regarding the effect of micro-station vibrations:
- Having a soft nano-hexapod allows to filter these vibrations between the suspensions 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 ref:fig:rotating_nass_effect_stage_vibration_vc).
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses %% Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
@ -4059,7 +4061,7 @@ exportFig('figs/rotating_nass_effect_floor_motion_pz.pdf', 'width', 'third', 'he
#+end_src #+end_src
#+name: fig:rotating_nass_effect_floor_motion #+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. #+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.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$} #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$}
@ -4254,7 +4256,7 @@ exportFig('figs/rotating_nass_effect_direct_forces_pz.pdf', 'width', 'third', 'h
#+end_src #+end_src
#+name: fig:rotating_nass_effect_direct_forces #+name: fig:rotating_nass_effect_direct_forces
#+caption: Effect of sample forces $f_{s,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low frequency. #+caption: Effect of sample forces $f_{s,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low-frequency.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$} #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$}
@ -4286,25 +4288,25 @@ In this study, the gyroscopic effects induced by the spindle's rotation have bee
Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms (Section ref:sec:rotating_iff_pure_int). Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms (Section ref:sec:rotating_iff_pure_int).
Two modifications of the classical acrshort:iff control have been proposed to overcome this issue. Two modifications of the classical acrshort:iff control have been proposed to overcome this issue.
The first modification concerns the controller and consists of adding a high pass filter to the pure integrators. The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators.
This is equivalent to moving the controller pole to the left along the real axis. This 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 ref:sec:rotating_iff_pseudo_int).
The second proposed modification concerns the mechanical system. The second proposed modification concerns the mechanical system.
Additional springs are added in parallel with the actuators and force sensors. 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 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 ref:sec:rotating_comp_act_damp.
While having very different implementations, both proposed modifications were found to be very similar when it comes to the attainable damping and the obtained closed loop system behavior. 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.
Then, 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 ref:sec:rotating_nano_hexapod).
As for the uniaxial model, three nano-hexapod stiffness are considered. 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 on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). 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).
Also, the attainable damping ratio of the soft nano-hexapod when using acrshort:iff is limited by 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 taken into account (Section ref:sec:rotating_nass). To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been considered (Section ref:sec:rotating_nass).
Results are similar to that of the uniaxial model except that come complexity is added for the soft nano-hexapod due to the spindle's rotation. 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 are only slightly affecting the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that was showing better results with the uniaxial model. 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.
* Bibliography :ignore: * Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}] #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

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@ -1,4 +1,4 @@
% Created 2024-04-29 Mon 21:05 % Created 2024-04-30 Tue 15:25
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -15,6 +15,8 @@
\newacronym{iff}{IFF}{Integral Force Feedback} \newacronym{iff}{IFF}{Integral Force Feedback}
\newacronym{rdc}{RDC}{Relative Damping Control} \newacronym{rdc}{RDC}{Relative Damping Control}
\newacronym{drga}{DRGA}{Dynamical Relative Gain Array} \newacronym{drga}{DRGA}{Dynamical Relative Gain Array}
\newacronym{hpf}{HPF}{high-pass filter}
\newacronym{lpf}{LPF}{low-pass filter}
\newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}} \newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}}
\newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}} \newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}}
\newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}} \newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}}
@ -40,28 +42,28 @@
\clearpage \clearpage
An important aspect of the \acrfull{nass} is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. 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. 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 \ref{sec:rotating_system_description})
This model is simple enough to be able to derive its dynamics analytically and to well understand its behavior, while still allowing to capture the important physical effects in play. 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 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 \ref{sec:rotating_iff_pure_int}).
Two modifications of the Integral Force Feedback are then proposed. Two modifications of the Integral Force Feedback are then proposed.
The first one consists of adding an 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 \ref{sec:rotating_iff_pseudo_int}).
It is shown that the \acrshort{iff} controller is stable for some values of the gain, and that damping can be added to the suspension modes. It is shown that the \acrshort{iff} controller is stable for some gain values, and that damping can be added to the suspension modes.
Optimal high pass filter cut-off frequency is computed. 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 \ref{sec:rotating_iff_parallel_stiffness}).
Under a certain condition, the unconditional stability of the the IFF controller is regained. Under certain conditions, the unconditional stability of the IFF controller is regained.
Optimal parallel stiffness is then computed. The optimal parallel stiffness is then computed.
This study of adapting \acrshort{iff} for the damping of rotating platforms was 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 \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}). 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, obtained damped plant and closed-loop compliance and transmissibility (Section \ref{sec:rotating_comp_act_damp}). 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}).
The previous analysis is applied on 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 optimal active damping controller are 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 \ref{sec:rotating_nano_hexapod}).
Up until this section, the study was performed on a very simplistic model that just captures the rotation aspect and the model parameters were not tuned to corresponds to the NASS. 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 \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 if the rotation imposes performance limitation for the NASS. The goal is to determine whether the rotation imposes performance limitation on the NASS.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -71,13 +73,13 @@ The goal is to determine if the rotation imposes performance limitation for the
\chapter{System Description and Analysis} \chapter{System Description and Analysis}
\label{sec:rotating_system_description} \label{sec:rotating_system_description}
The studied system 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 \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 \(F_u, F_v\).
A payload with a mass \(m\) in \(\si{\kilo\gram}\), is mounted on the (rotating) suspended platform. A payload with a mass \(m\) in \(\si{\kilo\gram}\), is mounted on the (rotating) suspended platform.
Two reference frames are used: an \emph{inertial} frame \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\) and a \emph{uniform rotating} frame \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\) rigidly fixed on top of the rotating stage with \(\vec{i}_w\) aligned with the rotation axis. Two reference frames are used: an \emph{inertial} frame \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\) and a \emph{uniform rotating} frame \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\) rigidly fixed on top of the rotating stage with \(\vec{i}_w\) aligned with the rotation axis.
The position of the payload is represented by \((d_u, d_v, 0)\) expressed in the rotating frame. The position of the payload is represented by \((d_u, d_v, 0)\) expressed in the rotating frame.
After the dynamics of this system is studied, the objective will be to damp the two suspension modes of the payload while the rotating stage performs a constant rotation. After the dynamics of this system is studied, the objective will be to dampen the two suspension modes of the payload while the rotating stage performs a constant rotation.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -89,7 +91,7 @@ After the dynamics of this system is studied, the objective will be to damp the
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 \ref{fig:rotating_3dof_model_schematic}, the Lagrangian equation \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 \eqref{eq:rotating_energy_functions_lagrange}. These terms are derived in \eqref{eq:rotating_energy_functions_lagrange}.
Note that the equation of motion corresponding to the constant rotation along \(\vec{i}_w\) is disregarded as this motion is considered to be imposed by the rotation stage. 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} \begin{equation}\label{eq:rotating_lagrangian_equations}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
@ -113,10 +115,10 @@ Substituting equations \eqref{eq:rotating_energy_functions_lagrange} into equati
The uniform rotation of the system induces two \emph{gyroscopic effects} as shown in equation \eqref{eq:rotating_eom_coupled}: The uniform rotation of the system induces two \emph{gyroscopic effects} as shown in equation \eqref{eq:rotating_eom_coupled}:
\begin{itemize} \begin{itemize}
\item \emph{Centrifugal forces}: that can been seen as an added \emph{negative stiffness} \(- m \Omega^2\) along \(\vec{i}_u\) and \(\vec{i}_v\) \item \emph{Centrifugal forces}: that can be seen as an added \emph{negative stiffness} \(- m \Omega^2\) along \(\vec{i}_u\) and \(\vec{i}_v\)
\item \emph{Coriolis forces}: that adds \emph{coupling} between the two orthogonal directions. \item \emph{Coriolis forces}: that adds \emph{coupling} between the two orthogonal directions.
\end{itemize} \end{itemize}
One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent to two \emph{uncoupled} one degree of freedom mass-spring-damper systems. One can verify that without rotation (\(\Omega = 0\)), the system becomes equivalent to two \emph{uncoupled} one 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. 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}. The four transfer functions in \(\mathbf{G}_d\) are shown in equation \eqref{eq:rotating_Gd_indiv_el}.
@ -133,7 +135,7 @@ The four transfer functions in \(\mathbf{G}_d\) are shown in equation \eqref{eq:
\end{subequations} \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. 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 transfer function matrix \(\mathbf{G}_d\) are now described by equation \eqref{eq:rotating_Gd_w0_xi_k}. The elements of the transfer function matrix \(\mathbf{G}_d\) are described by equation \eqref{eq:rotating_Gd_w0_xi_k}.
\begin{equation} \label{eq:rotating_xi_and_omega} \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}} \omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}}
\end{equation} \end{equation}
@ -185,9 +187,9 @@ Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal fo
\section{System Dynamics: Effect of rotation} \section{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 \(\mathbf{G}_d\) 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. 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 plot of these two terms are shown in Figure \ref{fig:rotating_bode_plot} for several rotational speeds \(\Omega\). The bode plots of these two terms are shown in Figure \ref{fig:rotating_bode_plot} for several rotational speeds \(\Omega\).
These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as \(\Omega\) increases. 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). 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).
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
@ -208,23 +210,23 @@ For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p
\chapter{Integral Force Feedback} \chapter{Integral Force Feedback}
\label{sec:rotating_iff_pure_int} \label{sec:rotating_iff_pure_int}
The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation. 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 active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat} and to make the plant easier to control for the high authority controller. As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat} and to make the plant easier to control for the high authority controller.
Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} and Direct Velocity Feedback (DVF) \cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}. Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} and Direct Velocity Feedback (DVF) \cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}.
In \cite{preumont91_activ}, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system. In \cite{preumont91_activ}, the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system.
When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitates to guarantee the stability of the closed loop system \cite{preumont02_force_feedb_versus_accel_feedb}. 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 latter shown that this property holds for multiple collated actuator/sensor pairs \cite{preumont08_trans_zeros_struc_contr_with}. It was later shown that this property holds for multiple collated actuator/sensor pairs \cite{preumont08_trans_zeros_struc_contr_with}.
The main advantages of IFF over other active damping techniques are the guaranteed stability even in presence of flexible dynamics, good performance and robustness properties \cite{preumont02_force_feedb_versus_accel_feedb}. The main advantages of IFF over other active damping techniques are the guaranteed stability even in the presence of flexible dynamics, good performance, and robustness properties \cite{preumont02_force_feedb_versus_accel_feedb}.
Several improvements of the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping \cite{teo15_optim_integ_force_feedb_activ_vibrat_contr} or adding an high pass filter to recover the loss of compliance at low frequency \cite{chesne16_enhan_dampin_flexib_struc_using_force_feedb}. Several improvements to the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping \cite{teo15_optim_integ_force_feedb_activ_vibrat_contr} or adding a high-pass filter to recover the loss of compliance at low-frequency \cite{chesne16_enhan_dampin_flexib_struc_using_force_feedb}.
Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to derive optimal gains for the IFF controller \cite{zhao19_optim_integ_force_feedb_contr}. \par Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to derive optimal gains for the IFF controller \cite{zhao19_optim_integ_force_feedb_contr}. \par
However, none of these study have been applied to a rotating system. However, none of these studies have been applied to rotating systems.
In this section, Integral Force Feedback strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alters 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.
\section{System and Equations of motion} \section{System and Equations of motion}
In order to apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure \ref{fig:rotating_3dof_model_schematic_iff}). 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. 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.
\begin{equation}\label{eq:rotating_iff_controller} \begin{equation}\label{eq:rotating_iff_controller}
@ -282,10 +284,10 @@ It is interesting to see that the frequency of the pair of complex conjugate zer
This is what usually gives the unconditional stability of IFF when collocated force sensors are used. 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 \emph{non-minimum phase behavior}. However, for non-null rotational speeds, the two real zeros \(z_r\) in equation \eqref{eq:rotating_iff_zero_real} are inducing a \emph{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 \(\SI{180}{\degree}\). 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 \(\SI{180}{\degree}\).
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 \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 taking into account the negative stiffness induced by the rotation). 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. 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.
\begin{equation}\label{eq:rotating_low_freq_gain_iff_plan} \begin{equation}\label{eq:rotating_low_freq_gain_iff_plan}
@ -295,14 +297,14 @@ This small displacement then increases the centrifugal force \(m\Omega^2d_u = \f
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
\section{Effect of the rotation speed on the IFF plant dynamics} \section{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 \ref{fig:rotating_iff_bode_plot_effect_rot}.
As was expected from the derived equations of motion: As expected from the derived equations of motion:
\begin{itemize} \begin{itemize}
\item when \(\Omega < \omega_0\): the low frequency gain is no longer zero and two (non-minimum phase) real zero appears at low frequency. \item 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 \(\Omega\). The low-frequency gain increases with \(\Omega\).
A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles that are split further apart as \(\Omega\) increases. A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles, which are split further apart as \(\Omega\) increases.
\item when \(\omega_0 < \Omega\): the low frequency pole becomes unstable. \item when \(\omega_0 < \Omega\): the low-frequency pole becomes unstable.
\end{itemize} \end{itemize}
\begin{figure}[htbp] \begin{figure}[htbp]
@ -322,9 +324,9 @@ A pair of (minimum phase) complex conjugate zeros appears between the two comple
\end{figure} \end{figure}
\section{Decentralized Integral Force Feedback} \section{Decentralized Integral Force Feedback}
The control diagram for decentralized Integral Force Feedback is shown in Figure \ref{fig:rotating_iff_diagram}. 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 decentralized IFF controller \(\bm{K}_F\) corresponds to a diagonal controller with integrators:
\begin{equation} \label{eq:rotating_Kf_pure_int} \begin{equation} \label{eq:rotating_Kf_pure_int}
\begin{aligned} \begin{aligned}
\mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\ \mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\
@ -332,21 +334,21 @@ The decentralized IFF controller \(\bm{K}_F\) corresponds to a diagonal controll
\end{aligned} \end{aligned}
\end{equation} \end{equation}
In order to see how the 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 \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 \(\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};\)) for \(g = 0\) and coincide with the transmission zeros (shown by \(\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];\)) as \(g \to \infty\). As explained in \cite{preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr}, the closed-loop poles start at the open-loop poles (shown by \(\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};\)) for \(g = 0\) and coincide with the transmission zeros (shown by \(\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];\)) 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 in non-null. Whereas collocated IFF is usually associated with unconditional stability \cite{preumont91_activ}, this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null.
This can be seen in the Root Locus plot (Figure \ref{fig:rotating_root_locus_iff_pure_int}) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. 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 like so: at low frequency, the loop gain is very large due to the pure integrator in \(K_{F}\) and the finite gain of the plant (Figure \ref{fig:rotating_iff_bode_plot_effect_rot}). Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in \(K_{F}\) and the finite gain of the plant (Figure \ref{fig:rotating_iff_bode_plot_effect_rot}).
The control system is thus canceling the spring forces which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. 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.
\chapter{Integral Force Feedback with an High Pass Filter} \chapter{Integral Force Feedback with a High-Pass Filter}
\label{sec:rotating_iff_pseudo_int} \label{sec:rotating_iff_pseudo_int}
As was explained in the previous section, the instability of the IFF controller applied on 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.
In order to limit the low frequency controller gain, an High Pass Filter (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 \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.
This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator \cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}. This modification of the IFF controller is typically performed to avoid saturation associated with the pure integrator \cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}.
This is however not the reason why this high pass filter is added here. This is however not the reason why this high-pass filter is added here.
\begin{equation}\label{eq:rotating_iff_lhf} \begin{equation}\label{eq:rotating_iff_lhf}
\boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} \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}}
@ -355,11 +357,11 @@ This is however not the reason why this high pass filter is added here.
\section{Modified Integral Force Feedback Controller} \section{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 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 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 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.
The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure \ref{fig:rotating_iff_root_locus_hpf_large}. 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 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}. 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). 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).
\begin{equation}\label{eq:rotating_gmax_iff_hpf} \begin{equation}\label{eq:rotating_gmax_iff_hpf}
\boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)} \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)}
@ -378,16 +380,16 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the contro
\end{center} \end{center}
\subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root Locus} \subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root Locus}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_iff_modified_loop_gain_root_locus}Comparison of the IFF with pure integrator and modified IFF with added high pass filter (\(\Omega = 0.1\omega_0\)). Loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with \(\omega_i = 0.1 \omega_0\) and \(g = 2\). Root Locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large})} \caption{\label{fig:rotating_iff_modified_loop_gain_root_locus}Comparison of the IFF with pure integrator and modified IFF with added high-pass filter (\(\Omega = 0.1\omega_0\)). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with \(\omega_i = 0.1 \omega_0\) and \(g = 2\). The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large})}
\end{figure} \end{figure}
\section{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)} \section{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 \eqref{eq:rotating_iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\).
The optimal values of \(\omega_i\) and \(g\) are here considered as the values for which the damping of all the closed-loop poles are 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.
In order 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}. 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}. 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}.
In order to study this trade off, the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) is computed as a function of \(\omega_i/\omega_0\). 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 \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 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.
@ -406,14 +408,14 @@ For larger values of \(\omega_i\), the attainable damping ratio decreases as a f
\end{center} \end{center}
\subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown} \subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as \(\omega_i\) increases which is confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})} \caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as \(\omega_i\) increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})}
\end{figure} \end{figure}
\section{Obtained Damped Plant} \section{Obtained Damped Plant}
In order 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 \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.
There is therefore 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 \(\omega_i\).
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 \ref{fig:rotating_iff_hpf_effect_wi_compliance}).
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
@ -451,14 +453,14 @@ The forces measured by the two force sensors represented in Figure \ref{fig:rota
\begin{bmatrix} d_u \\ d_v \end{bmatrix} \begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation} \end{equation}
In order 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 \eqref{eq:rotating_kp_alpha}.
\begin{equation}\label{eq:rotating_kp_alpha} \begin{equation}\label{eq:rotating_kp_alpha}
k_p = \alpha k, \quad k_a = (1 - \alpha) k k_p = \alpha k, \quad k_a = (1 - \alpha) k
\end{equation} \end{equation}
After the equations of motion derived and transformed in the Laplace domain, the transfer function matrix \(\mathbf{G}_k\) in Eq. \eqref{eq:rotating_Gk_mimo_tf} is computed. 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 Eq. \eqref{eq:rotating_Gk_diag} and \eqref{eq:rotating_Gk_off_diag}. Its elements are shown in Eqs. \eqref{eq:rotating_Gk_diag} and \eqref{eq:rotating_Gk_off_diag}.
\begin{equation}\label{eq:rotating_Gk_mimo_tf} \begin{equation}\label{eq:rotating_Gk_mimo_tf}
\begin{bmatrix} f_u \\ f_v \end{bmatrix} = \begin{bmatrix} f_u \\ f_v \end{bmatrix} =
@ -473,22 +475,22 @@ Its elements are shown in Eq. \eqref{eq:rotating_Gk_diag} and \eqref{eq:rotating
\end{align} \end{align}
\end{subequations} \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 are kept the same, the zeros of the diagonal terms have changed. 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. 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.
Thus, if the added \emph{parallel stiffness} \(k_p\) is higher than the \emph{negative stiffness} induced by centrifugal forces \(m \Omega^2\), the dynamics from actuator to its collocated force sensor will show \emph{minimum phase behavior}. Thus, if the added \emph{parallel stiffness} \(k_p\) is higher than the \emph{negative stiffness} induced by centrifugal forces \(m \Omega^2\), the dynamics from the actuator to its collocated force sensor will show \emph{minimum phase behavior}.
\begin{equation}\label{eq:rotating_kp_cond_cc_zeros} \begin{equation}\label{eq:rotating_kp_cond_cc_zeros}
\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}
\section{Effect of the parallel stiffness on the IFF plant} \section{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 \(k_p > m \Omega^2\).
The 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 \ref{fig:rotating_iff_effect_kp}.
One can see that the 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 \(k_p > m \Omega^2\).
In that case, the systems shows alternating complex conjugate poles and zeros as what is the case in the non-rotating case. 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 in Eq. \eqref{eq:rotating_Kf_pure_int}. 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}.
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 IFF is recovered. 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.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.55\linewidth} \begin{subfigure}{0.55\linewidth}
@ -503,14 +505,14 @@ It is shown that if the added stiffness is higher than the maximum negative stif
\end{center} \end{center}
\subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring} \subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_iff_plant_effect_kp}Effect of the parallel stiffness on the IFF plant} \caption{\label{fig:rotating_iff_plant_effect_kp}Effect of parallel stiffness on the IFF plant}
\end{figure} \end{figure}
\section{Effect of \(k_p\) on the attainable damping} \section{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 \cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the parallel stiffness \(k_p\) is foreseen to have some impact on the attainable damping. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros \cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the parallel stiffness \(k_p\) is expected to have some impact on the attainable damping.
To study this effect, Root Locus plots for several parallel stiffnesses \(k_p > m \Omega^2\) are shown in Figure \ref{fig:rotating_iff_kp_root_locus_effect_kp}. To study this effect, Root Locus plots for several parallel stiffnesses \(k_p > m \Omega^2\) are shown in Figure \ref{fig:rotating_iff_kp_root_locus_effect_kp}.
The frequencies of the transmission zeros of the system are increasing with an increase of the parallel stiffness \(k_p\) (thus getting closer to the poles) and the associated attainable damping is reduced. 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. 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 \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.
@ -519,24 +521,24 @@ This is confirmed by the Figure \ref{fig:rotating_iff_kp_optimal_gain} where the
\begin{center} \begin{center}
\includegraphics[scale=1,scale=1]{figs/rotating_iff_kp_root_locus_effect_kp.png} \includegraphics[scale=1,scale=1]{figs/rotating_iff_kp_root_locus_effect_kp.png}
\end{center} \end{center}
\subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of the parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$} \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.49\linewidth} \begin{subfigure}{0.49\linewidth}
\begin{center} \begin{center}
\includegraphics[scale=1,scale=0.9]{figs/rotating_iff_kp_optimal_gain.png} \includegraphics[scale=1,scale=0.9]{figs/rotating_iff_kp_optimal_gain.png}
\end{center} \end{center}
\subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. Corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown as the system is unstable.} \subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown because the system is unstable.}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_iff_optimal_kp}Effect of the parallel stiffness on the IFF plant} \caption{\label{fig:rotating_iff_optimal_kp}Effect of parallel stiffness on the IFF plant}
\end{figure} \end{figure}
\section{Damped plant} \section{Damped plant}
Let's choose a parallel stiffness equal to \(k_p = 2 m \Omega^2\) and compute the 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 \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 wanted. 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 due to the fact that ``pure'' integrators are used, and that the low frequency loop gains becomes large below some frequency. This is because ``pure'' integrators are used which are inducing large low-frequency loop gains.
In order to lower the low frequency gain, a high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): To lower the low-frequency gain, a high-pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane):
\begin{equation} \begin{equation}
K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix}
1 & 0 \\ 1 & 0 \\
@ -544,11 +546,11 @@ In order to lower the low frequency gain, a high pass filter is added to the IFF
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
In order to see how the high pass filter impacts the attainable damping, the controller gain \(g\) is kept constant while \(\omega_i\) is changed, and the minimum damping ratio of the damped plant is computed. To determine how the 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}. 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}). 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}. Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compare the obtained damped plant again with the undamped and with the ``pure'' IFF in Figure \ref{fig:rotating_iff_kp_added_hpf_damped_plant}.
The added high pass filter gives almost the same damping properties to the suspension while giving good low frequency behavior. The added high-pass filter gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.34\linewidth} \begin{subfigure}{0.34\linewidth}
@ -563,14 +565,14 @@ The added high pass filter gives almost the same damping properties to the suspe
\end{center} \end{center}
\subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF} \subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_iff_optimal_hpf}Effect of the high pass filter cut-off frequency on the obtained damping} \caption{\label{fig:rotating_iff_optimal_hpf}Effect of high-pass filter cut-off frequency on the obtained damping}
\end{figure} \end{figure}
\chapter{Relative Damping Control} \chapter{Relative Damping Control}
\label{sec:rotating_relative_damp_control} \label{sec:rotating_relative_damp_control}
In order 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 \ref{fig:rotating_3dof_model_schematic_rdc}.
Two controllers \(K_d\) are used to fed 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 \eqref{eq:rotating_rdc_controller}. These controllers are in principle pure derivators (\(K_d = s\)), but to be implemented in practice they are usually replaced by a high-pass filter \eqref{eq:rotating_rdc_controller}.
\begin{equation}\label{eq:rotating_rdc_controller} \begin{equation}\label{eq:rotating_rdc_controller}
K_d(s) = g \cdot \frac{s}{s + \omega_d} K_d(s) = g \cdot \frac{s}{s + \omega_d}
@ -597,7 +599,7 @@ The elements of \(\bm{G}_d\) were derived in Section \ref{sec:rotating_system_de
\end{align} \end{align}
\end{subequations} \end{subequations}
Neglecting the damping for simplicity (\(\xi \ll 1\)), the direct terms have two complex conjugate zeros which are 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 \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. 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} \begin{equation}\label{eq:rotating_rdc_zeros_poles}
@ -607,13 +609,13 @@ Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functi
\section{Decentralized Relative Damping Control} \section{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 \ref{sec:rotating_system_description} and are shown in Figure \ref{fig:rotating_bode_plot} (page \pageref{fig:rotating_bode_plot}).
In order 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 \ref{fig:rotating_rdc_root_locus}).
The closed-loop system is unconditionally stable as expected and the poles can be damped as much as wanted. The closed-loop system is unconditionally stable as expected and the poles can be damped as much as desired.
Let's select a reasonable ``Relative Damping Control'' gain, and compute the closed-loop damped system. 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 \ref{fig:rotating_rdc_damped_plant}.
The rotating aspect does not add any complexity for the use of Relative Damping Control. 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 Integral Force Feedback. It does not increase the low-frequency coupling as compared to the Integral Force Feedback.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.49\linewidth} \begin{subfigure}{0.49\linewidth}
@ -633,17 +635,17 @@ It does not increase the low frequency coupling as compared to Integral Force Fe
\chapter{Comparison of Active Damping Techniques} \chapter{Comparison of Active Damping Techniques}
\label{sec:rotating_comp_act_damp} \label{sec:rotating_comp_act_damp}
These two proposed IFF modifications as well as relative damping control are now 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 based on previous discussion about optimal parameters. These values are chosen one the basis of previous discussions about optimal parameters.
\section{Root Locus} \section{Root Locus}
Figure \ref{fig:rotating_comp_techniques_root_locus} shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control. Figure \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.
One can observe that the closed loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability. The closed-loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability.
This is not the case for the system where the controller is augmented with an HPF (in blue). This is not the case for the system in which the controller is augmented with an HPF (in blue).
It is interesting to note that the maximum added damping is very similar for both modified IFF techniques. It is interesting to note that the maximum added damping is very similar for both modified IFF techniques.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -665,19 +667,19 @@ It is interesting to note that the maximum added damping is very similar for bot
\section{Obtained Damped Plant} \section{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 \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.
Integral Force Feedback 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.
\section{Transmissibility And Compliance} \section{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 here defined as the transfer function from a 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 how much 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.
The compliance describes the displacement response of the payload to external forces applied to it. The compliance describes the displacement response of the payload to the external forces applied to it.
This is a useful metric when disturbances are directly applied to the payload. This is a useful metric when disturbances are directly applied to the payload.
It is here 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. 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 are 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 \ref{fig:rotating_comp_techniques_trans_compliance}).
Using IFF degrades the compliance at low frequency while using relative damping control degrades the transmissibility at high frequency. 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 holds when applied to rotating platforms. This is very well known characteristics of these common active damping techniques that hold when applied to rotating platforms.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
@ -692,21 +694,21 @@ This is very well known characteristics of these common active damping technique
\end{center} \end{center}
\subcaption{\label{fig:rotating_comp_techniques_compliance}Compliance} \subcaption{\label{fig:rotating_comp_techniques_compliance}Compliance}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_comp_techniques_trans_compliance}Comparison of the obtained transmissibilty (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques} \caption{\label{fig:rotating_comp_techniques_trans_compliance}Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques}
\end{figure} \end{figure}
\chapter{Rotating Nano-Hexapod} \chapter{Rotating Nano-Hexapod}
\label{sec:rotating_nano_hexapod} \label{sec:rotating_nano_hexapod}
The previous analysis is now applied on a model representing the rotating nano-hexapod. The previous analysis is now applied to a model representing a rotating nano-hexapod.
Three nano-hexapod stiffnesses are tested as for the uniaxial model: \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\). Three nano-hexapod stiffnesses are tested as for the uniaxial model: \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\).
Only the maximum rotating velocity is here considered (\(\Omega = \SI{60}{rpm}\)) with the light sample (\(m_s = \SI{1}{kg}\)) as 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.
\section{Nano-Active-Stabilization-System - Plant Dynamics} \section{Nano-Active-Stabilization-System - Plant Dynamics}
For the NASS, the maximum rotating velocity is \(\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}\) for a suspended mass on top of the nano-hexapod's actuators equal to \(m_n + m_s = \SI{16}{\kilo\gram}\). 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 parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \approx \SI{0.6}{\newton\per\mm}\).
The transfer functions from 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 \ref{fig:rotating_nano_hexapod_dynamics} for all three considered nano-hexapod stiffnesses.
The soft nano-hexapod is the most affected by the rotation. 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 for the stiffer nano-hexapods. This can be seen by the large shift of the resonance frequencies, and by the induced coupling, which is larger than that for the stiffer nano-hexapods.
The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix).
\begin{figure}[htbp] \begin{figure}[htbp]
@ -728,18 +730,18 @@ The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually es
\end{center} \end{center}
\subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$} \subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics. Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity (\(\Omega = 60\,\text{rpm}\)), and shaded lines are coupling terms at maximum rotating velocity} \caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity (\(\Omega = 60\,\text{rpm}\)), and shaded lines are coupling terms at maximum rotating velocity}
\end{figure} \end{figure}
\section{Optimal IFF with High Pass Filter} \section{Optimal IFF with a High-Pass Filter}
Integral Force Feedback with an added High Pass Filter is applied to the three nano-hexapods. 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 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 \ref{fig:rotating_iff_hpf_nass_optimal_gain}.
The IFF parameters are chosen as follow: The IFF parameters are chosen as follows:
\begin{itemize} \begin{itemize}
\item 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 the maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. \item 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.
This is done to have some control robustness. This is done to have some control robustness.
\item 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 obtained damping is \(\SI{95}{\percent}\) of the maximum achievable damping. \item 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 was shown in Section \ref{sec:rotating_iff_pseudo_int}. 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}.
\end{itemize} \end{itemize}
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}. 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}.
@ -762,11 +764,11 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\end{center} \end{center}
\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$} \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_iff_hpf_nass_optimal_gain}For each value of \(\omega_i\), the maximum damping ratio \(\xi\) is computed (blue) and the corresponding controller gain is shown (in red). The choosen controller parameters used for further analysis are shown by the large dots.} \caption{\label{fig:rotating_iff_hpf_nass_optimal_gain}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.}
\end{figure} \end{figure}
\begin{table}[htbp] \begin{table}[htbp]
\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters (\(\omega_i\) and \(g\)) for the modified IFF controller including a high pass filter. The corresponding achievable simultaneous damping of the two modes \(\xi\) is also shown.} \caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters (\(\omega_i\) and \(g\)) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping of the two modes \(\xi\) is also shown.}
\centering \centering
\begin{tabularx}{0.4\linewidth}{Xccc} \begin{tabularx}{0.4\linewidth}{Xccc}
\toprule \toprule
@ -781,14 +783,14 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\section{Optimal IFF with Parallel Stiffness} \section{Optimal IFF with Parallel Stiffness}
For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness). For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness).
In order 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). To keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased (\(k_a = k_n - k_p\), with \(k_n\) the total nano-hexapod stiffness).
A high pass filter is also added to limit the low frequency gain with a cut-off frequency \(\omega_i\) equal to one tenth of the system resonance (\(\omega_i = \omega_0/10\)). A high-pass filter is also added to limit the low-frequency gain with a cut-off frequency \(\omega_i\) equal to one tenth of the system resonance (\(\omega_i = \omega_0/10\)).
The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure \ref{fig:rotating_iff_kp_nass_optimal_gain}). The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure \ref{fig:rotating_iff_kp_nass_optimal_gain}).
It is shown that the soft nano-hexapod cannot yield good damping as the parallel stiffness cannot be made large enough compared to the negative stiffness induced by the rotation. 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 chosen too high as explained in Section \ref{sec:rotating_iff_parallel_stiffness}. 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 explain by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero \cite[chapt 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}. 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[chapt 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
This distance is larger for stiff nano-hexapod as the open-loop pole will be at higher frequencies while the open-loop zero, which depends on the value of the parallel stiffness, can only be made large for stiff nano-hexapods. 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. 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 \ref{tab:rotating_iff_kp_opt_iff_kp_params_nass}.
@ -817,7 +819,7 @@ The corresponding optimal controller gains and achievable damping are summarized
\section{Optimal Relative Motion Control} \section{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}). 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 is chosen such that 99\% of modal damping is obtained (obtained gains are summarized in Table \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 \ref{tab:rotating_rdc_opt_params_nass}).
\begin{minipage}[t]{0.49\linewidth} \begin{minipage}[t]{0.49\linewidth}
\begin{center} \begin{center}
@ -842,12 +844,12 @@ The gain is chosen is chosen such that 99\% of modal damping is obtained (obtain
\end{minipage} \end{minipage}
\section{Comparison of the obtained damped plants} \section{Comparison of the obtained damped plants}
Now that 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 \ref{fig:rotating_nass_damped_plant_comp}.
Similarly to what was concluded in previous analysis: Similar to what was concluded in the previous analysis:
\begin{itemize} \begin{itemize}
\item \acrshort{iff} adds coupling below the resonance frequency as compared to the open-loop and \acrshort{rdc} cases \item \acrshort{iff} adds more coupling below the resonance frequency as compared to the open-loop and \acrshort{rdc} cases
\item All three methods are yielding good damping, except for \acrshort{iff} applied on the soft nano-hexapod \item All three methods yield good damping, except for \acrshort{iff} applied on the soft nano-hexapod
\item Coupling is smaller for stiff nano-hexapods \item Coupling is smaller for stiff nano-hexapods
\end{itemize} \end{itemize}
@ -870,16 +872,16 @@ Similarly to what was concluded in previous analysis:
\end{center} \end{center}
\subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$} \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with \(k_p\) in red and RDC in yellow). The direct terms are shown by the solid lines and coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is \(\Omega = 60\,\text{rpm}\) and the suspended mass is \(m_n + m_s = \SI{16}{\kg}\).} \caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with \(k_p\) in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is \(\Omega = 60\,\text{rpm}\) and the suspended mass is \(m_n + m_s = \SI{16}{\kg}\).}
\end{figure} \end{figure}
\chapter{Nano-Active-Stabilization-System with rotation} \chapter{Nano-Active-Stabilization-System with rotation}
\label{sec:rotating_nass} \label{sec:rotating_nass}
Up 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 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 taken into account 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.
\section{Nano Active Stabilization System model} \section{Nano Active Stabilization System model}
In order to have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modelled 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 \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):
\begin{itemize} \begin{itemize}
\item the floor whose motion is imposed \item the floor whose motion is imposed
\item a 2-DoF granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}\), \(m_g = \SI{2500}{\kg}\)) \item a 2-DoF granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}\), \(m_g = \SI{2500}{\kg}\))
@ -898,14 +900,14 @@ A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the pay
\section{System dynamics} \section{System dynamics}
The dynamics of the un-damped and damped plants are identified using the optimal parameters found in Section \ref{sec:rotating_nano_hexapod}. 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 while the coupling terms are shown by the shaded ones. 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.
It can be observed that: It can be observed that:
\begin{itemize} \begin{itemize}
\item The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod \item The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod
\item Damping added by the three proposed techniques is quite high and the obtained plant is rather easy to control \item Damping added using the three proposed techniques is quite high, and the obtained plant is rather easy to control
\item There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz) \item There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz)
\item The two proposed IFF modification yields similar results \item The two proposed IFF modifications yield similar results
\end{itemize} \end{itemize}
\begin{figure}[htbp] \begin{figure}[htbp]
@ -933,18 +935,18 @@ It can be observed that:
\section{Effect of disturbances} \section{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 \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}).
Note that only the transfer function 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 as 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.
Conclusions are similar than with the uniaxial (non-rotating) model: Conclusions are similar than those of the uniaxial (non-rotating) model:
\begin{itemize} \begin{itemize}
\item Regarding the effect of floor motion and forces applied on the payload: \item Regarding the effect of floor motion and forces applied on the payload:
\begin{itemize} \begin{itemize}
\item 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 for the soft one \item 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
\item \acrshort{iff} degrades the performance at low frequency compared to \acrshort{rdc} \item \acrshort{iff} degrades the performance at low-frequency compared to \acrshort{rdc}
\end{itemize} \end{itemize}
\item Regarding the effect of micro-station vibrations: \item Regarding the effect of micro-station vibrations:
\begin{itemize} \begin{itemize}
\item Having a soft nano-hexapod allows to filter these vibrations between the suspensions 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}). \item 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}).
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
@ -967,7 +969,7 @@ Conclusions are similar than with the uniaxial (non-rotating) model:
\end{center} \end{center}
\subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$} \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_nass_effect_floor_motion}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.} \caption{\label{fig:rotating_nass_effect_floor_motion}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.}
\end{figure} \end{figure}
\begin{figure}[htbp] \begin{figure}[htbp]
@ -1012,7 +1014,7 @@ Conclusions are similar than with the uniaxial (non-rotating) model:
\end{center} \end{center}
\subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$} \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$}
\end{subfigure} \end{subfigure}
\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces \(f_{s,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low frequency.} \caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces \(f_{s,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low-frequency.}
\end{figure} \end{figure}
\chapter*{Conclusion} \chapter*{Conclusion}
@ -1020,25 +1022,25 @@ In this study, the gyroscopic effects induced by the spindle's rotation have bee
Decentralized \acrlong{iff} with pure integrators was shown to be unstable when applied to rotating platforms (Section \ref{sec:rotating_iff_pure_int}). Decentralized \acrlong{iff} with pure integrators was shown to be unstable when applied to rotating platforms (Section \ref{sec:rotating_iff_pure_int}).
Two modifications of the classical \acrshort{iff} control have been proposed to overcome this issue. Two modifications of the classical \acrshort{iff} control have been proposed to overcome this issue.
The first modification concerns the controller and consists of adding a high pass filter to the pure integrators. The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators.
This is equivalent to moving the controller pole to the left along the real axis. This 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 \ref{sec:rotating_iff_pseudo_int}).
The second proposed modification concerns the mechanical system. The second proposed modification concerns the mechanical system.
Additional springs are added in parallel with the actuators and force sensors. 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 \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 \ref{sec:rotating_comp_act_damp}.
While having very different implementations, both proposed modifications were found to be very similar when it comes to the attainable damping and the obtained closed loop system behavior. 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.
Then, 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 \ref{sec:rotating_nano_hexapod}).
As for the uniaxial model, three nano-hexapod stiffness are considered. 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 on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). 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).
Also, the attainable damping ratio of the soft nano-hexapod when using \acrshort{iff} is limited by 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 taken into account (Section \ref{sec:rotating_nass}). To be closer to the \acrlong{nass} dynamics, the limited compliance of the micro-station has been considered (Section \ref{sec:rotating_nass}).
Results are similar to that of the uniaxial model except that come complexity is added for the soft nano-hexapod due to the spindle's rotation. 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 are only slightly affecting the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that was showing better results with the uniaxial model. 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.
\printbibliography[heading=bibintoc,title={Bibliography}] \printbibliography[heading=bibintoc,title={Bibliography}]