Add XY-Stage picture

paper/paper.org

@@ -354,63 +354,58 @@ Either the control law can be change (Section ref:sec:iff_hpf) or the mechanical
 ** Modification of the Control Low
 # Reference to Preumont where its done
 
-In order to limit the low frequency loop gain, an high pass filter (HPF) can be added to the controller.
-The controller becomes
+In order to limit the low frequency controller gain, an high pass filter (HPF) can be added to the controller which becomes
 #+NAME: eq:IFF_LHF
 \begin{equation}
   \bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
 \end{equation}
-This is equivalent as to slightly shifting to pole to the left along the real axis.
+
+This is equivalent as to slightly shifting to 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.
+This is however not the case in this study as it will become in the next section.
 
 ** Feedback Analysis
 # Explain what do we mean for Loop Gain (loop gain for the decentralized loop)
+The loop gains for an individual decentralized controller $K_F(s)$ with and without the added HPF are shown in Figure ref:fig:loop_gain_modified_iff.
+The effect of the added HPF is a limitation of the low frequency gain.
 
-#+attr_latex: :options [c]{0.45\linewidth}
+# Explain how the root locus changes (the pole corresponding to the controller is moved to the left)
+The Root Loci for the decentralized IFF with and without the HPF are displayed in Figure ref:fig:root_locus_modified_iff.
+With the added HPF, the poles of the closed loop system are shown to be stable up to some value of the gain $g_\text{max}$
+#+NAME: eq:gmax_iff_hpf
+\begin{equation}
+  g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)
+\end{equation}
+This gain also corresponds as to when the low frequency loop gain reaches one.
+
+#+attr_latex: :options [b]{0.45\linewidth}
 #+begin_minipage
 #+name: fig:loop_gain_modified_iff
-#+caption: Bode Plot of the loop gain for IFF with and without the HPF with, $g = 2$, $\omega_i = 0.1 \omega_0$ and $\Omega = 0.1 \omega_0$
+#+caption: Modification of the loop gain with the added HFP, $g = 2$, $\omega_i = 0.1 \omega_0$ and $\Omega = 0.1 \omega_0$
 #+attr_latex: :scale 1 :float nil
 [[file:figs/loop_gain_modified_iff.pdf]]
 #+end_minipage
 \hfill
-#+attr_latex: :options [c]{0.5\linewidth}
+#+attr_latex: :options [b]{0.5\linewidth}
 #+begin_minipage
 #+name: fig:root_locus_modified_iff
-#+caption: Root Locus for IFF with and without the HPF, $\omega_i = 0.1 \omega_0$ and $\Omega = 0.1 \omega_0$
+#+caption: Modification of the Root Locus with the added HPF, $\omega_i = 0.1 \omega_0$ and $\Omega = 0.1 \omega_0$
 #+attr_latex: :scale 1 :float nil
 [[file:figs/root_locus_modified_iff.pdf]]
 #+end_minipage
 
-# Explain that now the low frequency loop gain does not reach a gain more than 1 (if g not so high)
-
-
-# Explain how the root locus changes (the pole corresponding to the controller is moved to the left)
-
-# Explain that it is stable for small values of $g$ but at some point, the system goes unstable
-# Explain what is the maximum value of the gain
-As shown in Figure ref:fig:root_locus_modified_iff, the poles of the closed loop system are stable for $g < g_\text{max}$
-\begin{equation}
-  g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)
-\end{equation}
-
 # Small rotational speeds allows to increase the control gain
 # Large wi allows more gain but less damping
 
-# Say that this corresponds as to have a low frequency gain of the loop gain less thank 1
-
 ** Optimal Control Parameters
 
 # Controller: two parameters: gain and wi
-Two parameters can be tuned for the controller eqref:eq:IFF_LHF, the gain $g$ and the frequency of the pole $\omega_i$.
+Two parameters can be tuned for the controller eqref:eq:IFF_LHF: the gain $g$ and the location of the pole $\omega_i$.
+The optimal values of $\omega_i$ and $g$ are considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
 
-# Try few wi
-Root Locus plots for several $\omega_i$ are shown in Figure ref:fig:root_locus_wi_modified_iff.
-
-# Small wi seems to allow more damping to be added
-# but the gain is limited to small values
-
-# Trade off
+# Root Loci
+The Root Loci for several $\omega_i$ are shown in Figure ref:fig:root_locus_wi_modified_iff.
+It is shown that even tough small $\omega_i$ seems to allow more damping to be added to the system resonances, the control gain $g$ may be limited to small values due to Eq. eqref:eq:gmax_iff_hpf.
 
 #+name: fig:root_locus_wi_modified_iff
 #+caption: Root Locus for several HPF cut-off frequencies $\omega_i$, $\Omega = 0.1 \omega_0$
@@ -418,7 +413,6 @@ Root Locus plots for several $\omega_i$ are shown in Figure ref:fig:root_locus_w
 [[file:figs/root_locus_wi_modified_iff.pdf]]
 
 # Study this trade-off
-The optimal values of $\omega_i$ and $g$ may be considered as the values for which the closed-loop poles are equally damped.
 
 # Explain how the figure is obtained
 
@@ -427,7 +421,7 @@ The optimal values of $\omega_i$ and $g$ may be considered as the values for whi
 # wi = 0.1 w0 is chosen
 
 #+name: fig:mod_iff_damping_wi
-#+caption: Attainable damping ratio $\xi_\text{cl}$ as a function of the HPF cut-off frequency. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown
+#+caption: Attainable damping ratio $\xi_\text{cl}$ as a function of the ratio $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown
 #+attr_latex: :scale 1
 [[file:figs/mod_iff_damping_wi.pdf]]
 
@@ -436,14 +430,25 @@ The optimal values of $\omega_i$ and $g$ may be considered as the values for whi
 ** Stiffness in Parallel with the Force Sensor
 # Zeros = remove force sensor
 # We want to have stable zeros => add stiffnesses in parallel
-Stiffness can be added in parallel to the force sensor to counteract the negative stiffness due to centrifugal forces.
-If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
+As was shown in the previous sections, the instability when using Decentralized IFF for rotating positioning platforms is due to Gyroscopic effects, more precisely to the negative stiffnesses induced by centrifugal forces.
+The idea in this section is to include additional springs in parallel with the force sensors to counteract the negative stiffness due to centrifugal forces.
+Such springs are schematically shown in Figure ref:fig:system_parallel_springs where $k_a$ is the stiffness of the actuator and $k_p$ the stiffness in parallel with the actuator and force sensor.
 
+#+attr_latex: :options [t]{0.55\linewidth}
+#+begin_minipage
 #+name: fig:system_parallel_springs
-#+caption:
-#+attr_latex: :scale 1
+#+caption: Studied system with additional springs in parallel with the actuators and force sensors
+#+attr_latex: :scale 1 :float nil
 [[file:figs/system_parallel_springs.pdf]]
-
+#+end_minipage
+\hfill
+#+attr_latex: :options [t]{0.40\linewidth}
+#+begin_minipage
+#+name: fig:cedrat_xy25xs
+#+caption: XY Piezoelectric Stage (XY25XS from Cedrat Technology)
+#+attr_latex: :width \linewidth :float nil
+[[file:figs/cedrat_xy25xs.png]]
+#+end_minipage
 
 # Sensed Force
 The forces measured by the force sensors are equal to:
@@ -456,9 +461,9 @@ The forces measured by the force sensors are equal to:
 
 # Maybe add the fact that this is equivalent to amplified piezo for instance
 # Add reference to cite:souleille18_concep_activ_mount_space_applic
-This could represent a system where
+This could represent a system where ref:fig:cedrat_xy25xs.
 
-** Plant Dynamics
+** Effect of the Parallel Stiffness on the Plant Dynamics
 
 We define an adimensional parameter $\alpha$, $0 \le \alpha < 1$, that describes the proportion of the stiffness in parallel with the actuator and force sensor:
 \begin{subequations}
@@ -493,7 +498,6 @@ The overall stiffness $k$ stays constant:
 # News terms with \alpha are added
 # w0 and xi are the same as before => only the zeros are changing and not the poles.
 
-** Effect of the Parallel Stiffness on the Plant Dynamics
 # Negative Stiffness due to rotation => the stiffness should be larger than that
 
 # For kp < negative stiffness => real zeros => non-minimum phase
@@ -505,6 +509,9 @@ The overall stiffness $k$ stays constant:
   \end{aligned}
 \end{equation}
 
+If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
+
+
 #+name: fig:plant_iff_kp
 #+caption: Bode Plot of $f_u/F_u$ without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
 #+attr_latex: :scale 1
@@ -572,6 +579,8 @@ The overall stiffness $k$ stays constant:
 * Conclusion
 <<sec:conclusion>>
 
+MIMO approach to study the coupling effects?
+
 * Acknowledgment
 :PROPERTIES:
 :UNNUMBERED: t

paper/paper.tex

@@ -1,4 +1,4 @@
-% Created 2020-07-01 mer. 10:48
+% Created 2020-07-01 mer. 14:20
 % Intended LaTeX compiler: pdflatex
 \documentclass{ISMA_USD2020}
 \usepackage[utf8]{inputenc}
@@ -53,7 +53,7 @@
 }
 
 \section{Introduction}
-\label{sec:org6a18e43}
+\label{sec:org5531002}
 \label{sec:introduction}
 Due to gyroscopic effects, the guaranteed robustness properties of Integral Force Feedback do not hold.
 Either the control architecture can be slightly modified or mechanical changes in the system can be performed.
@@ -61,10 +61,10 @@ This paper has been published
 The Matlab code that was use to obtain the results are available in \cite{dehaeze20_activ_dampin_rotat_posit_platf}.
 
 \section{Dynamics of Rotating Positioning Platforms}
-\label{sec:orgd72985f}
+\label{sec:org35aebb5}
 \label{sec:dynamics}
 \subsection{Model of a Rotating Positioning Platform}
-\label{sec:org617200e}
+\label{sec:org61dc9aa}
 In order to study how the rotation of a positioning platforms does affect the use of force feedback, a simple model of an X-Y positioning stage on top of a rotating stage is developed.
 
 The model is schematically represented in Figure \ref{fig:system} and forms the simplest system where gyroscopic forces can be studied.
@@ -95,7 +95,7 @@ The position of the payload is represented by \((d_u, d_v)\) expressed in the ro
 \end{figure}
 
 \subsection{Equations of Motion}
-\label{sec:orgb84906d}
+\label{sec:orgdd30881}
 To obtain of equation of motion for the system represented in Figure \ref{fig:system}, the Lagrangian equations are used:
 \begin{equation}
 \label{eq:lagrangian_equations}
@@ -139,7 +139,7 @@ One can verify that without rotation (\(\Omega = 0\)) the system becomes equival
 \end{subequations}
 
 \subsection{Transfer Functions in the Laplace domain}
-\label{sec:orgda6662b}
+\label{sec:org12fe22b}
 To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are transformed in the Laplace domain and the transfer function matrix from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) is obtained:
 \begin{equation}
 \label{eq:Gd_mimo_tf}
@@ -177,7 +177,7 @@ For all the numerical analysis in this study, \(\omega_0 = \SI{1}{\radian\per\se
 Even tough no system with such parameters will be encountered in practice, conclusions will be drawn relative to these parameters such that they can be generalized to any other parameter.
 
 \subsection{System Dynamics and Campbell Diagram}
-\label{sec:org006f494}
+\label{sec:org5558523}
 The poles of \(\bm{G}_d\) are the complex solutions \(p\) of
 \begin{equation}
   \left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0
@@ -232,10 +232,10 @@ For \(\Omega > \omega_0\), the low frequency complex conjugate poles \(p_{-}\) b
 \end{figure}
 
 \section{Decentralized Integral Force Feedback}
-\label{sec:org586ab0c}
+\label{sec:orga42580e}
 \label{sec:iff}
 \subsection{Force Sensors and Control Architecture}
-\label{sec:org56ae682}
+\label{sec:orgdd7381f}
 In order to apply Decentralized Integral Force Feedback to the system, force sensors are added in series with the two actuators (Figure \ref{fig:system_iff}).
 Two identical controllers \(K_F\) are added to feedback each of the sensed forces to its collocated actuator.
 The control diagram is shown in Figure \ref{fig:control_diagram_iff}.
@@ -255,7 +255,7 @@ The control diagram is shown in Figure \ref{fig:control_diagram_iff}.
 \end{minipage}
 
 \subsection{Plant Dynamics}
-\label{sec:org6b43274}
+\label{sec:org7bc5629}
 The forces measured by the force sensors are equal to:
 \begin{equation}
 \label{eq:measured_force}
@@ -310,7 +310,7 @@ This low frequency gain can be explained as follows: a constant force induces a
 \end{figure}
 
 \subsection{Decentralized Integral Force Feedback with Pure Integrators}
-\label{sec:orgaedf7bc}
+\label{sec:orgbe4061b}
 The two IFF controllers \(K_F\) are pure integrators
 \begin{equation}
   \bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_F(s) = g \cdot \frac{1}{s}
@@ -335,60 +335,55 @@ Two system modifications are proposed in the next sections to deal with this sta
 Either the control law can be change (Section \ref{sec:iff_hpf}) or the mechanical system slightly modified (Section \ref{sec:iff_kp}).
 
 \section{Integral Force Feedback with High Pass Filters}
-\label{sec:org34f8977}
+\label{sec:orgd914f89}
 \label{sec:iff_hpf}
 \subsection{Modification of the Control Low}
-\label{sec:org809db54}
-In order to limit the low frequency loop gain, an high pass filter (HPF) can be added to the controller.
-The controller becomes
+\label{sec:org9cbf657}
+In order to limit the low frequency controller gain, an high pass filter (HPF) can be added to the controller which becomes
 \begin{equation}
 \label{eq:IFF_LHF}
   \bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
 \end{equation}
-This is equivalent as to slightly shifting to pole to the left along the real axis.
+
+This is equivalent as to slightly shifting to 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}.
+This is however not the case in this study as it will become in the next section.
 
 \subsection{Feedback Analysis}
-\label{sec:orga2d434f}
-\begin{minipage}[c]{0.45\linewidth}
+\label{sec:org5571d5f}
+The loop gains for an individual decentralized controller \(K_F(s)\) with and without the added HPF are shown in Figure \ref{fig:loop_gain_modified_iff}.
+The effect of the added HPF is a limitation of the low frequency gain.
+
+The Root Loci for the decentralized IFF with and without the HPF are displayed in Figure \ref{fig:root_locus_modified_iff}.
+With the added HPF, the poles of the closed loop system are shown to be stable up to some value of the gain \(g_\text{max}\)
+\begin{equation}
+\label{eq:gmax_iff_hpf}
+  g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)
+\end{equation}
+This gain also corresponds as to when the low frequency loop gain reaches one.
+
+\begin{minipage}[b]{0.45\linewidth}
 \begin{center}
 \includegraphics[scale=1]{figs/loop_gain_modified_iff.pdf}
-\captionof{figure}{\label{fig:loop_gain_modified_iff}Bode Plot of the loop gain for IFF with and without the HPF with, \(g = 2\), \(\omega_i = 0.1 \omega_0\) and \(\Omega = 0.1 \omega_0\)}
+\captionof{figure}{\label{fig:loop_gain_modified_iff}Modification of the loop gain with the added HFP, \(g = 2\), \(\omega_i = 0.1 \omega_0\) and \(\Omega = 0.1 \omega_0\)}
 \end{center}
 \end{minipage}
 \hfill
-\begin{minipage}[c]{0.5\linewidth}
+\begin{minipage}[b]{0.5\linewidth}
 \begin{center}
 \includegraphics[scale=1]{figs/root_locus_modified_iff.pdf}
-\captionof{figure}{\label{fig:root_locus_modified_iff}Root Locus for IFF with and without the HPF, \(\omega_i = 0.1 \omega_0\) and \(\Omega = 0.1 \omega_0\)}
+\captionof{figure}{\label{fig:root_locus_modified_iff}Modification of the Root Locus with the added HPF, \(\omega_i = 0.1 \omega_0\) and \(\Omega = 0.1 \omega_0\)}
 \end{center}
 \end{minipage}
 
-
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/loop_gain_modified_iff.pdf}
-\caption{\label{fig:loop_gain_modified_iff}Bode Plot of the Loop Gain for IFF with and without the HPF with \(\omega_i = 0.1 \omega_0\), \(g = 2\) and \(\Omega = 0.1 \omega_0\)}
-\end{figure}
-
-As shown in Figure \ref{fig:root_locus_modified_iff}, the poles of the closed loop system are stable for \(g < g_\text{max}\)
-\begin{equation}
-  g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)
-\end{equation}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/root_locus_modified_iff.pdf}
-\label{fig:root_locus_modified_iff}
-\end{figure}
-
 \subsection{Optimal Control Parameters}
-\label{sec:org7cc60ca}
+\label{sec:org5313add}
 
-Two parameters can be tuned for the controller \eqref{eq:IFF_LHF}, the gain \(g\) and the frequency of the pole \(\omega_i\).
+Two parameters can be tuned for the controller \eqref{eq:IFF_LHF}: the gain \(g\) and the location of the pole \(\omega_i\).
+The optimal values of \(\omega_i\) and \(g\) are considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
 
-Root Locus plots for several \(\omega_i\) are shown in Figure \ref{fig:root_locus_wi_modified_iff}.
+The Root Loci for several \(\omega_i\) are shown in Figure \ref{fig:root_locus_wi_modified_iff}.
+It is shown that even tough small \(\omega_i\) seems to allow more damping to be added to the system resonances, the control gain \(g\) may be limited to small values due to Eq. \eqref{eq:gmax_iff_hpf}.
 
 \begin{figure}[htbp]
 \centering
@@ -396,27 +391,34 @@ Root Locus plots for several \(\omega_i\) are shown in Figure \ref{fig:root_locu
 \caption{\label{fig:root_locus_wi_modified_iff}Root Locus for several HPF cut-off frequencies \(\omega_i\), \(\Omega = 0.1 \omega_0\)}
 \end{figure}
 
-The optimal values of \(\omega_i\) and \(g\) may be considered as the values for which the closed-loop poles are equally damped.
-
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/mod_iff_damping_wi.pdf}
-\caption{\label{fig:mod_iff_damping_wi}Attainable damping ratio \(\xi_\text{cl}\) as a function of the HPF cut-off frequency. Corresponding control gain \(g_\text{opt}\) and \(g_\text{max}\) are also shown}
+\caption{\label{fig:mod_iff_damping_wi}Attainable damping ratio \(\xi_\text{cl}\) as a function of the ratio \(\omega_i/\omega_0\). Corresponding control gain \(g_\text{opt}\) and \(g_\text{max}\) are also shown}
 \end{figure}
 
 \section{Integral Force Feedback with Parallel Springs}
-\label{sec:org8ffeef1}
+\label{sec:org9bf4afa}
 \label{sec:iff_kp}
 \subsection{Stiffness in Parallel with the Force Sensor}
-\label{sec:org831f255}
-Stiffness can be added in parallel to the force sensor to counteract the negative stiffness due to centrifugal forces.
-If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
+\label{sec:org7253a28}
+As was shown in the previous sections, the instability when using Decentralized IFF for rotating positioning platforms is due to Gyroscopic effects, more precisely to the negative stiffnesses induced by centrifugal forces.
+The idea in this section is to include additional springs in parallel with the force sensors to counteract the negative stiffness due to centrifugal forces.
+Such springs are schematically shown in Figure \ref{fig:system_parallel_springs} where \(k_a\) is the stiffness of the actuator and \(k_p\) the stiffness in parallel with the actuator and force sensor.
 
-\begin{figure}[htbp]
-\centering
+\begin{minipage}[t]{0.55\linewidth}
+\begin{center}
 \includegraphics[scale=1]{figs/system_parallel_springs.pdf}
-\label{fig:system_parallel_springs}
-\end{figure}
+\captionof{figure}{\label{fig:system_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors}
+\end{center}
+\end{minipage}
+\hfill
+\begin{minipage}[t]{0.40\linewidth}
+\begin{center}
+\includegraphics[width=\linewidth]{figs/cedrat_xy25xs.png}
+\captionof{figure}{\label{fig:cedrat_xy25xs}XY Piezoelectric Stage (XY25XS from Cedrat-Technology)}
+\end{center}
+\end{minipage}
 
 
 The forces measured by the force sensors are equal to:
@@ -429,8 +431,8 @@ The forces measured by the force sensors are equal to:
 
 This could represent a system where
 
-\subsection{Plant Dynamics}
-\label{sec:org98721cc}
+\subsection{Effect of the Parallel Stiffness on the Plant Dynamics}
+\label{sec:orgf4cd5ba}
 
 We define an adimensional parameter \(\alpha\), \(0 \le \alpha < 1\), that describes the proportion of the stiffness in parallel with the actuator and force sensor:
 \begin{subequations}
@@ -461,8 +463,6 @@ The overall stiffness \(k\) stays constant:
 \end{bmatrix}
 \end{equation}
 
-\subsection{Effect of the Parallel Stiffness on the Plant Dynamics}
-\label{sec:org0b03ba2}
 \begin{equation}
   \begin{aligned}
     \alpha > \frac{\Omega^2}{{\omega_0}^2} \\
@@ -470,6 +470,9 @@ The overall stiffness \(k\) stays constant:
   \end{aligned}
 \end{equation}
 
+If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
+
+
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/plant_iff_kp.pdf}
@@ -483,7 +486,7 @@ The overall stiffness \(k\) stays constant:
 \end{figure}
 
 \subsection{Optimal Parallel Stiffness}
-\label{sec:org0aa60d8}
+\label{sec:orgeadc68d}
 \begin{figure}[htbp]
 \begin{subfigure}[c]{0.49\linewidth}
 \includegraphics[width=\linewidth]{figs/root_locus_iff_kps.pdf}
@@ -498,15 +501,15 @@ The overall stiffness \(k\) stays constant:
 \end{figure}
 
 \section{Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages}
-\label{sec:orgde2e2f0}
+\label{sec:org21311c2}
 \label{sec:comparison}
 \subsection{Physical Comparison}
-\label{sec:org661a4d0}
+\label{sec:orgff9353a}
 
 
 
 \subsection{Attainable Damping}
-\label{sec:orgb54beda}
+\label{sec:org43e21e8}
 
 \begin{figure}[htbp]
 \centering
@@ -515,7 +518,7 @@ The overall stiffness \(k\) stays constant:
 \end{figure}
 
 \subsection{Transmissibility and Compliance}
-\label{sec:org63e90d2}
+\label{sec:org4cd8990}
 \begin{figure}[htbp]
 \begin{subfigure}[c]{0.45\linewidth}
 \includegraphics[width=\linewidth]{figs/comp_compliance.pdf}
@@ -530,11 +533,13 @@ The overall stiffness \(k\) stays constant:
 \end{figure}
 
 \section{Conclusion}
-\label{sec:org646f615}
+\label{sec:org2c33141}
 \label{sec:conclusion}
 
+MIMO approach to study the coupling effects?
+
 \section*{Acknowledgment}
-\label{sec:org7bb2645}
+\label{sec:org33821c1}
 
 \bibliography{ref.bib}
 \end{document}