Update paper text

2020-06-22 13:35:01 +02:00
parent eb0635f11f
commit 2cb68a4d5f
4 changed files with 487 additions and 50 deletions
--- a/paper/paper.org
+++ b/paper/paper.org
@@ -1,4 +1,4 @@
-#+TITLE: Vibration control of a rotating Stewart platform
+#+TITLE: Active Damping of Rotating Positioning Platforms
 :DRAWER:
 #+LATEX_CLASS: ISMA_USD2020
 #+OPTIONS: toc:nil
@@ -53,20 +53,42 @@

 * Introduction
 <<sec:introduction>>
+*** Establish the importance of the research topic                 :ignore:
+# Active Damping + Rotating Systems

+*** Applications of active damping                                  :ignore:
+# Link to previous paper / tomography

-* Theory
-<<sec:theory>>
+cite:dehaeze18_sampl_stabil_for_tomog_exper

-** Rotating Positioning Stage
+*** Current active damping techniques                               :ignore:
+# IFF, DVF

-# Description of the system
+*** Describe a gap in the research                                 :ignore:
+# No literature on rotating systems => gyroscopic effects
+
+*** Describe the paper itself / the problem which is addressed     :ignore:
+
+*** Introduce Each part of the paper                               :ignore:
+
+* System Under Study
+** Rotating Positioning Platform
+# Simplest system where gyroscopic forces can be studied
+Consider the rotating X-Y stage of Figure [[fig:rotating_xy_platform]].
+
+# Present the system, parameters, assumptions
+
+# Small displacements
+
+# Constant rotating speed
+
+# Explain the frames (inertial frame x,y, rotating frame u,v)

 - $k$: Actuator's Stiffness [N/m]
 - $m$: Payload's mass [kg]
- $\omega_0 = \sqrt{\frac{k}{m}}$: Resonance of the (non-rotating) mass-spring system [rad/s]
 - $\Omega = \dot{\theta}$: rotation speed [rad/s]
-
+- $F_u$, $F_v$
+- $d_u$, $d_v$

 #+name: fig:rotating_xy_platform
 #+caption: Figure caption
@@ -74,34 +96,96 @@
 [[file:figs/rotating_xy_platform.pdf]]


+#+name: fig:cedrat_xy25xs
+#+caption: Figure caption
+#+attr_latex: :width 0.5\linewidth
+[[file:figs/cedrat_xy25xs.jpg]]
+
 ** Equation of Motion
+The system has two degrees of freedom and is thus fully described by the generalized coordinates $u$ and $v$.

 Let's express the kinetic energy $T$ and the potential energy $V$ of the mass $m$ (neglecting the rotational energy):
+
+Dissipation function $R$
+Kinetic energy $T$
+Potential energy $V$
 #+name: eq:energy_inertial_frame
 \begin{subequations}
  \begin{align}
-    T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
-    R & = \frac{1}{2} c \left( \dot{x}^2 + \dot{y}^2 \right) \\
-    V & = \frac{1}{2} k \left( x^2 + y^2 \right)
+    T & = \frac{1}{2} m \left( \left( \dot{u} - \Omega v \right)^2 + \left( \dot{v} + \Omega u \right)^2 \right) \\
+    R & = \frac{1}{2} c \left( \dot{u}^2 + \dot{v}^2 \right) \\
+    V & = \frac{1}{2} k \left( u^2 + v^2 \right)
  \end{align}
 \end{subequations}

 The Lagrangian is the kinetic energy minus the potential energy:
 #+name: eq:lagrangian_inertial_frame
 \begin{equation}
-L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
+L = T - V
 \end{equation}

-The external forces applied to the mass are:
+From the Lagrange's equations of the second kind eqref:eq:lagrange_second_kind, the equation of motion eqref:eq:eom_mixed is obtained ($q_1 = u$, $q_2 = v$).
+\begin{equation}
+  \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
+\end{equation}
+
+\begin{equation}
+  \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial R}{\partial \dot{q}_i} - \frac{\partial V}{\partial q_i} = Q_i
+\end{equation}
+
+with $Q_i$ is the generalized force associated with the generalized variable $q_i$ ($F_u$ and $F_v$).
+
 \begin{subequations}
  \begin{align}
-    F_{\text{ext}, x} &= F_u \cos{\theta} - F_v \sin{\theta}\\
-    F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta}
+    m \ddot{u} + c \dot{u} + ( k - m \Omega ) u &= F_u + 2 m \Omega \dot{v} \\
+    m \ddot{v} + c \dot{v} + ( k \underbrace{-\,m \Omega}_{\text{Centrif.}} ) v &= F_v \underbrace{-\,2 m \Omega \dot{u}}_{\text{Coriolis}}
+  \end{align}
+\end{subequations}
+
+# Explain Gyroscopic effects
+
+
+
+
+# Laplace Domain
+
+\begin{subequations}
+  \begin{align}
+    u &= \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_u +  \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_v \\
+    v &= \frac{-2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_u +  \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_v
+  \end{align}
+\end{subequations}
+
+# Change of variables
+\begin{equation}
+\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
+\bm{G}_d
+\begin{bmatrix} F_u \\ F_v \end{bmatrix}
+\end{equation}
+Where $\bm{G}_d$ is a $2 \times 2$ transfer function matrix.
+
+\begin{equation}
+\bm{G}_d = \frac{1}{k} \frac{1}{G_{dp}}
+\begin{bmatrix}
+   G_{dz} & G_{dc} \\
+  -G_{dc} & G_{dz}
+\end{bmatrix}
+\end{equation}
+With:
+\begin{subequations}
+  \begin{align}
+    G_{dp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
+    G_{dz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
+    G_{dc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
  \end{align}
 \end{subequations}


-From the Lagrange's equations of the second kind eqref:eq:lagrange_second_kind, the equation of motion eqref:eq:eom_mixed is obtained.
+
+- $\omega_0 = \sqrt{\frac{k}{m}}$: Natural frequency of the mass-spring system in $\si{\radian/\s}$
+- $\xi$ damping ratio
+
+

 #+name: eq:lagrange_second_kind
 \begin{equation}
@@ -186,18 +270,104 @@ As the rotation speed increases, one of the two resonant frequency goes to lower

 #+name: fig:campbell_diagram
 #+caption: Campbell Diagram
+#+attr_latex: :scale 1
 [[file:figs/campbell_diagram.pdf]]

 # Bode Plots for different ratio wr/w0

 The magnitude of the coupling terms are increasing with the rotation speed.

+#+name: fig:plant_compare_rotating_speed
+#+caption: Caption
+#+attr_latex: :scale 1
+[[file:figs/plant_compare_rotating_speed.pdf]]

-** Integral Force Feedback
+* Integral Force Feedback
+** Control Schematic

+** Equations

-** Direct Velocity Feedback
+** Plant Dynamics

+#+name: fig:root_locus_pure_iff
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/root_locus_pure_iff.pdf]]
+
+** Physical Interpretation
+
+* Integral Force Feedback with Low Pass Filters
+
+#+name: fig:loop_gain_modified_iff
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/loop_gain_modified_iff.pdf]]
+
+#+name: fig:root_locus_modified_iff
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/root_locus_modified_iff_bis.pdf]]
+
+#+name: fig:root_locus_wi_modified_iff
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/root_locus_wi_modified_iff.pdf]]
+
+* Integral Force Feedback with Parallel Springs
+
+#+name: fig:rotating_xy_platform_springs
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/rotating_xy_platform_springs.pdf]]
+
+#+name: fig:plant_iff_kp
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/plant_iff_kp.pdf]]
+
+#+name: fig:root_locus_iff_kps
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/root_locus_iff_kps.pdf]]
+
+#+name: fig:root_locus_iff_kp_bis
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/root_locus_iff_kp_bis.pdf]]
+
+#+name: fig:root_locus_opt_gain_iff_kp
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/root_locus_opt_gain_iff_kp.pdf]]
+
+#+name: fig:plant_iff_compare_rotating_speed
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/plant_iff_compare_rotating_speed.pdf]]
+
+* Direct Velocity Feedback
+
+#+name: fig:root_locus_dvf
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/root_locus_dvf.pdf]]
+
+* Comparison of the Proposed Active Damping Techniques
+
+#+name: fig:comp_root_locus
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/comp_root_locus.pdf]]
+
+#+name: fig:comp_compliance
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/comp_compliance.pdf]]
+
+#+name: fig:comp_transmissibility
+#+caption: Figure caption
+#+attr_latex: :scale 1
+[[file:figs/comp_transmissibility.pdf]]

 * Conclusion
 <<sec:conclusion>>
--- a/paper/paper.pdf
+++ b/paper/paper.pdf
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -1,4 +1,4 @@
-% Created 2020-06-08 lun. 11:40
+% Created 2020-06-22 lun. 13:27
 % Intended LaTeX compiler: pdflatex
 \documentclass{ISMA_USD2020}
 \usepackage[utf8]{inputenc}
@@ -31,15 +31,9 @@
 \affil[1] {Precision Mechatronics Laboratory\NewLineAffil University of Liege, Belgium \NewAffil}
 \affil[2] {BEAMS Department\NewLineAffil Free University of Brussels, Belgium \NewAffil}
 \affil[3] {European Synchrotron Radiation Facility \NewLineAffil Grenoble, France e-mail: \textbf{thomas.dehaeze@esrf.fr}}
+\bibliographystyle{IEEEtran}
 \date{}
-\title{Vibration control of a rotating Stewart platform}
-\hypersetup{
- pdfauthor={},
- pdftitle={Vibration control of a rotating Stewart platform},
- pdfkeywords={},
- pdfsubject={},
- pdfcreator={Emacs 27.0.91 (Org mode 9.4)}, 
- pdflang={English}}
+\title{Active Damping of Rotating Positioning Platforms}
 \begin{document}

 \maketitle
@@ -49,16 +43,23 @@
 }

 \section{Introduction}
-\label{sec:org335669b}
+\label{sec:orgd20252d}
 \label{sec:introduction}
+\cite{dehaeze18_sampl_stabil_for_tomog_exper}

+\section{System Under Study}
+\label{sec:orgacbe1ae}
+\subsection{Rotating Positioning Platform}
+\label{sec:org07e4fc8}
+Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.

-\section{Theory}
-\label{sec:org8b756e7}
-\label{sec:theory}
-
-\subsection{Rotating Positioning Stage}
-\label{sec:orgbf4a596}
+\begin{itemize}
+\item \(k\): Actuator's Stiffness [N/m]
+\item \(m\): Payload's mass [kg]
+\item \(\Omega = \dot{\theta}\): rotation speed [rad/s]
+\item \(F_u\), \(F_v\)
+\item \(d_u\), \(d_v\)
+\end{itemize}

 \begin{figure}[htbp]
 \centering
@@ -67,39 +68,304 @@
 \end{figure}


+\begin{figure}[htbp]
+\centering
+\includegraphics[width=0.5\linewidth]{figs/cedrat_xy25xs.jpg}
+\caption{\label{fig:cedrat_xy25xs}Figure caption}
+\end{figure}
+
 \subsection{Equation of Motion}
-\label{sec:orgaa8880a}
+\label{sec:orgac1a52a}
+The system has two degrees of freedom and is thus fully described by the generalized coordinates \(u\) and \(v\).

-Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
-\begin{align}
+Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\) (neglecting the rotational energy):
+
+Dissipation function \(R\)
+Kinetic energy \(T\)
+Potential energy \(V\)
+\begin{subequations}
 \label{eq:energy_inertial_frame}
-T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
-V & = \frac{1}{2} k \left( x^2 + y^2 \right)
-\end{align}
+  \begin{align}
+    T & = \frac{1}{2} m \left( \left( \dot{u} - \Omega v \right)^2 + \left( \dot{v} + \Omega u \right)^2 \right) \\
+    R & = \frac{1}{2} c \left( \dot{u}^2 + \dot{v}^2 \right) \\
+    V & = \frac{1}{2} k \left( u^2 + v^2 \right)
+  \end{align}
+\end{subequations}

-The Lagrangian is the kinetic energy minus the potential energy.
+The Lagrangian is the kinetic energy minus the potential energy:
 \begin{equation}
 \label{eq:lagrangian_inertial_frame}
-L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
+L = T - V
 \end{equation}

+From the Lagrange's equations of the second kind \eqref{eq:lagrange_second_kind}, the equation of motion \eqref{eq:eom_mixed} is obtained (\(q_1 = u\), \(q_2 = v\)).
+\begin{equation}
+  \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
+\end{equation}

-\subsection{Integral Force Feedback}
-\label{sec:org754b644}
+\begin{equation}
+  \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial R}{\partial \dot{q}_i} - \frac{\partial V}{\partial q_i} = Q_i
+\end{equation}
+
+with \(Q_i\) is the generalized force associated with the generalized variable \(q_i\) (\(F_u\) and \(F_v\)).
+
+\begin{subequations}
+  \begin{align}
+    m \ddot{u} + c \dot{u} + ( k - m \Omega ) u &= F_u + 2 m \Omega \dot{v} \\
+    m \ddot{v} + c \dot{v} + ( k \underbrace{-\,m \Omega}_{\text{Centrif.}} ) v &= F_v \underbrace{-\,2 m \Omega \dot{u}}_{\text{Coriolis}}
+  \end{align}
+\end{subequations}
+
+\begin{subequations}
+  \begin{align}
+    u &= \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_u +  \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_v \\
+    v &= \frac{-2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_u +  \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_v
+  \end{align}
+\end{subequations}
+
+\begin{equation}
+\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
+\bm{G}_d
+\begin{bmatrix} F_u \\ F_v \end{bmatrix}
+\end{equation}
+Where \(\bm{G}_d\) is a \(2 \times 2\) transfer function matrix.
+
+\begin{equation}
+\bm{G}_d = \frac{1}{k} \frac{1}{G_{dp}}
+\begin{bmatrix}
+   G_{dz} & G_{dc} \\
+  -G_{dc} & G_{dz}
+\end{bmatrix}
+\end{equation}
+With:
+\begin{subequations}
+  \begin{align}
+    G_{dp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
+    G_{dz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
+    G_{dc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
+  \end{align}
+\end{subequations}


-\subsection{Direct Velocity Feedback}
-\label{sec:org9cbf82a}

+\begin{itemize}
+\item \(\omega_0 = \sqrt{\frac{k}{m}}\): Natural frequency of the mass-spring system in \(\si{\radian/\s}\)
+\item \(\xi\) damping ratio
+\end{itemize}
+
+
+
+\begin{equation}
+\label{eq:lagrange_second_kind}
+  \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) = \frac{\partial L}{\partial q_j}
+\end{equation}
+
+\begin{subequations}
+\label{eq:eom_mixed}
+  \begin{align}
+    m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
+    m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
+  \end{align}
+\end{subequations}
+
+Performing the change coordinates from \((x, y)\) to \((d_x, d_y, \theta)\):
+\begin{subequations}
+  \begin{align}
+    x & = d_u \cos{\theta} - d_v \sin{\theta}\\
+    y & = d_u \sin{\theta} + d_v \cos{\theta}
+  \end{align}
+\end{subequations}
+
+Gives
+\begin{subequations}
+\label{eq:oem_coupled}
+  \begin{align}
+    m \ddot{d_u} + (k - m\dot{\theta}^2) d_u &= F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \label{eq:du_coupled} \\
+    m \ddot{d_v} + (k \underbrace{-\ m\dot{\theta}^2}_{\text{Centrif.}}) d_v &= F_v \underbrace{-\ 2 m\dot{d_u}\dot{\theta}}_{\text{Coriolis}} \underbrace{-\ m d_u\ddot{\theta}}_{\text{Euler}} \label{eq:dv_coupled}
+  \end{align}
+\end{subequations}
+
+We obtain two differential equations that are coupled through:
+\begin{itemize}
+\item \textbf{Euler forces}: \(m d_v \ddot{\theta}\)
+\item \textbf{Coriolis forces}: \(2 m \dot{d_v} \dot{\theta}\)
+\end{itemize}
+
+Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass \(m\) and stiffness \(k- m\dot{\theta}^2\).
+Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to \textbf{centrifugal forces}).
+
+\subsection{Constant Rotating Speed}
+\label{sec:org47aaeee}
+To simplify, let's consider a constant rotating speed \(\dot{\theta} = \Omega\) and thus \(\ddot{\theta} = 0\).
+
+\begin{equation}
+\label{eq:coupledplant}
+\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
+\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}
+\begin{bmatrix}
+  ms^2 + (k-m{\omega_0}^2) & 2 m \omega_0 s \\
+  -2 m \omega_0 s          & ms^2 + (k-m{\omega_0}^2) \\
+\end{bmatrix}
+\begin{bmatrix} F_u \\ F_v \end{bmatrix}
+\end{equation}
+
+\begin{equation}
+\label{eq:coupled_plant}
+\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
+\frac{\frac{1}{k}}{\left( \frac{s^2}{{\omega_0}^2} + (1 - \frac{{\Omega}^2}{{\omega_0}^2}) \right)^2 + \left( 2 \frac{{\Omega} s}{{\omega_0}^2} \right)^2}
+\begin{bmatrix}
+  \frac{s^2}{{\omega_0}^2} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} & 2 \frac{\Omega s}{{\omega_0}^2} \\
+  -2 \frac{\Omega s}{{\omega_0}^2}          & \frac{s^2}{{\omega_0}^2} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
+\end{bmatrix}
+\begin{bmatrix} F_u \\ F_v \end{bmatrix}
+\end{equation}
+
+When the rotation speed is null, the coupling terms are equal to zero and the diagonal terms corresponds to one degree of freedom mass spring system.
+\begin{equation}
+\label{eq:coupled_plant_no_rot}
+\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
+\frac{\frac{1}{k}}{\frac{s^2}{{\omega_0}^2} + 1}
+\begin{bmatrix}
+  1 & 0 \\
+  0 & 1
+\end{bmatrix}
+\begin{bmatrix} F_u \\ F_v \end{bmatrix}
+\end{equation}
+
+When the rotation speed in not null, the resonance frequency is duplicated into two pairs of complex conjugate poles.
+As the rotation speed increases, one of the two resonant frequency goes to lower frequencies as the other one goes to higher frequencies (Figure \ref{fig:campbell_diagram}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/campbell_diagram.pdf}
+\caption{\label{fig:campbell_diagram}Campbell Diagram}
+\end{figure}
+
+The magnitude of the coupling terms are increasing with the rotation speed.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/plant_compare_rotating_speed.pdf}
+\caption{\label{fig:plant_compare_rotating_speed}Caption}
+\end{figure}
+
+\section{Integral Force Feedback}
+\label{sec:org78c2eab}
+\subsection{Control Schematic}
+\label{sec:org6a00238}
+
+\subsection{Equations}
+\label{sec:org5480f1b}
+
+\subsection{Plant Dynamics}
+\label{sec:orgbb0952e}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_pure_iff.pdf}
+\caption{\label{fig:root_locus_pure_iff}Figure caption}
+\end{figure}
+
+\subsection{Physical Interpretation}
+\label{sec:orgdb25e2c}
+
+\section{Integral Force Feedback with Low Pass Filters}
+\label{sec:org2985d35}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/loop_gain_modified_iff.pdf}
+\caption{\label{fig:loop_gain_modified_iff}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_modified_iff_bis.pdf}
+\caption{\label{fig:root_locus_modified_iff}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_wi_modified_iff.pdf}
+\caption{\label{fig:root_locus_wi_modified_iff}Figure caption}
+\end{figure}
+
+\section{Integral Force Feedback with Parallel Springs}
+\label{sec:orga4142a5}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/rotating_xy_platform_springs.pdf}
+\caption{\label{fig:rotating_xy_platform_springs}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/plant_iff_kp.pdf}
+\caption{\label{fig:plant_iff_kp}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_iff_kps.pdf}
+\caption{\label{fig:root_locus_iff_kps}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_iff_kp_bis.pdf}
+\caption{\label{fig:root_locus_iff_kp_bis}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_opt_gain_iff_kp.pdf}
+\caption{\label{fig:root_locus_opt_gain_iff_kp}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/plant_iff_compare_rotating_speed.pdf}
+\caption{\label{fig:plant_iff_compare_rotating_speed}Figure caption}
+\end{figure}
+
+\section{Direct Velocity Feedback}
+\label{sec:org6a1be4f}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_dvf.pdf}
+\caption{\label{fig:root_locus_dvf}Figure caption}
+\end{figure}
+
+\section{Comparison of the Proposed Active Damping Techniques}
+\label{sec:orga9658c0}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_root_locus.pdf}
+\caption{\label{fig:comp_root_locus}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_compliance.pdf}
+\caption{\label{fig:comp_compliance}Figure caption}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_transmissibility.pdf}
+\caption{\label{fig:comp_transmissibility}Figure caption}
+\end{figure}

 \section{Conclusion}
-\label{sec:org8d24de3}
+\label{sec:orgcdf948f}
 \label{sec:conclusion}


-\section{Acknowledgment}
-\label{sec:orgb252937}
+\section*{Acknowledgment}
+\label{sec:org6c21e13}

-
-\bibliography{ref}
+\bibliography{ref.bib}
 \end{document}