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@ -20,6 +20,9 @@
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#+LATEX_HEADER: \usepackage{algorithmic, graphicx, textcomp}
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#+LATEX_HEADER: \usepackage{xcolor, import, hyperref}
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#+LATEX_HEADER: \usepackage[USenglish]{babel}
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#+LATEX_HEADER_EXTRA: \usepackage{tikz}
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#+LATEX_HEADER_EXTRA: \usetikzlibrary{shapes.misc}
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#+LATEX_HEADER: \setcounter{footnote}{1}
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#+LATEX_HEADER: \input{config.tex}
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@ -56,6 +59,10 @@
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*** Establish the importance of the research topic :ignore:
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# Active Damping + Rotating System
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Controller Poles are shown by black crosses (
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\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
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).
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*** Applications of active damping :ignore:
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# Link to previous paper / tomography
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@ -122,7 +129,7 @@ The Lagrangian is the kinetic energy minus the potential energy:
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L = T - V
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\end{equation}
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From the Lagrange's equations of the second kind eqref:eq:lagrange_second_kind, the equation of motion eqref:eq:eom_mixed is obtained ($q_1 = u$, $q_2 = v$).
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From the Lagrange's equations of the second kind, the equation of motion is obtained ($q_1 = u$, $q_2 = v$).
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\begin{equation}
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\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
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\end{equation}
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@ -319,7 +326,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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#+name: fig:root_locus_modified_iff
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#+caption: Figure caption
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#+attr_latex: :scale 1
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[[file:figs/root_locus_modified_iff_bis.pdf]]
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[[file:figs/root_locus_modified_iff_ter.pdf]]
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** Optimal Cut-Off Frequency
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@ -354,7 +361,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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#+name: fig:root_locus_iff_kp_bis
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#+caption: Figure caption
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#+attr_latex: :scale 1
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[[file:figs/root_locus_iff_kp_bis.pdf]]
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[[file:figs/root_locus_iff_kp_ter.pdf]]
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#+name: fig:root_locus_opt_gain_iff_kp
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#+caption: Figure caption
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paper/paper.pdf
@ -1,4 +1,4 @@
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% Created 2020-06-22 lun. 17:38
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% Created 2020-06-23 mar. 19:34
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% Intended LaTeX compiler: pdflatex
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\documentclass{ISMA_USD2020}
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\usepackage[utf8]{inputenc}
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@ -32,8 +32,17 @@
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\affil[2] {BEAMS Department\NewLineAffil Free University of Brussels, Belgium \NewAffil}
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\affil[3] {European Synchrotron Radiation Facility \NewLineAffil Grenoble, France e-mail: \textbf{thomas.dehaeze@esrf.fr}}
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\bibliographystyle{IEEEtran}
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\usepackage{tikz}
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\usetikzlibrary{shapes.misc}
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\date{}
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\title{Active Damping of Rotating Positioning Platforms}
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\hypersetup{
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pdfauthor={},
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pdftitle={Active Damping of Rotating Positioning Platforms},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 27.0.91 (Org mode 9.4)},
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pdflang={English}}
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\begin{document}
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\maketitle
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@ -43,14 +52,17 @@
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}
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\section{Introduction}
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\label{sec:org67e0a4e}
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\label{sec:org977317c}
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\label{sec:introduction}
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Controller Poles are shown by black crosses (
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\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
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).
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\cite{dehaeze18_sampl_stabil_for_tomog_exper}
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\section{System Under Study}
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\label{sec:org85bcde2}
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\label{sec:org042e800}
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\subsection{Rotating Positioning Platform}
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\label{sec:org4959a5e}
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\label{sec:org489e4b9}
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Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
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\begin{itemize}
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@ -75,7 +87,7 @@ Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
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\end{figure}
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\subsection{Equation of Motion}
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\label{sec:orgdb109d9}
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\label{sec:orgb1836d5}
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The system has two degrees of freedom and is thus fully described by the generalized coordinates \(u\) and \(v\).
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Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\) (neglecting the rotational energy):
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@ -96,7 +108,7 @@ The Lagrangian is the kinetic energy minus the potential energy:
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L = T - V
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\end{equation}
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From the Lagrange's equations of the second kind \eqref{eq:lagrange_second_kind}, the equation of motion \eqref{eq:eom_mixed} is obtained (\(q_1 = u\), \(q_2 = v\)).
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From the Lagrange's equations of the second kind, the equation of motion is obtained (\(q_1 = u\), \(q_2 = v\)).
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\begin{equation}
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\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
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\end{equation}
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@ -120,7 +132,7 @@ Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to \tex
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\subsection{Transfer Functions in the Laplace domain}
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\label{sec:orgfcd3def}
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\label{sec:orgb1002ed}
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\begin{subequations}
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\begin{align}
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@ -159,7 +171,7 @@ With:
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\subsection{Constant Rotating Speed}
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\label{sec:org81c7074}
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\label{sec:orga4faf60}
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To simplify, let's consider a constant rotating speed \(\dot{\theta} = \Omega\) and thus \(\ddot{\theta} = 0\).
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\begin{equation}
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@ -214,9 +226,9 @@ The magnitude of the coupling terms are increasing with the rotation speed.
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\end{figure}
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\section{Integral Force Feedback}
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\label{sec:orgc6c1b99}
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\label{sec:orgaf500b0}
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\subsection{Control Schematic}
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\label{sec:orgb93b297}
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\label{sec:orgbd9f859}
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Force Sensors are added in series with the actuators as shown in Figure \ref{fig:system_iff}.
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@ -227,7 +239,7 @@ Force Sensors are added in series with the actuators as shown in Figure \ref{fig
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\end{figure}
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\subsection{Equations}
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\label{sec:org4072ea4}
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\label{sec:org48206d5}
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The sensed forces are equal to:
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\begin{equation}
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\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
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@ -260,7 +272,7 @@ Which then gives:
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\subsection{Plant Dynamics}
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\label{sec:org0250ac0}
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\label{sec:orgec8431d}
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\begin{figure}[htbp]
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\centering
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@ -269,19 +281,18 @@ Which then gives:
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\end{figure}
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\subsection{Physical Interpretation}
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\label{sec:orgb2d79d2}
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\label{sec:org159680e}
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At low frequency, the gain is very large and thus no force is transmitted between the payload and the rotating stage.
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This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
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\section{Integral Force Feedback with High Pass Filters}
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\label{sec:orgabf7a6a}
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\label{sec:org694707d}
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\subsection{Modification of the Control Low}
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\label{sec:org4766bd6}
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\label{sec:org931fb10}
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\subsection{Close Loop Analysis}
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\label{sec:org4c639fd}
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\label{sec:org9de0aa7}
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\begin{figure}[htbp]
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\centering
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@ -291,12 +302,12 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/root_locus_modified_iff_bis.pdf}
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\includegraphics[scale=1]{figs/root_locus_modified_iff_ter.pdf}
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\caption{\label{fig:root_locus_modified_iff}Figure caption}
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\end{figure}
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\subsection{Optimal Cut-Off Frequency}
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\label{sec:orge829a45}
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\label{sec:org9808de1}
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\begin{figure}[htbp]
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\centering
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@ -312,7 +323,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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\end{figure}
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\section{Integral Force Feedback with Parallel Springs}
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\label{sec:orgd96ea25}
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\label{sec:orgd4915d5}
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\begin{figure}[htbp]
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\centering
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@ -334,7 +345,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/root_locus_iff_kp_bis.pdf}
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\includegraphics[scale=1]{figs/root_locus_iff_kp_ter.pdf}
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\caption{\label{fig:root_locus_iff_kp_bis}Figure caption}
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\end{figure}
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@ -351,7 +362,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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\end{figure}
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\section{Direct Velocity Feedback}
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\label{sec:org027d051}
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\label{sec:orgb0a5870}
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\begin{figure}[htbp]
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\centering
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@ -360,7 +371,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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\end{figure}
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\section{Comparison of the Proposed Active Damping Techniques}
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\label{sec:org1eaa959}
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\label{sec:org6097c1d}
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\begin{figure}[htbp]
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\centering
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@ -381,12 +392,12 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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\end{figure}
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\section{Conclusion}
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\label{sec:org1b2b4ae}
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\label{sec:org1624a6b}
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\label{sec:conclusion}
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\section*{Acknowledgment}
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\label{sec:org2ae16a5}
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\label{sec:org1b29790}
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\bibliography{ref.bib}
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\end{document}
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