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\author[1,3] {T. Dehaeze}
\author[1,2] {C. Collette}
\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}}
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\date{}
\title{Active Damping of Rotating Positioning Platforms}
\hypersetup{
 pdfauthor={},
 pdftitle={Active Damping of Rotating Positioning Platforms},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 27.0.91 (Org mode 9.4)}, 
 pdflang={English}}
\begin{document}

\maketitle

\abstract{
    Abstract text to be done
}

\section{Introduction}
\label{sec:org977317c}
\label{sec:introduction}
Controller Poles are shown by black crosses (
\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
).
\cite{dehaeze18_sampl_stabil_for_tomog_exper}

\section{System Under Study}
\label{sec:org042e800}
\subsection{Rotating Positioning Platform}
\label{sec:org489e4b9}
Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.

\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
\includegraphics[scale=1]{figs/system.pdf}
\caption{\label{fig:rotating_xy_platform}Figure caption}
\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:orgb1836d5}
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 \(V\)
\begin{subequations}
  \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:
\begin{equation}
L = T - V
\end{equation}

From the Lagrange's equations of the second kind, the equation of motion 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}
with \(Q_i\) is the generalized force associated with the generalized variable \(q_i\) (\(Q_1 = F_u\) and \(Q_2 = F_v\)).


\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{itemize}
\item Coriolis Forces: coupling
\item Centrifugal forces: negative stiffness
\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{Transfer Functions in the Laplace domain}
\label{sec:orgb1002ed}

\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}

\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}


\subsection{Constant Rotating Speed}
\label{sec:orga4faf60}
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:orgaf500b0}
\subsection{Control Schematic}
\label{sec:orgbd9f859}

Force Sensors are added in series with the actuators as shown in Figure \ref{fig:system_iff}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/system_iff.pdf}
\caption{\label{fig:system_iff}System with Force Sensors in Series with the Actuators. Decentralized Integral Force Feedback is used}
\end{figure}

\subsection{Equations}
\label{sec:org48206d5}
The sensed forces are equal to:
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k)
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation}

Which then gives:
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\bm{G}_{f}
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}

\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\frac{1}{G_{fp}}
\begin{bmatrix}
  G_{fz} & -G_{fc} \\
  G_{fc} &  G_{fz}
\end{bmatrix}
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}

\begin{align}
  G_{fp} &= \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_{fz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
  G_{fc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)
\end{align}


\subsection{Plant Dynamics}
\label{sec:orgec8431d}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_pure_iff.pdf}
\caption{\label{fig:root_locus_pure_iff}Root Locus}
\end{figure}

\subsection{Physical Interpretation}
\label{sec:org159680e}

At low frequency, the gain is very large and thus no force is transmitted between the payload and the rotating stage.
This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.

\section{Integral Force Feedback with High Pass Filters}
\label{sec:org694707d}
\subsection{Modification of the Control Low}
\label{sec:org931fb10}

\subsection{Close Loop Analysis}
\label{sec:org9de0aa7}

\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_ter.pdf}
\caption{\label{fig:root_locus_modified_iff}Figure caption}
\end{figure}

\subsection{Optimal Cut-Off Frequency}
\label{sec:org9808de1}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_wi_modified_iff_bis.pdf}
\caption{\label{fig:root_locus_wi_modified_iff}Figure caption}
\end{figure}


\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/mod_iff_damping_wi.pdf}
\caption{\label{fig:mod_iff_damping_wi}Figure caption}
\end{figure}

\section{Integral Force Feedback with Parallel Springs}
\label{sec:orgd4915d5}

\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_ter.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:orgb0a5870}

\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:org6097c1d}

\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:org1624a6b}
\label{sec:conclusion}


\section*{Acknowledgment}
\label{sec:org1b29790}

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