dehaeze26_decoupling_parallel_manipulators/paper/dehaeze26_decoupling.tex

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\journal{Journal of Sound and Vibration}
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\begin{document}

\begin{frontmatter}
  \title{Decoupling Control of Parallel Manipulators}
  
  \author[l1]{Thomas Dehaeze\corref{c1}}
  \ead{thomas.dehaeze@esrf.fr}
  \cortext[c1]{Corresponding author}
  \author[l2]{Mohit Verma}
  \author[l3]{Jennifer Watchi}
  \author[l3]{Christophe Collette}
  \address[l1]{ESRF, The European Synchrotron, Grenoble, France}
  \address[l2]{CSIR, Structural Engineering Research Centre, Taramani, Chennai, India.}
  \address[l3]{Precision Mechatronics Laboratory, University of Li\`{e}ge, Belgium.}

  \begin{abstract}
    abstract
  \end{abstract}

  \begin{keyword}
    singular value decomposition \sep{} decoupling \sep{} vibration isolation \sep{} active control
  \end{keyword}

\end{frontmatter}
\section{Introduction}

The control of parallel manipulators (and any MIMO system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this paper), followed by the application of SISO control for the decoupled plant.

When sensors are integrated within the struts, decentralized control may be applied, as the system is already well decoupled at low frequency.
For instance, \cite{furutani04_nanom_cuttin_machin_using_stewar} implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate PI controllers for each strut.
A similar control architecture was proposed in \cite{du14_piezo_actuat_high_precis_flexib} using strain gauge sensors integrated in each strut.

An alternative strategy involves decoupling the system in the Cartesian frame using Jacobian matrices.
As demonstrated during the study of Stewart platform kinematics, Jacobian matrices can be utilized to map actuator forces to forces and torques applied on the top platform.
This approach enables the implementation of controllers in a defined frame.
It has been applied with various sensor types including force sensors \cite{mcinroy00_desig_contr_flexur_joint_hexap}, relative displacement sensors \cite{kim00_robus_track_contr_desig_dof_paral_manip}, and inertial sensors \cite{li01_simul_vibrat_isolat_point_contr,abbas14_vibrat_stewar_platf}.
The Cartesian frame in which the system is decoupled is typically chosen at the point of interest (i.e., where the motion is of interest) or at the center of mass.

Modal decoupling represents another noteworthy decoupling strategy, wherein the ``local'' plant inputs and outputs are mapped to the modal space.
In this approach, multiple SISO plants, each corresponding to a single mode, can be controlled independently.
This decoupling strategy has been implemented for active damping applications \cite{holterman05_activ_dampin_based_decoup_colloc_contr}, which is logical as it is often desirable to dampen specific modes.
The strategy has also been employed in \cite{pu11_six_degree_of_freed_activ} for vibration isolation purposes using geophones, and in \cite{yang19_dynam_model_decoup_contr_flexib} using force sensors.

Another completely different strategy would be to implement a multivariable control directly on the coupled system.
\(\mathcal{H}_\infty\) and \(\mu\text{-synthesis}\) were applied to a Stewart platform model in \cite{lei08_multi_objec_robus_activ_vibrat}.
In \cite{xie17_model_contr_hybrid_passiv_activ}, decentralized force feedback was first applied, followed by \(\mathcal{H}_2\text{-synthesis}\) for vibration isolation based on accelerometers.
\(\mathcal{H}_\infty\text{-synthesis}\) was also employed in \cite{jiao18_dynam_model_exper_analy_stewar} for active damping based on accelerometers.
A comparative study between \(\mathcal{H}_\infty\text{-synthesis}\) and decentralized control in the frame of the struts was performed in \cite{thayer02_six_axis_vibrat_isolat_system}.
Their experimental closed-loop results indicated that the \(\mathcal{H}_\infty\) controller did not outperform the decentralized controller in the frame of the struts.
These limitations were attributed to the model's poor ability to predict off-diagonal dynamics, which is crucial for \(\mathcal{H}_\infty\text{-synthesis}\).

The purpose of this paper is to compare several methods for the decoupling of parallel manipulators, an analysis that appears to be lacking in the literature.
A simplified parallel manipulator model is introduced in Section \ref{ssec:detail_control_decoupling_model} as a test case for evaluating decoupling strategies.
The decentralized plant (transfer functions from actuators to sensors integrated in the struts) is examined in Section \ref{ssec:detail_control_decoupling_decentralized}.
Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section \ref{ssec:detail_control_decoupling_jacobian}), modal decoupling (Section \ref{ssec:detail_control_decoupling_modal}), and Singular Value Decomposition (SVD) decoupling (Section \ref{ssec:detail_control_decoupling_svd}).
Finally, a comparative analysis with concluding observations is provided in Section \ref{ssec:detail_control_decoupling_comp}.
\section{Test Model}
\label{ssec:detail_control_decoupling_model}

Instead of utilizing the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis.
The system illustrated in Figure \ref{fig:detail_control_decoupling_model_test} is used for this purpose.
It possesses three degrees of freedom (DoF) and incorporates three parallel struts.
Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform.

Two reference frames are defined within this model: frame \(\{M\}\) with origin \(O_M\) at the center of mass of the solid body, and frame \(\{K\}\) with origin \(O_K\) at the center of stiffness of the parallel manipulator.

\begin{minipage}[b]{0.60\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test.png}
\captionof{figure}{\label{fig:detail_control_decoupling_model_test}Model used to compare decoupling strategies}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.36\linewidth}
\begin{scriptsize}
\centering
\begin{tabularx}{\linewidth}{cXc}
\toprule
 & \textbf{Description} & \textbf{Value}\\
\midrule
\(l_a\) &  & \(0.5\,m\)\\
\(h_a\) &  & \(0.2\,m\)\\
\(k\) & Actuator stiffness & \(10\,N/\mu m\)\\
\(c\) & Actuator damping & \(200\,Ns/m\)\\
\(m\) & Payload mass & \(40\,\text{kg}\)\\
\(I\) & Payload \(R_z\) inertia & \(5\,\text{kg}m^2\)\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters}
\end{scriptsize}
\end{minipage}

The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass \eqref{eq:detail_control_decoupling_model_eom}, where \(\bm{\mathcal{X}}_{\{M\}}\) represents the two translations and one rotation with respect to the center of mass, and \(\bm{\mathcal{F}}_{\{M\}}\) denotes the forces and torque applied at the center of mass.

\begin{equation}\label{eq:detail_control_decoupling_model_eom}
  \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t), \quad
  \bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix}
    x \\
    y \\
    R_z
  \end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix}
    F_x \\
    F_y \\
    M_z
  \end{bmatrix}
\end{equation}

The Jacobian matrix \(\bm{J}_{\{M\}}\) is employed to map the spring, damping, and actuator forces to XY forces and Z torque expressed at the center of mass \eqref{eq:detail_control_decoupling_jacobian_CoM}.

\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM}
  \bm{J}_{\{M\}} = \begin{bmatrix}
    1 & 0 &  h_a \\
    0 & 1 & -l_a \\
    0 & 1 &  l_a \\
  \end{bmatrix}
\end{equation}

Subsequently, the equation of motion relating the actuator forces \(\tau\) to the motion of the mass \(\bm{\mathcal{X}}_{\{M\}}\) is derived \eqref{eq:detail_control_decoupling_plant_cartesian}.

\begin{equation}\label{eq:detail_control_decoupling_plant_cartesian}
  \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t)
\end{equation}

The matrices representing the payload inertia, actuator stiffness, and damping are shown in \eqref{eq:detail_control_decoupling_system_matrices}.

\begin{equation}\label{eq:detail_control_decoupling_system_matrices}
  \bm{M}_{\{M\}} = \begin{bmatrix}
    m & 0 & 0 \\
    0 & m & 0 \\
    0 & 0 & I
  \end{bmatrix}, \quad
  \bm{\mathcal{K}} = \begin{bmatrix}
    k & 0 & 0 \\
    0 & k & 0 \\
    0 & 0 & k
  \end{bmatrix}, \quad
  \bm{\mathcal{C}} = \begin{bmatrix}
    c & 0 & 0 \\
    0 & c & 0 \\
    0 & 0 & c
  \end{bmatrix}
\end{equation}

The parameters employed for the subsequent analysis are summarized in Table \ref{tab:detail_control_decoupling_test_model_params}, which includes values for geometric parameters (\(l_a\), \(h_a\)), mechanical properties (actuator stiffness \(k\) and damping \(c\)), and inertial characteristics (payload mass \(m\) and rotational inertia \(I\)).
\section{Control in the frame of the struts}
\label{ssec:detail_control_decoupling_decentralized}

The dynamics in the frame of the struts are first examined.
The equation of motion relating actuator forces \(\bm{\mathcal{\tau}}\) to strut relative motion \(\bm{\mathcal{L}}\) is derived from equation \eqref{eq:detail_control_decoupling_plant_cartesian} by mapping the Cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix \(\bm{J}_{\{M\}}\) defined in \eqref{eq:detail_control_decoupling_jacobian_CoM}.
The obtained transfer function from \(\bm{\mathcal{\tau}}\) to \(\bm{\mathcal{L}}\) is shown in \eqref{eq:detail_control_decoupling_plant_decentralized}.

\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized}
  \frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1}
\end{equation}

At low frequencies, the plant converges to a diagonal constant matrix whose diagonal elements are equal to the actuator stiffnesses \eqref{eq:detail_control_decoupling_plant_decentralized_low_freq}.
At high frequencies, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal.

\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq}
  \bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}}
\end{equation}

The magnitude of the coupled plant \(\bm{G}_{\mathcal{L}}\) is illustrated in Figure \ref{fig:detail_control_decoupling_coupled_plant_bode}.
This representation confirms that at low frequencies (below the first suspension mode), the plant is well decoupled.
Depending on the symmetry present in the system, certain diagonal elements may exhibit identical values, as demonstrated for struts 2 and 3 in this example.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_coupled_plant_bode.png}
\caption{\label{fig:detail_control_decoupling_coupled_plant_bode}Model dynamics from actuator forces to relative displacement sensor of each strut.}
\end{figure}
\section{Jacobian Decoupling}
\label{ssec:detail_control_decoupling_jacobian}
\subsection{Jacobian Matrix}

The Jacobian matrix \(\bm{J}_{\{O\}}\) serves a dual purpose in the decoupling process: it converts strut velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\), and it transforms actuator forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\), as expressed in equation \eqref{eq:detail_control_decoupling_jacobian}.

\begin{subequations}\label{eq:detail_control_decoupling_jacobian}
  \begin{align}
    \dot{\bm{\mathcal{X}}}_{\{O\}} &= \bm{J}_{\{O\}} \dot{\bm{\mathcal{L}}}, \quad \dot{\bm{\mathcal{L}}} = \bm{J}_{\{O\}}^{-1} \dot{\bm{\mathcal{X}}}_{\{O\}} \\
    \bm{\mathcal{F}}_{\{O\}}       &= \bm{J}_{\{O\}}^{\intercal} \bm{\tau},            \quad \bm{\tau} = \bm{J}_{\{O\}}^{-\intercal} \bm{\mathcal{F}}_{\{O\}}
  \end{align}
\end{subequations}

The resulting plant (Figure \ref{fig:detail_control_jacobian_decoupling_arch}) have inputs and outputs with clear physical interpretations:
\begin{itemize}
\item \(\bm{\mathcal{F}}_{\{O\}}\) represents forces/torques applied on the payload at the origin of frame \(\{O\}\)
\item \(\bm{\mathcal{X}}_{\{O\}}\) represents translations/rotation of the payload expressed in frame \(\{O\}\)
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_control_jacobian.png}
\caption{\label{fig:detail_control_jacobian_decoupling_arch}Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{O\}}\) to \(\bm{\mathcal{X}}_{\{O\}}\)}
\end{figure}

The transfer function from \(\bm{\mathcal{F}}_{\{O\}\) to \(\bm{\mathcal{X}}_{\{O\}}\), denoted \(\bm{G}_{\{O\}}(s)\) can be computed using \eqref{eq:detail_control_decoupling_plant_jacobian}.

\begin{equation}\label{eq:detail_control_decoupling_plant_jacobian}
  \frac{\bm{\mathcal{X}}_{\{O\}}}{\bm{\mathcal{F}}_{\{O\}}}(s) = \bm{G}_{\{O\}}(s) = \left( \bm{J}_{\{O\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{O\}} s^2 + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{O\}} s + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{O\}} \right)^{-1}
\end{equation}

The frame \(\{O\}\) can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice.
Two natural reference frames are particularly relevant: the center of mass and the center of stiffness.
\subsection{Center Of Mass}

When the decoupling frame is located at the center of mass (frame \(\{M\}\) in Figure \ref{fig:detail_control_decoupling_model_test}), the Jacobian matrix and its inverse are expressed as in \eqref{eq:detail_control_decoupling_jacobian_CoM_inverse}.

\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse}
  \bm{J}_{\{M\}} = \begin{bmatrix}
    1 & 0 &  h_a \\
    0 & 1 & -l_a \\
    0 & 1 &  l_a \\
  \end{bmatrix}, \quad \bm{J}_{\{M\}}^{-1} = \begin{bmatrix}
    1 & \frac{h_a}{2 l_a} & \frac{-h_a}{2 l_a} \\
    0 & \frac{1}{2} & \frac{1}{2} \\
    0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} \\
  \end{bmatrix}
\end{equation}

Analytical formula of the plant \(\bm{G}_{\{M\}}(s)\) is derived \eqref{eq:detail_control_decoupling_plant_CoM}.

\begin{equation}\label{eq:detail_control_decoupling_plant_CoM}
  \frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1}
\end{equation}

At high frequencies, the plant converges to the inverse of the mass matrix, which is a diagonal matrix \eqref{eq:detail_control_decoupling_plant_CoM_high_freq}.

\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq}
  \bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix}
    1/m & 0   & 0 \\
    0   & 1/m & 0 \\
    0   & 0   & 1/I
  \end{bmatrix}
\end{equation}

Consequently, the plant exhibits effective decoupling at frequencies above the highest suspension mode as shown in Figure \ref{fig:detail_control_decoupling_jacobian_plant_CoM}.
This strategy is typically employed in systems with low-frequency suspension modes \cite{butler11_posit_contr_lithog_equip}, where the plant approximates decoupled mass lines.

The low-frequency coupling observed in this configuration has a clear physical interpretation.
When a static force is applied at the center of mass, the suspended mass rotates around the center of stiffness.
This rotation is due to torque induced by the stiffness of the first actuator (i.e. the one on the left side), which is not aligned with the force application point.
This phenomenon is illustrated in Figure \ref{fig:detail_control_decoupling_model_test_CoM}.

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_jacobian_plant_CoM.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoM}Dynamics at the CoM}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test_CoM.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_model_test_CoM}Static force applied at the CoM}
\end{subfigure}
\caption{\label{fig:detail_control_jacobian_decoupling_plant_CoM_results}Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_decoupling_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoM}).}
\end{figure}
\subsection{Center Of Stiffness}

When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in \eqref{eq:detail_control_decoupling_jacobian_CoK_inverse}.

\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoK_inverse}
  \bm{J}_{\{K\}} = \begin{bmatrix}
    1 & 0 &  0   \\
    0 & 1 & -l_a \\
    0 & 1 &  l_a
  \end{bmatrix}, \quad \bm{J}_{\{K\}}^{-1} = \begin{bmatrix}
    1 & 0                & 0   \\
    0 & \frac{1}{2}      & \frac{1}{2} \\
    0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a}
  \end{bmatrix}
\end{equation}

The frame \(\{K\}\) was selected based on physical reasoning, positioned in line with the side strut and equidistant between the two vertical struts.
However, it could alternatively be determined through analytical methods to ensure that \(\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}\) forms a diagonal matrix.
It should be noted that the existence of such a center of stiffness (i.e. a frame \(\{K\}\) for which \(\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}\) is diagonal) is not guaranteed for arbitrary systems.
This property is typically achievable only in systems exhibiting specific symmetrical characteristics, as is the case in the present example.

The analytical expression for the plant in this configuration was then computed \eqref{eq:detail_control_decoupling_plant_CoK}.

\begin{equation}\label{eq:detail_control_decoupling_plant_CoK}
  \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1}
\end{equation}

Figure \ref{fig:detail_control_decoupling_jacobian_plant_CoK_results} presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the center of stiffness.
The plant is well decoupled below the suspension mode with the lowest frequency \eqref{eq:detail_control_decoupling_plant_CoK_low_freq}, making it particularly suitable for systems with high stiffness.

\begin{equation}\label{eq:detail_control_decoupling_plant_CoK_low_freq}
  \bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-\intercal}
\end{equation}

The physical reason for high-frequency coupling is illustrated in Figure \ref{fig:detail_control_decoupling_model_test_CoK}.
When a high-frequency force is applied at a point not aligned with the center of mass, it induces rotation around the center of mass.

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_jacobian_plant_CoK.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test_CoK.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK}
\end{subfigure}
\caption{\label{fig:detail_control_decoupling_jacobian_plant_CoK_results}Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).}
\end{figure}
\section{Modal Decoupling}
\label{ssec:detail_control_decoupling_modal}
Modal decoupling represents an approach based on the principle that a mechanical system's behavior can be understood as a combination of contributions from various modes \cite{rankers98_machin}.
To convert the dynamics in the modal space, the equation of motion are first written with respect to the center of mass \eqref{eq:detail_control_decoupling_equation_motion_CoM}.

\begin{equation}\label{eq:detail_control_decoupling_equation_motion_CoM}
  \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t)
\end{equation}

For modal decoupling, a change of variables is introduced \eqref{eq:detail_control_decoupling_modal_coordinates} where \(\bm{\mathcal{X}}_{m}\) represents the modal amplitudes and \(\bm{\Phi}\) is a \(n \times n\)\footnote{\(n\) corresponds to the number of degrees of freedom, here \(n = 3\)} matrix whose columns correspond to the mode shapes of the system, computed from \(\bm{M}_{\{M\}}\) and \(\bm{K}_{\{M\}}\).

\begin{equation}\label{eq:detail_control_decoupling_modal_coordinates}
  \bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m}
\end{equation}

By pre-multiplying equation \eqref{eq:detail_control_decoupling_equation_motion_CoM} by \(\bm{\Phi}^{\intercal}\) and applying the change of variable \eqref{eq:detail_control_decoupling_modal_coordinates}, a new set of equations of motion is obtained \eqref{eq:detail_control_decoupling_equation_modal_coordinates} where \(\bm{\tau}_m\) represents the modal input, while \(\bm{M}_m\), \(\bm{C}_m\), and \(\bm{K}_m\) denote the modal mass, damping, and stiffness matrices respectively.

\begin{equation}\label{eq:detail_control_decoupling_equation_modal_coordinates}
  \underbrace{\bm{\Phi}^{\intercal} \bm{M} \bm{\Phi}}_{\bm{M}_m} \bm{\ddot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{C} \bm{\Phi}}_{\bm{C}_m} \bm{\dot{\mathcal{X}}}_m(t)  + \underbrace{\bm{\Phi}^{\intercal} \bm{K} \bm{\Phi}}_{\bm{K}_m} \bm{\mathcal{X}}_m(t) = \underbrace{\bm{\Phi}^{\intercal} \bm{J}^{\intercal} \bm{\tau}(t)}_{\bm{\tau}_m(t)}
\end{equation}
The inherent mathematical structure of the mass, damping, and stiffness matrices \cite[, chapt. 8]{lang17_under} ensures that modal matrices are diagonal \cite[, chapt. 2.3]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
This diagonalization transforms equation \eqref{eq:detail_control_decoupling_equation_modal_coordinates} into a set of \(n\) decoupled equations, enabling independent control of each mode without cross-interaction.

To implement this approach from a decentralized plant, the architecture shown in Figure \ref{fig:detail_control_decoupling_modal} is employed.
Inputs of the decoupling plant are the modal modal inputs \(\bm{\tau}_m\) and the outputs are the modal amplitudes \(\bm{\mathcal{X}}_m\).
This implementation requires knowledge of the system's equations of motion, from which the mode shapes matrix \(\bm{\Phi}\) is derived.
The resulting decoupled system features diagonal elements each representing second-order resonant systems that are straightforward to control individually.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_modal.png}
\caption{\label{fig:detail_control_decoupling_modal}Modal Decoupling Architecture}
\end{figure}

Modal decoupling was then applied to the test model.
First, the eigenvectors \(\bm{\Phi}\) of \(\bm{M}_{\{M\}}^{-1}\bm{K}_{\{M\}}\) were computed \eqref{eq:detail_control_decoupling_modal_eigenvectors}.
While analytical derivation of eigenvectors could be obtained for such a simple system, they are typically computed numerically for practical applications.

\begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors}
  \bm{\Phi} = \begin{bmatrix}
    \frac{I - h_a^2 m - 2 l_a^2 m - \alpha}{2 h_a m} & 0 & \frac{I - h_a^2 m - 2 l_a^2 m + \alpha}{2 h_a m} \\
    0                                                & 1 & 0                                                \\
    1                                                & 0 & 1
  \end{bmatrix},\ \alpha = \sqrt{\left( I + m (h_a^2 - 2 l_a^2) \right)^2 + 8 m^2 h_a^2 l_a^2}
\end{equation}

The numerical values for the eigenvector matrix and its inverse are shown in \eqref{eq:detail_control_decoupling_modal_eigenvectors_matrices}.

\begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors_matrices}
  \bm{\Phi} = \begin{bmatrix}
    -0.905 & 0 & -0.058 \\
     0     & 1 &  0     \\
     0.424 & 0 & -0.998
  \end{bmatrix}, \quad
  \bm{\Phi}^{-1} = \begin{bmatrix}
    -1.075 & 0 &  0.063 \\
     0     & 1 &  0     \\
    -0.457 & 0 & -0.975
  \end{bmatrix}
\end{equation}

The two computed matrices were implemented in the control architecture of Figure \ref{fig:detail_control_decoupling_modal}, resulting in three distinct second order plants as depicted in Figure \ref{fig:detail_control_decoupling_modal_plant}.
Each of these diagonal elements corresponds to a specific mode, as shown in Figure \ref{fig:detail_control_decoupling_model_test_modal}, resulting in a perfectly decoupled system.

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_modal_plant.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_model_test_modal.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes}
\end{subfigure}
\caption{\label{fig:detail_control_decoupling_modal_plant_modes}Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to invidiually address different modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal})}
\end{figure}
\section{SVD Decoupling}
\label{ssec:detail_control_decoupling_svd}
\subsection{Singular Value Decomposition}

Singular Value Decomposition (SVD) represents a powerful mathematical tool with extensive applications in data analysis \cite[, chapt. 1]{brunton22_data} and multivariable control systems where it is particularly valuable for analyzing directional properties in multivariable systems \cite{skogestad07_multiv_feedb_contr}.

The SVD constitutes a unique matrix decomposition applicable to any complex matrix \(\bm{X} \in \mathbb{C}^{n \times m}\), expressed as:

\begin{equation}\label{eq:detail_control_svd}
  \bm{X} = \bm{U} \bm{\Sigma} \bm{V}^H
\end{equation}

where \(\bm{U} \in \mathbb{C}^{n \times n}\) and \(\bm{V} \in \mathbb{C}^{m \times m}\) are unitary matrices with orthonormal columns, and \(\bm{\Sigma} \in \mathbb{R}^{n \times n}\) is a diagonal matrix with real, non-negative entries.
For real matrices \(\bm{X}\), the resulting \(\bm{U}\) and \(\bm{V}\) matrices are also real, making them suitable for decoupling applications.
\subsection{Decoupling using the SVD}

The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a Frequency Response Function (FRF), which yields a complex matrix \(\bm{G}(\omega_i)\) for multiple frequency points \(\omega_i\).
A specific frequency is then selected for optimal decoupling, with the targeted crossover frequency \(\omega_c\) often serving as an appropriate choice.

Since real matrices are required for the decoupling transformation, a real approximation of the complex measured response at the selected frequency must be computed.
In this work, the method proposed in \cite{kouvaritakis79_theor_pract_charac_locus_desig_method} was used as it preserves maximal orthogonality in the directional properties of the input complex matrix.

Following this approximation, a real matrix \(\tilde{\bm{G}}(\omega_c)\) is obtained, and SVD is performed on this matrix.
The resulting (real) unitary matrices \(\bm{U}\) and \(\bm{V}\) are structured such that \(\bm{V}^{-\intercal} \tilde{\bm{G}}(\omega_c) \bm{U}^{-1}\) forms a diagonal matrix.
These singular input and output matrices are then applied to decouple the system as illustrated in Figure \ref{fig:detail_control_decoupling_svd}, and the decoupled plant is described by \eqref{eq:detail_control_decoupling_plant_svd}.

\begin{equation}\label{eq:detail_control_decoupling_plant_svd}
  \bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-\intercal}
\end{equation}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_svd.png}
\caption{\label{fig:detail_control_decoupling_svd}Decoupled plant \(\bm{G}_{\text{SVD}}\) using the Singular Value Decomposition}
\end{figure}

Implementation of SVD decoupling requires access to the system's FRF, at least in the vicinity of the desired decoupling frequency.
This information can be obtained either experimentally or derived from a model.
While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency.
Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping.
\subsection{Example}

Plant decoupling using the Singular Value Decomposition was then applied on the test model.
A decoupling frequency of \(100\,\text{Hz}\) was used.
The plant response at that frequency, as well as its real approximation and the obtained \(\bm{U}\) and \(\bm{V}\) matrices are shown in \eqref{eq:detail_control_decoupling_svd_example}.

\begin{equation}\label{eq:detail_control_decoupling_svd_example}
\begin{align}
  & \bm{G}_{\{\mathcal{L}\}}(\omega_c = 2\pi \cdot 100) = 10^{-9} \begin{bmatrix}
    -99 - j 2.6 &  74  + j 4.2 & -74  - j 4.2 \\
     74 + j 4.2 & -247 - j 9.7 &  102 + j 7.0 \\
    -74 - j 4.2 &  102 + j 7.0 & -247 - j 9.7
  \end{bmatrix} \\
  & \xrightarrow[\text{approximation}]{\text{real}} \tilde{\bm{G}}_{\{\mathcal{L}\}}(\omega_c) = 10^{-9} \begin{bmatrix}
    -99 &  74  & -74  \\
     74 & -247 &  102 \\
    -74 &  102 & -247
  \end{bmatrix} \\
  & \xrightarrow[\text{SVD}]{\phantom{\text{approximation}}} \bm{U} = \begin{bmatrix}
    0.34 & 0    &  0.94 \\
   -0.66 & 0.71 &  0.24 \\
    0.66 & 0.71 & -0.24
  \end{bmatrix}, \ \bm{V} = \begin{bmatrix}
    -0.34 &  0    & -0.94 \\
     0.66 & -0.71 & -0.24 \\
    -0.66 & -0.71 &  0.24
  \end{bmatrix}
\end{align}
\end{equation}

Using these \(\bm{U}\) and \(\bm{V}\) matrices, the decoupled plant is computed according to equation \eqref{eq:detail_control_decoupling_plant_svd}.
The resulting plant, depicted in Figure \ref{fig:detail_control_decoupling_svd_plant}, exhibits remarkable decoupling across a broad frequency range, extending well beyond the vicinity of \(\omega_c\).
Additionally, the diagonal terms manifest as second-order dynamic systems, facilitating straightforward controller design.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_svd_plant.png}
\caption{\label{fig:detail_control_decoupling_svd_plant}Plant dynamics \(\bm{G}_{\text{SVD}}(s)\) obtained after decoupling using Singular Value Decomposition}
\end{figure}

As it was surprising to obtain such a good decoupling at all frequencies, a variant system with identical dynamics but different sensor configurations was examined.
Instead of using relative motion sensors collocated with the struts, three relative motion sensors were positioned as shown in Figure \ref{fig:detail_control_decoupling_model_test_alt}.
Although Jacobian matrices could theoretically be used to map these sensors to the frame of the struts, application of the same SVD decoupling procedure yielded the plant response shown in Figure \ref{fig:detail_control_decoupling_svd_alt_plant}, which exhibits significantly greater coupling.
Notably, the coupling demonstrates local minima near the decoupling frequency, consistent with the fact that the decoupling matrices were derived specifically for that frequency point.

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test_alt.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_model_test_alt}Alternative location of sensors}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_svd_alt_plant.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant}
\end{subfigure}
\caption{\label{fig:detail_control_svd_decoupling_not_symmetrical}Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}).}
\end{figure}

The exceptional performance of SVD decoupling on the plant with collocated sensors warrants further investigation.
This effectiveness may be attributed to the symmetrical properties of the plant, as evidenced in the Bode plots of the decentralized plant shown in Figure \ref{fig:detail_control_decoupling_coupled_plant_bode}.
The phenomenon potentially relates to previous research on SVD controllers applied to systems with specific symmetrical characteristics \cite{hovd97_svd_contr_contr}.
\section{Comparison of decoupling strategies}
\label{ssec:detail_control_decoupling_comp}

While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table \ref{tab:detail_control_decoupling_strategies_comp}.

Each method employs a distinct conceptual framework: Jacobian decoupling is ``topology-driven'', relying on the geometric configuration of the system; modal decoupling is ``physics-driven'', based on the system's dynamical equations; and SVD decoupling is ``data-driven'', utilizing measured frequency response functions.

The physical interpretation of decoupled plant inputs and outputs varies considerably among these methods.
With Jacobian decoupling, inputs and outputs retain clear physical meaning, corresponding to forces/torques and translations/rotations in a specified reference frame.
Modal decoupling arranges inputs to excite individual modes, with outputs combined to measure these modes separately.
For SVD decoupling, inputs and outputs represent special directions ordered by decreasing controllability and observability at the chosen frequency, though physical interpretation becomes challenging for parallel manipulators.

This difference in interpretation relates directly to the ``control space'' in which the controllers operate.
When these ``control spaces'' meaningfully relate to the control objectives, controllers can be tuned to directly match specific requirements.
For Jacobian decoupling, the controller typically operates in a frame positioned at the point where motion needs to be controlled, for instance where the light is focused in the NASS application.
Modal decoupling provides a natural framework when specific vibrational modes require targeted control.
SVD decoupling generally results in a loss of physical meaning for the ``control space'', potentially complicating the process of relating controller design to practical system requirements.

The quality of decoupling achieved through these methods also exhibits distinct characteristics.
Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the center of stiffness, or at high frequencies when aligned with the center of mass.
Systems designed with coincident centers of mass and stiffness may achieve excellent decoupling using this approach.
Modal decoupling offers good decoupling across all frequencies, though its effectiveness relies on the model accuracy, with discrepancies potentially resulting in significant off-diagonal elements.
SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point.

\begin{table}[htbp]
\caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies}
\centering
\scriptsize
\begin{tabularx}{\linewidth}{lXXX}
\toprule
 & \textbf{Jacobian} & \textbf{Modal} & \textbf{SVD}\\
\midrule
\textbf{Philosophy} & Topology Driven & Physics Driven & Data Driven\\
\midrule
\textbf{Requirements} & Known geometry & Known equations of motion & Identified FRF\\
\midrule
\textbf{Decoupling Matrices} & Jacobian matrix \(\bm{J}_{\{O\}}\) & Eigenvectors \(\bm{\Phi}\) & SVD matrices \(\bm{U}\) and \(\bm{V}\)\\
\midrule
\textbf{Decoupled Plant} & \(\bm{G}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-1} \bm{G}_{\mathcal{L}}(s) \bm{J}_{\{O\}}^{-\intercal}\) & \(\bm{G}_m(s) = \bm{\Phi}^{-1} \bm{G}_{\mathcal{X}}(s) \bm{\Phi}^{-\intercal}\) & \(\bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}(s) \bm{V}^{-\intercal}\)\\
\midrule
\textbf{Controller} & \(\bm{K}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-\intercal} \bm{K}_{d}(s) \bm{J}_{\{O\}}^{-1}\) & \(\bm{K}_m(s) = \bm{\Phi}^{-\intercal} \bm{K}_{d}(s) \bm{\Phi}^{-1}\) & \(\bm{K}_{\text{SVD}}(s) = \bm{V}^{-\intercal} \bm{K}_{d}(s) \bm{U}^{-1}\)\\
\midrule
\textbf{Interpretation} & Forces/Torques to Displacement/Rotation in chosen frame & Inputs (resp. outputs) to excite (resp. sense) individual modes & Directions of max to min controllability/observability\\
\midrule
\textbf{Effectiveness} & Decoupling at low or high frequency depending on the chosen frame & Good decoupling at all frequencies & Good decoupling near the chosen frequency\\
\midrule
\textbf{Pros} & Retain physical meaning of inputs / outputs. Controller acts on a meaningfully ``frame'' & Ability to target specific modes. Simple \(2^{nd}\) order diagonal plants & Good Decoupling near the crossover. Very General and requires no model\\
\midrule
\textbf{Cons} & Good decoupling at all frequency can only be obtained for specific mechanical architecture & Relies on the accuracy of equation of motions. Robustness to unmodelled dynamics may be poor & Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor\\
\bottomrule
\end{tabularx}
\end{table}
\section{Acknowledgments}
\printbibliography
\end{document}