phd-simscape-nano-hexapod/simscape-nano-hexapod.tex

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\author{Dehaeze Thomas}
\date{\today}
\title{Simscape Model - Nano Hexapod}
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\begin{document}

\maketitle
\tableofcontents

\clearpage

Now that the multi-body model of the micro-station has been developed and validated using dynamical measurements, a model of the active vibration platform can be integrated.

First, the mechanical architecture of the active platform needs to be carefully chosen.
In Section \ref{sec:nhexa_platform_review}, a quick review of active vibration platforms is performed.

The chosen architecture is the Stewart platform, which is presented in Section \ref{sec:nhexa_stewart_platform}.
It is a parallel manipulator that require the use of specific tools to study its kinematics.

However, to study the dynamics of the Stewart platform, the use of analytical equations is very complex.
Instead, a multi-body model of the Stewart platform is developed (Section \ref{sec:nhexa_model}), that can then be easily integrated on top of the micro-station's model.

From a control point of view, the Stewart platform is a MIMO system with complex dynamics.
To control such system, it requires several tools to study interaction (Section \ref{sec:nhexa_control}).

\chapter{Active Vibration Platforms}
\label{sec:nhexa_platform_review}
\textbf{Goals}:
\begin{itemize}
\item Quick review of active vibration platforms (5 or 6DoF) similar to NASS
\item Explain why Stewart platform architecture is chosen

\item Wanted controlled DOF: Y, Z, Ry
\item But because of continuous rotation (key specificity): X,Y,Z,Rx,Ry in the frame of the active platform

\item Literature review? (\textbf{maybe more suited for chapter 2})
\begin{itemize}
\item \url{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}
\item Talk about flexible joint? Maybe not so much as it should be topic of second chapter.
Just say that we must of flexible joints that can be defined as 3 to 6DoF joints, and it will be optimize in chapter 2.
\end{itemize}
\item \cite{taghirad13_paral}

\item For some systems, just XYZ control (stack stages), example: holler
\item For other systems, Stewart platform (ID16a), piezo based
\item Examples of Stewart platforms for general vibration control, some with Piezo, other with Voice coil. IFF, \ldots{}
Show different geometry configuration
\item DCM: tripod?
\end{itemize}
\section{Active vibration control of sample stages}

\href{file:///home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/nass-introduction.org}{Review of stages with online metrology for Synchrotrons}

\begin{itemize}
\item[{$\square$}] Talk about external metrology?
Maybe not the topic here.
\item[{$\square$}] Talk about control architecture?
\item[{$\square$}] Comparison with the micro-station / NASS
\end{itemize}

\section{Serial and Parallel Manipulators}

\textbf{Goal}:
\begin{itemize}
\item Explain why a parallel manipulator is here preferred
\item Compact, 6DoF, higher control bandwidth, linear, simpler

\item Show some example of serial and parallel manipulators

\item A review of Stewart platform will be given in Chapter related to the detailed design of the Nano-Hexapod
\end{itemize}


\begin{table}[htbp]
\caption{\label{tab:nhexa_serial_vs_parallel}Advantages and Disadvantages of both serial and parallel robots}
\centering
\begin{tabularx}{\linewidth}{lXX}
\toprule
 & \textbf{Serial Robots} & \textbf{Parallel Robots}\\
\midrule
Advantages & Large Workspace & High Stiffness\\
Disadvantages & Low Stiffness & Small Workspace\\
Kinematic Struture & Open & Closed-loop\\
\bottomrule
\end{tabularx}
\end{table}

\chapter{The Stewart platform}
\label{sec:nhexa_stewart_platform}
The Stewart platform, first introduced by Stewart in 1965 \cite{stewart65_platf_with_six_degrees_freed} for flight simulation applications, represents a significant milestone in parallel manipulator design.
This mechanical architecture has evolved far beyond its original purpose, finding applications across diverse fields from precision positioning systems to robotic surgery.
The fundamental design consists of two platforms connected by six adjustable struts in parallel, creating a fully parallel manipulator capable of six degrees of freedom motion.

Unlike serial manipulators where errors worsen through the kinematic chain, parallel architectures distribute loads across multiple actuators, leading to enhanced mechanical stiffness and improved positioning accuracy.
This parallel configuration also results in superior dynamic performance, as the actuators directly contribute to the platform's motion without intermediate linkages.
These characteristics of Stewart platforms have made them particularly valuable in applications requiring high precision and stiffness.

For the NASS application, the Stewart platform architecture presents three key advantages.
First, as a fully parallel manipulator, all motion errors of the micro-station can be compensated through the coordinated action of the six actuators.
Second, its compact design compared to serial manipulators makes it ideal for integration on top micro-station where only \(95\,mm\) of height is available.
Third, the good dynamical properties should enable high bandwidth positioning control.

While Stewart platforms excel in precision and stiffness, they typically exhibit a relatively limited workspace compared to serial manipulators.
However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station which are in the order of \(10\,\mu m\).

This section provides a comprehensive analysis of the Stewart platform's properties, focusing on aspects crucial for precision positioning applications.
The analysis encompasses the platform's kinematic relationships (Section \ref{ssec:nhexa_stewart_platform_kinematics}), the use of the Jacobian matrix (Section \ref{ssec:nhexa_stewart_platform_jacobian}), static behavior (Section \ref{ssec:nhexa_stewart_platform_static}), and dynamic characteristics (Section \ref{ssec:nhexa_stewart_platform_dynamics}).
These theoretical foundations form the basis for subsequent design decisions and control strategies, which will be elaborated in later sections.
\section{Mechanical Architecture}
\label{ssec:nhexa_stewart_platform_architecture}

The Stewart platform consists of two rigid platforms connected by six struts arranged in parallel (Figure \ref{fig:nhexa_stewart_architecture}).
Each strut incorporates an active prismatic joint that enables controlled length variation, with its ends attached to the fixed and mobile platforms through joints.
The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom\footnote{Different architecture exists, typically referred as ``6-SPS'' (Spherical, Prismatic, Spherical) or ``6-UPS'' (Universal, Prismatic, Spherical)}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_architecture.png}
\caption{\label{fig:nhexa_stewart_architecture}Schematical representation of the Stewart platform architecture.}
\end{figure}

To facilitate rigorous analysis of the Stewart platform, four reference frames are defined:
\begin{itemize}
\item The fixed base frame \(\{F\}\), located at the center of the base platform's bottom surface, serves as the mounting reference for the support structure.
\item The mobile frame \(\{M\}\), situated at the center of the top platform's upper surface, provides a reference for payload mounting.
\item The point-of-interest frame \(\{A\}\), fixed to the base but positioned at the workspace center.
\item The moving point-of-interest frame \(\{B\}\), attached to the mobile platform and coincident with frame \(\{A\}\) in the home position.
\end{itemize}

Frames \(\{F\}\) and \(\{M\}\) serve primarily to define the joint locations.
On the other hand, frames \(\{A\}\) and \(\{B\}\) are used to describe the relative motion of the two platforms through the position vector \({}^A\bm{P}_B\) of frame \(\{B\}\) expressed in frame \(\{A\}\) and the rotation matrix \({}^A\bm{R}_B\) expressing the orientation of \(\{B\}\) with respect to \(\{A\}\).
For the nano-hexapod, frames \(\{A\}\) and \(\{B\}\) are chosen to be located at the theoretical focus point of the X-ray light which is \(150\,mm\) above the top platform, i.e. above \(\{M\}\).

Location of the joints and orientation and length of the struts are crucial for subsequent kinematic, static, and dynamic analyses of the Stewart platform.
The center of rotation for the joint fixed to the base is noted \(\bm{a}_i\), while \(b_i\) is used for the top platform joints.
The struts orientation are represented by the unit vectors \(\hat{\bm{s}}_i\) and their lengths by the scalars \(l_i\).
This is summarized in Figure \ref{fig:nhexa_stewart_notations}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_notations.png}
\caption{\label{fig:nhexa_stewart_notations}Frame and key notations for the Stewart platform}
\end{figure}

\section{Kinematic Analysis}
\label{ssec:nhexa_stewart_platform_kinematics}
The kinematic analysis of the Stewart platform involves understanding the geometric relationships between the platform position/orientation and the actuator lengths, without considering the forces involved.
\paragraph{Loop Closure}

The foundation of the kinematic analysis lies in the geometric constraints imposed by each strut, which can be expressed through loop closure equations.
For each strut \(i\) (illustrated in Figure \ref{fig:nhexa_stewart_loop_closure}), the loop closure equation \eqref{eq:nhexa_loop_closure} can be written.

\begin{equation}\label{eq:nhexa_loop_closure}
  {}^A\bm{P}_B = {}^A\bm{a}_i + l_i{}^A\hat{\bm{s}}_i - \underbrace{{}^B\bm{b}_i}_{{}^A\bm{R}_B {}^B\bm{b}_i} \quad \text{for } i=1 \text{ to } 6
\end{equation}

Such equation links the pose variables \({}^A\bm{P}\) and \({}^A\bm{R}_B\), the position vectors describing the known geometry of the base and of the moving platform, \(\bm{a}_i\) and \(\bm{b}_i\), and the strut vector \(l_i {}^A\hat{\bm{s}}_i\):

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_loop_closure.png}
\caption{\label{fig:nhexa_stewart_loop_closure}Notations to compute the kinematic loop closure}
\end{figure}

\paragraph{Inverse Kinematics}

The inverse kinematic problem involves determining the required strut lengths \(\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^T\) for a desired platform pose \(\bm{\mathcal{X}}\) (i.e. position \({}^A\bm{P}\) and orientation \({}^A\bm{R}_B\)).
This problem can be solved analytically using the loop closure equations \eqref{eq:nhexa_loop_closure}.
The obtain strut lengths are given by \eqref{eq:nhexa_inverse_kinematics}.

\begin{equation}\label{eq:nhexa_inverse_kinematics}
    l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
\end{equation}

If the position and orientation of the platform lie in the feasible workspace, the solution is unique.
Otherwise, the solution gives complex numbers.

\paragraph{Forward Kinematics}

The forward kinematic problem seeks to determine the platform pose \(\bm{\mathcal{X}}\) given a set of strut lengths \(\bm{\mathcal{L}}\).
Unlike the inverse kinematics, this presents a significant challenge as it requires solving a system of nonlinear equations.
While various numerical methods exist for solving this problem, they can be computationally intensive and may not guarantee convergence to the correct solution.

For the nano-hexapod application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative.
This approximation, developed in subsequent sections through the Jacobian matrix analysis, proves particularly useful for real-time control applications.


\section{The Jacobian Matrix}
\label{ssec:nhexa_stewart_platform_jacobian}
The Jacobian matrix plays a central role in analyzing the Stewart platform's behavior, providing a linear mapping between platform and actuator velocities.
While the previously derived kinematic relationships are essential for position analysis, the Jacobian enables velocity analysis and forms the foundation for both static and dynamic studies.
\paragraph{Jacobian Computation - Velocity Loop Closure}

As was shown in Section \ref{ssec:nhexa_stewart_platform_kinematics}, the strut lengths \(\bm{\mathcal{L}}\) and the platform pose \(\bm{\mathcal{X}}\) are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts.

By taking the time derivative of the position loop close \eqref{eq:nhexa_loop_closure}, the \emph{velocity loop closure} is obtained \eqref{eq:nhexa_loop_closure_velocity}.

\begin{equation}\label{eq:nhexa_loop_closure_velocity}
  {}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{a}_i}_{=0}
\end{equation}

Moreover, we have:
\begin{itemize}
\item \({}^A\dot{\bm{R}}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{R}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{b}_i\) in which \({}^A\bm{\omega}\) denotes the angular velocity of the moving platform expressed in the fixed frame \(\{\bm{A}\}\).
\item \(l_i {}^A\dot{\hat{\bm{s}}}_i = l_i \left( {}^A\bm{\omega}_i \times \hat{\bm{s}}_i \right)\) in which \({}^A\bm{\omega}_i\) is the angular velocity of strut \(i\) express in fixed frame \(\{\bm{A}\}\).
\end{itemize}

By multiplying both sides by \({}^A\hat{s}_i\), \eqref{eq:nhexa_loop_closure_velocity_bis} is obtained.

\begin{equation}\label{eq:nhexa_loop_closure_velocity_bis}
  {}^A\hat{\bm{s}}_i {}^A\bm{v}_p + \underbrace{{}^A\hat{\bm{s}}_i ({}^A\bm{\omega} \times {}^A\bm{b}_i)}_{=({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) {}^A\bm{\omega}} = \dot{l}_i + \underbrace{{}^A\hat{s}_i l_i \left( {}^A\bm{\omega}_i \times {}^A\hat{\bm{s}}_i \right)}_{=0}
\end{equation}

Equation \eqref{eq:nhexa_loop_closure_velocity_bis} can be rearranged in a matrix form to obtain \eqref{eq:nhexa_jacobian_velocities}, with \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1 \ \dots \ \dot{l}_6 ]^T\) the vector of strut velocities, and \(\dot{\bm{\mathcal{X}}} = [{}^A\bm{v}_p ,\  {}^A\bm{\omega}]^T\) the vector of platform velocity and angular velocity.

\begin{equation}\label{eq:nhexa_jacobian_velocities}
  \boxed{\dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}}}
\end{equation}

The matrix \(\bm{J}\) is called the Jacobian matrix, and is defined by \eqref{eq:nhexa_jacobian}, with:
\begin{itemize}
\item \({}^A\hat{\bm{s}}_i\) the orientation of the struts expressed in \(\{A\}\)
\item \({}^A\bm{b}_i\) the position of the joints with respect to \(O_B\) and express in \(\{A\}\)
\end{itemize}

\begin{equation}\label{eq:nhexa_jacobian}
  \bm{J} = \begin{bmatrix}
    {{}^A\hat{\bm{s}}_1}^T & ({}^A\bm{b}_1 \times {}^A\hat{\bm{s}}_1)^T \\
    {{}^A\hat{\bm{s}}_2}^T & ({}^A\bm{b}_2 \times {}^A\hat{\bm{s}}_2)^T \\
    {{}^A\hat{\bm{s}}_3}^T & ({}^A\bm{b}_3 \times {}^A\hat{\bm{s}}_3)^T \\
    {{}^A\hat{\bm{s}}_4}^T & ({}^A\bm{b}_4 \times {}^A\hat{\bm{s}}_4)^T \\
    {{}^A\hat{\bm{s}}_5}^T & ({}^A\bm{b}_5 \times {}^A\hat{\bm{s}}_5)^T \\
    {{}^A\hat{\bm{s}}_6}^T & ({}^A\bm{b}_6 \times {}^A\hat{\bm{s}}_6)^T
  \end{bmatrix}
\end{equation}

This Jacobian matrix \(\bm{J}\) therefore links the rate of change of strut length to the velocity and angular velocity of the top platform with respect to the fixed base through a set of linear equations.
However, \(\bm{J}\) needs to be recomputed for every Stewart platform pose as it depends on the actual pose of of the manipulator.

\paragraph{Approximate solution of the Forward and Inverse Kinematic problems}

For small displacements \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\) around an operating point \(\bm{\mathcal{X}}_0\) (for which the Jacobian was computed), the associated joint displacement \(\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^T\) can be computed using the Jacobian (approximate solution of the inverse kinematic problem):
\begin{equation}\label{eq:nhexa_inverse_kinematics_approximate}
  \boxed{\delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}}
\end{equation}

Similarly, for small joint displacements \(\delta\bm{\mathcal{L}}\), it is possible to find the induced small displacement of the mobile platform (approximate solution of the forward kinematic problem):
\begin{equation}\label{eq:nhexa_forward_kinematics_approximate}
  \boxed{\delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}}}
\end{equation}

These two relations solve the forward and inverse kinematic problems for small displacement in a \emph{approximate} way.
As the inverse kinematic can be easily solved exactly this is not much useful, however, as the forward kinematic problem is difficult to solve, this approximation can be very useful for small displacements.

\paragraph{Range validity of the approximate inverse kinematics}

The accuracy of the Jacobian-based forward kinematics solution was estimated through a systematic error analysis.
For a series of platform positions along the \$x\$-axis, the exact strut lengths are computed using the analytical inverse kinematics equation \eqref{eq:nhexa_inverse_kinematics}.
These strut lengths are then used with the Jacobian to estimate the platform pose, from which the error between the estimated and true poses can be calculated.

The estimation errors in the \(x\), \(y\), and \(z\) directions are shown in Figure \ref{fig:nhexa_forward_kinematics_approximate_errors}.
The results demonstrate that for displacements up to approximately \(1\,\%\) of the hexapod's size (which corresponds to \(100\,\mu m\) as the size of the Stewart platform is here \(\approx 100\,mm\)), the Jacobian approximation provides excellent accuracy.

This finding has particular significance for the Nano-hexapod application.
Since the maximum required stroke (\(\approx 100\,\mu m\)) is three orders of magnitude smaller than the stewart platform size (\(\approx 100\,mm\)), the Jacobian matrix can be considered constant throughout the workspace.
It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_forward_kinematics_approximate_errors.png}
\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with an height of \(100\,mm\) was used to perform this analysis}
\end{figure}

\paragraph{Static Forces}

The static force analysis of the Stewart platform can be elegantly performed using the principle of virtual work.
This principle states that, for a system in static equilibrium, the total virtual work of all forces acting on the system must be zero for any virtual displacement compatible with the system's constraints.

Let \(\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T\) represent the vector of actuator forces applied in each strut, and \(\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T\) denote the external wrench (combined force \(\bm{f}\) and torque \(\bm{n}\)) acting on the mobile platform at point \(\bm{O}_B\).
The virtual work \(\delta W\) consists of two contributions:
\begin{itemize}
\item The work performed by the actuator forces through virtual strut displacements \(\delta \bm{\mathcal{L}}\): \(\bm{\tau}^T \delta \bm{\mathcal{L}}\)
\item The work performed by the external wrench through virtual platform displacements \(\delta \bm{\mathcal{X}}\): \(-\bm{\mathcal{F}}^T \delta \bm{\mathcal{X}}\)
\end{itemize}

The principle of virtual work can thus be expressed as:
\begin{equation}
\delta W = \bm{\tau}^T \delta \bm{\mathcal{L}} - \bm{\mathcal{F}}^T \delta \bm{\mathcal{X}} = 0
\end{equation}

Using the Jacobian relationship that links virtual displacements \eqref{eq:nhexa_inverse_kinematics_approximate}, this equation becomes:
\begin{equation}
\left( \bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T \right) \delta \bm{\mathcal{X}} = 0
\end{equation}

Since this equation must hold for any virtual displacement \(\delta \bm{\mathcal{X}}\), the following force mapping relationships can be derived:

\begin{equation}\label{eq:nhexa_jacobian_forces}
\bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \quad \Rightarrow \quad \boxed{\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}} \quad \text{and} \quad \boxed{\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}}
\end{equation}

These equations establish that the transpose of the Jacobian matrix maps actuator forces to platform forces and torques, while its inverse transpose maps platform forces and torques to required actuator forces.

\section{Static Analysis}
\label{ssec:nhexa_stewart_platform_static}

The static stiffness characteristics of the Stewart platform play a crucial role in its performance, particularly for precision positioning applications.
These characteristics are fundamentally determined by both the actuator properties and the platform geometry.

Starting from the individual actuators, the relationship between applied force \(\delta \tau_i\) and resulting displacement \(\delta l_i\) for each strut \(i\) is characterized by its stiffness \(k_i\):
\begin{equation}
\tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6
\end{equation}

These individual relationships can be combined into a matrix form using the diagonal stiffness matrix \(\mathcal{K}\):
\begin{equation}
\bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}}, \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right]
\end{equation}

By applying the force mapping relationships \eqref{eq:nhexa_jacobian_forces} derived in the previous section and the Jacobian relationship for small displacements \eqref{eq:nhexa_forward_kinematics_approximate}, the relationship between applied wrench \(\bm{\mathcal{F}}\) and resulting platform displacement \(\delta \bm{\mathcal{X}}\) is obtained \eqref{eq:nhexa_stiffness_matrix}.

\begin{equation}\label{eq:nhexa_stiffness_matrix}
\bm{\mathcal{F}} = \underbrace{\bm{J}^T \mathcal{K} \bm{J}}_{\bm{K}} \delta \bm{\mathcal{X}}
\end{equation}
where \(\bm{K} = \bm{J}^T \mathcal{K} \bm{J}\) is identified as the platform stiffness matrix.

The inverse relationship is given by the compliance matrix \(\bm{C}\):

\begin{equation}
\delta \bm{\mathcal{X}} = \underbrace{(\bm{J}^T \mathcal{K} \bm{J})^{-1}}_{\bm{C}} \bm{\mathcal{F}}
\end{equation}

These relationships reveal that the overall platform stiffness and compliance characteristics are determined by two factors:
\begin{itemize}
\item The individual actuator stiffnesses represented by \(\mathcal{K}\)
\item The geometric configuration embodied in the Jacobian matrix \(\bm{J}\)
\end{itemize}

This geometric dependency means that the platform's stiffness varies throughout its workspace, as the Jacobian matrix changes with the platform's position and orientation.
For the NASS application, where the workspace is relatively small compared to the platform dimensions, these variations can be considered minimal.
However, the initial geometric configuration significantly impacts the overall stiffness characteristics.
The relationship between maximum stroke and stiffness presents another important design consideration.
As both parameters are influenced by the geometric configuration, their optimization involves inherent trade-offs that must be carefully balanced based on application requirements.
The optimization of this configuration to achieve desired stiffness properties while having enough stroke will be addressed during the detailed design phase.

\section{Dynamic Analysis}
\label{ssec:nhexa_stewart_platform_dynamics}

The dynamic behavior of a Stewart platform can be analyzed through various approaches, depending on the desired level of model fidelity.
For initial analysis, we consider a simplified model with the following assumptions:
\begin{itemize}
\item Massless struts
\item Ideal joints without friction or compliance
\item Rigid platform and base
\end{itemize}

Under these assumptions, the system dynamics can be expressed in the Cartesian space as:
\begin{equation}
M s^2 \mathcal{X} = \Sigma \mathcal{F}
\end{equation}

where \(M\) represents the platform mass matrix, \(\mathcal{X}\) the platform pose, and \(\Sigma \mathcal{F}\) the sum of forces acting on the platform.

The primary forces acting on the system are actuator forces \(\bm{\tau}\), elastic forces due to strut stiffness \(-\mathcal{K} \mathcal{L}\) and damping forces in the struts \(\mathcal{C} \dot{\mathcal{L}}\).

\begin{equation}
\Sigma \bm{\mathcal{F}} = \bm{J}^T (\tau - \mathcal{K} \mathcal{L} - s \mathcal{C} \mathcal{L}), \quad \mathcal{K} = \text{diag}(k_1\,\dots\,k_6),\  \mathcal{C} = \text{diag}(c_1\,\dots\,c_6)
\end{equation}

Combining these forces and using \eqref{eq:nhexa_forward_kinematics_approximate} yields the complete dynamic equation \eqref{eq:nhexa_dynamical_equations}.

\begin{equation}\label{eq:nhexa_dynamical_equations}
\bm{M} s^2 \bm{\mathcal{X}} = \bm{\mathcal{F}} - \bm{J}^T \bm{\mathcal{K}} \bm{J} \bm{\mathcal{X}} - \bm{J}^T \bm{\mathcal{C}} \bm{J} s \bm{\mathcal{X}}
\end{equation}

The transfer function in the Cartesian frame becomes \eqref{eq:nhexa_transfer_function_cart}.

\begin{equation}\label{eq:nhexa_transfer_function_cart}
\frac{\mathcal{X}}{\mathcal{F}}(s) = (  M s^2 + \bm{J}^{T} \mathcal{C} J s + \bm{J}^{T} \mathcal{K} J )^{-1}
\end{equation}

Through coordinate transformation using the Jacobian matrix, the dynamics in the actuator space is obtained \eqref{eq:nhexa_transfer_function_struts}.

\begin{equation}\label{eq:nhexa_transfer_function_struts}
\frac{\mathcal{L}}{\tau}(s) = ( \bm{J}^{-T} M \bm{J}^{-1} s^2 + \mathcal{C} + \mathcal{K} )^{-1}
\end{equation}

While this simplified model provides useful insights, real Stewart platforms exhibit more complex behaviors.
Several factors significantly increase model complexity:
\begin{itemize}
\item Strut dynamics, including mass distribution and internal resonances
\item Joint compliance and friction effects
\item Supporting structure dynamics and payload dynamics, which are both very critical for NASS
\end{itemize}

These additional effects make analytical modeling impractical for complete system analysis.

\section*{Conclusion}
The fundamental characteristics of the Stewart platform have been analyzed in this chapter.
Essential kinematic relationships were developed through loop closure equations, from which both exact and approximate solutions for the inverse and forward kinematic problems were derived.
The Jacobian matrix was established as a central mathematical tool, through which crucial insights into velocity relationships, static force transmission, and dynamic behavior of the platform were obtained.

For the NASS application, where displacements are typically limited to the micrometer range, the accuracy of linearized models using a constant Jacobian matrix has been demonstrated, by which both analysis and control can be significantly simplified.
However, additional complexities such as strut masses, joint compliance, and supporting structure dynamics must be considered in the full dynamic behavior.
This will be performed in the next section using a multi-body model.

All these characteristics (maneuverability, stiffness, dynamics, etc.) are fundamentally determined by the platform's geometry.
While a reasonable geometric configuration will be used to validate the NASS during this conceptual phase, the optimization of these geometric parameters will be explored during the detailed design phase.

\chapter{Multi-Body Model}
\label{sec:nhexa_model}
The dynamic modeling of Stewart platforms has traditionally relied on analytical approaches.
However, these analytical models become increasingly complex when the full dynamic behavior of struts and joints must be captured.
To overcome these limitations, a flexible multi-body approach has been developed that can be readily integrated into the broader NASS system model.
Through this multi-body modeling approach, each component model (including joints, actuators, and sensors) can be progressively refined.

The analysis is structured in three parts.
First, the multi-body model is developed, wherein detailed geometric parameters, inertial properties, and actuator characteristics are established (Section \ref{ssec:nhexa_model_def}).
The model is then validated through comparison with analytical equations in a simplified configuration (Section \ref{ssec:nhexa_model_validation}).
Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section \ref{ssec:nhexa_model_dynamics}).
\section{Model Definition}
\label{ssec:nhexa_model_def}
\paragraph{Geometry}

The Stewart platform's geometry is defined by two principal coordinate frames (Figure \ref{fig:nhexa_stewart_model_def}): a fixed base frame \(\{F\}\) and a moving platform frame \(\{M\}\).
The joints connecting the actuators to these frames are located at positions \({}^Fa_i\) and \({}^Mb_i\) respectively.
The point of interest, denoted by frame \(\{A\}\), is situated \(150\,mm\) above the moving platform frame \(\{M\}\).

The geometric parameters of the nano-hexapod are summarized in Table \ref{tab:nhexa_stewart_model_geometry}.
These parameters define the positions of all connection points in their respective coordinate frames.
From these parameters, key kinematic properties can be derived: the strut orientations \(\hat{s}_i\), strut lengths \(l_i\), and the system's Jacobian matrix \(\bm{J}\).

\begin{minipage}[b]{0.6\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_stewart_model_def.png}
\captionof{figure}{\label{fig:nhexa_stewart_model_def}Geometry of the stewart platform}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.38\linewidth}
\begin{scriptsize}
\centering
\begin{tabularx}{0.75\linewidth}{Xrrr}
\toprule
 & \(\bm{x}\) & \(\bm{y}\) & \(\bm{z}\)\\
\midrule
\({}^MO_B\) & \(0\) & \(0\) & \(150\)\\
\({}^FO_M\) & \(0\) & \(0\) & \(95\)\\
\({}^Fa_1\) & \(-92\) & \(-77\) & \(20\)\\
\({}^Fa_2\) & \(92\) & \(-77\) & \(20\)\\
\({}^Fa_3\) & \(113\) & \(-41\) & \(20\)\\
\({}^Fa_4\) & \(21\) & \(118\) & \(20\)\\
\({}^Fa_5\) & \(-21\) & \(118\) & \(20\)\\
\({}^Fa_6\) & \(-113\) & \(-41\) & \(20\)\\
\({}^Mb_1\) & \(-28\) & \(-106\) & \(-20\)\\
\({}^Mb_2\) & \(28\) & \(-106\) & \(-20\)\\
\({}^Mb_3\) & \(106\) & \(28\) & \(-20\)\\
\({}^Mb_4\) & \(78\) & \(78\) & \(-20\)\\
\({}^Mb_5\) & \(-78\) & \(78\) & \(-20\)\\
\({}^Mb_6\) & \(-106\) & \(28\) & \(-20\)\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]}
\end{scriptsize}
\end{minipage}

\paragraph{Inertia of Plates}

The fixed base and moving platform are modeled as solid cylindrical bodies.
The base platform is characterized by a radius of \(120\,mm\) and thickness of \(15\,mm\), matching the dimensions of the micro-hexapod's top platform.
The moving platform is similarly modeled with a radius of \(110\,mm\) and thickness of \(15\,mm\).
Both platforms are assigned a mass of \(5\,kg\).

\paragraph{Joints}

The platform's joints play a crucial role in its dynamic behavior.
At both the upper and lower connection points, various degrees of freedom can be modeled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components.
For each degree of freedom, stiffness characteristics can be incorporated into the model.

In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modeled as two-degree-of-freedom universal joints, while the top joints are represented as three-degree-of-freedom spherical joints.
These joints are considered massless and exhibit no stiffness along their degrees of freedom.

\paragraph{Actuators}

The actuator model comprises several key elements (Figure \ref{fig:nhexa_actuator_model}).
At its core, each actuator is modeled as a prismatic joint with internal stiffness \(k_a\) and damping \(c_a\), driven by a force source \(f\).
Similarly to what was found using the rotating 3-DoF model, a parallel stiffness \(k_p\) is added in parallel with the force sensor to ensure stability when considering spindle rotation effects.

Each actuator is equipped with two sensors: a force sensor providing measurements \(f_m\) and a relative motion sensor measuring displacement \(d_L\).
The actuator parameters used in the conceptual phase are presented in Table \ref{tab:nhexa_actuator_parameters}.

This modular approach to actuator modeling allows for future refinements as the design evolves, enabling the incorporation of additional dynamic effects or sensor characteristics as needed.

\begin{minipage}[b]{0.6\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_actuator_model.png}
\captionof{figure}{\label{fig:nhexa_actuator_model}Model of the nano-hexapod actuators}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.38\linewidth}
\begin{scriptsize}
\centering
\begin{tabularx}{0.4\linewidth}{Xl}
\toprule
 & Value\\
\midrule
\(k_a\) & \(1\,N/\mu m\)\\
\(c_a\) & \(50\,N/(m/s)\)\\
\(k_p\) & \(0.05\,N/\mu m\)\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters}
\end{scriptsize}
\end{minipage}

\section{Validation of the multi-body model}
\label{ssec:nhexa_model_validation}

The developed multi-body model of the Stewart platform is represented schematically in Figure \ref{fig:nhexa_stewart_model_input_outputs}, highlighting the key inputs and outputs: actuator forces \(\bm{f}\), force sensor measurements \(\bm{f}_m\), and relative displacement measurements \(\bm{d}_L\).
The frames \(\{F\}\) and \(\{M\}\) serve as interfaces for integration with other elements in the multi-body system.
A three-dimensional visualization of the model is presented in Figure \ref{fig:nhexa_simscape_screenshot}.

\begin{minipage}[b]{0.6\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_stewart_model_input_outputs.png}
\captionof{figure}{\label{fig:nhexa_stewart_model_input_outputs}Nano-Hexapod plant with inputs and outputs. Frames \(\{F\}\) and \(\{M\}\) can be connected to other elements in the multi-body models.}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.35\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.90\linewidth]{figs/nhexa_simscape_screenshot.jpg}
\captionof{figure}{\label{fig:nhexa_simscape_screenshot}3D representation of the multi-body model}
\end{center}
\end{minipage}

The validation of the multi-body model is performed using the simplest Stewart platform configuration, enabling direct comparison with the analytical transfer functions derived in Section \ref{ssec:nhexa_stewart_platform_dynamics}.
This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness \(k_a = 1\,\text{N}/\mu\text{m}\) and damping \(c_a = 10\,\text{N}/({\text{m}/\text{s}})\).
The geometric parameters remain as specified in Table \ref{tab:nhexa_actuator_parameters}.

While the moving platform itself is considered massless, a \(10\,\text{kg}\) cylindrical payload is mounted on top with a radius of \(r = 110\,mm\) and a height \(h = 300\,mm\).

For the analytical model, the stiffness, damping and mass matrices are defined in \eqref{eq:nhexa_analytical_matrices}.

\begin{subequations}\label{eq:nhexa_analytical_matrices}
\begin{align}
\bm{\mathcal{K}} &= \text{diag}(k_a,\ k_a,\ k_a,\ k_a,\ k_a,\ k_a) \\
\bm{\mathcal{C}} &= \text{diag}(c_a,\ c_a,\ c_a,\ c_a,\ c_a,\ c_a) \\
\bm{M} &= \text{diag}\left(m,\ m,\ m,\ \frac{1}{12}m(3r^2 + h^2),\ \frac{1}{12}m(3r^2 + h^2),\ \frac{1}{2}mr^2\right)
\end{align}
\end{subequations}

The transfer functions from actuator forces to strut displacements are computed using these matrices according to equation \eqref{eq:nhexa_transfer_function_struts}.
These analytical transfer functions are then compared with those extracted from the multi-body model.
The multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform.

Figure \ref{fig:nhexa_comp_multi_body_analytical} presents a comparison between the analytical and multi-body transfer functions, specifically showing the response from the first actuator force to all six strut displacements.
The close agreement between both approaches across the frequency spectrum validates the multi-body model's accuracy in capturing the system's dynamic behavior.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_comp_multi_body_analytical.png}
\caption{\label{fig:nhexa_comp_multi_body_analytical}Comparison of the analytical transfer functions and the multi-body model}
\end{figure}

\section{Nano Hexapod Dynamics}
\label{ssec:nhexa_model_dynamics}

Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics has been performed.
The model parameters are set according to the specifications outlined in Section \ref{ssec:nhexa_model_def}, with a payload mass of \(10\,kg\).
Transfer functions from actuator forces \(\bm{f}\) to both strut displacements \(\bm{d}_L\) and force measurements \(\bm{f}_m\) are derived from the multi-body model.

The transfer functions relating actuator forces to strut displacements are presented in Figure \ref{fig:nhexa_multi_body_plant_dL}.
Due to the system's symmetrical design and identical strut configurations, all diagonal terms (transfer functions from force \(f_i\) to displacement \(d_{Li}\) of the same strut) exhibit identical behavior.
While the system possesses six degrees of freedom, only four distinct resonance frequencies are observed in the frequency response.
This reduction from six to four observable modes is attributed to the system's symmetry, where two pairs of resonances occur at identical frequencies.

The system's behavior can be characterized in three frequency regions.
At low frequencies, well below the first resonance, the plant demonstrates good decoupling between actuators, with the response dominated by the strut stiffness: \(G(j\omega) \xrightarrow[\omega \to 0]{} \mathcal{K}^{-1}\).
In the mid-frequency range, the system exhibits coupled dynamics through its resonant modes, reflecting the complex interactions between the platform's degrees of freedom.
At high frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: \(G(j\omega) \xrightarrow[\omega \to \infty]{} J M^{-T} J^T \frac{-1}{\omega^2}\)

The force sensor transfer functions, shown in Figure \ref{fig:nhexa_multi_body_plant_fm}, display characteristics typical of collocated actuator-sensor pairs.
Each actuator's transfer function to its associated force sensor exhibits alternating complex conjugate poles and zeros.
The inclusion of parallel stiffness introduces an additional complex conjugate zero at low frequency, a feature previously observed in the three-degree-of-freedom rotating model.

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nhexa_multi_body_plant_dL.png}
\end{center}
\subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{d}_{L}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nhexa_multi_body_plant_fm.png}
\end{center}
\subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{m}$}
\end{subfigure}
\caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed from the nano-hexapod multi-body model}
\end{figure}

\section*{Conclusion}
The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system.
Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform.

A key advantage of this modeling approach lies in its flexibility for future refinements.
While the current implementation employs idealized joints for the conceptual design phase, the framework readily accommodates the incorporation of joint stiffness and other non-ideal effects.
The joint stiffness, known to impact the performance of decentralized IFF control strategy \cite{preumont07_six_axis_singl_stage_activ}, can be studied as the design evolved and will be optimized during the detail design phase.
The validated multi-body model will serve as a valuable tool for predicting system behavior and evaluating control performance throughout the design process.

\chapter{Control of Stewart Platforms}
\label{sec:nhexa_control}
\begin{itemize}
\item Contrary to what was done with the uniaxial model SISO control => MIMO control: much more complex than because of interaction
\item Possible to ignore interaction when good decoupling is achieved: important to have tools to study interaction.
\item Different ways to try to decouple a MIMO plant
\item Here, just basic strategy, similar to what was done with the uniaxial model is used to validate the concept
\item Control will be optimized during the detailed design phase
\item Reference book: \cite{skogestad07_multiv_feedb_contr}
\end{itemize}
\section{Centralized and Decentralized Control}
\label{ssec:nhexa_control_centralized_decentralized}

\begin{itemize}
\item Explain what is centralized and decentralized:
\begin{itemize}
\item linked to the sensor position relative to the actuators
\item linked to the fact that sensors and actuators pairs are ``independent'' or each other (related to the control architecture, not because there is no coupling)
This does not mean there is no coupling
Decentralized = The controller state depends on one sensor only and will impact one actuator signal only
\end{itemize}
\item When can decentralized control be used and when centralized control is necessary?
Study of interaction: RGA
\item IFF: Decentralized (Section \ref{ssec:nhexa_control_iff})
\item HAC: Centralized (Section \ref{ssec:nhexa_control_hac_lac})
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_decentralized_control.png}
\caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.}
\end{figure}

\section{Choice of the control space}
\label{ssec:nhexa_control_space}

\begin{itemize}
\item Suppose an external metrology measures the pose of frame \(\{B\}\) with respect to \(\{A\}\), noted \(\bm{\mathcal{X}}\).
The goal is to position the top platform to follow some reference signal \(\bm{r}_\mathcal{X}\).

One strategy is to use the Jacobian matrix to perform an approximate inverse kinematics in real time to map the error in the frame of the struts \(\bm{\epsilon}_\mathcal{L}\), and then a diagonal controller is used to control the position of each strut by output forces to be applied on each strut \(\bm{\tau}\).

Another strategy is to have the controller get the cartesian errors as input \(\bm{\epsilon}_{\mathcal{L}}\) and output forces and torques to apply to the top platform \(\bm{\mathcal{F}}\).
The Jacobian is then used to map these forces and torque to force to be applied by each strut.
\end{itemize}

\begin{figure}[htbp]
\begin{subfigure}{0.98\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_control_strut.png}
\end{center}
\subcaption{\label{fig:nhexa_control_strut}Control in the frame of the struts. $\bm{J}$ is used to project errors in the frame of the struts}
\end{subfigure}

\bigskip
\begin{subfigure}{0.98\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_control_cartesian.png}
\end{center}
\subcaption{\label{fig:nhexa_control_cartesian}Control in the Cartesian frame. $\bm{J}^{-T}$ is used to project force and torques on each strut}
\end{subfigure}
\caption{\label{fig:nhexa_control_frame}Two control strategies}
\end{figure}

\begin{itemize}
\item Trade-off for both strategies from looking at the obtained plant.
\begin{itemize}
\item The plant in the frame of the struts is shown in Figure \ref{fig:nhexa_control_strut}
equal diagonal plant elements: just one controller to design, well decoupled at low frequency
\item The plant in Cartesian frame is shown in Figure \ref{fig:nhexa_control_cartesian}
less visible modes in some directions: vertical plant: second order plant, same for Rz
But Coupling: \(\epsilon_{R_x}/\mathcal{F}_y\), \(\epsilon_{R_y}/\mathcal{F}_x\), \(\epsilon_{D_x}/\mathcal{M}_y\), \(\epsilon_{D_y}/\mathcal{M}_x\)
can choose the bandwidth for different DoF, but coupling may be present at low frequency
\end{itemize}

\item Say that in order to validate the conceptual design, the control will be performed in the frame of the struts for simplicity

\item There are much to discuss about controlling a Stewart platform, this will be done during the detail design phase.
\end{itemize}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nhexa_plant_frame_struts.png}
\end{center}
\subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nhexa_plant_frame_cartesian.png}
\end{center}
\subcaption{\label{fig:nhexa_plant_frame_cartesian}Plant in the Cartesian Frame}
\end{subfigure}
\caption{\label{fig:nhexa_plant_frame}Bode plot of the transfer functions computed from the nano-hexapod multi-body model}
\end{figure}

\section{Active Damping with Decentralized IFF}
\label{ssec:nhexa_control_iff}

Integral Force Feedback is implemented in a decentralized way (i.e. similarly to what is shown in Figure \ref{fig:nhexa_stewart_decentralized_control}, but using force sensors instead of relative motion sensors).

Block diagram is shown in Figure \ref{fig:nhexa_decentralized_iff_schematic}, with controller \(\bm{K}_{\text{IFF}}(s)\) being a diagonal controller \eqref{eq:nhexa_kiff} (i.e. one independent controller for each strut) with pure integrators on the diagonal.

\begin{equation}\label{eq:nhexa_kiff}
    \bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
        K_{\text{IFF}}(s) & & 0 \\
        & \ddots & \\
        0 & & K_{\text{IFF}}(s)
    \end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s}
\end{equation}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_decentralized_iff_schematic.png}
\caption{\label{fig:nhexa_decentralized_iff_schematic}Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs \(\bm{f}^{\prime}\)}
\end{figure}

Note that here, we are not considering stiffness in parallel with the force sensors are the Stewart platform is not rotating (this will be studied in the next section when the Stewart platform will be located on top of the micro-station).

Similarly to what was done with the 3DoF model, the Root Locus plot is computed by estimating the poles of the closed-loop system as the controller gain \(g\) is varied from \(0\) to \(\infty\).


\begin{itemize}
\item[{$\square$}] Interaction around resonances is very high: show that with RGA (encoder outputs)
\item[{$\square$}] But guaranteed stability with decentralized IFF \cite{preumont08_trans_zeros_struc_contr_with}
\begin{itemize}
\item[{$\square$}] I think there is another paper about that
\end{itemize}
\item[{$\square$}] nice way to have some control authority around that frequency, which would be impossible with positioning sensors
\end{itemize}

For decentralized control: ``MIMO root locus'' can be used to estimate the damping / optimal gain
Poles and converging towards \emph{transmission zeros}
=> Already explain in 3DoF model

Show effect of changed payload mass? (no maybe NASS section)

Compute:
\begin{itemize}
\item[{$\square$}] Plant dynamics (already shown earlier)
\item[{$\square$}] Root Locus with decentralized IFF (only pure integrator?)
\item[{$\square$}] show the poles for one value of the gain => How to optimize the added damping to all modes?
\begin{itemize}
\item[{$\square$}] Add some papers citations
\end{itemize}
\item[{$\square$}] Effect of rotation and added parallel stiffness? Or maybe in next section (NASS + Spindle)?
\end{itemize}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_iff_loop_gain.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root Locus}
\end{subfigure}
\caption{\label{fig:nhexa_decentralized_iff_results}Decentralized IFF}
\end{figure}

\section{MIMO High-Authority Control - Low-Authority Control}
\label{ssec:nhexa_control_hac_lac}

The HAC-IFF architecture is shown in Figure \ref{fig:nhexa_hac_iff_schematic}.
The reference signal \(\bm{r}_{\mathcal{X}}\) is compared with the measured pose \(\bm{\mathcal{X}}\).
The Jacobian matrix is used to solve the approximate inverse kinematics in real time.
Finally, the (diagonal) High Authority Controller \(\bm{K}_{\text{HAC}}\) is doing the doing in the frame of the struts.


\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_hac_iff_schematic.png}
\caption{\label{fig:nhexa_hac_iff_schematic}HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts}
\end{figure}

The transfer functions from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) (i.e. without the Decentralized IFF being implemented) are compared with the transfer functions from \(\bm{f}^{\prime}\) to \(\bm{\epsilon}_{\mathcal{L}}\) (i.e. with the Decentralized IFF being implemented).

\begin{itemize}
\item[{$\square$}] Maybe two subfigures for undamped and damped
\end{itemize}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/nhexa_decentralized_hac_iff_plant_undamped.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/nhexa_decentralized_hac_iff_plant_damped.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped with Decentralized IFF}
\end{subfigure}
\caption{\label{fig:nhexa_decentralized_hac_iff_plant}Plant in the frame of the strut for the High Authority Controller.}
\end{figure}

From the obtained damped plant, the High Authority Controller is developed.

\begin{equation}\label{eq:nhexa_khac}
    \bm{K}_{\text{HAC}}(s) = \begin{bmatrix}
        K_{\text{HAC}}(s) & & 0 \\
        & \ddots & \\
        0 & & K_{\text{HAC}}(s)
    \end{bmatrix}, \quad K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}
\end{equation}

\begin{itemize}
\item In order to check the stability of the feedback MIMO loop:
\begin{itemize}
\item Characteristic Loci: Eigenvalues of \(\bm{G}(j\omega)\bm{K}(j\omega)\) plotted in the complex plane
\item Generalized Nyquist Criterion: If \(G(s)\) has \(p_0\) unstable poles, then the closed-loop system with return ratio \(kG(s)\) is stable if and only if the characteristic loci of \(kG(s)\), taken together, encircle the point \(-1\), \(p_0\) times anti-clockwise, assuming there are no hidden modes
\end{itemize}
\end{itemize}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_hac_iff_loop_gain.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_loop_gain}Loop Gain}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_hac_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Root Locus}
\end{subfigure}
\caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF}
\end{figure}













\begin{itemize}
\item[{$\square$}] Show some performance metric? For instance compliance?
\end{itemize}

\section*{Conclusion}


\chapter*{Conclusion}
\label{sec:nhexa_conclusion}

\begin{itemize}
\item Configurable Stewart platform model
\item Will be included in the multi-body model of the micro-station => nass multi body model
\item Control: complex problem, try to use simplest architecture
\end{itemize}

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