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}).
\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}
The main disadvantage of Stewart platforms is the small workspace when compare the serial manipulators which is not a problem here.
\section{Mechanical Architecture}
\label{ssec:nhexa_stewart_platform_architecture}
A Stewart manipulator consists of two platforms connected by six struts (Figure \ref{fig:nhexa_stewart_architecture}).
Each strut is connected to the fixed and the mobile platforms with a joint.
Typically, a universal joint is used on one side while a spherical joint is used on the other side\footnote{Different architecture exists, typically referred as ``6-SPS'' (Spherical, Prismatic, Spherical) or ``6-UPS'' (Universal, Prismatic, Spherical)}.
In the strut, there is an active element working as a prismatic joint.
Frames \(\{F\}\) and \(\{M\}\) are useful to describe the location of the joints in a meaningful frame.
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 \(\{B\}\) expressed in \(\{A\}\) and the rotation matrix \({}^A\bm{R}_B\) expressing the orientation of \(\{B\}\) with respect to \(\{A\}\).
For the nano-hexapod, these frames are chosen to be located at the theoretical focus point of the X-ray light (\(150\,mm\) above the top platform, i.e. above \(\{M\}\)).
Location of the joints and orientation and length of the struts are very important for the study of the Stewart platform as well.
The center of rotation for the joint fixed to the base is noted \(\bm{a}_i\), while \(b_i\) is used for the top joints.
The struts orientation are indicated 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}.
Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces that cause the motion.
\paragraph{Loop Closure}
At the displacement level, the \emph{closure} of each kinematic loop (illustrated in Figure \ref{fig:nhexa_stewart_loop_closure}) can be express in the vector form as
\begin{equation}
\vec{ab} = \vec{aa_i} + \vec{a_ib_i} - \vec{bb_i}\quad\text{for } i = 1 \text{ to } 6
\end{equation}
in which \(\vec{aa_i}\) and \(\vec{bb_i}\) can be easily obtained from the geometry of the attachment points in the base and in the moving platform.
The loop closure can be written as the unknown 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 limb vector \(l_i {}^A\hat{\bm{s}}_i\):
For \emph{inverse kinematic analysis}, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables \(\bm{\mathcal{L}}=\left[ l_1, l_2, l_3, l_4, l_5, l_6\right]\).
This problem can be easily solved using the loop closures \eqref{eq:nhexa_loop_close}.
In \emph{forward kinematic analysis}, it is assumed that the vector of limb lengths \(\bm{\mathcal{L}}\) is given and the problem is to find the position \({}^A\bm{P}\) and the orientation \({}^A\bm{R}_B\).
This is a difficult problem that requires to solve nonlinear equations.
In a next section, an approximate solution of the forward kinematics problem is proposed for small displacements.
\section{The Jacobian Matrix}
In vector calculus, the Jacobian matrix represents the best linear approximation of a vector-valued function near a working point.
Consider a function \(\bm{f}: \mathbb{R}^n \rightarrow\mathbb{R}^m\) with continuous first-order partial derivatives.
For any input \(\bm{x}\in\mathbb{R}^n\), this function produces an output \(\bm{f}(\bm{x})\in\mathbb{R}^m\).
The Jacobian matrix \(\bm{J}\) of \(\bm{f}\) at point \(\bm{x}\) is the \(m \times n\) matrix whose \((i,j)\) entry is:
\(J_{ij}=\frac{\partial f_i}{\partial x_j}\)
This matrix represents the linear transformation that best approximates \(\bm{f}\) in a neighborhood of \(\bm{x}\).
In other words, for points sufficiently close to \(\bm{x}\), the function \(\bm{f}\) behaves approximately like its Jacobian matrix.
\(\bm{\mathcal{L}}\) and \(\bm{\mathcal{X}}\) are related through a system of \emph{nonlinear algebraic equations} representing the \emph{kinematic constraints imposed by the struts}, which can be generally written as \(f(\bm{\mathcal{L}}, \bm{\mathcal{X}})=0\).
\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 limb \(i\) express in fixed frame \(\{\bm{A}\}\).
\item\(\hat{\bm{s}}_i\) the orientation of the limbs expressed in \(\{A\}\)
\item\(\bm{b}_i\) the position of the joints with respect to \(O_B\) and express in \(\{A\}\)
\end{itemize}
The Jacobian matrix links the rate of change of strut length to the velocity and angular velocity of the top platform with respect to the fixed base.
This Jacobian matrix needs to be recomputed for every Stewart platform pose.
\paragraph{Approximate solution of the Forward and Inverse Kinematic problems}
For small displacements mobile platform displacement \(\delta\bm{\mathcal{X}}=[\delta x, \delta y, \delta z, \delta\theta_x, \delta\theta_y, \delta\theta_z ]^T\) around \(\bm{\mathcal{X}}_0\), the associated joint displacement can be computed using the Jacobian (approximate solution of the inverse kinematic problem):
Similarly, for small joint displacements \(\delta\bm{\mathcal{L}}=[\delta l_1,\ \dots,\ \delta l_6]^T\) around \(\bm{\mathcal{L}}_0\), it is possible to find the induced small displacement of the mobile platform (approximate solution of the forward kinematic problem):
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}
As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix.
For small displacements, the approximate solution is expected to work well.
We would like here to determine up to what displacement this approximation can be considered as correct.
Then, we can determine the range for which the approximate inverse kinematic is valid.
This will also gives us the range for which the approximate forward kinematic is valid.
\begin{itemize}
\item[{$\square$}]\href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org}{Estimation of the range validity of the approximate inverse kinematics}
\end{itemize}
Let's first compare the perfect and approximate solution of the inverse for pure \(x\) translations.
The approximate and exact required strut stroke to have the wanted mobile platform \(x\) displacement are computed.
The estimated error is shown in Figure etc\ldots{}
For small wanted displacements (up to \(\approx1\%\) of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct.
In the case of the Nano-hexapod, the maximum stroke is estimate to the around \(100\,\mu m\) while its size is around \(100\,mm\), therefore the fixed Jacobian matrix is a very good approximate for the forward and inverse kinematics.
\paragraph{Static Forces}
Let's note \(\bm{\tau}=[\tau_1, \tau_2, \cdots, \tau_6]^T\) the vector of actuator forces applied in each strut and \(\bm{\mathcal{F}}=[\bm{f}, \bm{n}]^T\) external force/torque action on the mobile platform at \(\bm{O}_B\).
The \emph{principle of virtual work} states that the total virtual work \(\delta W\), done by all actuators and external forces is equal to zero:
\begin{equation}
\delta W = \bm{\tau}^T \delta\bm{\mathcal{L}} - \bm{\mathcal{F}}^T \delta\bm{\mathcal{X}} = 0
\end{equation}
From the definition of the Jacobian (\(\delta\bm{\mathcal{L}}=\bm{J}\cdot\delta\bm{\mathcal{X}}\)), we have \(\left(\bm{\tau}^T \bm{J}-\bm{\mathcal{F}}^T \right)\delta\bm{\mathcal{X}}=0\) that holds for any \(\delta\bm{\mathcal{X}}\), hence:
Therefore, the same Jacobian matrix can also be used to map actuator forces to forces and torques applied on the mobile platform at the defined frame \(\{B\}\).
Reasonable choice of geometry is made in chapter 1.
Optimization of the geometry will be made in chapter 2.
\chapter{Multi-Body Model}
\label{sec:nhexa_model}
\textbf{Goal}:
\begin{itemize}
\item Study the dynamics of Stewart platform
\item Instead of working with complex analytical models: a multi-body model is used.
Complex because has to model the inertia of the struts.
Cite papers that tries to model the stewart platform analytically
Advantage: it will be easily included in the model of the NASS
\item Mention the Toolbox (maybe make a DOI for that)
\item[{$\square$}] Have a table somewhere that summarizes the main characteristics of the nano-hexapod model
\begin{itemize}
\item location of joints
\item size / mass of platforms, etc\ldots{}
\end{itemize}
\end{itemize}
\section{Model Definition}
\label{ssec:nhexa_model_def}
\begin{itemize}
\item[{$\square$}] Make a schematic of the definition process (for instance knowing the ai, bi points + \{A\} and \{B\} allows to compute Jacobian, etc\ldots{})
\item What is important for the model:
\begin{itemize}
\item Inertia of plates and struts
\item Positions of joints / Orientation of struts
\item Definition of frames (for Jacobian, stiffness analysis, etc\ldots{})
\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)
\end{itemize}
\item When can decentralized control be used and when centralized control is necessary?
\item Jacobian matrices, CoK, CoM, control in the frame of the struts, SVD, Modal, \ldots{}
\item Combined CoM and CoK => Discussion of cubic architecture ? (quick, as it is going to be in detailed in chapter 2)
\item Explain also the link with the setpoint: it is interesting to have the controller in the frame of the performance variables
Also speak about disturbances? (and how disturbances can be mixed to different outputs due to control and interaction)
\item Table that summarizes the trade-off for each strategy
\item Say that in this study, we will do the control in the frame of the struts for simplicity (even though control in the cartesian frame was also tested)
\item Characteristic Loci: Eigenvalues of \(G(j\omega)\) 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}
\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
