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\author[shortname]{Thomas Dehaeze \inst{1,2}, Julien Bonnefoy \inst{1} \and Christophe Collette \inst{2,3}}
\institute[shortinst]{\inst{1} European Synchrotron Radiation Facility, Grenoble, France \and %
\inst{2} Precision Mechatronics Laboratory, University of Liege, Belgium \and %
\inst{3} BEAMS Department, Free University of Brussels, Belgium}
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\author{Dehaeze Thomas, Bonnefoy Julien and Collette Christophe}
\date{}
\title{Mechatronics Approach for the Development of a Nano-Active-Stabilization-System}
\subtitle{MEDSI2020, July 26-29, 2021}
\hypersetup{
 pdfauthor={Dehaeze Thomas, Bonnefoy Julien and Collette Christophe},
 pdftitle={Mechatronics Approach for the Development of a Nano-Active-Stabilization-System},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 27.2 (Org mode 9.5)}, 
 pdflang={English}}
\begin{document}

\maketitle
\section*{Introduction}
\begin{frame}[label={sec:org4b0545d}]{The ID31 Micro Station}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/assemblage.png}
\end{center}

\textbf{Objective}: Position samples along complex trajectories with high precision\newline
\textbf{Stacked Positioning Stages}: \(\approx 10\,\mu m\) precision limited by stages vibrations, thermal effects, ground motion, \ldots{}
\end{frame}

\begin{frame}[label={sec:orgb7c2959}]{Introduction - The Nano Active Stabilization System}
\textbf{Objective}: Improve the position accuracy from \(\approx 10\,\mu m\) down to \(\approx 10\,nm\) \newline
\textbf{Design approach}: ``Model based design'' (extensive use of models and test benches)

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nass-concept.pdf}
\end{center}
\end{frame}

\begin{frame}[label={sec:orgaf8e005}]{The Nano-Hexapod - Why such mechanical architecture?}
\begin{itemize}
\item Why stewart architecture
\begin{itemize}
\item 6 DoF to control / 6 actuators
\end{itemize}
\item Only flexible elements
\begin{itemize}
\item no backlash
\item no play
\end{itemize}
\item How it is working
\begin{itemize}
\item Jacobian matrix both for actuation and sensing
\end{itemize}
\item Forward / Inverse kinematics : meaning. Easy to compute for small displacements
\item[{$\square$}] Schematic of Stewart platform
\end{itemize}

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nano_hexapod_elements.red.pdf}
\end{center}
\end{frame}

\begin{frame}[label={sec:orgb9ee458}]{Stewart Platforms Architecture}
\vspace{-2em}

\begin{columns}
\begin{column}{0.5\columnwidth}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=\linewidth]{figs/stewart_schematic.png}
\caption{\label{fig:stewart_schematic}Geometry of a Stewart Platform}
\end{figure}

\vspace{-1em}

\begin{tcolorbox}[title=Advantages]
\begin{itemize}
\item Compact
\item Allows 6dof motion
\item Can be ``monolithic''
\end{itemize}
\end{tcolorbox}
\end{column}

\begin{column}{0.5\columnwidth}
\begin{tcolorbox}[title=Definition of the Geomtry]
\begin{itemize}
\item \(\bm{a}_i\): position of the attachment points on the fixed base
\item \(\bm{b}_i\): position of moving attachment points
\item \(l_i\): length of each limb
\item \(\hat{\bm{s}}_i\): unit vector representing the direction of each limb
\end{itemize}
\end{tcolorbox}
\end{column}
\end{columns}
\end{frame}

\begin{frame}[label={sec:org09981cc}]{Stewart Platform Architecture - Kinematics}
\begin{itemize}
\item \(\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^T\): vector of actuated joint coordinates
\item \(\bm{\mathcal{X}} = \left[ {}^A\bm{P}, \bm{}^A\hat{\bm{s}} \right]^T\): vector of platform motion variables
\end{itemize}

\begin{align*}
\bm{\mathcal{X}} & \xrightarrow[\text{Simple}]{\text{Inverse Kinematics}}  \bm{\mathcal{L}} \\
\bm{\mathcal{L}} & \xrightarrow[\text{Complex}]{\text{Forward Kinematics}} \bm{\mathcal{X}}
\end{align*}

For small displacements: \textbf{Jacobian} matrix
\begin{equation*}
  \bm{J} = \begin{bmatrix}
    {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\
    {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\
    {\hat{\bm{s}}_3}^T & (\bm{b}_3 \times \hat{\bm{s}}_3)^T \\
    {\hat{\bm{s}}_4}^T & (\bm{b}_4 \times \hat{\bm{s}}_4)^T \\
    {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\
    {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T
  \end{bmatrix}
\end{equation*}

\begin{align*}
  \delta\bm{\mathcal{L}} &= \bm{J}      \delta\bm{\mathcal{X}} \\
  \delta\bm{\mathcal{X}} &= \bm{J}^{-1} \delta\bm{\mathcal{L}}
\end{align*}

As an example, for the Nano-Hexapod:
\begin{equation*}
\bm{J} = \begin{bmatrix*}[r]
 0.69 & -0.38 & 0.61 & -0.13 & -0.10 &  0.08 \\
-0.69 & -0.38 & 0.61 & -0.13 &  0.10 & -0.08 \\
-0.02 &  0.80 & 0.61 &  0.15 & -0.06 &  0.08 \\
 0.68 & -0.41 & 0.61 & -0.02 & -0.16 & -0.08 \\
-0.68 & -0.41 & 0.61 & -0.02 &  0.16 &  0.08 \\
 0.02 &  0.80 & 0.61 &  0.15 &  0.06 & -0.08
\end{bmatrix*}, \quad \bm{J}^{-1} = \begin{bmatrix*}[r]
 0.84 & -0.84 & -1.00 & -0.15 &  0.15 &  1.00 \\
 0.66 &  0.66 &  0.39 & -1.06 & -1.06 &  0.39 \\
 0.27 &  0.27 &  0.27 &  0.27 &  0.27 &  0.27 \\
-4.51 & -4.51 &  0.12 &  4.39 &  4.39 &  0.12 \\
 2.46 & -2.46 & -5.14 & -2.67 &  2.67 &  5.14 \\
 1.96 & -1.96 &  1.96 & -1.96 &  1.96 & -1.96
\end{bmatrix*}
\end{equation*}

\begin{itemize}
\item[{$\square$}] Control architecture in the frame of the legs with the Jacobian matrix
\end{itemize}
\end{frame}

\begin{frame}[label={sec:orgec6681e}]{Control Challenges - Analogy}
\vspace{-2em}

\only<1>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/001_Room.pdf}
\end{center}

}\only<2>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/002_Analogies.pdf}
\end{center}

}\only<3>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/003_Laser.pdf}
\end{center}

}\only<4>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/004_Top-Platform.pdf}
\end{center}

}\only<5>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/005_Candle.pdf}
\end{center}

}\only<6>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/006_Objective and Challenges.pdf}
\end{center}

}\only<7>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/007_Truck.pdf}
\end{center}

}\only<8>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/008_Trampoline.pdf}
\end{center}

}\only<9>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/009_Spindle.pdf}
\end{center}

}\only<10>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/010_Metrology.pdf}
\end{center}

}\only<11>{

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/011_Flame.pdf}
\end{center}

}
\end{frame}

\begin{frame}[label={sec:orgd25c78f}]{Overview of the Mechatronic Approach - Model Based Design}
\vspace{-1em}

\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nass-mechatronics-approach.png}
\end{center}
\end{frame}

\begin{frame}[label={sec:org4f4bab3}]{Outline}
\tableofcontents
\end{frame}
\end{document}