381 lines
20 KiB
TeX
381 lines
20 KiB
TeX
% Created 2025-04-21 Mon 19:46
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% Intended LaTeX compiler: pdflatex
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\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
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\input{preamble.tex}
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\input{preamble_extra.tex}
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\bibliography{nass-design.bib}
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\author{Dehaeze Thomas}
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\date{\today}
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\title{Nano Hexapod - Obtained Design}
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\hypersetup{
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pdfauthor={Dehaeze Thomas},
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pdftitle={Nano Hexapod - Obtained Design},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 30.1 (Org mode 9.7.26)},
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pdflang={English}}
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\usepackage{biblatex}
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\begin{document}
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\maketitle
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\tableofcontents
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\clearpage
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_nano_hexapod_elements.png}
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\caption{\label{fig:detail_design_nano_hexapod_elements}Obtained mechanical design of the Active platform, the ``nano-hexapod''}
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\end{figure}
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Detail design phase:
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\begin{itemize}
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\item key elements were optimized such as: actuator and flexible joints
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\item relative motion sensor (an encoder) was also selected
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\item specific kinematics of the Stewart platform (i.e. position of joints and orientation of struts) was not found to be too critical for this application.
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Yet, the geometry was fixed in Section [\ldots{}]
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\end{itemize}
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In this section, the mechanical design of the active platform, shown in Figure \ref{fig:detail_design_nano_hexapod_elements}, is detailed.
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The main design objectives are:
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\begin{itemize}
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\item Well defined kinematics: Good positioning of the top flexible joint rotation point \(\bm{b}_i\) and correct orientation of the struts \(\hat{\bm{s}}_i\).
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The goal is to have a well defined geometry such that the Jacobian matrix is well defined.
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\item Space constrains: it should fit within a cylinder with radius of \(120\,\text{mm}\) and height of \(95\,\text{mm}\)
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\item As good performances were obtained with the multi-body model.
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The final design should behave as close as possible to ``perfect'' stewart platform.
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This means that the frequency of flexible modes that could be problematic for control must be made as high as possible.
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\item Easy mounting and alignment.
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\item Easy maintenance: the struts should be easily changed in case for failure.
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\end{itemize}
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\chapter{Mechanical Design}
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\label{sec:detail_design_mechanics}
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\subsubsection{Struts}
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The strut design is shown in Figure \ref{fig:detail_design_strut}.
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The design of the struts was driven by:
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\begin{itemize}
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\item having stiff interface between the amplified piezoelectric actuator and the two flexible joints
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\item having stiff interface between the flexible joints and the two places (discussed afterwards)
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\item Because the angular stroke of the flexible joints is fairly limited, it is important to be able to mount the strut such that the two cylindrical interfaces are coaxial.
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Do to so:
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\begin{itemize}
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\item A mounting bench was designed
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The mounting procedure will be described in Section [\ldots{}]
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\item Cylindrical washers, shown in Figure \ref{fig:detail_design_strut_without_enc}, were integrated to allow for adjustments.
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The issue was that the flatness between the two interface planes of the APA shown in Figure \ref{fig:detail_design_apa} could not be guaranteed.
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With the added cylindrical washers and the mounting tool, it should be possible to well align the struts even in the presence of machining inaccuracies.
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\end{itemize}
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\item Possibility to fix the encoder parallel to the strut, as shown in Figure \ref{fig:detail_design_strut_with_enc}
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\end{itemize}
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\begin{figure}[htbp]
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.9]{figs/detail_design_strut_without_enc.png}
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\end{center}
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\subcaption{\label{fig:detail_design_strut_without_enc}Before encoder integration}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.9]{figs/detail_design_strut_with_enc.png}
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\end{center}
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\subcaption{\label{fig:detail_design_strut_with_enc}With the mounted encoder}
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\end{subfigure}
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\caption{\label{fig:detail_design_strut}Design of the Nano-Hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration.}
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\end{figure}
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The flexible joints are manufactured using wire-cut electrical discharge machining, allowing for:
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\begin{itemize}
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\item very tight tolerances:
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\begin{itemize}
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\item allowing good location of the center of rotation with respect to the plate interfaces (red surfaces shown in Figure \ref{fig:detail_design_flexible_joint})
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\item allowing correct neck dimension to have the wanted properties (stiffness and angular stroke)
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\end{itemize}
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\item Such part is fragile, mainly due to its small ``neck'' dimension of only \(0.25\,\text{mm}\)
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Such machining technique has little to no cutting forces.
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\end{itemize}
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The flexible joints are made from a stainless steel referenced as ``X5CrNiCuNb16-4'' (also called ``F16Ph'').
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This material is chosen for:
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\begin{itemize}
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\item its high yield strength: specified >1GPa using heat treatment.
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\item its high fatigue resistance
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\end{itemize}
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Figure \ref{fig:detail_design_flexible_joint}
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\begin{itemize}
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\item Interface with the APA has a cylindrical shape to allow the use of cylindrical washers
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A slotted hole has been added to align the flexible joint with the APA using a dowel pin.
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\item Two threaded holes on the sides can be used to mount the encoders
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\item The interface with the plate will be latter described.
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\end{itemize}
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The amplified piezoelectric actuators are APA300ML.
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Modification of the mechanical interfaces were asked to the manufacturer.
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Two planes surfaces and a dowel hole were used, as shown in Figure \ref{fig:detail_design_apa}.
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The amplifying structure, is also made of stainless steel.
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\begin{figure}[htbp]
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_design_flexible_joint.png}
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\end{center}
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\subcaption{\label{fig:detail_design_flexible_joint}Flexible joint}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_design_apa.png}
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\end{center}
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\subcaption{\label{fig:detail_design_apa}Amplified Piezoelectric Actuator}
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\end{subfigure}
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\caption{\label{fig:detail_design_apa_joints}Two main components of the struts: the flexible joint (\subref{fig:detail_design_flexible_joint}) and the amplified piezoelectric actuator (\subref{fig:detail_design_apa}).}
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\end{figure}
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To correctly measure the relative motion of each strut, the encoders need to measure the relative motion between the two flexible joint's rotational centers.
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Two interface parts, made of aluminum, are used to fix the encoder and ruler to the two fleible joints as shown in Figure \ref{fig:detail_design_strut_with_enc}.
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\subsubsection{Plates}
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The two plates of the active platform were designed to:
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\begin{itemize}
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\item Maximize the frequency of flexible modes
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\item have good positioning of the top flexible joints, and good/known orientation of the struts.
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\end{itemize}
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To maximize the flexible joints, finite element analysis were used iteratively.
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While topology optimization could have been used, a network of reinforcing ribs was used as shown in Figure \ref{fig:detail_design_top_plate}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,scale=1]{figs/detail_design_top_plate.png}
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\caption{\label{fig:detail_design_top_plate}The mechanical design for the top platform incorporates precisely positioned V-grooves for the joint interfaces (displayed in red). The purpose of the encoder interface (shown in green) is detailed later.}
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\end{figure}
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The fixation interface for the joints and ``V-grooves''.
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The cylindrical part of the flexible joint is located (or constrained) within the V-groove via two distinct line contacts (Figure \ref{fig:detail_design_fixation_flexible_joints}).
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Therefore, these grooves are defining the initial strut orientation
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High machining accuracy is required, such that during the mounting of the active platform, the flexible joints are that ``rest'' position
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The ``flat'' interface of each top flexible joint is also in contact with the top platform, as shown in Figure \ref{fig:detail_design_location_top_flexible_joints}, such that the center of rotation of the top flexible joints \(\bm{b}_i\) are well located with respect to the top platform.
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The bottom flexible joints are not Figure \ref{fig:detail_design_location_bot_flex}
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The two plates are made with a martensitic stainless steel ``X30Cr13'':
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\begin{itemize}
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\item It has high hardness, such that the reference surfaces to not deform when fixing the flexible joints
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\item This should allow to assemble and disassemble the struts many times if necessary
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\end{itemize}
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\begin{figure}
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\begin{subfigure}{0.33\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.99\linewidth]{figs/detail_design_fixation_flexible_joints.png}
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\end{center}
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\subcaption{\label{fig:detail_design_fixation_flexible_joints}Flexible Joint Clamping}
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\end{subfigure}
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\begin{subfigure}{0.33\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.99\linewidth]{figs/detail_design_location_top_flexible_joints.png}
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\end{center}
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\subcaption{\label{fig:detail_design_location_top_flexible_joints}Top positioning}
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\end{subfigure}
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\begin{subfigure}{0.33\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.99\linewidth]{figs/detail_design_location_bot_flex.png}
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\end{center}
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\subcaption{\label{fig:detail_design_location_bot_flex}Bottom Positioning}
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\end{subfigure}
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\caption{\label{fig:detail_design_fixation_flexible_joints}Fixation of the flexible points to the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joint is in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joint is just oriented (\subref{fig:detail_design_location_bot_flex}).}
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\end{figure}
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\subsubsection{Finite Element Analysis}
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Finite element analysis of the complete active platform was performed to identify problematic modes (Figure \ref{fig:detail_design_fem_nano_hexapod}):
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\begin{itemize}
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\item First six modes were found to be ``suspension'' modes were the top plate moves as a rigid body, and the six struts are only moving axially (Figure \ref{fig:detail_design_fem_rigid_body_mode})
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\item Then, between \(205\,\text{Hz}\) and \(420\,\text{Hz}\) many ``local'' modes of the struts were observed.
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On is represented in Figure \ref{fig:detail_design_fem_strut_mode}.
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While these modes seem not to induce any motion of the top platform, it induces a relative displacement of the encoder with respect to the ruler.
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Therefore, when controlling the position of the active platform using the encoders, such modes could be problematic.
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Whether these modes are problematic is difficult to estimate at this point as:
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\begin{itemize}
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\item it is not known if the APA will ``excite'' these modes
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\item theoretically, if the struts are well aligned, these modes should not be observed
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\end{itemize}
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Then, flexible modes of the top plate are appearing above \(650\,\text{Hz}\) (Figure \ref{fig:detail_design_fem_plate_mode})
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\end{itemize}
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\begin{figure}[htbp]
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\begin{subfigure}{0.36\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_fem_rigid_body_mode.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension mode}
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\end{subfigure}
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\begin{subfigure}{0.36\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_fem_strut_mode.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_fem_strut_mode}Strut - Local mode}
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\end{subfigure}
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\begin{subfigure}{0.26\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_fem_plate_mode.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_fem_plate_mode}Top plate mode}
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\end{subfigure}
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\caption{\label{fig:detail_design_fem_nano_hexapod}Measurement of strut flexible modes. First six modes are ``suspension'' modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from \(205\,\text{Hz}\) to \(420\,\text{Hz}\) (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at \(650\,\text{Hz}\) (\subref{fig:detail_design_fem_plate_mode})}
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\end{figure}
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\subsubsection{Alternative Encoder Placement}
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To anticipate potential issue with local modes of the struts, an alternative fixation for the encoder is planned:
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\begin{itemize}
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\item Instead of being fixed to the struts, the encoders are fixed to the plates instead, as shown in Figure \ref{fig:detail_design_enc_plates_design}.
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\item The support are made of aluminum, and it is verified that the natural modes are at high enough frequency (Figure \ref{fig:detail_design_enc_support_modes}).
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\item The positioning of the encoders are made using pockets in both plates as shown in Figure \ref{fig:detail_design_top_plate}.
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\item The encoders are aligned parallel to the struts, but yet they don't exactly measure the relative motion of each strut.
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\item This means that if relative motion of the active platform is performed based on the encoders, the accuracy of the motion may be affected.
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The issue is that the Kinematics may not be correct.
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\end{itemize}
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\begin{figure}[htbp]
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\begin{subfigure}{0.59\textwidth}
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\begin{center}
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\includegraphics[scale=1,height=5cm]{figs/detail_design_enc_plates.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_enc_plates}Nano-Hexapod with encoders fixed to the plates}
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\end{subfigure}
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\begin{subfigure}{0.39\textwidth}
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\begin{center}
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\includegraphics[scale=1,height=5cm]{figs/detail_design_encoders_plates.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_encoders_plates}Zoom on encoder fixation}
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\end{subfigure}
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\caption{\label{fig:detail_design_enc_plates_design}Alternative way of using the encoders: they are fixed directly to the plates.}
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\end{figure}
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\begin{figure}[htbp]
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\begin{subfigure}{0.33\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.5]{figs/detail_design_enc_support_mode_1.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_enc_support_mode_1}$1^{\text{st}}$ mode at $1120\,\text{Hz}$}
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\end{subfigure}
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\begin{subfigure}{0.33\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.5]{figs/detail_design_enc_support_mode_2.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_enc_support_mode_2}$2^{\text{nd}}$ mode at $2020\,\text{Hz}$}
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\end{subfigure}
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\begin{subfigure}{0.33\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.5]{figs/detail_design_enc_support_mode_3.jpg}
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\end{center}
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\subcaption{\label{fig:detail_design_enc_support_mode_3}$3^{\text{rd}}$ mode at $2080\,\text{Hz}$}
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\end{subfigure}
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\caption{\label{fig:detail_design_enc_support_modes}Finite Element Analysis of the encoder supports. Encoder inertia was taken into account.}
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\end{figure}
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\chapter{Multi-Body Model}
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\label{sec:detail_design_model}
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Before all the mechanical parts were ordered, the multi-body model of the active platform was refined using the design parts.
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Two configurations, displayed in Figure \ref{fig:detail_design_simscape_encoder}, were considered:
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\begin{itemize}
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\item Encoders fixed to the struts
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\item Encoders fixed to the plates
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\end{itemize}
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Plates were modelled as rigid bodies, with inertia computed from the 3D shape.
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\begin{figure}[htbp]
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_simscape_encoder_struts.png}
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\end{center}
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\subcaption{\label{fig:detail_design_simscape_encoder_struts}Encoders fixed to the struts}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_simscape_encoder_plates.png}
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\end{center}
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\subcaption{\label{fig:detail_design_simscape_encoder_plates}Encoders fixed to the plates}
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\end{subfigure}
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\caption{\label{fig:detail_design_simscape_encoder}3D representation of the multi-body model. There are two configurations: encoders fixed to the struts (\subref{fig:detail_design_simscape_encoder_struts}) and encoders fixed to the plates (\subref{fig:detail_design_simscape_encoder_plates}).}
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\end{figure}
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\subsubsection{Flexible Joints}
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Different models of the flexible joints where considered:
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\begin{itemize}
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\item 2DoF: only bending stiffnesses
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\item 3DoF: added torsional stiffness
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\item 4DoF: added axial stiffness
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\end{itemize}
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The multi-body model for the 4DoF configuration is shown in Figure \ref{fig:detail_design_simscape_model_flexible_joint}.
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It is composed of three solid bodies connected by joints whose stiffnesses are computed from the finite element model.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,scale=1]{figs/detail_design_simscape_model_flexible_joint.png}
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\caption{\label{fig:detail_design_simscape_model_flexible_joint}Multi-Body (using the Simscape software) model of the flexible joints. A 4-DoFs model is shown.}
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\end{figure}
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\subsubsection{Amplified Piezoelectric Actuators}
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The amplified piezoelectric actuators are modelled as explained in Section [..].
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Two different models can be used in the multi-body model:
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\begin{itemize}
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\item a 2DoF ``axial'' model
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\item a ``super-element'' extracted from the finite element model
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\end{itemize}
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\subsubsection{Encoders}
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Up to now, relative displacement sensors were implemented as a relative distance measurement between \(\bm{a}_i\) and \(\bm{b}_i\).
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As shown in the previous section, flexible modes of the struts may negatively impact the encoder signal.
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It was therefore necessary to better model the encoder.
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The optical encoder works:
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\begin{itemize}
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\item Encoder heads contains a light source shine on the ruler, and a photo-diode.
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This is represented by frame \(\{E\}\) in Figure \ref{fig:detail_design_simscape_encoder}.
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\item ruler or scale with a grating (here with a \(20\,\mu m\) pitch). A reference frame is indicated by \(\{R\}\)
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\end{itemize}
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Therefore, the measured displacement is the relative position of \(\{E\}\) (i.e. there the light ``hits'' the scale) with respect to frame \(\{R\}\), in the direction of the scale.
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In that case, a rotation of the encoder, as shown in figure \ref{fig:detail_design_simscape_encoder_disp} induces a measured displacement.
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\begin{figure}[htbp]
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_design_simscape_encoder.png}
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\end{center}
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\subcaption{\label{fig:detail_design_simscape_encoder}Aligned encoder and ruler}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_design_simscape_encoder_disp.png}
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\end{center}
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\subcaption{\label{fig:detail_design_simscape_encoder_disp}Rotation of the encoder head}
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\end{subfigure}
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\caption{\label{fig:detail_design_simscape_encoder_model}Representation of the encoder model in the multi-body model. Measurement \(d_i\) corresponds to the \(x\) position of the encoder frame \(\{E\}\) expresssed in the ruller frame \(\{R\}\) (\subref{fig:detail_design_simscape_encoder}). A rotation of the encoder therefore induces a measured displacement (\subref{fig:detail_design_simscape_encoder_disp}).}
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\end{figure}
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\subsubsection{Simulation}
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Based on this refined model:
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\begin{itemize}
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\item the active platform could be integrated on top of the micro-station's model.
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\item the obtained dynamics was considered good
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\item simulations of tomography experiments were performed, and similar performance were obtained as during the conceptual design
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\item this is not presented here as results are very similar to the simulations performed in Section [\ldots{}]
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\end{itemize}
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\chapter{Conclusion}
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\label{sec:detail_design_conclusion}
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\printbibliography[heading=bibintoc,title={Bibliography}]
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\end{document}
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