Commit before modification for PhD thesis

.gitignore

@@ -1,52 +1,260 @@
-auto/
-*.tex
-*.bbl
-*.synctex.gz
-.auctex-auto/
-_minted*
-
-# Emacs
-auto/
-
-# Simulink Real Time
-*bio.m
-*pt.m
-*ref.m
-*ri.m
-*xcp.m
-*.mldatx
-*.slxc
-*.xml
-*_slrt_rtw/
-
-# data
-data/
-
-# Windows default autosave extension
-*.asv
-
-# OSX / *nix default autosave extension
-*.m~
-
-# Compiled MEX binaries (all platforms)
-*.mex*
-
-# Packaged app and toolbox files
-*.mlappinstall
-*.mltbx
-
-# Generated helpsearch folders
-helpsearch*/
-
-# Simulink code generation folders
+mat/
+figures/
+ltximg/
 slprj/
-sccprj/
+matlab/slprj/
+*.slxc
 
-# Matlab code generation folders
-codegen/
 
-# Simulink autosave extension
-*.autosave
+# ============================================================
+# ============================================================
+# LATEX
+# ============================================================
+# ============================================================
 
-# Octave session info
-octave-workspace
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Comment the next line if you want to keep your tikz graphics files
+*.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+*.ist
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# todonotes
+*.tdo
+
+# easy-todo
+*.lod
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices
+*.xyc
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# KBibTeX
+*~[0-9]*
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta

.latexmkrc

@@ -0,0 +1,111 @@
+#!/bin/env perl
+
+# Shebang is only to get syntax highlighting right across GitLab, GitHub and IDEs.
+# This file is not meant to be run, but read by `latexmk`.
+
+# ======================================================================================
+# Perl `latexmk` configuration file
+# ======================================================================================
+
+# ======================================================================================
+# PDF Generation/Building/Compilation
+# ======================================================================================
+
+@default_files=('test-bench-nass-spindle.tex');
+
+# PDF-generating modes are:
+# 1: pdflatex, as specified by $pdflatex variable (still largely in use)
+# 2: postscript conversion, as specified by the $ps2pdf variable (useless)
+# 3: dvi conversion, as specified by the $dvipdf variable (useless)
+# 4: lualatex, as specified by the $lualatex variable (best)
+# 5: xelatex, as specified by the $xelatex variable (second best)
+$pdf_mode = 1;
+
+# Treat undefined references and citations as well as multiply defined references as
+# ERRORS instead of WARNINGS.
+# This is only checked in the *last* run, since naturally, there are undefined references
+# in initial runs.
+# This setting is potentially annoying when debugging/editing, but highly desirable
+# in the CI pipeline, where such a warning should result in a failed pipeline, since the
+# final document is incomplete/corrupted.
+#
+# However, I could not eradicate all warnings, so that `latexmk` currently fails with
+# this option enabled.
+# Specifically, `microtype` fails together with `fontawesome`/`fontawesome5`, see:
+# https://tex.stackexchange.com/a/547514/120853
+# The fix in that answer did not help.
+# Setting `verbose=silent` to mute `microtype` warnings did not work.
+# Switching between `fontawesome` and `fontawesome5` did not help.
+$warnings_as_errors = 0;
+
+# Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853
+$show_time = 1;
+
+# Default is 5; we seem to need more owed to the complexity of the document.
+# Actual documents probably don't need this many since they won't use all features,
+# plus won't be compiling from cold each time.
+$max_repeat=7;
+
+# --shell-escape option (execution of code outside of latex) is required for the
+#'svg' package.
+# It converts raw SVG files to the PDF+PDF_TEX combo using InkScape.
+#
+# SyncTeX allows to jump between source (code) and output (PDF) in IDEs with support
+# (many have it). A value of `1` is enabled (gzipped), `-1` is enabled but uncompressed,
+# `0` is off.
+# Testing in VSCode w/ LaTeX Workshop only worked for the compressed version.
+# Adjust this as needed. Of course, only relevant for local use, no effect on a remote
+# CI pipeline (except for slower compilation, probably).
+#
+# %O and %S will forward Options and the Source file, respectively, given to latexmk.
+#
+# `set_tex_cmds` applies to all *latex commands (latex, xelatex, lualatex, ...), so
+# no need to specify these each. This allows to simply change `$pdf_mode` to get a
+# different engine. Check if this works with `latexmk --commands`.
+set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S");
+
+# Use default pdf viewer
+$pdf_previewer = 'zathura';
+
+# option 2 is same as 1 (run biber when necessary), but also deletes the
+# regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original
+# bib file is not available!
+$bibtex_use = 2;  # default: 1
+
+# Change default `biber` call, help catch errors faster/clearer. See
+# https://web.archive.org/web/20200526101657/https://www.semipol.de/2018/06/12/latex-best-practices.html#database-entries
+$biber = "biber --validate-datamodel %O %S";
+
+# Glossaries
+add_cus_dep('glo', 'gls', 0, 'run_makeglossaries');
+add_cus_dep('acn', 'acr', 0, 'run_makeglossaries');
+
+sub run_makeglossaries {
+  if ( $silent ) {
+    system "makeglossaries -q -s '$_[0].ist' '$_[0]'";
+  }
+  else {
+    system "makeglossaries -s '$_[0].ist' '$_[0]'";
+  };
+}
+
+# ======================================================================================
+# Auxiliary Files
+# ======================================================================================
+
+# Let latexmk know about generated files, so they can be used to detect if a
+# rerun is required, or be deleted in a cleanup.
+# loe: List of Examples (KOMAScript)
+# lol: List of Listings (`listings` and `minted` packages)
+# run.xml: biber runs
+# glg: glossaries log
+# glstex: generated from glossaries-extra
+push @generated_exts, 'loe', 'lol', 'run.xml', 'glstex', 'glo', 'gls', 'glg', 'acn', 'acr', 'alg';
+
+# Also delete the *.glstex files from package glossaries-extra. Problem is,
+# that that package generates files of the form "basename-digit.glstex" if
+# multiple glossaries are present. Latexmk looks for "basename.glstex" and so
+# does not find those. For that purpose, use wildcard.
+# Also delete files generated by gnuplot/pgfplots contour plots
+# (.dat, .script, .table).
+$clean_ext = "%R-*.glstex %R_contourtmp*.*";

figs/LION_metrology_interferometers.svg

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preamble.tex

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+\usepackage{float}
+
+\usepackage{caption,tabularx,booktabs}
+\usepackage{bm}
+
+\usepackage{xpatch} % Recommanded for biblatex
+\usepackage[        % use biblatex for bibliography
+    backend=biber,  % use biber backend (bibtex replacement) or bibtex
+    style=ieee,     % bib style
+    hyperref=true,  % activate hyperref support
+    backref=true,   % activate backrefs
+    isbn=false,     % don't show isbn tags
+    url=false,      % don't show url tags
+    doi=false,      % don't show doi tags
+    urldate=long,   % display type for dates
+    maxnames=3,     %
+    minnames=1,     %
+    maxbibnames=5,  %
+    minbibnames=3,  %
+    maxcitenames=2, %
+    mincitenames=1  %
+    ]{biblatex}
+
+\setlength\bibitemsep{1.1\itemsep}
+
+% \renewcommand*{\bibfont}{\footnotesize}
+
+\usepackage{fontawesome}
+
+\usepackage{caption}
+\usepackage{subcaption}
+
+\captionsetup[figure]{labelfont=bf}
+\captionsetup[subfigure]{labelfont=bf}
+\captionsetup[listing]{labelfont=bf}
+\captionsetup[table]{labelfont=bf}
+
+\usepackage{xcolor}
+
+\definecolor{my-blue}{HTML}{6b7adb}
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+\definecolor{my-pale-turq}{HTML}{e6f6f9}
+
+\usepackage{inconsolata}
+
+\usepackage[newfloat=true, chapter]{minted}
+\usemintedstyle{autumn}
+
+\setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey}
+\setminted[matlab]{label=Matlab}
+\setminted[latex]{label=LaTeX}
+\setminted[bash]{label=Bash}
+\setminted[python]{label=Python}
+\setminted[text]{label=Results}
+\setminted[md]{label=Org Mode}
+
+\setmintedinline{fontsize=\normalsize,bgcolor=my-pale-grey}
+
+\usepackage[most]{tcolorbox}
+
+\tcbuselibrary{minted}
+
+\newtcolorbox{seealso}{   enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=See Also}
+\newtcolorbox{hint}{      enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint}
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+  borderline west={4pt}{0pt}{my-grey}}
+
+\renewenvironment{verse}{\begin{my-verse}}{\end{my-verse}}
+
+\usepackage{environ}% http://ctan.org/pkg/environ
+\NewEnviron{aside}{%
+  \marginpar{\BODY}
+}
+
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+\usepackage{soul}
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+\makeatletter
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+
+\usepackage{hyperref}
+\hypersetup{
+  colorlinks = true,
+  allcolors = my-blue
+}
+
+\usepackage{hypcap}

test-bench-nass-spindle.tex

@@ -0,0 +1,508 @@
+% Created 2024-03-19 Tue 11:19
+% Intended LaTeX compiler: pdflatex
+\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
+
+\input{preamble.tex}
+\bibliography{test-bench-nass-spindle.bib}
+\author{Dehaeze Thomas}
+\date{\today}
+\title{Nano-Hexapod on top of a Spindle - Test Bench}
+\hypersetup{
+ pdfauthor={Dehaeze Thomas},
+ pdftitle={Nano-Hexapod on top of a Spindle - Test Bench},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 29.2 (Org mode 9.7)}, 
+ pdflang={English}}
+\usepackage{biblatex}
+
+\begin{document}
+
+\maketitle
+\tableofcontents
+
+\clearpage
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/IMG_20221220_152429.jpg}
+\caption{\label{fig:picture_setup}Setup with the Spindle, nano-hexapod and metrology}
+\end{figure}
+\chapter{Test-Bench Description}
+\begin{note}
+Here are the documentation of the equipment used for this test bench:
+\begin{itemize}
+\item Voltage Amplifier: PiezoDrive \href{doc/PD200-V7-R1.pdf}{PD200}
+\item Amplified Piezoelectric Actuator: Cedrat \href{doc/APA300ML.pdf}{APA300ML}
+\item DAC/ADC: Speedgoat \href{doc/IO131-OEM-Datasheet.pdf}{IO131}
+\item Encoder: Renishaw \href{doc/L-9517-9678-05-A\_Data\_sheet\_VIONiC\_series\_en.pdf}{Vionic} and used \href{doc/L-9517-9862-01-C\_Data\_sheet\_RKLC\_EN.pdf}{Ruler}
+\item LION Precision \href{doc/Catalog-CPL190290.pdf}{CPL290}
+\item Spindle: Lab Motion \href{doc/RS250S.pdf}{RT250S} with \href{doc/DB36\_connections\_english\_V2.pdf}{Drivebox 3.6} controller
+\end{itemize}
+\end{note}
+\section{Alignment}
+
+Procedure:
+\begin{enumerate}
+\item Align bottom sphere with the spindle rotation axis (\textasciitilde{} 10um)
+\item Align top sphere with the spindle rotation axis (\textasciitilde{} 10um)
+\end{enumerate}
+\section{Short Range metrology system}
+
+There are 5 interferometers pointing at 2 spheres as shown in Figure \ref{fig:LION_metrology_interferometers}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.5\linewidth]{figs/IMG_20221216_181305.jpg}
+\caption{\label{fig:picture_metrology}Metrology system with LION sphere (1 inch diameter) and 5 interferometers fixed to their individual tip-tilts}
+\end{figure}
+
+\begin{center}
+\begin{tabular}{ll}
+ & Value\\
+\hline
+Sphere Diameter & 25.4mm\\
+Distance between the spheres & 76.2mm\\
+\end{tabular}
+\end{center}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.5\linewidth]{figs/LION_metrology_interferometers.png}
+\caption{\label{fig:LION_metrology_interferometers}Schematic of the measurement system}
+\end{figure}
+
+\textbf{Assumptions}:
+\begin{itemize}
+\item Interferometers are perfectly positioned / oriented
+\item Sphere is perfect
+\end{itemize}
+
+\textbf{Compute the Jacobian matrix}:
+\begin{itemize}
+\item From pure X-Y-Z-Rx-Ry small motions, compute the effect on the 5 measured distances
+\item Compute the matrix
+\item Inverse the matrix
+\item Verify that it is working with simple example (for example using Solidworkds)
+\end{itemize}
+
+We have the following set of equations:
+\begin{align}
+d_1 &= -D_y + l_2 R_x \\
+d_2 &= -D_y - l_1 R_x \\
+d_3 &= -D_x - l_2 R_y \\
+d_4 &= -D_x + l_1 R_y \\
+d_5 &= -D_z
+\end{align}
+
+That can be written as a linear transformation:
+\begin{equation}
+\begin{bmatrix}
+  d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5
+\end{bmatrix} = \begin{bmatrix}
+   0 & -1 &  0 &  l_2 &  0   \\
+   0 & -1 &  0 & -l_1 &  0   \\
+  -1 &  0 &  0 &  0   & -l_2 \\
+  -1 &  0 &  0 &  0   &  l_1 \\
+   0 &  0 & -1 &  0   &  0
+\end{bmatrix} \cdot \begin{bmatrix}
+  D_x \\ D_y \\ D_z \\ R_x \\ R_y
+\end{bmatrix}
+\end{equation}
+
+By inverting the matrix, we obtain the Jacobian relation:
+\begin{equation}
+\begin{bmatrix}
+  D_x \\ D_y \\ D_z \\ R_x \\ R_y
+\end{bmatrix} = \begin{bmatrix}
+   0 & -1 &  0 &  l_2 &  0   \\
+   0 & -1 &  0 & -l_1 &  0   \\
+  -1 &  0 &  0 &  0   & -l_2 \\
+  -1 &  0 &  0 &  0   &  l_1 \\
+   0 &  0 & -1 &  0   &  0
+\end{bmatrix}^{-1} \cdot \begin{bmatrix}
+  d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5
+\end{bmatrix}
+\end{equation}
+
+\begin{table}[htbp]
+\caption{\label{tab:jacobian_metrology}Jacobian matrix for the metrology system}
+\centering
+\begin{tabularx}{\linewidth}{cXXXXX}
+\toprule
+ & \textbf{d1} & \textbf{d2} & \textbf{d3} & \textbf{d4} & \textbf{d5}\\
+\midrule
+\textbf{Dx} & 0.0 & 0.0 & -0.79 & -0.21 & 0.0\\
+\textbf{Dy} & -0.79 & -0.21 & -0.0 & -0.0 & 0.0\\
+\textbf{Dz} & 0.0 & 0.0 & 0.0 & 0.0 & -1.0\\
+\textbf{Rx} & 13.12 & -13.12 & 0.0 & -0.0 & 0.0\\
+\textbf{Ry} & 0.0 & 0.0 & -13.12 & 13.12 & 0.0\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Spindle errors}
+The spindle is rotated at 60rpm during 10 turns.
+The signal of all 5 interferometers are recorded.
+\subsection{Errors in \(D_x\) and \(D_y\)}
+
+Because of the eccentricity of the reference surfaces (the spheres), we expect the motion in the X-Y plane to be a circle as a first approximation.
+We can first see that in Figure \ref{fig:dx_dy_motion_rotation} that shows the measured \(D_x\) and \(D_y\) motion as a function of the \(R_z\) angle.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dx_dy_motion_rotation.png}
+\caption{\label{fig:dx_dy_motion_rotation}Dx and Dy motion during the rotation}
+\end{figure}
+
+A circle is fit, and the obtained radius of the circle (i.e. the excentricity) is estimated to be:
+
+\begin{verbatim}
+Error linked to excentricity = 19 um
+\end{verbatim}
+
+
+The motion in the X-Y plane as well as the circle fit and the residual motion (circle fit subtracted from the measured motion) are shown in Figure \ref{fig:dx_dy_spindle_rotation}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dx_dy_spindle_rotation.png}
+\caption{\label{fig:dx_dy_spindle_rotation}Dx and Dy motion during the spindle rotation}
+\end{figure}
+
+Let's now analyse the frequency content in the signal.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dx_dy_spindle_rotation_asd.png}
+\caption{\label{fig:dx_dy_spindle_rotation_asd}Amplitude Spectral Density of the measured Dx and Dy motion}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dx_dy_spindle_rotation_cas.png}
+\caption{\label{fig:dx_dy_spindle_rotation_cas}Cumulative Amplitude Spectrum of the measured Dx and Dy motion}
+\end{figure}
+\subsection{Errors in vertical motion \(D_z\)}
+
+The top interferometer is measuring the vertical motion of the sphere.
+
+However, if the top sphere is not perfectly aligned with the spindle axis, there will also measure some vertical motion due to this excentricity.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dz_motion_rotation.png}
+\caption{\label{fig:dz_motion_rotation}Dz motion during the rotation}
+\end{figure}
+
+Let's fit a sinus with a period of one turn.
+
+\begin{verbatim}
+Errors linked to excentricity = 410 [nm]
+\end{verbatim}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dz_motion_rotation_excentricity.png}
+\caption{\label{fig:dz_motion_rotation_excentricity}Effect of the excentricity and remaining Dz motion}
+\end{figure}
+
+If we look at the remaining motion after removing the effect of the eccentricity (Figure \ref{fig:dz_motion_rotation_excentricity}, right), we can see a signal with 20 periods every turn.
+Let's fit this.
+
+\begin{verbatim}
+Errors linked to spindle motor = 58 [nm]
+\end{verbatim}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dz_motion_rotation_poles.png}
+\caption{\label{fig:dz_motion_rotation_poles}Effect of the magnetic pole pairs and remaining Dz motion}
+\end{figure}
+
+Let's look at the signal in the frequency domain.
+
+On top of the peak at 1Hz (excentricity) and at 20Hz (number of pole pairs), we can observe a frequency of 126Hz (i.e. 126 periods per turn, approx 2.85 deg).
+
+\begin{quote}
+Could this be related to the air bearing system?
+\end{quote}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dz_spindle_rotation_asd.png}
+\caption{\label{fig:dz_spindle_rotation_asd}Amplitude Spectral Density of the measured Dz motion}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/dz_spindle_rotation_cas.png}
+\caption{\label{fig:dz_spindle_rotation_cas}Cumulative Amplitude Spectrum of the measured Dz motion}
+\end{figure}
+\subsection{Angle errors in \(R_x\) and \(R_y\)}
+
+\begin{verbatim}
+amplitude = 281 urad
+\end{verbatim}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/rx_ry_spindle_rotation.png}
+\caption{\label{fig:rx_ry_spindle_rotation}Rx and Ry motion during the spindle rotation}
+\end{figure}
+
+Let's now analyse the frequency content in the signal.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/rx_ry_spindle_rotation_asd.png}
+\caption{\label{fig:rx_ry_spindle_rotation_asd}Amplitude Spectral Density of the measured Rx and Ry motion}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/rx_ry_spindle_rotation_cas.png}
+\caption{\label{fig:rx_ry_spindle_rotation_cas}Cumulative Amplitude Spectrum of the measured Rx and Ry motion}
+\end{figure}
+\chapter{Simscape Model}
+A 3D view of the Simscape model is shown in Figure \ref{fig:simscape_model_spindle_bench}.
+The Spindle is represented by a \emph{Bushing joint}.
+Axial, radial and tilt stiffnesses are taken from the Spindle datasheet (see Table).
+
+\begin{table}[htbp]
+\caption{\label{tab:spindle_stiffnesses}Spindle stiffnesses}
+\centering
+\begin{tabularx}{\linewidth}{lXX}
+\toprule
+\textbf{Stiffness} & \textbf{Value} & \textbf{Unit}\\
+\midrule
+Axial & 402 & \(N/\mu m\)\\
+Radial & 226 & \(N/\mu m\)\\
+Tilt & 2380 & \(Nm/mrad\)\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+The metrology system consists of 5 distance measurements (represented by the red lines in Figure \ref{fig:simscape_model_spindle_bench}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.5\linewidth]{figs/simscape_model_spindle_bench.jpg}
+\caption{\label{fig:simscape_model_spindle_bench}Screenshot of the 3D view of the Simscape model}
+\end{figure}
+\section{Simscape model parameters}
+The nano-hexapod is initialized.
+
+The Jacobian matrix that computes the \([x, y, z, R_x, R_y]\) motion of the sample from the 5 interferometers is defined below.
+\section{Control Architecture}
+
+Let's note:
+\begin{itemize}
+\item \(d\mathcal{L}_m = [d_{\mathcal{L}_1},\ d_{\mathcal{L}_2},\ d_{\mathcal{L}_3},\ d_{\mathcal{L}_4},\ d_{\mathcal{L}_5},\ d_{\mathcal{L}_6}]\) the measurement of the 6 encoders fixed to the nano-hexapod
+\item \(\bm{\tau}_m = [\tau_{m_1},\ \tau_{m_2},\ \tau_{m_3},\ \tau_{m_4},\ \tau_{m_5},\ \tau_{m_6}]\) the voltages measured by the 6 force sensors
+\item \(\bm{u} = [u_1,\ u_2,\ u_3,\ u_4,\ u_5,\ u_6]\) the voltages send to the voltage amplifiers for the 6 piezoelectric actuators
+\item \(R_z\) the spindle measured angle (encoder)
+\item \(\bm{d}_m = [d_1,\ d_2,\ d_3,\ d_4,\ d_5]\) the distances measured by the 5 interferometers (see Figure \ref{fig:LION_metrology_interferometers_bis})
+\end{itemize}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.5\linewidth]{figs/LION_metrology_interferometers.png}
+\caption{\label{fig:LION_metrology_interferometers_bis}Schematic of the measurement system}
+\end{figure}
+\section{Computation of the strut errors from the external metrology}
+
+The following frames are defined:
+\begin{itemize}
+\item \(\{ W \}\): the frame that represents the wanted pose of the sample
+\item \(\{ M \}\): the frame that represents the measured pose of the sample (estimated from the 5 interferometers and the spindle encoder)
+\item \(\{ G \}\): the frame fixed to the granite and positioned at the sample's center
+\item \(\{ H \}\): the frame fixed to the the spindle rotor, and positioned at the sample's center
+\end{itemize}
+
+We can express several homogeneous transformation matrices.
+
+Frame fixed to the spindle rotor (centered on the sample's position), expressed in the frame of the granite:
+\begin{equation}
+{}^{G}\bm{T}_H = \begin{bmatrix}
+cos(R_z) & -sin(R_z) & 0 & 0 \\
+sin(R_z) &  cos(R_z) & 0 & 0 \\
+0        &  0        & 1 & 0 \\
+0        &  0        & 0 & 1
+\end{bmatrix}
+\end{equation}
+with \(R_z\) the spindle encoder.
+
+
+Wanted position expressed in the frame of the granite:
+\begin{equation}
+{}^{G}\bm{T}_W = \begin{bmatrix}
+ &                                 & & r_{D_x} \\
+ & \bm{R}_x(r_{R_x}) \bm{R}_y(r_{R_y}) \bm{R}_z(r_{R_z}) & & r_{D_y} \\
+ &                                 & & r_{D_z} \\
+0        &  0                      & 0 & 1
+\end{bmatrix}
+\end{equation}
+with \(\bm{R}(r_{R_x}, r_{R_y}, r_{R_z})\) representing the wanted orientation of the sample with respect to the granite.
+Typically, \(r_{R_x} = 0\), \(r_{R_y} = 0\) and \(r_{R_z}\) corresponds to the spindle encoder \(R_z\).
+
+
+Measured position of the sample with respect to the granite:
+\begin{equation}
+{}^{G}\bm{T}_M = \begin{bmatrix}
+ &                                 & & y_{D_x} \\
+ & \bm{R}_x(y_{R_x}) \bm{R}_y(y_{R_y}) \bm{R}_z(R_z) & & y_{D_y} \\
+ &                                 & & y_{D_z} \\
+0        &  0                      & 0 & 1
+\end{bmatrix}
+\end{equation}
+with \(R_z\) the spindle encoder, and \([y_{D_x},\ y_{D_y},\ y_{D_z},\ y_{R_x},\ y_{R_y}]\) are obtained from the 5 interferometers:
+\begin{equation}
+\begin{bmatrix}
+  y_{D_x} \\ y_{D_y} \\ y_{D_z} \\ y_{R_x} \\ y_{R_y}
+\end{bmatrix} = \begin{bmatrix}
+   0 & -1 &  0 &  l_2 &  0   \\
+   0 & -1 &  0 & -l_1 &  0   \\
+  -1 &  0 &  0 &  0   & -l_2 \\
+  -1 &  0 &  0 &  0   &  l_1 \\
+   0 &  0 & -1 &  0   &  0
+\end{bmatrix}^{-1} \cdot \begin{bmatrix}
+  d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5
+\end{bmatrix}
+\end{equation}
+
+
+In order to have the \textbf{position error in the frame of the nano-hexapod}, we have to compute \({}^M\bm{T}_W\):
+\begin{align}
+{}^M\bm{T}_W &= {}^M\bm{T}_G \cdot {}^G\bm{T}_W \\
+             &= {{}^G\bm{T}_M}^{-1} \cdot {}^G\bm{T}_W
+\end{align}
+
+The \textbf{inverse of the transformation matrix} can be obtained by
+\begin{equation}
+  {}^B\bm{T}_A = {}^A\bm{T}_B^{-1} =
+  \left[ \begin{array}{ccc|c}
+      &   &   & \\
+      & {}^A\bm{R}_B^T  &   & -{}^A \bm{R}_B^T {}^A\bm{P}_{O_B} \\
+      &   &   & \cr
+    \hline
+    0 & 0 & 0 & 1 \\
+  \end{array} \right]
+\end{equation}
+
+The position errors \(\bm{\epsilon}_{\mathcal{X}} = [\epsilon_{D_x},\ \epsilon_{D_y},\ \epsilon_{D_z},\ \epsilon_{R_x},\ \epsilon_{R_y},\ \epsilon_{R_z}]\) expressed in a frame fixed to the nano-hexapod can be extracted from \({}^W\bm{T}_M\):
+\begin{itemize}
+\item \(\epsilon_{D_x} = {}^M\bm{T}_W(1,4)\)
+\item \(\epsilon_{D_y} = {}^M\bm{T}_W(2,4)\)
+\item \(\epsilon_{D_z} = {}^M\bm{T}_W(3,4)\)
+\item \(\epsilon_{R_y} = \text{atan2}({}^M\bm{T}_W(1,3), \sqrt{{}^M\bm{T}_W(1,1)^2 + {}^M\bm{T}_W(1,2)^2})\)
+\item \(\epsilon_{R_x} = \text{atan2}(\frac{-{}^M\bm{T}_W(2,3)}{\cos(\epsilon_{R_y})}, \frac{{}^M\bm{T}_W(3,3)}{\cos(\epsilon_{R_y})})\)
+\item \(\epsilon_{R_z} = \text{atan2}(\frac{-{}^M\bm{T}_W(1,2)}{\cos(\epsilon_{R_y})}, \frac{{}^M\bm{T}_W(1,1)}{\cos(\epsilon_{R_y})})\)
+\end{itemize}
+
+Finally, the strut errors \(\bm{\epsilon}_{\mathcal{L}} = [\epsilon_{\matcal{L}_1},\ \epsilon_{\matcal{L}_2},\ \epsilon_{\matcal{L}_3},\ \epsilon_{\matcal{L}_4},\ \epsilon_{\matcal{L}_5},\ \epsilon_{\matcal{L}_6}]\) can be computed from:
+\begin{equation}
+\bm{\epsilon}_\mathcal{L} = \bm{J} \cdot \bm{\epsilon}_\mathcal{X}
+\end{equation}
+\section{IFF Plant}
+\section{DVF Plant}
+\section{HAC Plant}
+The transfer functions from the 6 actuator inputs to the 6 estimated strut errors are extracted from the Simscape model.
+
+The obtained transfer functions are shown in Figure \ref{fig:simscape_model_hac_plant}.
+
+We can see that the system is well decoupled at low frequency (i.e. below the first resonance of the Nano-Hexapod).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/simscape_model_hac_plant.png}
+\caption{\label{fig:simscape_model_hac_plant}HAC plant obtained on the Simscape model}
+\end{figure}
+\chapter{Control Experiment}
+\section{IFF Plant}
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/G_iif_exp_no_rotation.png}
+\caption{\label{fig:G_iif_exp_no_rotation}Obtained transfer function from generated voltages to measured voltages on the piezoelectric force sensor}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/G_iif_exp_comp_no_rotation.png}
+\caption{\label{fig:G_iif_exp_comp_no_rotation}Comparison with the model}
+\end{figure}
+\section{IFF Controller}
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/root_locus_iff_no_payload.png}
+\caption{\label{fig:root_locus_iff_no_payload}Root Locus for IFF}
+\end{figure}
+\section{Open Loop Plant}
+Here the \(R_z\) motion of the Hexapod is estimated from the encoders.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/G_damp_exp_no_rotation.png}
+\caption{\label{fig:G_damp_exp_no_rotation}Obtained transfer function from generated voltages to estimated strut motion}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_hac_plant_exp_simscape.png}
+\caption{\label{fig:comp_hac_plant_exp_simscape}Comparison of the open-loop plant measured experimentally and extracted from Simscape}
+\end{figure}
+\section{Damped Plant}
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/G_damp_damped_exp_no_rotation.png}
+\caption{\label{fig:G_damp_damped_exp_no_rotation}Obtained transfer function from generated voltages to estimated strut motion}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_damped_undamped_plant.png}
+\caption{\label{fig:comp_damped_undamped_plant}Comparison of the undamped and damped plant with IFF}
+\end{figure}
+\section{HAC Controller}
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/first_hac_K_exp_loop_gain.png}
+\caption{\label{fig:first_hac_K_exp_loop_gain}Loop gain for the HAC}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/first_hac_K_exp_root_locus.png}
+\caption{\label{fig:first_hac_K_exp_root_locus}Obtained Root Locus}
+\end{figure}
+\section{Compare dynamics seen by interferometers and by encoders}
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_dynamics_int_ext_metrology.png}
+\caption{\label{fig:comp_dynamics_int_ext_metrology}Comparison of the identified dynamic by the internal metrology (encoders) and by the external metrology (interferometers)}
+\end{figure}
+\section{Compare dynamics obtained with different Rz estimations}
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_plant_encoders_Va.png}
+\caption{\label{fig:comp_plant_encoders_Va}Comparison of the obtained plant using the Encoders or using the output Voltages to estimate Rz}
+\end{figure}
+\chapter{Closed-Loop Results}
+\section{Open and Closed loop results}
+\begin{center}
+\includegraphics[scale=1]{figs/spindle_errors_1rpm_ol.png}
+\end{center}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/spindle_errors_1rpm_op_cl.png}
+\caption{\label{fig:spindle_errors_1rpm_op_cl}Comparison of the Open-Loop and Closed-Loop spindle errors}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/spindle_errors_1rpm_op_cl_rot.png}
+\caption{\label{fig:spindle_errors_1rpm_op_cl_rot}Comparison of the Open-Loop and Closed-Loop spindle errors - Rotation}
+\end{figure}
+\end{document}