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