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\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}

\input{preamble.tex}
\bibliography{test-bench-struts.bib}
\author{Dehaeze Thomas}
\date{\today}
\title{Test Bench - Nano-Hexapod Struts}
\hypersetup{
 pdfauthor={Dehaeze Thomas},
 pdftitle={Test Bench - Nano-Hexapod Struts},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 29.3 (Org mode 9.7)}, 
 pdflang={English}}
\usepackage{biblatex}

\begin{document}

\maketitle
\tableofcontents

\clearpage
In this document, a test-bench is used to characterize the struts of the nano-hexapod.

Each strut includes (Figure \ref{fig:test_struts_picture_strut}):
\begin{itemize}
\item 2 flexible joints at each ends.
These flexible joints have been characterized in a separate test bench (see \ldots{}).
\item 1 Amplified Piezoelectric Actuator (APA300ML) (described in Section \ldots{}).
Two stacks are used as an actuator and one stack as a (force) sensor.
\item 1 encoder (Renishaw Vionic) that has been characterized in a separate test bench (see \ldots{}).
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_struts_picture_strut.jpg}
\caption{\label{fig:test_struts_picture_strut}One strut including two flexible joints, an amplified piezoelectric actuator and an encoder}
\end{figure}

Then the struts are mounted (procedure described in Section \ref{sec:test_struts_mounting}), and are fixed to the same measurement bench.
The goals are to:
\begin{itemize}
\item Section \ref{sec:test_struts_dynamical_meas}: Identify the dynamics from the generated DAC voltage to:
\begin{itemize}
\item the sensors stack generated voltage
\item the measured displacement by the encoder
\item the measured displacement by the interferometer (representing encoders that would be fixed to the nano-hexapod's plates instead of the struts)
\end{itemize}
\item Section \ref{sec:test_struts_simscape}: Compare the measurements with the Simscape model of the struts and tune the models
\end{itemize}

The final goal of the work presented in this document is to have an accurate Simscape model of the struts that can then be included in the Simscape model of the nano-hexapod.

\begin{table}[htbp]
\caption{\label{tab:test_struts_section_matlab_code}Report sections and corresponding Matlab files}
\centering
\begin{tabularx}{0.6\linewidth}{lX}
\toprule
\textbf{Sections} & \textbf{Matlab File}\\
\midrule
Section \ref{sec:test_struts_flexible_modes} & \texttt{test\_struts\_1\_flexible\_modes.m}\\
Section \ref{sec:test_struts_dynamical_meas} & \texttt{test\_struts\_2\_dynamical\_meas.m}\\
Section \ref{sec:test_struts_simscape} & \texttt{test\_struts\_3\_simscape\_model.m}\\
\bottomrule
\end{tabularx}
\end{table}
\chapter{Mounting Procedure}
\label{sec:test_struts_mounting}
\section{Mounting Bench}

A mounting bench is used to greatly simply the mounting of the struts as well as ensuring the correct strut length and coaxiality of the flexible joint's interfaces.
This is very important in order to not loose any stroke when the struts will be mounted on the nano-hexapod.

A CAD view of the mounting bench is shown in Figure \ref{fig:test_struts_mounting_bench_first_concept}.

Faro arm\footnote{Faro Arm Platinum 4ft, accuracy of \(\pm 13\mu m\)}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.6\linewidth]{figs/test_struts_mounting_bench_first_concept.png}
\caption{\label{fig:test_struts_mounting_bench_first_concept}CAD view of the mounting bench}
\end{figure}

The main part of the bench is here to ensure both the correct strut length and strut coaxiality as shown in Figure \ref{fig:test_struts_mounting_step_0}.

\begin{figure}[htbp]
\begin{subfigure}{0.56\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/test_struts_mounting_step_0.jpg}
\end{center}
\subcaption{\label{fig:test_struts_mounting_step_0}Useful features of the main mounting element}
\end{subfigure}
\begin{subfigure}{0.43\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/test_struts_check_dimensions_bench.jpg}
\end{center}
\subcaption{\label{fig:test_struts_check_dimensions_bench}Dimensional check}
\end{subfigure}
\caption{\label{fig:test_struts_mounting_base_part}Caption\ldots{}, add foot note with Faro arm}
\end{figure}

The tight tolerances of this element has been verified as shown in Figure \ref{fig:test_struts_check_dimensions_bench} and were found to comply with the requirements.

The flexible joints are rigidly fixed to cylindrical tools shown in Figures \ref{fig:test_struts_cylindrical_mounting_part_top} and \ref{fig:test_struts_cylindrical_mounting_part_bot} which are then mounted on the mounting tool shown in Figure \ref{fig:test_struts_mounting_step_0}.
This cylindrical tool is here to protect the flexible joints when tightening the screws and therefore applying large torque.
\section{Mounting Procedure}

The mounting procedure is as follows:
\begin{enumerate}
\item Screw flexible joints inside the cylindrical interface element shown in Figure \ref{fig:test_struts_cylindrical_mounting}
\item Fix the two interface elements. One of the two should be clamped, the other one should have its axial rotation free.
Visually align the clamped one horizontally. (Figure \ref{fig:test_struts_mounting_step_1})
\item Put cylindrical washers, APA and interface pieces on top of the flexible joints (Figure \ref{fig:test_struts_mounting_step_2})
\item Put the 4 screws just in contact such that everything is correctly positioned and such that the ``free'' flexible joint is correctly oriented
\item Put the 8 lateral screws in contact
\item Tighten the 4 screws to fix the APA on the two flexible joints (using a torque screwdriver)
\item Remove the 4 laterals screws
\item (optional) Put the APA horizontally and fix the encoder and align it to maximize the contrast (Figure \ref{fig:test_struts_mounting_step_3})
\item Disassemble to have an properly mounted strut (Figure \ref{fig:test_struts_mounting_step_4}) for which the coaxiality between the two flexible joint's interfaces is good
\end{enumerate}

\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/test_struts_cylindrical_mounting_part_top.jpg}
\end{center}
\subcaption{\label{fig:test_struts_cylindrical_mounting_part_top}Cylindral Interface (Top)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/test_struts_cylindrical_mounting_part_bot.jpg}
\end{center}
\subcaption{\label{fig:test_struts_cylindrical_mounting_part_bot}Cylindrlcal Interface (Bottom)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/test_struts_mounting_joints.jpg}
\end{center}
\subcaption{\label{fig:test_struts_mounting_joints}Mounted flexible joints}
\end{subfigure}
\caption{\label{fig:test_struts_cylindrical_mounting}Preparation of the flexible joints by fixing them in their cylindrical interface}
\end{figure}

\begin{figure}[htbp]
\begin{subfigure}{0.5\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_1.jpg}
\end{center}
\subcaption{\label{fig:test_struts_mounting_step_1}Step 1}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_2.jpg}
\end{center}
\subcaption{\label{fig:test_struts_mounting_step_2}Step 2}
\end{subfigure}

\bigskip
\begin{subfigure}{0.5\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_3.jpg}
\end{center}
\subcaption{\label{fig:test_struts_mounting_step_3}Step 3}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_4.jpg}
\end{center}
\subcaption{\label{fig:test_struts_mounting_step_4}Step 4}
\end{subfigure}
\caption{\label{fig:test_struts_mounting_steps}Steps for mounting the struts.}
\end{figure}
\chapter{Measurement of flexible modes}
\label{sec:test_struts_flexible_modes}
\section{Introduction}

From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure \ref{fig:test_struts_mode_shapes}): an ``X-bending'' mode at 189Hz, a ``Y-bending'' mode at 285Hz and a ``Z-torsion'' mode at 400Hz.

\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_mode_shapes_1.png}
\end{center}
\subcaption{\label{fig:test_struts_mode_shapes_3}X-bending mode (189Hz)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_mode_shapes_2.png}
\end{center}
\subcaption{\label{fig:test_struts_mode_shapes_3}Y-bending mode (285Hz)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_mode_shapes_3.png}
\end{center}
\subcaption{\label{fig:test_struts_mode_shapes_3}Z-torsion mode (400Hz)}
\end{subfigure}
\caption{\label{fig:test_struts_mode_shapes}Spurious resonances of the struts estimated from a Finite Element Model}
\end{figure}
\section{Measurement Setup}

A Laser vibrometer is measuring the difference of motion between two beam path (red points in Figure \ref{fig:test_struts_meas_modes}).
The strut is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.

The ``X-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_x_bending}.
The ``Y-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_y_bending}.
Finally, the ``Z-torsion'' is measured as shown in Figure \ref{fig:test_struts_meas_z_torsion}.

This is done with and without the encoder fixed to the strut.

\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_meas_x_bending.jpg}
\end{center}
\subcaption{\label{fig:test_struts_meas_x_bending}X-bending mode}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_meas_y_bending.jpg}
\end{center}
\subcaption{\label{fig:test_struts_meas_y_bending}Y-bending mode}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_meas_z_torsion.jpg}
\end{center}
\subcaption{\label{fig:test_struts_meas_z_torsion}Z-torsion mode}
\end{subfigure}
\caption{\label{fig:test_struts_meas_modes}Measurement of strut flexible modes}
\end{figure}
\section{Measured results}
The obtained frequency response functions are shown in Figure \ref{fig:test_struts_spur_res_frf}.

\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/test_struts_spur_res_frf_no_enc.png}
\end{center}
\subcaption{\label{fig:test_struts_spur_res_frf_no_enc}without encoder}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/test_struts_spur_res_frf_enc.png}
\end{center}
\subcaption{\label{fig:test_struts_spur_res_frf_enc}with the encoder}
\end{subfigure}
\caption{\label{fig:test_struts_spur_res_frf}Measured frequency response functions without the encoder \ref{fig:test_struts_spur_res_frf} and with the encoder \ref{fig:test_struts_spur_res_frf_enc}}
\end{figure}
Table \ref{tab:test_struts_spur_mode_freqs} summarizes the measured resonance frequencies as well as the computed ones using the Finite Element Model.
It is shown that:
\begin{itemize}
\item the resonance frequencies of the 3 modes are only slightly increasing when the encoder is removed
\item the computed resonance frequencies from the FEM are very close to the measured one when the encoder is fixed to the strut
\end{itemize}

\begin{table}[htbp]
\caption{\label{tab:test_struts_spur_mode_freqs}Measured frequency of the strut spurious modes}
\centering
\begin{tabularx}{0.7\linewidth}{Xccc}
\toprule
\textbf{Mode} & \textbf{Struts (FEM)} & \textbf{Struts (exp)} & \textbf{Plates (exp)}\\
\midrule
X-Bending & 189Hz & 198Hz & 226Hz\\
Y-Bending & 285Hz & 293Hz & 337Hz\\
Z-Torsion & 400Hz & 381Hz & 398Hz\\
\bottomrule
\end{tabularx}
\end{table}
\chapter{Dynamical measurements}
\label{sec:test_struts_dynamical_meas}
The bench is shown in Figure \ref{fig:test_struts_bench_leg_overview}.
Measurements are performed either when no encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_front}) or when one encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_coder}).

\begin{figure}[htbp]
\begin{subfigure}{0.35\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_bench_leg_overview.jpg}
\end{center}
\subcaption{\label{fig:test_struts_bench_leg_overview}Overview}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_bench_leg_front.jpg}
\end{center}
\subcaption{\label{fig:test_struts_bench_leg_front}Strut without encoder}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_bench_leg_coder.jpg}
\end{center}
\subcaption{\label{fig:test_struts_bench_leg_coder}Strut with encoder}
\end{subfigure}
\caption{\label{fig:test_struts_bench_leg}Experimental setup to measured the dynamics of the struts.}
\end{figure}

First, the effect of the encoder on the measured dynamics is studied in Section \ref{ssec:test_struts_effect_encoder}.
Then, the dynamics seen by the encoder and by the interferometers are compared in Section \ref{ssec:test_struts_comp_enc_int}.
Finally, all the measured struts are compared in terms of dynamics in Section \ref{ssec:test_struts_comp_all_struts}.
\section{Effect of the Encoder on the measured dynamics}
\label{ssec:test_struts_effect_encoder}

Figure \ref{fig:test_struts_effect_encoder_int}
Same goes for the transfer function from excitation voltage \(u\) to the axial motion of the strut \(d_a\) as measured by the interferometer ().

The transfer function from the excitation voltage \(u\) to the generated voltage \(V_s\) by the sensor stack is not influence by the fixation of the encoder (Figure \ref{fig:test_struts_effect_encoder_iff}).
This means that the IFF control strategy should be as effective whether or not the encoders are fixed to the struts.

\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_encoder_int.png}
\end{center}
\subcaption{\label{fig:test_struts_effect_encoder_int}$u$ to $d_a$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_encoder_iff.png}
\end{center}
\subcaption{\label{fig:test_struts_effect_encoder_iff}$u$ to $V_s$}
\end{subfigure}
\caption{\label{fig:test_struts_effect_encoder}Effect of having the encoder fixed to the struts on the measured dynamics from \(u\) to \(d_a\) (\subref{fig:test_struts_effect_encoder_int}) and from \(u\) to \(V_s\) \cref{fig:test_struts_effect_encoder_iff}}
\end{figure}
\section{Comparison of the encoder and interferometer}
\label{ssec:test_struts_comp_enc_int}

The dynamics as measured by the encoder and by the interferometers are compared in Figure \ref{fig:test_struts_comp_enc_int}.

The dynamics from the excitation voltage \(u\) to the measured displacement by the encoder \(d_e\) presents much more complicated behavior than the transfer function to the displacement as measured by the Interferometer (compared in Figure \ref{fig:test_struts_comp_enc_int}).
It will be further investigated why the two dynamics as so different and what are causing all these resonances.

As shown in Figure \ref{fig:test_struts_comp_enc_int}, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.
These resonances correspond to parasitic resonances of the strut itself that was estimated using a finite element model of the strut (Figure \ref{fig:test_struts_mode_shapes}):
\begin{itemize}
\item Mode in X-bending at 189Hz
\item Mode in Y-bending at 285Hz
\item Mode in Z-torsion at 400Hz
\end{itemize}

The good news is that these resonances are not seen on the interferometer (they are therefore not impacting the axial motion of the strut).
But these resonances are making the use of encoder fixed to the strut difficult.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_struts_comp_enc_int.png}
\caption{\label{fig:test_struts_comp_enc_int}Comparison of the transfer functions from excitation voltage \(u\) to either the encoder \(d_e\) or the interferometer \(d_a\)}
\end{figure}
\section{Comparison of all the Struts}
\label{ssec:test_struts_comp_all_struts}

Then, the transfer function from the DAC output voltage \(u\) to the measured displacement by the Attocube is computed for all the struts and shown in Figure \ref{fig:test_struts_comp_interf_plants}.
All the struts are giving very similar FRF.

\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/test_struts_comp_interf_plants.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/test_struts_comp_iff_plants.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$}
\end{subfigure}
\caption{\label{fig:test_struts_comp_plants}Comparison of the measured plants}
\end{figure}

There is a very large variability of the dynamics as measured by the encoder as shown in Figure \ref{fig:test_struts_comp_enc_plants}.
Even-though the same peaks are seen for all of the struts (95Hz, 200Hz, 300Hz, 400Hz), the amplitude of the peaks are not the same.
Moreover, the location or even the presence of complex conjugate zeros is changing from one strut to the other.

All of this will be studied in Section \ref{sec:test_struts_simscape} using the Simscape model.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_struts_comp_enc_plants.png}
\caption{\label{fig:test_struts_comp_enc_plants}Estimated FRF for the DVF plant (transfer function from \(u\) to the encoder \(d_e\))}
\end{figure}
\begin{important}
All the struts are giving very consistent behavior from the excitation voltage \(u\) to the force sensor generated voltage \(V_s\) and to the interferometer measured displacement \(d_a\).
However, the dynamics from \(u\) to the encoder measurement \(d_e\) is much more complex and variable from one strut to the other most likely due to poor alignment of the APA with respect to the flexible joints.
\end{important}
\chapter{Simscape Model}
\label{sec:test_struts_simscape}
However, now the full strut is put instead of only the APA (see Figure \ref{fig:test_struts_simscape_model}).

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/test_struts_simscape_model.png}
\caption{\label{fig:test_struts_simscape_model}Screenshot of the Simscape model of the strut fixed to the bench}
\end{figure}

This Simscape model is used to:
\begin{itemize}
\item compare the measured FRF with the modelled FRF
\item help the correct understanding/interpretation of the results
\item tune the model of the struts (APA, flexible joints, encoder)
\end{itemize}

This study is structured as follow:
\begin{itemize}
\item Section \ref{ssec:test_struts_comp_model}: the measured FRF are compared with the Simscape model.
\item Section \ref{ssec:test_struts_effect_misalignment}: the flexible APA model is used, and the effect of a misalignment of the APA and flexible joints is studied.
It is found that the misalignment has a large impact on the dynamics from \(u\) to \(d_e\).
\item Section \ref{ssec:test_struts_effect_joint_stiffness}: the effect of the flexible joint's stiffness on the dynamics is studied.
It is found that the axial stiffness of the joints has a large impact on the location of the zeros on the transfer function from \(V_s\) to \(d_e\).
\end{itemize}
\section{Comparison with the Model}
\label{ssec:test_struts_comp_model}
Two models of the APA300ML are used here for comparison:
\begin{itemize}
\item a simple two degrees of freedom model
\item a model using a super element extracted from a finite element model
\end{itemize}

These two models of the APA300ML were tuned to best match measured frequency response functions of the APA alone.
The flexible joints are here modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness).

These two models are compared with the measured frequency responses in Figure \ref{fig:test_struts_comp_frf_flexible_model}.

The model dynamics from DAC voltage \(u\) to the axial motion of the strut \(d_a\) (Figure \ref{fig:test_struts_comp_frf_flexible_model_int}) and from DAC voltage \(u\) to the force sensor voltage \(V_s\) (Figure \ref{fig:test_struts_comp_frf_flexible_model_iff}) are well matching the experimental identification.

However, the transfer function from \(u\) to encoder displacement \(d_e\) are not well matching for both models.
For the 2DoF model, this is normal as the resonances affecting the dynamics are not modelled at all (the APA300ML is modelled as infinitely rigid in all directions except the translation along it's actuation axis).
For the flexible model, it will be shown in the next section that by adding some misalignment betwen the flexible joints and the APA300ML, this model can better represent the observed dynamics.

\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_comp_frf_flexible_model_int.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_comp_frf_flexible_model_enc.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_frf_flexible_model_enc}$u$ to $d_e$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_comp_frf_flexible_model_iff.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$}
\end{subfigure}
\caption{\label{fig:test_struts_comp_frf_flexible_model}Comparison of the measured dynamics and of the Simscape dynamics using the ``flexible'' APA300ML model (Super-Element extracted from a Finite Element Model).}
\end{figure}
\section{Effect of a misalignment of the APA and flexible joints on the transfer function from actuator to encoder}
\label{ssec:test_struts_effect_misalignment}
As shown in Figure \ref{fig:test_struts_comp_enc_plants}, the dynamics from actuator to encoder for all the struts is very different.

This could be explained by a large variability in the alignment of the flexible joints and the APA (at the time, the alignment pins were not used).

Depending on the alignment, the spurious resonances of the struts (Figure \ref{fig:test_struts_mode_shapes}) can be excited differently.

For instance, consider Figure \ref{fig:test_struts_misalign_schematic} where there is a misalignment in the \(y\) direction.
In such case, the mode at 200Hz is foreseen to be more excited as the misalignment \(d_y\) increases and therefore the dynamics from the actuator to the encoder should also change around 200Hz.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_struts_misalign_schematic.png}
\caption{\label{fig:test_struts_misalign_schematic}Mis-alignement between the joints and the APA}
\end{figure}

If the misalignment is in the \(x\) direction, the mode at 285Hz should be more affected whereas a misalignment in the \(z\) direction should not affect these resonances.

Such statement is studied in this section.
\subsection{Measured misalignment of the APA and flexible joints}

The misalignment between the two flexible joints and the APA has been measured for all the struts:
\begin{itemize}
\item the strut is fixed to the mounting bench
\item using an indicator, the height difference from the flexible joints and the APA is measured both for the top and bottom joints and on both sides
\item then it is possible to obtain the misalignment for both flexible joints
\end{itemize}

The raw measurements are shown in Table \ref{tab:meas_misalignment_struts_raw}.

As the flexible joint's ``thickness'' is 1mm larger than the APA ``thickness'', ideally (i.e. if it were perfectly centered) we would measure \texttt{-0.50mm} each time.

\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_raw}Measured misalignments of the struts (\texttt{R} means ``red'' side, and \texttt{B} means ``black side'') in [mm]}
\centering
\begin{tabularx}{0.4\linewidth}{cllll}
\toprule
\textbf{Strut} & \textbf{R Top} & \textbf{B Top} & \textbf{R Bot} & \textbf{B Bot}\\
\midrule
1 & -0.4 & -0.6 & -0.16 & -0.82\\
2 & -0.67 & -0.3 & -0.34 & -0.63\\
3 & -0.07 & -0.88 & -0.16 & -0.79\\
4 & -0.48 & -0.46 & 0.07 & -1.0\\
5 & -0.33 & -0.64 & -0.48 & -0.52\\
\bottomrule
\end{tabularx}
\end{table}

Also, the sum of the measured distances on each side should be 1mm (equal to the thickness difference between the flexible joint and the APA).
This is verified in Table \ref{tab:meas_misalignment_struts_thickness}.

\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_thickness}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.2\linewidth}{cll}
\toprule
\textbf{Strut} & \textbf{Top} & \textbf{Bot}\\
\midrule
1 & -1.0 & -0.98\\
2 & -0.97 & -0.97\\
3 & -0.95 & -0.95\\
4 & -0.94 & -0.93\\
5 & -0.97 & -1.0\\
\bottomrule
\end{tabularx}
\end{table}

The differences of the measured distances on each side corresponds to the misalignment on that same side (Table \ref{tab:meas_misalignment_struts_results}).

\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_results}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.25\linewidth}{cll}
\toprule
\textbf{Strut} & \textbf{Top} & \textbf{Bot}\\
\midrule
1 & 0.1 & 0.33\\
2 & -0.185 & 0.145\\
3 & 0.405 & 0.315\\
4 & -0.01 & 0.535\\
5 & 0.155 & 0.02\\
\bottomrule
\end{tabularx}
\end{table}

\begin{important}
The misalignment of the APA and flexible joints is quite large and variable from one strut to the other.
\end{important}
\subsection{Perfectly aligned APA}
Let's first consider that the strut is perfectly mounted such that the two flexible joints and the APA are aligned.
And define the inputs and outputs of the models:
\begin{itemize}
\item Input: voltage generated by the DAC
\item Output: measured displacement by the encoder
\end{itemize}

The transfer function is identified and shown in Figure \ref{fig:comp_enc_frf_align_perfect}.
From Figure \ref{fig:comp_enc_frf_align_perfect}, it is clear that:
\begin{enumerate}
\item The model with perfect alignment is not matching the measured FRF
\item The mode at 200Hz is not present in the identified dynamics of the Simscape model
\item The measured FRF have different shapes
\end{enumerate}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/comp_enc_frf_align_perfect.png}
\caption{\label{fig:comp_enc_frf_align_perfect}Comparison of the model with a perfectly aligned APA and flexible joints with the measured FRF from actuator to encoder}
\end{figure}

\begin{question}
Why is the flexible mode of the strut at 200Hz is not seen in the model in Figure \ref{fig:comp_enc_frf_align_perfect}?

Probably because the presence of this mode is not due because of the ``unbalanced'' mass of the encoder, but rather because of the misalignment of the APA with respect to the two flexible joints.
This will be verified in the next sections.
\end{question}
\subsection{Effect of a misalignment in y}
Let's compute the transfer function from output DAC voltage \(V_s\) to the measured displacement by the encoder \(d_e\) for several misalignment in the \(y\) direction:
The obtained dynamics are shown in Figure \ref{fig:effect_misalignment_y}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/effect_misalignment_y.png}
\caption{\label{fig:effect_misalignment_y}Effect of a misalignement in the \(y\) direction}
\end{figure}

\begin{important}
The alignment of the APA with the flexible joints as a \textbf{huge} influence on the dynamics from actuator voltage to measured displacement by the encoder.
The misalignment in the \(y\) direction mostly influences:
\begin{itemize}
\item the presence of the flexible mode at 200Hz
\item the location of the complex conjugate zero between the first two resonances:
\begin{itemize}
\item if \(d_y < 0\): there is no zero between the two resonances and possibly not even between the second and third ones
\item if \(d_y > 0\): there is a complex conjugate zero between the first two resonances
\end{itemize}
\item the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
\end{itemize}
\end{important}
\subsection{Effect of a misalignment in x}
Let's compute the transfer function from output DAC voltage to the measured displacement by the encoder for several misalignment in the \(x\) direction:
The obtained dynamics are shown in Figure \ref{fig:effect_misalignment_x}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/effect_misalignment_x.png}
\caption{\label{fig:effect_misalignment_x}Effect of a misalignement in the \(x\) direction}
\end{figure}

\begin{important}
The misalignment in the \(x\) direction mostly influences the presence of the flexible mode at 300Hz.
\end{important}
\subsection{Comparison with identified misalignment}
\subsection{Find the misalignment of each strut}
From the previous analysis on the effect of a \(x\) and \(y\) misalignment, it is possible to estimate the \(x,y\) misalignment of the measured struts.

The misalignment that gives the best match for the FRF are defined below.
For each misalignment, the dynamics from the DAC voltage to the encoder measurement is identified.
The results are shown in Figure \ref{fig:comp_all_struts_corrected_misalign}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/comp_all_struts_corrected_misalign.png}
\caption{\label{fig:comp_all_struts_corrected_misalign}Comparison (model and measurements) of the FRF from DAC voltage u to measured displacement by the encoders for all the struts}
\end{figure}

\begin{important}
By tuning the misalignment of the APA with respect to the flexible joints, it is possible to obtain a good fit between the model and the measurements (Figure \ref{fig:comp_all_struts_corrected_misalign}).

If encoders are to be used when fixed on the struts, it is therefore very important to properly align the APA and the flexible joints when mounting the struts.

In the future, a ``pin'' will be used to better align the APA with the flexible joints.
We can expect the amplitude of the spurious resonances to decrease.
\end{important}
\section{Comparison of all the (re-aligned) Struts}
\label{sec:test_struts_meas_all_aligned_struts}
\begin{itemize}
\item[{$\square$}] Should this be included here?
\end{itemize}

The struts are re-aligned and measured using the same test bench.
\subsection{Measured misalignment of the APA and flexible joints}
The misalignment between the APA and the flexible joints are measured.

The results are defined below and summarized in Table \ref{tab:meas_misalignment_struts_new_raw}.

\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_new_raw}Measured misalignments of the struts (\texttt{R} means ``red'' side, and \texttt{B} means ``black side'') in [mm]}
\centering
\begin{tabularx}{0.45\linewidth}{cllll}
\toprule
\textbf{Strut} & \textbf{R Top} & \textbf{B Top} & \textbf{R Bot} & \textbf{B Bot}\\
\midrule
1 & -0.54 & -0.5 & -0.5 & -0.52\\
2 & -0.44 & -0.55 & -0.49 & -0.49\\
3 & -0.48 & -0.5 & -0.5 & -0.46\\
4 & -0.45 & -0.51 & -0.51 & -0.45\\
5 & -0.5 & -0.5 & -0.5 & -0.5\\
6 & -0.5 & -0.49 & -0.43 & -0.54\\
\bottomrule
\end{tabularx}
\end{table}

Also, the sum of the measured distances on each side should be 1mm (equal to the thickness difference between the flexible joint and the APA).
This is verified in Table \ref{tab:meas_misalignment_struts_new_thickness}.

\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_new_thickness}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.2\linewidth}{cll}
\toprule
\textbf{APA} & \textbf{Top} & \textbf{Bot}\\
\midrule
1 & -1.04 & -1.02\\
2 & -0.99 & -0.98\\
4 & -0.98 & -0.96\\
5 & -0.96 & -0.96\\
6 & -1.0 & -1.0\\
8 & -0.99 & -0.97\\
\bottomrule
\end{tabularx}
\end{table}

The differences of the measured distances on each side corresponds to the misalignment on that same side (Table \ref{tab:meas_misalignment_struts_new_results}).

\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_new_results}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.25\linewidth}{cll}
\toprule
\textbf{APA} & \textbf{Top} & \textbf{Bot}\\
\midrule
1 & -0.02 & 0.01\\
2 & 0.055 & 0.0\\
4 & 0.01 & -0.02\\
5 & 0.03 & -0.03\\
6 & 0.0 & 0.0\\
8 & -0.005 & 0.055\\
\bottomrule
\end{tabularx}
\end{table}

\begin{important}
After using the alignment pins, the misalignment of the APA and flexible joints are much smaller (\(< 50\,\mu m\) for all the struts).
\end{important}
\subsection{FRF Identification - Setup}
The excitation signal is a low pass filtered white noise.
Both the encoder and the force sensor voltage are measured.

Here are the leg numbers that have been measured.
We get the frequency vector that will be the same for all the frequency domain analysis.
\subsection{FRF Identification - Encoder}
In this section, the dynamics from \(u\) to \(d_e\) (encoder) is identified.

Then, the transfer function from the DAC output voltage \(u\) to the measured displacement by the encoder \(d_e\) is computed:
The obtained transfer functions are shown in Figure \ref{fig:struts_align_frf_dvf_plant_tf}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/struts_align_frf_dvf_plant_tf.png}
\caption{\label{fig:struts_align_frf_dvf_plant_tf}Estimated FRF for the DVF plant (transfer function from \(u\) to the encoder \(d_e\))}
\end{figure}

\begin{important}
Even though the struts are much better aligned, we still observe high variability between the struts for the transfer function from \(u\) to \(d_e\).
\end{important}
\subsection{Conclusion}

\begin{important}
Having the struts well aligned does not change significantly the obtained dynamics.
\end{important}

The measured FRF are now saved for further use.
\section{Effect of flexible joint's characteristics}
\label{ssec:test_struts_effect_joint_stiffness}
As the struts are composed of one APA and two flexible joints, it is obvious that the flexible joint characteristics will change the dynamic behavior of the struts.

Using the Simscape model, the effect of the flexible joint's characteristics on the dynamics as measured on the test bench are studied:
\begin{itemize}
\item Section \ref{sec:struts_effect_bending_stiff_joints}: the effects of a change of bending stiffness is studied
\item Section \ref{sec:struts_effect_axial_stiff_joints}: the effects of a change of axial stiffness is studied
\item Section \ref{sec:struts_effect_bending_damping_joints}: the effects of a change of bending damping is studied
\end{itemize}

The studied dynamics is between \(u\) and the encoder displacement \(d_e\).
\subsection{Effect of bending stiffness of the flexible joints}
\label{sec:struts_effect_bending_stiff_joints}

Let's initialize an APA which is a little bit misaligned.
The bending stiffnesses for which the dynamics is identified are defined below.
Then the identification is performed for all the values of the bending stiffnesses.
The obtained dynamics from DAC voltage to encoder measurements are compared in Figure \ref{fig:effect_enc_bending_stiff}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/effect_enc_bending_stiff.png}
\caption{\label{fig:effect_enc_bending_stiff}Dynamics from DAC output to encoder for several bending stiffnesses}
\end{figure}

\begin{important}
The bending stiffness of the joints has little impact on the transfer function from \(u\) to \(d_e\).
\end{important}
\subsection{Effect of axial stiffness of the flexible joints}
\label{sec:struts_effect_axial_stiff_joints}

The axial stiffnesses for which the dynamics is identified are defined below.
Then the identification is performed for all the values of the bending stiffnesses.
The obtained dynamics from DAC voltage to encoder measurements are compared in Figure \ref{fig:effect_enc_axial_stiff}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/effect_enc_axial_stiff.png}
\caption{\label{fig:effect_enc_axial_stiff}Dynamics from DAC output to encoder for several axial stiffnesses}
\end{figure}

\begin{important}
The axial stiffness of the flexible joint has a large impact on the frequency of the complex conjugate zero.
Using the measured FRF on the test-bench, if is therefore possible to estimate the axial stiffness of the flexible joints from the location of the zero.

This method gives nice match between the measured FRF and the one extracted from the simscape model, however it could give not so accurate values of the joint's axial stiffness as other factors are also influencing the location of the zero.

Using this method, an axial stiffness of \(70 N/\mu m\) is found to give good results (and is reasonable based on the finite element models).
\end{important}
\subsection{Effect of bending damping}
\label{sec:struts_effect_bending_damping_joints}
Now let's study the effect of the bending damping of the flexible joints.

The tested bending damping are defined below:
Then the identification is performed for all the values of the bending damping.
The results are shown in Figure \ref{fig:effect_enc_bending_damp}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/effect_enc_bending_damp.png}
\caption{\label{fig:effect_enc_bending_damp}Dynamics from DAC output to encoder for several bending damping}
\end{figure}
\chapter{Conclusion}
\label{sec:test_struts_conclusion}
\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}
