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test-bench-flexible-joints.tex

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+% Created 2024-03-19 Tue 17:35
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 \tableofcontents
 
 \clearpage
+In this document, we present a test-bench that has been developed in order to measure the bending stiffness of flexible joints.
+
+It is structured as follow:
+\begin{itemize}
+\item Section \ref{sec:flexible_joints}: the geometry of the flexible joints and the expected stiffness and stroke are presented
+\item Section \ref{sec:flex_dim_meas}: each flexible joint is measured using a profile projector
+\item Section \ref{sec:test_bench_desc}: the stiffness measurement bench is presented
+\item Section \ref{sec:error_budget}: an error budget is performed in order to estimate the accuracy of the measured stiffness
+\item Section \ref{sec:first_measurements}: first measurements are performed
+\item Section \ref{sec:bending_stiffness_meas}: the bending stiffness of the flexible joints are measured
+\end{itemize}
+
 \begin{table}[htbp]
 \caption{\label{tab:flexible_joints_section_matlab_code}Report sections and corresponding Matlab files}
 \centering
@@ -33,8 +45,786 @@ Section \ref{sec:flexible_joints}\_ & \texttt{flexible\_joints\_1\_.m}\\
 \bottomrule
 \end{tabularx}
 \end{table}
-\chapter{Mechanical Inspection}
-\label{sec:flexible_joints_mechanics}
+\chapter{Flexible Joints}
+\label{sec:flexible_joints}
+
+The flexible joints that are going to be measured in this document have been design to be used with a Nano-Hexapod (Figure \ref{fig:nano_hexapod}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.7\linewidth]{figs/nano_hexapod.png}
+\caption{\label{fig:nano_hexapod}CAD view of the Nano-Hexapod containing the flexible joints}
+\end{figure}
+
+Ideally, these flexible joints would behave as perfect ball joints, that is to say:
+\begin{itemize}
+\item no bending and torsional stiffnesses
+\item infinite shear and axial stiffnesses
+\item un-limited bending and torsional stroke
+\item no friction, no backlash
+\end{itemize}
+
+
+The real characteristics of the flexible joints will influence the dynamics of the Nano-Hexapod.
+Using a multi-body dynamical model of the nano-hexapod, the specifications in term of stiffness and stroke of the flexible joints have been determined and summarized in Table \ref{tab:flexible_joints_specs}.
+
+\begin{table}[htbp]
+\caption{\label{tab:flexible_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model}
+\centering
+\begin{tabularx}{0.5\linewidth}{Xcc}
+\toprule
+ & \textbf{Specification} & \textbf{FEM}\\
+\midrule
+Axial Stiffness & > 100 [N/um] & 94\\
+Shear Stiffness & > 1 [N/um] & 13\\
+Bending Stiffness & < 100 [Nm/rad] & 5\\
+Torsion Stiffness & < 500 [Nm/rad] & 260\\
+Bending Stroke & > 1 [mrad] & 24.5\\
+Torsion Stroke & > 5 [urad] & \\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+Then, the classical geometry of a flexible ball joint shown in Figure \ref{fig:flexible_joint_fem_geometry} has been optimized in order to meet the requirements.
+This has been done using a Finite Element Software and the obtained joint's characteristics are summarized in Table \ref{tab:flexible_joints_specs}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.5\linewidth]{figs/flexible_joint_fem_geometry.png}
+\caption{\label{fig:flexible_joint_fem_geometry}Flexible part of the Joint used for FEM - CAD view}
+\end{figure}
+
+The obtained geometry are defined in the \href{doc/flex\_joints.pdf}{drawings of the flexible joints}.
+The material is a special kind of stainless steel called ``F16PH''.
+
+The flexible joints can be seen on Figure \ref{fig:received_flex}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/IMG_20210302_173619.jpg}
+\caption{\label{fig:received_flex}15 of the 16 flexible joints}
+\end{figure}
+\chapter{Dimensional Measurements}
+\label{sec:flex_dim_meas}
+\section{Measurement Bench}
+
+The axis corresponding to the flexible joints are defined in Figure \ref{fig:flexible_joint_axis}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.3\linewidth]{figs/flexible_joint_axis.png}
+\caption{\label{fig:flexible_joint_axis}Define axis for the flexible joints}
+\end{figure}
+
+The dimensions of the flexible part in the Y-Z plane will contribute to the X-bending stiffness.
+Similarly, the dimensions of the flexible part in the X-Z plane will contribute to the Y-bending stiffness.
+
+The setup to measure the dimension of the ``X'' flexible beam is shown in Figure \ref{fig:flexible_joint_y_flex_meas_setup}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=1.0\linewidth]{figs/flexible_joint_y_flex_meas_setup.png}
+\caption{\label{fig:flexible_joint_y_flex_meas_setup}Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness}
+\end{figure}
+
+What we typically observe is shown in Figure \ref{fig:soft_measure_flex_size}.
+It is then possible to estimate to dimension of the flexible beam with an accuracy of \(\approx 5\,\mu m\),
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=1.0\linewidth]{figs/soft_measure_flex_size.jpg}
+\caption{\label{fig:soft_measure_flex_size}Image used to measure the flexible joint's dimensions}
+\end{figure}
+\section{Measurement Results}
+The expected flexible beam thickness is \(250\,\mu m\).
+However, it is more important that the thickness of all beams are close to each other.
+
+The dimension of the beams are been measured at each end to be able to estimate the mean of the beam thickness.
+
+All the measured dimensions are summarized in Table \ref{tab:flex_dim}.
+
+\begin{table}[htbp]
+\caption{\label{tab:flex_dim}Measured Dimensions of the flexible beams in \(\mu m\)}
+\centering
+\begin{tabularx}{0.4\linewidth}{Xcccc}
+\toprule
+ & Y1 & Y2 & X1 & X2\\
+\midrule
+1 & 223 & 226 & 224 & 214\\
+2 & 229 & 231 & 237 & 224\\
+3 & 234 & 230 & 239 & 231\\
+4 & 233 & 227 & 229 & 232\\
+5 & 225 & 212 & 228 & 228\\
+6 & 220 & 221 & 224 & 220\\
+7 & 206 & 207 & 228 & 226\\
+8 & 230 & 224 & 224 & 223\\
+9 & 223 & 231 & 228 & 233\\
+10 & 228 & 230 & 235 & 231\\
+11 & 197 & 207 & 211 & 204\\
+12 & 227 & 226 & 225 & 226\\
+13 & 215 & 228 & 231 & 220\\
+14 & 216 & 224 & 224 & 221\\
+15 & 209 & 214 & 220 & 221\\
+16 & 213 & 210 & 230 & 229\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+An histogram of these measured dimensions is shown in Figure \ref{fig:beam_dim_histogram}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/beam_dim_histogram.png}
+\caption{\label{fig:beam_dim_histogram}Histogram for the (16x2) measured beams' thickness}
+\end{figure}
+\chapter{Measurement Test Bench - Bending Stiffness}
+\label{sec:test_bench_desc}
+The most important characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\).
+
+To do so, we have to apply a torque \(T_x\) on the flexible joint and measure its angular deflection \(\theta_x\).
+The stiffness is then
+\begin{equation}
+k_{R_x} = \frac{T_x}{\theta_x}
+\end{equation}
+
+As it is quite difficult to apply a pure torque, a force will be applied instead.
+The application point of the force should far enough from the flexible part such that the obtained bending is much larger than the displacement in shear.
+
+The working principle of the bench is schematically shown in Figure \ref{fig:test_bench_principle}.
+One part of the flexible joint is fixed. On the mobile part, a force \(F_x\) is applied which is equivalent to a torque applied on the flexible joint center.
+The induced rotation is measured with a displacement sensor \(d_x\).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/test_bench_principle.png}
+\caption{\label{fig:test_bench_principle}Test Bench - working principle}
+\end{figure}
+
+
+This test-bench will be used to have a first approximation of the bending stiffnesss and stroke of the flexible joints.
+Another test-bench, better engineered will be used to measure the flexible joint's characteristics with better accuracy.
+\section{Flexible joint Geometry}
+The flexible joint used for the Nano-Hexapod is shown in Figure \ref{fig:flexible_joint_geometry}.
+Its bending stiffness is foreseen to be \(k_{R_y}\approx 5\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 25\,mrad\).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/flexible_joint_geometry.png}
+\caption{\label{fig:flexible_joint_geometry}Geometry of the flexible joint}
+\end{figure}
+
+The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 20\,mm\).
+
+Let's define the parameters on Matlab.
+\section{Required external applied force}
+
+The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is:
+\begin{equation}
+  \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
+\end{equation}
+
+Therefore, the applied force to test the full range of the flexible joint is:
+\begin{equation}
+  F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
+\end{equation}
+
+And we obtain:
+\begin{equation} F_{x,max} = 6.2\, [N] \end{equation}
+
+The measurement range of the force sensor should then be higher than \(6.2\,N\).
+\section{Required actuator stroke and sensors range}
+
+The flexible joint is designed to allow a bending motion of \(\pm 25\,mrad\).
+The corresponding stroke at the location of the force sensor is:
+\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
+
+\begin{equation} d_{max} = 0.5\, [mm] \end{equation}
+
+In order to test the full range of the flexible joint, the stroke of the translation stage used to move the force sensor should be higher than \(0.5\,mm\).
+Similarly, the measurement range of the displacement sensor should also be higher than \(0.5\,mm\).
+\section{Test Bench}
+
+A CAD view of the measurement bench is shown in Figure \ref{fig:test_bench_flex_overview}.
+
+\begin{note}
+Here are the different elements used in this bench:
+\begin{itemize}
+\item \textbf{Translation Stage}: \href{doc/V-408-Datasheet.pdf}{V-408}
+\item \textbf{Load Cells}: \href{doc/A700000007147087.pdf}{FC2231-0000-0010-L}
+\item \textbf{Encoder}: \href{doc/L-9517-9448-05-B\_Data\_sheet\_RESOLUTE\_BiSS\_en.pdf}{Renishaw Resolute 1nm}
+\end{itemize}
+\end{note}
+
+Both the measured force and displacement are acquired at the same time using a Speedgoat machine.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_bench_flex_overview.png}
+\caption{\label{fig:test_bench_flex_overview}Schematic of the test bench to measure the bending stiffness of the flexible joints}
+\end{figure}
+
+A side view of the bench with the important quantities are shown in Figure \ref{fig:test_bench_flex_side}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.25\linewidth]{figs/test_bench_flex_side.png}
+\caption{\label{fig:test_bench_flex_side}Schematic of the test bench to measure the bending stiffness of the flexible joints}
+\end{figure}
+\chapter{Error budget}
+\label{sec:error_budget}
+Many things can impact the accuracy of the measured bending stiffness such as:
+\begin{itemize}
+\item Errors in the force and displacement measurement
+\item Shear effects
+\item Deflection of the Force sensor
+\item Errors in the geometry of the bench
+\end{itemize}
+
+In this section, we wish to estimate the attainable accuracy with the current bench, and identified the limiting factors.
+\section{Finite Element Model}
+From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables \ref{tab:axial_shear_characteristics} and \ref{tab:bending_torsion_characteristics}.
+
+\begin{table}[htbp]
+\caption{\label{tab:axial_shear_characteristics}Axial/Shear characteristics}
+\centering
+\begin{tabularx}{0.6\linewidth}{Xccc}
+\toprule
+ & Stiffness [N/um] & Max Force [N] & Stroke [um]\\
+\midrule
+Axial & 94 & 469 & 5\\
+Shear & 13 & 242 & 19\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:bending_torsion_characteristics}Bending/Torsion characteristics}
+\centering
+\begin{tabularx}{0.7\linewidth}{Xccc}
+\toprule
+ & Stiffness [Nm/rad] & Max Torque [Nmm] & Stroke [mrad]\\
+\midrule
+Bending & 5 & 118 & 24\\
+Torsional & 260 & 1508 & 6\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Setup}
+
+The setup is schematically represented in Figure \ref{fig:test_bench_flex_side_bis}.
+
+The force is applied on top of the flexible joint with a distance \(h\) with the joint's center.
+The displacement of the flexible joint is also measured at the same height.
+
+The height between the joint's center and the force application point is:
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.25\linewidth]{figs/test_bench_flex_side.png}
+\caption{\label{fig:test_bench_flex_side_bis}Schematic of the test bench to measure the bending stiffness of the flexible joints}
+\end{figure}
+\section{Effect of Bending}
+The torque applied is:
+\begin{equation}
+  M_y = F_x \cdot h
+\end{equation}
+
+The flexible joint is experiencing a rotation \(\theta_y\) due to the torque \(M_y\):
+\begin{equation}
+  \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x \cdot h}{k_{R_y}}
+\end{equation}
+
+This rotation is then measured by the displacement sensor.
+The measured displacement is:
+\begin{equation}
+  D_b = h \tan(\theta_y) = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) \label{eq:bending_stiffness_formula}
+\end{equation}
+\section{Computation of the bending stiffness}
+From equation \eqref{eq:bending_stiffness_formula}, we can compute the bending stiffness:
+\begin{equation}
+  k_{R_y} = \frac{F_x \cdot h}{\tan^{-1}\left( \frac{D_b}{h} \right)}
+\end{equation}
+
+For small displacement, we have
+\begin{equation}
+  \boxed{k_{R_y} \approx h^2 \frac{F_x}{d_x}}
+\end{equation}
+
+And therefore, to precisely measure \(k_{R_y}\), we need to:
+\begin{itemize}
+\item precisely measure the motion \(d_x\)
+\item precisely measure the applied force \(F_x\)
+\item precisely now the height of the force application point \(h\)
+\end{itemize}
+\section{Estimation error due to force and displacement sensors accuracy}
+The maximum error on the measured displacement with the encoder is 40 nm.
+This quite negligible compared to the measurement range of 0.5 mm.
+
+The accuracy of the force sensor is around 1\% and therefore, we should expect to have an accuracy on the measured stiffness of at most 1\%.
+\section{Estimation error due to Shear}
+The effect of Shear on the measured displacement is simply:
+\begin{equation}
+  D_s = \frac{F_x}{k_s}
+\end{equation}
+
+The measured displacement will be the effect of shear + effect of bending
+\begin{equation}
+  d_x = D_b + D_s = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_s} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_s} \right)
+\end{equation}
+
+The estimated bending stiffness \(k_{\text{est}}\) will then be:
+\begin{equation}
+  k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}}
+\end{equation}
+
+\begin{verbatim}
+The measurement error due to Shear is 0.1 %
+\end{verbatim}
+\section{Estimation error due to force sensor compression}
+The measured displacement is not done directly at the joint's location.
+The force sensor compression will then induce an error on the joint's stiffness.
+
+The force sensor stiffness \(k_F\) is estimated to be around:
+\begin{verbatim}
+k_F = 1.0e+06 [N/m]
+\end{verbatim}
+
+
+The measured displacement will be the sum of the displacement induced by the bending and by the compression of the force sensor:
+\begin{equation}
+  d_x = D_b + \frac{F_x}{k_F} = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_F} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_F} \right)
+\end{equation}
+
+The estimated bending stiffness \(k_{\text{est}}\) will then be:
+\begin{equation}
+  k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}}
+\end{equation}
+
+\begin{verbatim}
+The measurement error due to height estimation errors is 0.8 %
+\end{verbatim}
+\section{Estimation error due to height estimation error}
+Let's consider an error in the estimation of the height from the application of the force to the joint's center:
+\begin{equation}
+h_{\text{est}} = h (1 + \epsilon)
+\end{equation}
+
+The computed bending stiffness will be:
+\begin{equation}
+  k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x}
+\end{equation}
+
+And the stiffness estimation error is:
+\begin{equation}
+\frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2
+\end{equation}
+
+\begin{verbatim}
+The measurement error due to height estimation errors of 0.2 [mm] is 1.6 %
+\end{verbatim}
+\section{Conclusion}
+Based on the above analysis, we should expect no better than few percent of accuracy using the current test-bench.
+This is well enough for a first estimation of the bending stiffness of the flexible joints.
+
+Another measurement bench allowing better accuracy will be developed.
+\chapter{First Measurements}
+\label{sec:first_measurements}
+\begin{itemize}
+\item Section \ref{sec:test_meas_probe}:
+\item Section \ref{sec:meas_probe_stiffness}:
+\end{itemize}
+\section{Agreement between the probe and the encoder}
+\label{sec:test_meas_probe}
+\begin{note}
+\begin{itemize}
+\item \textbf{Encoder}: \href{doc/L-9517-9448-05-B\_Data\_sheet\_RESOLUTE\_BiSS\_en.pdf}{Renishaw Resolute 1nm}
+\item \textbf{Displacement Probe}: \href{doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf}{Millimar C1216 electronics} and \href{doc/tmp3m0cvmue\_7888038c-cdc8-48d8-a837-35de02760685.pdf}{Millimar 1318 probe}
+\end{itemize}
+\end{note}
+
+The measurement setup is made such that the probe measured the translation table displacement.
+It should then measure the same displacement as the encoder.
+Using this setup, we should be able to compare the probe and the encoder.
+
+Let's load the measurements.
+The time domain measured displacement by the probe and by the encoder is shown in Figure \ref{fig:comp_encoder_probe_time}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_encoder_probe_time.png}
+\caption{\label{fig:comp_encoder_probe_time}Time domain measurement}
+\end{figure}
+
+If we zoom, we see that there is some delay between the encoder and the probe (Figure \ref{fig:comp_encoder_probe_time_zoom}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_encoder_probe_time_zoom.png}
+\caption{\label{fig:comp_encoder_probe_time_zoom}Time domain measurement (Zoom)}
+\end{figure}
+
+This delay is estimated using the \texttt{finddelay} command.
+
+\begin{verbatim}
+The time delay is approximately 15.8 [ms]
+\end{verbatim}
+
+
+The measured mismatch between the encoder and the probe with and without compensating for the time delay are shown in Figure \ref{fig:comp_encoder_probe_mismatch}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_encoder_probe_mismatch.png}
+\caption{\label{fig:comp_encoder_probe_mismatch}Measurement mismatch, with and without delay compensation}
+\end{figure}
+
+Finally, the displacement of the probe is shown as a function of the displacement of the encoder and a linear fit is made (Figure \ref{fig:comp_encoder_probe_linear_fit}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/comp_encoder_probe_linear_fit.png}
+\caption{\label{fig:comp_encoder_probe_linear_fit}Measured displacement by the probe as a function of the measured displacement by the encoder}
+\end{figure}
+
+\begin{important}
+From the measurement, it is shown that the probe is well calibrated.
+However, there is some time delay of tens of milliseconds that could induce some measurement errors.
+\end{important}
+\section{Measurement of the Millimar 1318 probe stiffness}
+\label{sec:meas_probe_stiffness}
+
+\begin{note}
+\begin{itemize}
+\item \textbf{Translation Stage}: \href{doc/V-408-Datasheet.pdf}{V-408}
+\item \textbf{Load Cell}: \href{doc/A700000007147087.pdf}{FC2231-0000-0010-L}
+\item \textbf{Encoder}: \href{doc/L-9517-9448-05-B\_Data\_sheet\_RESOLUTE\_BiSS\_en.pdf}{Renishaw Resolute 1nm}
+\item \textbf{Displacement Probe}: \href{doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf}{Millimar C1216 electronics} and \href{doc/tmp3m0cvmue\_7888038c-cdc8-48d8-a837-35de02760685.pdf}{Millimar 1318 probe}
+\end{itemize}
+\end{note}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/setup_mahr_stiff_meas_side.jpg}
+\caption{\label{fig:setup_mahr_stiff_meas_side}Setup - Side View}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/setup_mahr_stiff_meas_top.jpg}
+\caption{\label{fig:setup_mahr_stiff_meas_top}Setup - Top View}
+\end{figure}
+
+Let's load the measurement results.
+The time domain measured force and displacement are shown in Figure \ref{fig:mahr_time_domain}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/mahr_time_domain.png}
+\caption{\label{fig:mahr_time_domain}Time domain measurements}
+\end{figure}
+
+
+Now we can estimate the stiffness with a linear fit.
+
+This is very close to the 0.04 [N/mm] written in the \href{doc/tmp3m0cvmue\_7888038c-cdc8-48d8-a837-35de02760685.pdf}{Millimar 1318 probe datasheet}.
+
+And compare the linear fit with the raw measurement data (Figure \ref{fig:mahr_stiffness_f_d_plot}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/mahr_stiffness_f_d_plot.png}
+\caption{\label{fig:mahr_stiffness_f_d_plot}Measured displacement as a function of the measured force. Raw data and linear fit}
+\end{figure}
+
+\begin{summary}
+The Millimar 1318 probe has a stiffness of \(\approx 0.04\,[N/mm]\).
+\end{summary}
+\section{Force Sensor Calibration}
+\begin{note}
+\textbf{Load Cells}:
+\begin{itemize}
+\item \href{doc/A700000007147087.pdf}{FC2231-0000-0010-L}
+\item \href{doc/FRE\_DS\_XFL212R\_FR\_A3.pdf}{XFL212R}
+\end{itemize}
+\end{note}
+
+There are both specified to have \(\pm 1 \%\) of non-linearity over the full range.
+
+The XFL212R has a spherical interface while the FC2231 has a flat surface.
+Therefore, we should have a nice point contact when using the two force sensors as shown in Figure \ref{fig:force_sensor_calibration_setup}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.8\linewidth]{figs/IMG_20210309_145333.jpg}
+\caption{\label{fig:force_sensor_calibration_setup}Zoom on the two force sensors in contact}
+\end{figure}
+
+The two force sensors are therefore measuring the exact same force, and we can compare the two measurements.
+
+Let's load the measured force of both sensors.
+We remove any offset such that they are both measuring no force when not in contact.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/force_calibration_time.png}
+\caption{\label{fig:force_calibration_time}Measured force using both sensors as a function of time}
+\end{figure}
+
+Let's select only the first part from the moment they are in contact until the maximum force is reached.
+
+Then, let's make a linear fit between the two measured forces.
+
+The two forces are plotted against each other as well as the linear fit in Figure \ref{fig:calibrated_force_dit}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/calibrated_force_dit.png}
+\caption{\label{fig:calibrated_force_dit}Measured two forces and linear fit}
+\end{figure}
+
+The measurement error between the two sensors is shown in Figure \ref{fig:force_meas_error}.
+It is below 0.1N for the full measurement range.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/force_meas_error.png}
+\caption{\label{fig:force_meas_error}Error in Newtons}
+\end{figure}
+
+The same error is shown in percentage in Figure \ref{fig:force_meas_error_percentage}.
+The error is less than 1\% when the measured force is above 5N.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/force_meas_error_percentage.png}
+\caption{\label{fig:force_meas_error_percentage}Error in percentage}
+\end{figure}
+\section{Force Sensor Noise}
+The objective of this measurement is to estimate the noise of the force sensor \href{doc/A700000007147087.pdf}{FC2231-0000-0010-L}.
+To do so, we don't apply any force to the sensor, and we measure its output for 100s.
+
+Let's load the measurement data.
+
+The measured force is shown in Figure \ref{fig:force_noise_time}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/force_noise_time.png}
+\caption{\label{fig:force_noise_time}Measured force}
+\end{figure}
+
+Let's now compute the Amplitude Spectral Density of the measured force.
+
+The results is shown in Figure \ref{fig:force_noise_asd}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/force_noise_asd.png}
+\caption{\label{fig:force_noise_asd}Amplitude Spectral Density of the meaured force}
+\end{figure}
+\section{Force Sensor Stiffness}
+The objective of this measurement is to estimate the stiffness of the force sensor \href{doc/A700000007147087.pdf}{FC2231-0000-0010-L}.
+
+To do so, a very stiff element is fixed in front of the force sensor as shown in Figure \ref{fig:setup_meas_force_sensor_stiffness}.
+
+Then, we apply a force on the stiff element through the force sensor.
+We measure the deflection of the force sensor using an encoder.
+
+Then, having the force and the deflection, we should be able to estimate the stiffness of the force sensor supposing the stiffness of the other elements are much larger.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.6\linewidth]{figs/IMG_20210309_145242.jpg}
+\caption{\label{fig:setup_meas_force_sensor_stiffness}Bench used to measured the stiffness of the force sensor}
+\end{figure}
+
+From the documentation, the deflection of the sensor at the maximum load (50N) is 0.05mm, the stiffness is therefore foreseen to be around \(1\,N/\mu m\).
+
+Let's load the measured force as well as the measured displacement.
+Some pre-processing is applied on the data.
+The linear fit is performed.
+The displacement as a function of the force as well as the linear fit are shown in Figure \ref{fig:force_sensor_stiffness_fit}.
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/force_sensor_stiffness_fit.png}
+\caption{\label{fig:force_sensor_stiffness_fit}Displacement as a function of the measured force}
+\end{figure}
+
+And we obtain the following stiffness:
+\begin{verbatim}
+k = 0.76 [N/um]
+\end{verbatim}
+\chapter{Bending Stiffness Measurement}
+\label{sec:bending_stiffness_meas}
+\section{Introduction}
+
+A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure \ref{fig:picture_bending_x_meas_side_overview}.
+A closer view on flexible joint is shown in Figure \ref{fig:picture_bending_x_meas_side_close} and a zoom on the force sensor tip is shown in Figure \ref{fig:picture_bending_x_meas_side_zoom}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/picture_bending_x_meas_side_overview.jpg}
+\caption{\label{fig:picture_bending_x_meas_side_overview}Side view of the flexible joint stiffness bench. X-Bending stiffness is measured.}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/picture_bending_x_meas_side_close.jpg}
+\caption{\label{fig:picture_bending_x_meas_side_close}Zoom on the flexible joint - Side view}
+\end{figure}
+
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.4\linewidth]{figs/picture_bending_x_meas_side_zoom.jpg}
+\caption{\label{fig:picture_bending_x_meas_side_zoom}Zoom on the tip of the force sensor}
+\end{figure}
+
+The same bench used to measure the Y-bending stiffness of the flexible joint is shown in Figure \ref{fig:picture_bending_y_meas_side_close}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/picture_bending_y_meas_side_close.jpg}
+\caption{\label{fig:picture_bending_y_meas_side_close}Stiffness measurement bench - Y-d bending measurement}
+\end{figure}
+\section{Analysis of one measurement}
+
+In this section is shown how the data are analysis in order to measured:
+\begin{itemize}
+\item the bending stiffness
+\item the bending stroke
+\item the stiffness once the mechanical stops are in contact
+\end{itemize}
+
+
+The height from the flexible joint's center and the point of application force \(h\) is defined below:
+The obtained time domain measurements are shown in Figure \ref{fig:flex_joint_meas_example_time_domain}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/flex_joint_meas_example_time_domain.png}
+\caption{\label{fig:flex_joint_meas_example_time_domain}Typical time domain measurements}
+\end{figure}
+
+The displacement as a function of the force is then shown in Figure \ref{fig:flex_joint_meas_example_F_d}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/flex_joint_meas_example_F_d.png}
+\caption{\label{fig:flex_joint_meas_example_F_d}Typical measurement of the diplacement as a function of the applied force}
+\end{figure}
+
+The bending stiffness can be estimated by computing the slope of the curve in Figure \ref{fig:flex_joint_meas_example_F_d}.
+The bending stroke and the stiffness when touching the mechanical stop can also be estimated from the same figure.
+
+The raw data as well as the fit corresponding to the two stiffnesses are shown in Figure \ref{fig:flex_joint_meas_example_F_d_lin_fit}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/flex_joint_meas_example_F_d_lin_fit.png}
+\caption{\label{fig:flex_joint_meas_example_F_d_lin_fit}Typical measurement of the diplacement as a function of the applied force with estimated linear fits}
+\end{figure}
+
+Then, the bending stroke is estimated as crossing point between the two fitted lines:
+The obtained characteristics are summarized in Table \ref{tab:obtained_caracteristics_flex_1_x}.
+
+\begin{table}[htbp]
+\caption{\label{tab:obtained_caracteristics_flex_1_x}Estimated characteristics of the flexible joint number 1 for the X-direction}
+\centering
+\begin{tabularx}{0.5\linewidth}{lc}
+\toprule
+Bending Stiffness [Nm/rad] & 5.5\\
+Bending Stiffness @ stop [Nm/rad] & 173.6\\
+Bending Stroke [mrad] & 18.9\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Bending stiffness and bending stroke of all the flexible joints}
+
+Now, let's estimate the bending stiffness and stroke for all the flexible joints.
+
+The results are summarized in Table \ref{tab:meas_flexible_joints_x_dir} for the X direction and in Table \ref{tab:meas_flexible_joints_y_dir} for the Y direction.
+
+\begin{table}[htbp]
+\caption{\label{tab:meas_flexible_joints_x_dir}Measured characteristics of the flexible joints in the X direction}
+\centering
+\begin{tabularx}{0.6\linewidth}{cccc}
+\toprule
+ & \(R_{R_x}\) {[}Nm/rad] & \(k_{R_x,s}\) {[}Nm/rad] & \(R_{x,\text{max}}\) {[}mrad]\\
+\midrule
+1 & 5.5 & 173.6 & 18.9\\
+2 & 6.1 & 195.0 & 17.6\\
+3 & 6.1 & 191.3 & 17.7\\
+4 & 5.8 & 136.7 & 18.3\\
+5 & 5.7 & 88.9 & 22.0\\
+6 & 5.7 & 183.9 & 18.7\\
+7 & 5.7 & 157.9 & 17.9\\
+8 & 5.8 & 166.1 & 17.9\\
+9 & 5.8 & 159.5 & 18.2\\
+10 & 6.0 & 143.6 & 18.1\\
+11 & 5.0 & 163.8 & 17.7\\
+12 & 6.1 & 111.9 & 17.0\\
+13 & 6.0 & 142.0 & 17.4\\
+14 & 5.8 & 130.1 & 17.9\\
+15 & 5.7 & 170.7 & 18.6\\
+16 & 6.0 & 148.7 & 17.5\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\begin{table}[htbp]
+\caption{\label{tab:meas_flexible_joints_y_dir}Measured characteristics of the flexible joints in the Y direction}
+\centering
+\begin{tabularx}{0.6\linewidth}{cccc}
+\toprule
+ & \(R_{R_y}\) {[}Nm/rad] & \(k_{R_y,s}\) {[}Nm/rad] & \(R_{y,\text{may}}\) {[}mrad]\\
+\midrule
+1 & 5.7 & 323.5 & 17.9\\
+2 & 5.9 & 306.0 & 17.2\\
+3 & 6.0 & 224.4 & 16.8\\
+4 & 5.7 & 247.3 & 17.8\\
+5 & 5.8 & 250.9 & 13.0\\
+6 & 5.8 & 244.5 & 17.8\\
+7 & 5.3 & 214.8 & 18.1\\
+8 & 5.8 & 217.2 & 17.6\\
+9 & 5.7 & 225.0 & 17.6\\
+10 & 6.0 & 254.7 & 17.3\\
+11 & 4.9 & 261.1 & 18.4\\
+12 & 5.9 & 161.5 & 16.7\\
+13 & 6.1 & 227.6 & 16.8\\
+14 & 5.9 & 221.3 & 17.8\\
+15 & 5.4 & 241.5 & 17.8\\
+16 & 5.3 & 291.1 & 17.7\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Analysis}
+The dispersion of the measured bending stiffness is shown in Figure \ref{fig:bending_stiffness_histogram} and of the bending stroke in Figure \ref{fig:bending_stroke_histogram}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/bending_stiffness_histogram.png}
+\caption{\label{fig:bending_stiffness_histogram}Histogram of the measured bending stiffness}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/bending_stroke_histogram.png}
+\caption{\label{fig:bending_stroke_histogram}Histogram of the measured bending stroke}
+\end{figure}
+
+The relation between the measured beam thickness and the measured bending stiffness is shown in Figure \ref{fig:flex_thickness_vs_bending_stiff}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/flex_thickness_vs_bending_stiff.png}
+\caption{\label{fig:flex_thickness_vs_bending_stiff}Measured bending stiffness as a function of the estimated flexible beam thickness}
+\end{figure}
+\section{Conclusion}
+\begin{important}
+The measured bending stiffness and bending stroke of the flexible joints are very close to the estimated one using a Finite Element Model.
+
+The characteristics of all the flexible joints are also quite close to each other.
+This should allow us to model them with unique parameters.
+\end{important}
 \chapter{Conclusion}
 \label{sec:flexible_joints_conclusion}
 \printbibliography[heading=bibintoc,title={Bibliography}]