Better estimate bending stiffness

test-bench-flexible-joints.org

@@ -345,7 +345,9 @@ meas_flex = [[223, 226, 224, 214];
 #+begin_src matlab :exports none
 %% Histogram of the measured gap
 figure;
+hold on;
 histogram([(meas_flex(:,1)+meas_flex(:,2))/2,(meas_flex(:,3)+meas_flex(:,4))/2], 7)
+hold off;
 xlabel("Measured beam thickness [$\mu m$]");
 xticks([200, 205, 210, 215, 220, 225, 230, 235])
 #+end_src
@@ -731,7 +733,6 @@ A closer view on the force sensor tip is shown in Figure ref:fig:test_joints_pic
 #+end_src
 
 ** Load Cell Calibration
-
 In order to estimate the measured errors of the load cell "FC2231", it is compared against another load cell[fn:5].
 The two load cells are measured simultaneously while they are pushed against each other (see Figure ref:fig:test_joints_force_sensor_calib_picture).
 The contact between the two load cells is well defined as one has a spherical interface while the other has a flat surface.
@@ -780,7 +781,7 @@ exportFig('figs/test_joints_force_sensor_calib_fit.pdf', 'width', 'half', 'heigh
 #+end_src
 
 #+name: fig:test_joints_force_sensor_calib
-#+caption: Caption with reference to sub figure (\subref{fig:test_joints_force_sensor_calib_picture}),  (\subref{fig:test_joints_force_sensor_calib_fit})
+#+caption: Estimation of the load cell accuracy. A picture of the measurement bench is shown in (\subref{fig:test_joints_force_sensor_calib_picture}). Comparison of the two measured forces is made in (\subref{fig:test_joints_force_sensor_calib_fit})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:test_joints_force_sensor_calib_picture}Zoom on the two force sensors in contact}
@@ -838,10 +839,10 @@ exportFig('figs/test_joints_force_sensor_stiffness_fit.pdf', 'width', 'half', 'h
 #+end_src
 
 #+name: fig:test_joints_meas_force_sensor_stiffness
-#+caption: Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}). Measurement results is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_fit}).
+#+caption: Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}). Measurement results is shown in (\subref{fig:test_joints_force_sensor_stiffness_fit}).
 #+attr_latex: :options [htbp]
 #+begin_figure
-#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_force_sensor_stiffness_picture}Picture of the test}
+#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_force_sensor_stiffness_picture}Picture of the measurement bench}
 #+attr_latex: :options {0.49\textwidth}
 #+begin_subfigure
 #+attr_latex: :height 5.5cm
@@ -866,62 +867,52 @@ The bending stiffness of the flexible joint can be estimated by computing the sl
 The bending stroke can also be estimated as shown in Figure ref:fig:test_joints_meas_F_d_lin_fit and is found to be $\theta_{y,\text{max}} = 20.9\,\text{mrad}$.
 
 #+begin_src matlab
-%% Load Measured Data for first flexible joint
+%% Estimate the bending stiffness and stroke from the measurement - First Flexible joint
+% Load Measured Data for first flexible joint
 load('meas_stiff_flex_1_x.mat', 't', 'F', 'd');
 
-%% Zero the force
+% Start measurement at t = 0.2 s
-F = F - mean(F(t > 0.1 & t < 0.3));
-
-%% Start measurement at t = 0.2 s
 d = d(t > 0.2);
 F = F(t > 0.2);
 t = t(t > 0.2); t = t - t(1);
 
-%% Compute torque and angular displacement
+% Zero the force
-h = 22.5e-3; % Height [m]
+F = F - mean(F(t < 0.2));
-Ty = h * F; % Applied torque in [Nm]
-thetay = atan2(d, h); % Measured angle in [rad]
 
-%% Reset the measured angular stroke such that it is equal at zero when the load cell touches the flexible joint
 % Find when the force sensor touches the flexible joint
 i_l_start = find(F > 0.3, 1, 'first');
-% Reset the measured diplacement at that point
-% thetay = thetay - thetay(i_l_start);
 
-%% Determine the linear region and region when touching the mechanical stop
+% Compute torque and angular displacement
+h = 22.5e-3; % Height [m]
+Tx = h * F; % Applied torque in [Nm]
+thetax = atan2(d - d(i_l_start), h); % Measured angle in [rad]
+
 % Find then the maximum torque is applied
-[~, i_s_stop] = max(Ty);
+[~, i_s_stop] = max(Tx);
 % Linear region stops ~ when 90% of the stroke is reached
-i_l_stop = find(thetay > 0.9*thetay(i_s_stop), 1, 'first');
+i_l_stop = find(thetax > 0.9*thetax(i_s_stop), 1, 'first');
 % Mechanical "Stop" region start ~20Nmm before maximum torque is applied
-i_s_start = find(Ty > max(Ty)-20e-3, 1, 'first');
+i_s_start = find(Tx > max(Tx)-20e-3, 1, 'first');
 
-%% Define variables for the two regions
+% Linear fit in the "linear" region
-% Linear Region
+fit_l = polyfit(Tx(i_l_start:i_l_stop), thetax(i_l_start:i_l_stop), 1);
-Ty_l = Ty(i_l_start:i_l_stop);
-thetay_l = thetay(i_l_start:i_l_stop);
 
-% "Mechanical stop" Region
+% Linear fit in the "mechanical stop" region
-Ty_s = Ty(i_s_start:i_s_stop);
+fit_s = polyfit(Tx(i_s_start:i_s_stop), thetax(i_s_start:i_s_stop), 1);
-thetay_s = thetay(i_s_start:i_s_stop);
 
-%% Fit the best straight line for the two regions
+% Reset displacement more precisely based on fit
-fit_l = polyfit(Ty_l, thetay_l, 1);
+thetax = thetax - fit_l(2);
-fit_s = polyfit(Ty_s, thetay_s, 1);
-
-%% Reset displacement more precisely based on fit
-thetay = thetay - fit_l(2);
 fit_s(2) = fit_s(2) - fit_l(2);
 fit_l(2) = 0;
 
 %% Estimation of the bending stiffness
-kRy_l = 1/fit_l(1); % Bending Stiffness [Nm/rad]
+kRx_l = 1/fit_l(1); % Bending Stiffness [Nm/rad]
-kRy_s = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad]
+kRx_s = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad]
 
 %% Estimation of the bending stroke
 % This is done by finding the intersection of the two linear fits
 theta_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1)); % Maximum angular stroke [rad]
-Ty_at_theta_max = (fit_s(2) - fit_l(2))/(fit_l(1) - fit_s(1));
+Tx_at_theta_max = (fit_s(2) - fit_l(2))/(fit_l(1) - fit_s(1));
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -937,14 +928,14 @@ plot(0.4*cos(0:0.01:2*pi)+3, ...
      1.1*sin(0:0.01:2*pi)+4.8, 'k--');
 text(3.5, 4.8, sprintf('Mechanical\nStop'), 'horizontalalignment', 'left');
 hold off;
-ylabel('Force $F_x$ [N]');
+ylabel('Force $F_y$ [N]');
 ylim([-4, 6])
 ylimr = get(gca,'Ylim');
 
 yyaxis right
 plot(t, 1e3*d);
 xlabel('Time [s]');
-ylabel('Displacement $d_x$ [mm]');
+ylabel('Displacement $d_y$ [mm]');
 xlim([0,5]);
 % Make the force and displacement superimpose
 ylim([0.364 - 4*(0.8315-0.364)/4.095, 0.364 + 6*(0.8315-0.364)/4.095])
@@ -956,22 +947,23 @@ exportFig('figs/test_joints_meas_bend_time.pdf', 'width', 'half', 'height', 'nor
 
 #+begin_src matlab :exports none
 % Regions where to plot the fitted data
-Ty_l_fit = [0, Ty_at_theta_max];
+Tx_l_fit = [0, Tx_at_theta_max];
-Ty_s_fit = [Ty_at_theta_max, Ty(i_s_stop)];
+Tx_s_fit = [Tx_at_theta_max, Tx(i_s_stop)];
 
 figure;
 hold on;
-plot(Ty(1:i_s_stop), 1e3*thetay(1:i_s_stop), '-k', 'DisplayName', 'Raw data')
+plot(Tx(1:i_s_stop), 1e3*thetax(1:i_s_stop), '-k', 'DisplayName', 'Raw data')
-plot(Ty_l_fit, 1e3*(Ty_l_fit*fit_l(1) + fit_l(2)), '--', 'DisplayName', sprintf('$k_{R_y} = %.1f$ Nm/rad', kRy_l))
+plot(Tx_l_fit, 1e3*(Tx_l_fit*fit_l(1) + fit_l(2)), '--', 'DisplayName', sprintf('$k_{R_x} = %.1f$ Nm/rad', kRx_l))
-plot(Ty_s_fit, 1e3*(Ty_s_fit*fit_s(1) + fit_s(2)), '--', 'DisplayName', sprintf('$k_{R_y,stop} = %.0f$ Nm/rad', kRy_s))
+plot(Tx_s_fit, 1e3*(Tx_s_fit*fit_s(1) + fit_s(2)), '--', 'DisplayName', sprintf('$k_{R_x,stop} = %.0f$ Nm/rad', kRx_s))
-plot([0, Ty_at_theta_max], [1e3*theta_max, 1e3*theta_max], 'k--', 'HandleVisibility', 'off')
+plot([0, Tx_at_theta_max], [1e3*theta_max, 1e3*theta_max], 'k--', 'HandleVisibility', 'off')
+plot([0, Tx_at_theta_max], [0, 0], 'k--', 'HandleVisibility', 'off')
 anArrow = annotation('doublearrow', 'LineWidth', 0.5);
 anArrow.Parent = gca;
 anArrow.Position = [0.05, 0, 0, 1e3*fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1))];
-text(0.052, 0.4*1e3*fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1)), sprintf('$\\theta_{y,\\max} = %.1f$ mrad', 1e3*theta_max), 'horizontalalignment', 'left');
+text(0.052, 0.4*1e3*fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1)), sprintf('$\\theta_{x,\\max} = %.1f$ mrad', 1e3*theta_max), 'horizontalalignment', 'left');
 hold off;
-xlabel('Torque $T_y$ [Nm]');
+xlabel('Torque $T_x$ [Nm]');
-ylabel('Angle $\theta_y$ [mrad]');
+ylabel('Angle $\theta_x$ [mrad]');
 xlim([0, 0.15]);
 ylim([-5,25]);
 leg = legend('location', 'southeast', 'FontSize', 8);
@@ -983,7 +975,7 @@ exportFig('figs/test_joints_meas_F_d_lin_fit.pdf', 'width', 'half', 'height', 'n
 #+end_src
 
 #+name: fig:test_joints_meas_example
-#+caption: Results obtained on the first flexible joint. Measured force and displacement are shown in (\subref{fig:test_joints_meas_bend_time}). The estimated angular displacement $\theta_y$ as a function of the estimated applied torque $T_{y}$ is shown in (\subref{fig:test_joints_meas_F_d_lin_fit}). The bending stiffness $k_{R_y}$ of the flexible joint can be estimated by computing a best linear fit (red dashed line).
+#+caption: Results obtained on the first flexible joint. Measured force and displacement are shown in (\subref{fig:test_joints_meas_bend_time}). The estimated angular displacement $\theta_x$ as a function of the estimated applied torque $T_{x}$ is shown in (\subref{fig:test_joints_meas_F_d_lin_fit}). The bending stiffness $k_{R_x}$ of the flexible joint can be estimated by computing a best linear fit (red dashed line).
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_bend_time}Force and displacement measured as a function of time}
@@ -1000,210 +992,208 @@ exportFig('figs/test_joints_meas_F_d_lin_fit.pdf', 'width', 'half', 'height', 'n
 #+end_subfigure
 #+end_figure
 
-** TODO Comparison of the measured Stiffnesses
+** Measured flexible joint stiffnesses
-Now, let's estimate the bending stiffness and stroke for all the flexible joints.
+The same measurement is performed for all the 16 flexible joints, both in the $x$ and $y$ directions.
+The measured angular motion as a function of the applied torque are shown in Figure ref:fig:test_joints_meas_bending_all_raw_data for all the 16 flexible joints.
+This gives a first idea of the dispersion of the measured bending stiffnesses (i.e. slope of the linear region) and of the angular stroke.
+
+An histogram of the measured bending stiffnesses is show in Figure ref:fig:test_joints_bend_stiff_hist.
+Most of the bending stiffnesses are between $4.6\,Nm/rad$ and $5.0\,Nm/rad$.
 
 #+begin_src matlab :exports none
-%% Initialize variables
+%% Measure the bending stiffness and Stroke for all the flexible joints
-kRx = zeros(1,16);
+figure;
-kSx = zeros(1,16);
+hold on;
-Rmx = zeros(1,16);
+
+%% Start with the X-bending
+% Initialize variables
+Rx = zeros(1,16); % Bending stiffnesses [Nm/rad]
+kSx = zeros(1,16); % Bending stiffnesses at "stop" [Nm/rad]
+Rmx = zeros(1,16); % Bending stroke [rad]
 
 for i = 1:16
-    %% Load the data
+    % Load the data
-    load(['meas_stiff_flex_' num2str(i) '_x.mat'], 't', 'F', 'd');
+    load(sprintf('meas_stiff_flex_%i_x.mat', i), 't', 'F', 'd');
 
-    %% Automatic Zero of the force
+    % Start measurement at t = 0.2 s
-    F = F - mean(F(t > 0.1 & t < 0.3));
-
-    %% Start measurement at t = 0.2 s
     d = d(t > 0.2);
     F = F(t > 0.2);
     t = t(t > 0.2); t = t - t(1);
 
-    %% Estimate linear region and "stop" region
+    % Zero the force
+    F = F - mean(F(t < 0.2));
+
+    % Find when the force sensor touches the flexible joint
     i_l_start = find(F > 0.3, 1, 'first');
+
+    % Zero the displacement when it comes in contact
     d = d - d(i_l_start);
-    [~, i_s_stop] = max(F);
-    i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
-    i_s_start = find(F > max(F)-1, 1, 'first');
 
-    F_l = F(i_l_start:i_l_stop);
+    % Compute torque and angular displacement
-    d_l = d(i_l_start:i_l_stop);
+    h = 22.5e-3; % Height [m]
+    Tx = h * F; % Applied torque in [Nm]
+    thetax = atan2(d, h); % Measured angle in [rad]
 
-    F_s = F(i_s_start:i_s_stop);
+    % Find then the maximum torque is applied
-    d_s = d(i_s_start:i_s_stop);
+    [~, i_s_stop] = max(Tx);
+    % Linear region stops ~ when 90% of the stroke is reached
+    i_l_stop = find(thetax > 0.9*thetax(i_s_stop), 1, 'first');
+    % Mechanical "Stop" region start ~20Nmm before maximum torque is applied
+    i_s_start = find(Tx > max(Tx)-20e-3, 1, 'first');
 
-    %% Straight line fit
+    % Linear fit in the "linear" region
-    fit_l = polyfit(F_l, d_l, 1);
+    fit_l = polyfit(Tx(i_l_start:i_l_stop), thetax(i_l_start:i_l_stop), 1);
-    fit_s = polyfit(F_s, d_s, 1);
 
-    %% Reset displacement based on fit
+    % Linear fit in the "mechanical stop" region
-    d = d - fit_l(2);
+    fit_s = polyfit(Tx(i_s_start:i_s_stop), thetax(i_s_start:i_s_stop), 1);
+
+    % Reset displacement more precisely based on fit
+    thetax = thetax - fit_l(2);
     fit_s(2) = fit_s(2) - fit_l(2);
     fit_l(2) = 0;
 
-    %% Estimated Stroke
+    % Estimation of the bending stiffness and bending stroke
-    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
+    kRx(i) = 1/fit_l(1); % Bending Stiffness [Nm/rad]
-
+    kSx(i) = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad]
-    %% Save stiffnesses and stroke
+    Rmx(i) = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1)); % Maximum angular stroke [rad]
-    kRx(i) = (h)^2/fit_l(1);
-    kSx(i) = (h)^2/fit_s(1);
-    Rmx(i) = atan2(d_max,h);
 
+    if i == 1
+        plot(Tx(1:10:i_s_stop), 1e3*thetax(1:10:i_s_stop), '-', 'color', [colors(1,:), 0.4], ...
+            'DisplayName', '$k_{R_x}$')
+    else
+        plot(Tx(1:10:i_s_stop), 1e3*thetax(1:10:i_s_stop), '-', 'color', [colors(1,:), 0.4], ...
+            'HandleVisibility', 'off')
+    end
 end
-#+end_src
 
-#+begin_src matlab :exports none
+%% Continue with the Y-bending
-%% Initialize variables
+% Initialize variables
-kRy = zeros(1,16);
+kRy = zeros(1,16); % Bending stiffnesses [Nm/rad]
-kSy = zeros(1,16);
+kSy = zeros(1,16); % Bending stiffnesses at "stop" [Nm/rad]
-Rmy = zeros(1,16);
+Rmy = zeros(1,16); % Bending stroke [rad]
 
 for i = 1:16
-    %% Load the data
+    % Load the data
-    load(['meas_stiff_flex_' num2str(i) '_y.mat'], 't', 'F', 'd');
+    load(sprintf('meas_stiff_flex_%i_y.mat', i), 't', 'F', 'd');
 
-    %% Automatic Zero of the force
+    % Start measurement at t = 0.2 s
-    F = F - mean(F(t > 0.1 & t < 0.3));
-
-    %% Start measurement at t = 0.2 s
     d = d(t > 0.2);
     F = F(t > 0.2);
     t = t(t > 0.2); t = t - t(1);
 
-    %% Estimate linear region and "stop" region
+    % Zero the force
+    F = F - mean(F(t < 0.2));
+
+    % Find when the force sensor touches the flexible joint
     i_l_start = find(F > 0.3, 1, 'first');
+
+    % Zero the displacement when it comes in contact
     d = d - d(i_l_start);
-    [~, i_s_stop] = max(F);
-    i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
-    i_s_start = find(F > max(F)-1, 1, 'first');
 
-    F_l = F(i_l_start:i_l_stop);
+    % Compute torque and angular displacement
-    d_l = d(i_l_start:i_l_stop);
+    h = 22.5e-3; % Height [m]
+    Ty = h * F; % Applied torque in [Nm]
+    thetay = atan2(d, h); % Measured angle in [rad]
 
-    F_s = F(i_s_start:i_s_stop);
+    % Find then the maximum torque is applied
-    d_s = d(i_s_start:i_s_stop);
+    [~, i_s_stop] = max(Ty);
+    % Linear region stops ~ when 90% of the stroke is reached
+    i_l_stop = find(thetay > 0.9*thetay(i_s_stop), 1, 'first');
+    % Mechanical "Stop" region start ~20Nmm before maximum torque is applied
+    i_s_start = find(Ty > max(Ty)-20e-3, 1, 'first');
 
-    %% Straight line fit
+    % Linear fit in the "linear" region
-    fit_l = polyfit(F_l, d_l, 1);
+    fit_l = polyfit(Ty(i_l_start:i_l_stop), thetay(i_l_start:i_l_stop), 1);
-    fit_s = polyfit(F_s, d_s, 1);
 
-    %% Reset displacement based on fit
+    % Linear fit in the "mechanical stop" region
-    d = d - fit_l(2);
+    fit_s = polyfit(Ty(i_s_start:i_s_stop), thetay(i_s_start:i_s_stop), 1);
+
+    % Reset displacement more precisely based on fit
+    thetay = thetay - fit_l(2);
     fit_s(2) = fit_s(2) - fit_l(2);
     fit_l(2) = 0;
 
-    %% Estimated Stroke
+    % Estimation of the bending stiffness and bending stroke
-    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
+    kRy(i) = 1/fit_l(1); % Bending Stiffness [Nm/rad]
-
+    kSy(i) = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad]
-    %% Save stiffnesses and stroke
+    Rmy(i) = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1)); % Maximum angular stroke [rad]
-    kRy(i) = (h)^2/fit_l(1);
-    kSy(i) = (h)^2/fit_s(1);
-    Rmy(i) = atan2(d_max,h);
 
+    if i == 1
+        plot(Ty(1:10:i_s_stop), 1e3*thetay(1:10:i_s_stop), '-', 'color', [colors(2,:), 0.4], ...
+            'DisplayName', '$k_{R_y}$')
+    else
+        plot(Ty(1:10:i_s_stop), 1e3*thetay(1:10:i_s_stop), '-', 'color', [colors(2,:), 0.4], ...
+            'HandleVisibility', 'off')
     end
+end
+
+xlabel('Torque $T$ [Nm]');
+ylabel('Angle $\theta$ [mrad]');
+xlim([0, 0.15]);
+ylim([-5,25]);
+legend('location', 'southeast', 'FontSize', 8);
 #+end_src
 
-The results are summarized in Table ref:tab:test_joints_meas_results_x_dir for the X direction and in Table ref:tab:test_joints_meas_results_y_dir for the Y direction.
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/test_joints_meas_bending_all_raw_data.pdf', 'width', 'half', 'height', 'normal');
+#+end_src
+
+#+begin_src matlab :exports none
+figure;
+histogram([kRx, kRy], [4:0.2:5])
+xlabel('Bending stiffness [Nm/rad]')
+xticks([4:0.2:5])
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/test_joints_bend_stiff_hist.pdf', 'width', 'half', 'height', 'normal');
+#+end_src
+
+#+name: fig:test_joints_meas_bending_results
+#+caption: Result of measured $k_{R_x}$ and $k_{R_y}$ stiffnesses for all the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). An histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist})
+#+attr_latex: :options [htbp]
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_bending_all_raw_data}Measured torque and angular motion for all the flexible joints}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :height 5.3cm
+[[file:figs/test_joints_meas_bending_all_raw_data.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness in the x and y directions}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :height 5.3cm
+[[file:figs/test_joints_bend_stiff_hist.png]]
+#+end_subfigure
+#+end_figure
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-data2orgtable([kRx; kSx; 1e3*Rmx]', {'1','2','3','4','5','6','7','8','9','10','11','12','13','14','15','16'}, {'$R_{R_x}$ [Nm/rad]', '$k_{R_x,s}$ [Nm/rad]', '$R_{x,\text{max}}$ [mrad]'}, ' %.1f ');
+data2orgtable([kRx; kRy]', {'1','2','3','4','5','6','7','8','9','10','11','12','13','14','15','16'}, {'$R_{R_x}$ [Nm/rad]', '$k_{R_y}$ [Nm/rad]'}, ' %.1f ');
 #+end_src
 
 #+name: tab:test_joints_meas_results_x_dir
-#+caption: Measured characteristics of the flexible joints in the X direction
+#+caption: Measured bending stiffnesses
-#+attr_latex: :environment tabularx :width 0.6\linewidth :align cccc
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xccc
 #+attr_latex: :center t :booktabs t :float t
 #+RESULTS:
-|    | $R_{R_x}$ [Nm/rad] | $k_{R_x,s}$ [Nm/rad] | $R_{x,\text{max}}$ [mrad] |
+|    | $R_{R_x}$ [Nm/rad] | $k_{R_y}$ [Nm/rad] |   |
-|----+--------------------+----------------------+---------------------------|
+|----+--------------------+--------------------+---|
-|  1 |                5.5 |                173.6 |                      18.9 |
+|  1 |                4.4 |                4.6 | X |
-|  2 |                6.1 |                195.0 |                      17.6 |
+|  2 |                5.0 |                4.8 | X |
-|  3 |                6.1 |                191.3 |                      17.7 |
+|  3 |                5.0 |                4.8 | X |
-|  4 |                5.8 |                136.7 |                      18.3 |
+|  4 |                4.7 |                4.6 | X |
-|  5 |                5.7 |                 88.9 |                      22.0 |
+|  5 |                4.6 |                4.7 | X |
-|  6 |                5.7 |                183.9 |                      18.7 |
+|  6 |                4.6 |                4.7 | X |
-|  7 |                5.7 |                157.9 |                      17.9 |
+|  7 |                4.7 |                4.3 |   |
-|  8 |                5.8 |                166.1 |                      17.9 |
+|  8 |                4.7 |                4.7 | X |
-|  9 |                5.8 |                159.5 |                      18.2 |
+|  9 |                4.7 |                4.6 | X |
-| 10 |                6.0 |                143.6 |                      18.1 |
+| 10 |                4.9 |                4.8 | X |
-| 11 |                5.0 |                163.8 |                      17.7 |
+| 11 |                4.0 |                3.9 |   |
-| 12 |                6.1 |                111.9 |                      17.0 |
+| 12 |                4.9 |                4.8 | X |
-| 13 |                6.0 |                142.0 |                      17.4 |
+| 13 |                4.9 |                4.9 | X |
-| 14 |                5.8 |                130.1 |                      17.9 |
+| 14 |                4.7 |                4.7 | X |
-| 15 |                5.7 |                170.7 |                      18.6 |
+| 15 |                4.6 |                4.4 |   |
-| 16 |                6.0 |                148.7 |                      17.5 |
+| 16 |                4.9 |                4.3 |   |
-
-#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-data2orgtable([kRy; kSy; 1e3*Rmy]', {'1','2','3','4','5','6','7','8','9','10','11','12','13','14','15','16'}, {'$R_{R_y}$ [Nm/rad]', '$k_{R_y,s}$ [Nm/rad]', '$R_{y,\text{may}}$ [mrad]'}, ' %.1f ');
-#+end_src
-
-#+name: tab:test_joints_meas_results_y_dir
-#+caption: Measured characteristics of the flexible joints in the Y direction
-#+attr_latex: :environment tabularx :width 0.6\linewidth :align cccc
-#+attr_latex: :center t :booktabs t :float t
-#+RESULTS:
-|    | $R_{R_y}$ [Nm/rad] | $k_{R_y,s}$ [Nm/rad] | $R_{y,\text{may}}$ [mrad] |
-|----+--------------------+----------------------+---------------------------|
-|  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 |
-
-The dispersion of the measured bending stiffness is shown in Figure ref:fig:test_joints_bend_stiff_hist and of the bending stroke in Figure ref:fig:test_joints_bend_stroke_hist.
-
-#+begin_src matlab :exports none
-figure;
-hold on;
-histogram(kRx, 'DisplayName', '$k_{R_x}$')
-histogram(kRy, 'DisplayName', '$k_{R_y}$')
-hold off;
-xlabel('Bending Stiffness [Nm/rad]')
-legend('FontSize', 8);
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/test_joints_bend_stiff_hist.pdf', 'width', 'wide', 'height', 'normal');
-#+end_src
-
-#+name: fig:test_joints_bend_stiff_hist
-#+caption: Histogram of the measured bending stiffness
-#+RESULTS:
-[[file:figs/test_joints_bend_stiff_hist.png]]
-
-#+begin_src matlab :exports none
-figure;
-hold on;
-histogram(1e3*Rmx, 'DisplayName', '$k_{R_x}$')
-histogram(1e3*Rmy, 'DisplayName', '$k_{R_y}$')
-hold off;
-xlabel('Bending Stroke [mrad]')
-legend('FontSize', 8);
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/test_joints_bend_stroke_hist.pdf', 'width', 'wide', 'height', 'normal');
-#+end_src
-
-#+name: fig:test_joints_bend_stroke_hist
-#+caption: Histogram of the measured bending stroke
-#+RESULTS:
-[[file:figs/test_joints_bend_stroke_hist.png]]
-
-The relation between the measured beam thickness and the measured bending stiffness is shown in Figure ref:fig:test_joints_thickness_stiffness.
 
 #+begin_src matlab :exports none
 load('flex_meas_dim.mat', 'meas_flex');

test-bench-flexible-joints.tex

@@ -1,4 +1,4 @@
-% Created 2024-04-05 Fri 17:52
+% Created 2024-04-06 Sat 00:18
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -29,6 +29,7 @@ Deviations from this ideal properties will impact the dynamics of the Nano-Hexap
 During the detailed design phase, specifications in term of stiffness and stroke have been determined and are summarized in Table \ref{tab:test_joints_specs}.
 
 \begin{table}[htbp]
+\caption{\label{tab:test_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model}
 \centering
 \begin{tabularx}{0.5\linewidth}{Xcc}
 \toprule
@@ -41,8 +42,6 @@ Torsion Stiffness & \(< 500\,Nm/\text{rad}\) & 260\\
 Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:test_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model}
-
 \end{table}
 
 After optimization using a finite element model, the geometry shown in Figure \ref{fig:test_joints_schematic} has been obtained and the corresponding flexible joints characteristics are summarized in Table \ref{tab:test_joints_specs}.
@@ -104,6 +103,7 @@ Finally, the test bench is manufacturer and used to measure the bending stiffnes
 Results are shown in Section \ref{sec:test_joints_bending_stiffness_meas}
 
 \begin{table}[htbp]
+\caption{\label{tab:test_joints_section_matlab_code}Report sections and corresponding Matlab files}
 \centering
 \begin{tabularx}{0.6\linewidth}{lX}
 \toprule
@@ -114,8 +114,6 @@ Section \ref{sec:test_joints_test_bench_desc} & \texttt{test\_joints\_2\_bench\_
 Section \ref{sec:test_joints_bending_stiffness_meas} & \texttt{test\_joints\_3\_bending\_stiff\_meas.m}\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:test_joints_section_matlab_code}Report sections and corresponding Matlab files}
-
 \end{table}
 \chapter{Dimensional Measurements}
 \label{sec:test_joints_flex_dim_meas}
@@ -359,6 +357,7 @@ The most important source of error comes from estimation error of the distance b
 An overall accuracy of \(\approx 5\,\%\) can be expected with this measurement bench, which should be enough for a first estimation of the bending stiffness of the flexible joints.
 
 \begin{table}[htbp]
+\caption{\label{tab:test_joints_error_budget}Summary of the error budget for the estimation of the bending stiffness}
 \centering
 \begin{tabularx}{0.4\linewidth}{lX}
 \toprule
@@ -371,8 +370,6 @@ Displacement sensor & \(\epsilon_d < 0.01\,\%\)\\
 Force sensor & \(\epsilon_F < 1\,\%\)\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:test_joints_error_budget}Summary of the error budget for the estimation of the bending stiffness}
-
 \end{table}
 \chapter{Bending Stiffness Measurement}
 \label{sec:test_joints_bending_stiffness_meas}
@@ -397,7 +394,6 @@ A closer view on the force sensor tip is shown in Figure \ref{fig:test_joints_pi
 \caption{\label{fig:test_joints_picture_bench}Caption with reference to sub figure (\subref{fig:fig_label_a})}
 \end{figure}
 \section{Load Cell Calibration}
-
 In order to estimate the measured errors of the load cell ``FC2231'', it is compared against another load cell\footnote{XFL212R-50N from TE Connectivity. Measurement range is \(50\,N\). Specified accuracy is \(1\,\%\) of the full range}.
 The two load cells are measured simultaneously while they are pushed against each other (see Figure \ref{fig:test_joints_force_sensor_calib_picture}).
 The contact between the two load cells is well defined as one has a spherical interface while the other has a flat surface.
@@ -423,7 +419,7 @@ However, the estimated non-linearity is bellow \(1\,\%\) for forces between \(0.
 \end{figure}
 \section{Load Cell Stiffness}
 The objective of this measurement is to estimate the stiffness \(k_F\) of the force sensor.
-To do so, a stiff element (much stiffer than the estimated \(k_F \approx 1\,N/\mu m\)) is fixed in front of the force sensor as shown in Figure \ref{fig:test_joints_meas_force_sensor_stiffness}.
+To do so, a stiff element (much stiffer than the estimated \(k_F \approx 1\,N/\mu m\)) is fixed in front of the force sensor as shown in Figure \ref{fig:test_joints_meas_force_sensor_stiffness_picture}.
 Then, the force sensor is pushed again this stiff element while the force and the encoder displacement are measured.
 Measured displacement as a function of the force is shown in Figure \ref{fig:test_joints_force_sensor_stiffness_fit}.
 The load cell stiffness can then be estimated by computing a linear fit, and is found to be \(k_F \approx 0.75\,N/\mu m\).
@@ -441,10 +437,9 @@ The load cell stiffness can then be estimated by computing a linear fit, and is
 \end{center}
 \subcaption{\label{fig:test_joints_force_sensor_stiffness_fit}Measured displacement as a function of the force}
 \end{subfigure}
-\caption{\label{fig:test_joints_meas_force_sensor_stiffness}Estimation of the load cell stiffness. (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}) (\subref{fig:test_joints_meas_force_sensor_stiffness_fit})}
+\caption{\label{fig:test_joints_meas_force_sensor_stiffness}Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}). Measurement results is shown in (\subref{fig:test_joints_force_sensor_stiffness_fit}).}
 \end{figure}
-\section{Analysis of one measurement}
+\section{Bending Stiffness estimation}
-
 The actual stiffness measurement in now performed by manually moving the translation stage from a start position where the force sensor is not yet in contact with the flexible joint to a position where flexible joint is on its mechanical stop.
 
 The measured force and displacement as a function of time are shown in Figure \ref{fig:test_joints_meas_bend_time}.
@@ -467,85 +462,54 @@ The bending stroke can also be estimated as shown in Figure \ref{fig:test_joints
 \end{center}
 \subcaption{\label{fig:test_joints_meas_F_d_lin_fit}Angular displacement measured as a function of the applied torque}
 \end{subfigure}
-\caption{\label{fig:test_joints_meas_example}Results obtained on the first flexible joint. Measured force and displacement are shown in (\subref{fig:test_joints_meas_bend_time}). The estimated angular displacement \(\theta_y\) as a function of the estimated applied torque \(T_{y}\) is shown in (\subref{fig:test_joints_meas_F_d_lin_fit}). The bending stiffness \(k_{R_y}\) of the flexible joint can be estimated by computing a best linear fit (red dashed line).}
+\caption{\label{fig:test_joints_meas_example}Results obtained on the first flexible joint. Measured force and displacement are shown in (\subref{fig:test_joints_meas_bend_time}). The estimated angular displacement \(\theta_x\) as a function of the estimated applied torque \(T_{x}\) is shown in (\subref{fig:test_joints_meas_F_d_lin_fit}). The bending stiffness \(k_{R_x}\) of the flexible joint can be estimated by computing a best linear fit (red dashed line).}
 \end{figure}
-\section{Bending stiffness and bending stroke of all the flexible joints}
+\section{Measured flexible joint stiffnesses}
-
+The same measurement is performed for all the 16 flexible joints, both in the \(x\) and \(y\) directions.
-Now, let's estimate the bending stiffness and stroke for all the flexible joints.
+The measured torque and angular motion are all shown in Figure \ref{fig:test_joints_meas_bending_all_raw_data}.
-
-The results are summarized in Table \ref{tab:test_joints_meas_results_x_dir} for the X direction and in Table \ref{tab:test_joints_meas_results_y_dir} for the Y direction.
-
-\begin{table}[htbp]
-\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}
-\caption{\label{tab:test_joints_meas_results_x_dir}Measured characteristics of the flexible joints in the X direction}
-
-\end{table}
-
-\begin{table}[htbp]
-\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}
-\caption{\label{tab:test_joints_meas_results_y_dir}Measured characteristics of the flexible joints in the Y direction}
-
-\end{table}
-\section{Analysis}
-The dispersion of the measured bending stiffness is shown in Figure \ref{fig:test_joints_bend_stiff_hist} and of the bending stroke in Figure \ref{fig:test_joints_bend_stroke_hist}.
 
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{0.49\textwidth}
-\includegraphics[scale=1]{figs/test_joints_bend_stiff_hist.png}
+\begin{center}
-\caption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness}
+\includegraphics[scale=1,height=5.3cm]{figs/test_joints_meas_bending_all_raw_data.png}
+\end{center}
+\subcaption{\label{fig:test_joints_meas_bending_all_raw_data}Measured torque and angular motion for all the flexible joints}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,height=5.3cm]{figs/test_joints_bend_stiff_hist.png}
+\end{center}
+\subcaption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness in the x and y directions}
+\end{subfigure}
+\caption{\label{fig:test_joints_meas_bending_results}Result of measured stiffnesses for all the flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). An histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist})}
 \end{figure}
 
-\begin{figure}[htbp]
+\begin{table}[htbp]
+\caption{\label{tab:test_joints_meas_results_x_dir}Measured bending stiffnesses}
 \centering
-\includegraphics[scale=1]{figs/test_joints_bend_stroke_hist.png}
+\begin{tabularx}{0.6\linewidth}{Xccc}
-\caption{\label{fig:test_joints_bend_stroke_hist}Histogram of the measured bending stroke}
+\toprule
-\end{figure}
+ & \(R_{R_x}\) {[}Nm/rad] & \(k_{R_y}\) {[}Nm/rad] & \\
-
+\midrule
-The relation between the measured beam thickness and the measured bending stiffness is shown in Figure \ref{fig:test_joints_thickness_stiffness}.
+1 & 4.4 & 4.6 & X\\
+2 & 5.0 & 4.8 & X\\
+3 & 5.0 & 4.8 & X\\
+4 & 4.7 & 4.6 & X\\
+5 & 4.6 & 4.7 & X\\
+6 & 4.6 & 4.7 & X\\
+7 & 4.7 & 4.3 & \\
+8 & 4.7 & 4.7 & X\\
+9 & 4.7 & 4.6 & X\\
+10 & 4.9 & 4.8 & X\\
+11 & 4.0 & 3.9 & \\
+12 & 4.9 & 4.8 & X\\
+13 & 4.9 & 4.9 & X\\
+14 & 4.7 & 4.7 & X\\
+15 & 4.6 & 4.4 & \\
+16 & 4.9 & 4.3 & \\
+\bottomrule
+\end{tabularx}
+\end{table}
 
 \begin{figure}[htbp]
 \centering