test-bench-flexible-joints.org

@@ -345,9 +345,7 @@ 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
@@ -733,6 +731,7 @@ 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.
@@ -781,7 +780,7 @@ exportFig('figs/test_joints_force_sensor_calib_fit.pdf', 'width', 'half', 'heigh
 #+end_src
 
 #+name: fig:test_joints_force_sensor_calib
-#+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})
+#+caption: Caption with reference to sub figure (\subref{fig:test_joints_force_sensor_calib_picture}),  (\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}
@@ -839,10 +838,10 @@ 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_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_meas_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 measurement bench}
+#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_force_sensor_stiffness_picture}Picture of the test}
 #+attr_latex: :options {0.49\textwidth}
 #+begin_subfigure
 #+attr_latex: :height 5.5cm
@@ -867,52 +866,62 @@ 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
-%% Estimate the bending stiffness and stroke from the measurement - First Flexible joint
+%% Load Measured Data for first flexible joint
-% Load Measured Data for first flexible joint
 load('meas_stiff_flex_1_x.mat', 't', 'F', 'd');
 
-% Start measurement at t = 0.2 s
+%% Zero the force
+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);
 
-% Zero the force
+%% Compute torque and angular displacement
-F = F - mean(F(t < 0.2));
+h = 22.5e-3; % Height [m]
+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);
 
-% Compute torque and angular displacement
+%% Determine the linear region and region when touching the mechanical stop
-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(Tx);
+[~, i_s_stop] = max(Ty);
 % Linear region stops ~ when 90% of the stroke is reached
-i_l_stop = find(thetax > 0.9*thetax(i_s_stop), 1, 'first');
+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(Tx > max(Tx)-20e-3, 1, 'first');
+i_s_start = find(Ty > max(Ty)-20e-3, 1, 'first');
 
-% Linear fit in the "linear" region
+%% Define variables for the two regions
-fit_l = polyfit(Tx(i_l_start:i_l_stop), thetax(i_l_start:i_l_stop), 1);
+% Linear Region
+Ty_l = Ty(i_l_start:i_l_stop);
+thetay_l = thetay(i_l_start:i_l_stop);
 
-% Linear fit in the "mechanical stop" region
+% "Mechanical stop" Region
-fit_s = polyfit(Tx(i_s_start:i_s_stop), thetax(i_s_start:i_s_stop), 1);
+Ty_s = Ty(i_s_start:i_s_stop);
+thetay_s = thetay(i_s_start:i_s_stop);
 
-% Reset displacement more precisely based on fit
+%% Fit the best straight line for the two regions
-thetax = thetax - fit_l(2);
+fit_l = polyfit(Ty_l, thetay_l, 1);
+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
-kRx_l = 1/fit_l(1); % Bending Stiffness [Nm/rad]
+kRy_l = 1/fit_l(1); % Bending Stiffness [Nm/rad]
-kRx_s = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad]
+kRy_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]
-Tx_at_theta_max = (fit_s(2) - fit_l(2))/(fit_l(1) - fit_s(1));
+Ty_at_theta_max = (fit_s(2) - fit_l(2))/(fit_l(1) - fit_s(1));
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -928,14 +937,14 @@ 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 $F_y$ [N]');
+ylabel('Force $F_x$ [N]');
 ylim([-4, 6])
 ylimr = get(gca,'Ylim');
 
 yyaxis right
 plot(t, 1e3*d);
 xlabel('Time [s]');
-ylabel('Displacement $d_y$ [mm]');
+ylabel('Displacement $d_x$ [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])
@@ -947,23 +956,22 @@ exportFig('figs/test_joints_meas_bend_time.pdf', 'width', 'half', 'height', 'nor
 
 #+begin_src matlab :exports none
 % Regions where to plot the fitted data
-Tx_l_fit = [0, Tx_at_theta_max];
+Ty_l_fit = [0, Ty_at_theta_max];
-Tx_s_fit = [Tx_at_theta_max, Tx(i_s_stop)];
+Ty_s_fit = [Ty_at_theta_max, Ty(i_s_stop)];
 
 figure;
 hold on;
-plot(Tx(1:i_s_stop), 1e3*thetax(1:i_s_stop), '-k', 'DisplayName', 'Raw data')
+plot(Ty(1:i_s_stop), 1e3*thetay(1:i_s_stop), '-k', 'DisplayName', 'Raw data')
-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_l_fit, 1e3*(Ty_l_fit*fit_l(1) + fit_l(2)), '--', 'DisplayName', sprintf('$k_{R_y} = %.1f$ Nm/rad', kRy_l))
-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(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([0, Tx_at_theta_max], [1e3*theta_max, 1e3*theta_max], 'k--', 'HandleVisibility', 'off')
+plot([0, Ty_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('$\\theta_{x,\\max} = %.1f$ 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('$\\theta_{y,\\max} = %.1f$ mrad', 1e3*theta_max), 'horizontalalignment', 'left');
 hold off;
-xlabel('Torque $T_x$ [Nm]');
+xlabel('Torque $T_y$ [Nm]');
-ylabel('Angle $\theta_x$ [mrad]');
+ylabel('Angle $\theta_y$ [mrad]');
 xlim([0, 0.15]);
 ylim([-5,25]);
 leg = legend('location', 'southeast', 'FontSize', 8);
@@ -975,7 +983,7 @@ 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 $\theta_x$ 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).
+#+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).
 #+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}
@@ -992,190 +1000,243 @@ exportFig('figs/test_joints_meas_F_d_lin_fit.pdf', 'width', 'half', 'height', 'n
 #+end_subfigure
 #+end_figure
 
-** Measured flexible joint stiffnesses
+** TODO Comparison of the measured 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 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.
+#+begin_src matlab :exports none
-
+%% Initialize variables
-An histogram of the measured bending stiffnesses is show in Figure ref:fig:test_joints_bend_stiff_hist.
+kRx = zeros(1,16);
-Most of the bending stiffnesses are between $4.6\,Nm/rad$ and $5.0\,Nm/rad$.
+kSx = zeros(1,16);
+Rmx = zeros(1,16);
+
+for i = 1:16
+    %% Load the data
+    load(['meas_stiff_flex_' num2str(i) '_x.mat'], 't', 'F', 'd');
+
+    %% Automatic Zero of the force
+    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
+    i_l_start = find(F > 0.3, 1, 'first');
+    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);
+    d_l = d(i_l_start:i_l_stop);
+
+    F_s = F(i_s_start:i_s_stop);
+    d_s = d(i_s_start:i_s_stop);
+
+    %% Straight line fit
+    fit_l = polyfit(F_l, d_l, 1);
+    fit_s = polyfit(F_s, d_s, 1);
+
+    %% Reset displacement based on fit
+    d = d - fit_l(2);
+    fit_s(2) = fit_s(2) - fit_l(2);
+    fit_l(2) = 0;
+
+    %% Estimated Stroke
+    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
+
+    %% Save stiffnesses and stroke
+    kRx(i) = (h)^2/fit_l(1);
+    kSx(i) = (h)^2/fit_s(1);
+    Rmx(i) = atan2(d_max,h);
+
+end
+#+end_src
+
+#+begin_src matlab :exports none
+%% Initialize variables
+kRy = zeros(1,16);
+kSy = zeros(1,16);
+Rmy = zeros(1,16);
+
+for i = 1:16
+    %% Load the data
+    load(['meas_stiff_flex_' num2str(i) '_y.mat'], 't', 'F', 'd');
+
+    %% Automatic Zero of the force
+    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
+    i_l_start = find(F > 0.3, 1, 'first');
+    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);
+    d_l = d(i_l_start:i_l_stop);
+
+    F_s = F(i_s_start:i_s_stop);
+    d_s = d(i_s_start:i_s_stop);
+
+    %% Straight line fit
+    fit_l = polyfit(F_l, d_l, 1);
+    fit_s = polyfit(F_s, d_s, 1);
+
+    %% Reset displacement based on fit
+    d = d - fit_l(2);
+    fit_s(2) = fit_s(2) - fit_l(2);
+    fit_l(2) = 0;
+
+    %% Estimated Stroke
+    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
+
+    %% Save stiffnesses and stroke
+    kRy(i) = (h)^2/fit_l(1);
+    kSy(i) = (h)^2/fit_s(1);
+    Rmy(i) = atan2(d_max,h);
+
+end
+#+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 :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 ');
+#+end_src
+
+#+name: tab:test_joints_meas_results_x_dir
+#+caption: Measured characteristics of the flexible joints in the X direction
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align cccc
+#+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] |
+|----+--------------------+----------------------+---------------------------|
+|  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 |
+
+#+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
-%% Measure the bending stiffness and Stroke for all the flexible joints
 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
 
-%% Start with the X-bending
+#+begin_src matlab :tangle no :exports results :results file replace
-% Initialize variables
+exportFig('figs/test_joints_bend_stiff_hist.pdf', 'width', 'wide', 'height', 'normal');
-Rx = zeros(1,16); % Bending stiffnesses [Nm/rad]
+#+end_src
-kSx = zeros(1,16); % Bending stiffnesses at "stop" [Nm/rad]
-Rmx = zeros(1,16); % Bending stroke [rad]
 
-for i = 1:16
+#+name: fig:test_joints_bend_stiff_hist
-    % Load the data
+#+caption: Histogram of the measured bending stiffness
-    load(sprintf('meas_stiff_flex_%i_x.mat', i), 't', 'F', 'd');
+#+RESULTS:
+[[file:figs/test_joints_bend_stiff_hist.png]]
 
-    % Start measurement at t = 0.2 s
+#+begin_src matlab :exports none
-    d = d(t > 0.2);
+figure;
-    F = F(t > 0.2);
+hold on;
-    t = t(t > 0.2); t = t - t(1);
+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
 
-    % Zero the force
+#+begin_src matlab :tangle no :exports results :results file replace
-    F = F - mean(F(t < 0.2));
+exportFig('figs/test_joints_bend_stroke_hist.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
 
-    % Find when the force sensor touches the flexible joint
+#+name: fig:test_joints_bend_stroke_hist
-    i_l_start = find(F > 0.3, 1, 'first');
+#+caption: Histogram of the measured bending stroke
+#+RESULTS:
+[[file:figs/test_joints_bend_stroke_hist.png]]
 
-    % Zero the displacement when it comes in contact
+The relation between the measured beam thickness and the measured bending stiffness is shown in Figure ref:fig:test_joints_thickness_stiffness.
-    d = d - d(i_l_start);
 
-    % Compute torque and angular displacement
+#+begin_src matlab :exports none
-    h = 22.5e-3; % Height [m]
+load('flex_meas_dim.mat', 'meas_flex');
-    Tx = h * F; % Applied torque in [Nm]
+figure;
-    thetax = atan2(d, h); % Measured angle in [rad]
+hold on;
-
+plot((meas_flex(:,1)+meas_flex(:,2))/2, kRx, 'o', 'DisplayName', '$x$')
-    % Find then the maximum torque is applied
+plot((meas_flex(:,3)+meas_flex(:,4))/2, kRy, 'o', 'DisplayName', '$y$')
-    [~, i_s_stop] = max(Tx);
+hold off;
-    % Linear region stops ~ when 90% of the stroke is reached
+xlabel('Flexible Beam Thickness [$\mu m$]');
-    i_l_stop = find(thetax > 0.9*thetax(i_s_stop), 1, 'first');
+ylabel('Bending Stiffness [Nm/rad]');
-    % Mechanical "Stop" region start ~20Nmm before maximum torque is applied
-    i_s_start = find(Tx > max(Tx)-20e-3, 1, 'first');
-
-    % Linear fit in the "linear" region
-    fit_l = polyfit(Tx(i_l_start:i_l_stop), thetax(i_l_start:i_l_stop), 1);
-
-    % Linear fit in the "mechanical stop" region
-    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;
-
-    % Estimation of the bending stiffness and bending stroke
-    kRx(i) = 1/fit_l(1); % Bending Stiffness [Nm/rad]
-    kSx(i) = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad]
-    Rmx(i) = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1)); % Maximum angular stroke [rad]
-
-    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
-
-%% Continue with the Y-bending
-% Initialize variables
-kRy = zeros(1,16); % Bending stiffnesses [Nm/rad]
-kSy = zeros(1,16); % Bending stiffnesses at "stop" [Nm/rad]
-Rmy = zeros(1,16); % Bending stroke [rad]
-
-for i = 1:16
-    % Load the data
-    load(sprintf('meas_stiff_flex_%i_y.mat', i), 't', 'F', 'd');
-
-    % 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);
-
-    % 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);
-
-    % Compute torque and angular displacement
-    h = 22.5e-3; % Height [m]
-    Ty = h * F; % Applied torque in [Nm]
-    thetay = atan2(d, h); % Measured angle in [rad]
-
-    % Find then the maximum torque is applied
-    [~, 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');
-
-    % Linear fit in the "linear" region
-    fit_l = polyfit(Ty(i_l_start:i_l_stop), thetay(i_l_start:i_l_stop), 1);
-
-    % Linear fit in the "mechanical stop" region
-    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;
-
-    % Estimation of the bending stiffness and bending stroke
-    kRy(i) = 1/fit_l(1); % Bending Stiffness [Nm/rad]
-    kSy(i) = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad]
-    Rmy(i) = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1)); % Maximum angular stroke [rad]
-
-    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
 
 #+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/test_joints_meas_bending_all_raw_data.pdf', 'width', 'half', 'height', 'normal');
+exportFig('figs/test_joints_thickness_stiffness.pdf', 'width', 'wide', 'height', 'normal');
 #+end_src
 
-#+begin_src matlab :exports none
+#+name: fig:test_joints_thickness_stiffness
-figure;
+#+caption: Measured bending stiffness as a function of the estimated flexible beam thickness
-histogram([kRx, kRy], [4:0.2:5])
+#+RESULTS:
-xlabel('Bending stiffness [Nm/rad]')
+[[file:figs/test_joints_thickness_stiffness.png]]
-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
 
 ** Conclusion
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
 
+#+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
 
 * Conclusion
 <<sec:test_joints_conclusion>>

test-bench-flexible-joints.tex

@@ -1,4 +1,4 @@
-% Created 2024-04-06 Sat 00:30
+% Created 2024-04-05 Fri 17:52
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -29,7 +29,6 @@ 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
@@ -42,6 +41,8 @@ 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}.
@@ -103,7 +104,6 @@ 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
@@ -114,6 +114,8 @@ 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}
@@ -357,7 +359,6 @@ 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
@@ -370,6 +371,8 @@ 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}
@@ -394,6 +397,7 @@ 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.
@@ -415,11 +419,11 @@ However, the estimated non-linearity is bellow \(1\,\%\) for forces between \(0.
 \end{center}
 \subcaption{\label{fig:test_joints_force_sensor_calib_fit}Measured two force and linear fit}
 \end{subfigure}
-\caption{\label{fig:test_joints_force_sensor_calib}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})}
+\caption{\label{fig:test_joints_force_sensor_calib}Caption with reference to sub figure (\subref{fig:test_joints_force_sensor_calib_picture}),  (\subref{fig:test_joints_force_sensor_calib_fit})}
 \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_picture}.
+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}.
 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 \(k_F \approx 0.75\,N/\mu m\).
@@ -429,7 +433,7 @@ The load cell stiffness can then be estimated by computing a linear fit, and is
 \begin{center}
 \includegraphics[scale=1,height=5.5cm]{figs/test_joints_meas_force_sensor_stiffness_picture.jpg}
 \end{center}
-\subcaption{\label{fig:test_joints_meas_force_sensor_stiffness_picture}Picture of the measurement bench}
+\subcaption{\label{fig:test_joints_meas_force_sensor_stiffness_picture}Picture of the test}
 \end{subfigure}
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
@@ -437,9 +441,10 @@ 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. 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}).}
+\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})}
 \end{figure}
-\section{Bending Stiffness estimation}
+\section{Analysis of one measurement}
+
 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}.
@@ -462,36 +467,98 @@ 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 \(\theta_x\) 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).}
+\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).}
 \end{figure}
-\section{Measured flexible joint stiffnesses}
+\section{Bending stiffness and bending stroke of 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}.
+Now, let's estimate the bending stiffness and stroke for all the flexible joints.
-Most of the bending stiffnesses are between \(4.6\,Nm/rad\) and \(5.0\,Nm/rad\).
+
+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]
-\begin{subfigure}{0.49\textwidth}
+\centering
-\begin{center}
+\includegraphics[scale=1]{figs/test_joints_bend_stiff_hist.png}
-\includegraphics[scale=1,height=5.3cm]{figs/test_joints_meas_bending_all_raw_data.png}
+\caption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness}
-\end{center}
+\end{figure}
-\subcaption{\label{fig:test_joints_meas_bending_all_raw_data}Measured torque and angular motion for all the flexible joints}
+
-\end{subfigure}
+\begin{figure}[htbp]
-\begin{subfigure}{0.49\textwidth}
+\centering
-\begin{center}
+\includegraphics[scale=1]{figs/test_joints_bend_stroke_hist.png}
-\includegraphics[scale=1,height=5.3cm]{figs/test_joints_bend_stiff_hist.png}
+\caption{\label{fig:test_joints_bend_stroke_hist}Histogram of the measured bending stroke}
-\end{center}
+\end{figure}
-\subcaption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness in the x and y directions}
+
-\end{subfigure}
+The relation between the measured beam thickness and the measured bending stiffness is shown in Figure \ref{fig:test_joints_thickness_stiffness}.
-\caption{\label{fig:test_joints_meas_bending_results}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})}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/test_joints_thickness_stiffness.png}
+\caption{\label{fig:test_joints_thickness_stiffness}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:test_joints_conclusion}
 \printbibliography[heading=bibintoc,title={Bibliography}]