From 57bf1e0b8085614677cd51bbca2573dc043b8c46 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Sat, 6 Apr 2024 00:27:19 +0200 Subject: [PATCH] Better estimate bending stiffness --- figs/test_joints_bend_stiff_hist.pdf | Bin 17080 -> 12758 bytes figs/test_joints_bend_stiff_hist.png | Bin 12110 -> 10417 bytes figs/test_joints_meas_F_d_lin_fit.pdf | Bin 128019 -> 128305 bytes figs/test_joints_meas_F_d_lin_fit.png | Bin 29193 -> 29100 bytes figs/test_joints_meas_bend_time.pdf | Bin 60630 -> 60652 bytes figs/test_joints_meas_bend_time.png | Bin 42272 -> 42470 bytes .../test_joints_meas_bending_all_raw_data.pdf | Bin 0 -> 996025 bytes .../test_joints_meas_bending_all_raw_data.png | Bin 0 -> 39947 bytes figs/test_joints_thickness_stiffness.pdf | Bin 20459 -> 21213 bytes figs/test_joints_thickness_stiffness.png | Bin 32354 -> 34311 bytes test-bench-flexible-joints.org | 398 +++++++++--------- test-bench-flexible-joints.pdf | Bin 5222292 -> 5994402 bytes test-bench-flexible-joints.tex | 134 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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 -F = F - mean(F(t > 0.1 & t < 0.3)); - -%% Start measurement at t = 0.2 s +% 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 -h = 22.5e-3; % Height [m] -Ty = h * F; % Applied torque in [Nm] -thetay = atan2(d, h); % Measured angle in [rad] +% Zero the force +F = F - mean(F(t < 0.2)); -%% 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 Region -Ty_l = Ty(i_l_start:i_l_stop); -thetay_l = thetay(i_l_start:i_l_stop); +% Linear fit in the "linear" region +fit_l = polyfit(Tx(i_l_start:i_l_stop), thetax(i_l_start:i_l_stop), 1); -% "Mechanical stop" Region -Ty_s = Ty(i_s_start:i_s_stop); -thetay_s = thetay(i_s_start:i_s_stop); +% 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); -%% Fit the best straight line for the two regions -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); +% 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 -kRy_l = 1/fit_l(1); % Bending Stiffness [Nm/rad] -kRy_s = 1/fit_s(1); % Mechanical "Stop" Stiffness [Nm/rad] +kRx_l = 1/fit_l(1); % Bending 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]; -Ty_s_fit = [Ty_at_theta_max, Ty(i_s_stop)]; +Tx_l_fit = [0, Tx_at_theta_max]; +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(Ty_l_fit, 1e3*(Ty_l_fit*fit_l(1) + fit_l(2)), '--', 'DisplayName', sprintf('$k_{R_y} = %.1f$ Nm/rad', kRy_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([0, Ty_at_theta_max], [1e3*theta_max, 1e3*theta_max], 'k--', 'HandleVisibility', 'off') +plot(Tx(1:i_s_stop), 1e3*thetax(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(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, 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]'); -ylabel('Angle $\theta_y$ [mrad]'); +xlabel('Torque $T_x$ [Nm]'); +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 -Now, let's estimate the bending stiffness and stroke for all the flexible joints. +** Measured flexible joint stiffnesses +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 -kRx = zeros(1,16); -kSx = zeros(1,16); -Rmx = zeros(1,16); +%% Measure the bending stiffness and Stroke for all the flexible joints +figure; +hold on; + +%% 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(['meas_stiff_flex_' num2str(i) '_x.mat'], 't', 'F', 'd'); + % Load the data + load(sprintf('meas_stiff_flex_%i_x.mat', i), '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 + % 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); - d_l = d(i_l_start:i_l_stop); + % Compute torque and angular displacement + 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); - d_s = d(i_s_start:i_s_stop); + % Find then the maximum torque is applied + [~, 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 - fit_l = polyfit(F_l, d_l, 1); - fit_s = polyfit(F_s, d_s, 1); + % Linear fit in the "linear" region + fit_l = polyfit(Tx(i_l_start:i_l_stop), thetax(i_l_start:i_l_stop), 1); - %% Reset displacement based on fit - d = d - fit_l(2); + % 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; - %% 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); + % 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 -#+end_src -#+begin_src matlab :exports none -%% Initialize variables -kRy = zeros(1,16); -kSy = zeros(1,16); -Rmy = zeros(1,16); +%% 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(['meas_stiff_flex_' num2str(i) '_y.mat'], 't', 'F', 'd'); + % Load the data + load(sprintf('meas_stiff_flex_%i_y.mat', i), '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 + % 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); - d_l = d(i_l_start:i_l_stop); + % 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] - F_s = F(i_s_start:i_s_stop); - d_s = d(i_s_start:i_s_stop); + % 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'); - %% Straight line fit - fit_l = polyfit(F_l, d_l, 1); - fit_s = polyfit(F_s, d_s, 1); + % Linear fit in the "linear" region + fit_l = polyfit(Ty(i_l_start:i_l_stop), thetay(i_l_start:i_l_stop), 1); - %% Reset displacement based on fit - d = d - fit_l(2); + % 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; - %% 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); + % 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 -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 -#+attr_latex: :environment tabularx :width 0.6\linewidth :align cccc +#+caption: Measured bending stiffnesses +#+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] | -|----+--------------------+----------------------+---------------------------| -| 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 -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. +| | $R_{R_x}$ [Nm/rad] | $k_{R_y}$ [Nm/rad] | | +|----+--------------------+--------------------+---| +| 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 | | #+begin_src matlab :exports none load('flex_meas_dim.mat', 'meas_flex'); diff --git a/test-bench-flexible-joints.pdf b/test-bench-flexible-joints.pdf index 544d85fa16f467077f7fbd488b1d8e2a933b2906..2d5c1383b2428103cce68f68970c25517cb9aa55 100644 GIT binary patch delta 1059942 zcmZsibx>SQ`=)VscXxLf9D=(BcY?e7;O-CxcXtc!F2UVhf)gNUV0qtfzu#_c?H^ao zdAiR$eV?A1>h8Pyr0(=U6kcCK)a%28^QE9e&;#S*gb)YV(83?zqY2gP+1s;09?e!^ z(`!QoHqBu#L6>CjK7_%+SqpdDlaoRS)A`0JP<|eMjU({5= zui*4lAKuq4ee*2BYswwB+=;|3l7=Trpb*v9G!!WT zIU~qof`YGQFd=!_IRAZy%XCH8_0MOxenTKW3eohR>=0j25V)k?OS|ujfVjfTYEUc_ z>9sYdYhcguVB-wU#?d3Ooojl%s-@Q@i@t(wHiNdCFSteINS3ilY6+Ja zne~d#y`htr2c=2g_`j7rpdifXOrdLIj!#H2L_-nqd1o>~i%bJ!*kedxLXbLFw 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fde2a06..2ecfc31 100644 --- a/test-bench-flexible-joints.tex +++ b/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} - -Now, let's estimate the bending stiffness and stroke for all the flexible joints. - -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}. +\section{Measured flexible joint stiffnesses} +The same measurement is performed for all the 16 flexible joints, both in the \(x\) and \(y\) directions. +The measured torque and angular motion are all shown in Figure \ref{fig:test_joints_meas_bending_all_raw_data}. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_joints_bend_stiff_hist.png} -\caption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\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} -\caption{\label{fig:test_joints_bend_stroke_hist}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:test_joints_thickness_stiffness}. +\begin{tabularx}{0.6\linewidth}{Xccc} +\toprule + & \(R_{R_x}\) {[}Nm/rad] & \(k_{R_y}\) {[}Nm/rad] & \\ +\midrule +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

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