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C2-test-bench-flexible-joints/test_joints_1_dim_meas.m

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+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./mat/'); % Path for data
+
+%% Colors for the figures
+colors = colororder;
+
+%% Measured gap for the 16 flexible joints
+meas_flex = [[223, 226, 224, 214];
+             [229, 231, 237, 224];
+             [234, 230, 239, 231];
+             [233, 227, 229, 232];
+             [225, 212, 228, 228];
+             [220, 221, 224, 220];
+             [206, 207, 228, 226];
+             [230, 224, 224, 223];
+             [223, 231, 228, 233];
+             [228, 230, 235, 231];
+             [197, 207, 211, 204];
+             [227, 226, 225, 226];
+             [215, 228, 231, 220];
+             [216, 224, 224, 221];
+             [209, 214, 220, 221];
+             [213, 210, 230, 229]];
+
+%% 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])
+
+%% Save beam sizes
+save('./mat/flex_meas_dim.mat', 'meas_flex');

C2-test-bench-flexible-joints/test_joints_2_bench_dimensioning.m

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+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./mat/'); % Path for data
+
+%% Colors for the figures
+colors = colororder;
+
+%% Parameters for study
+kRx = 5;       % Bending Stiffness [Nm/rad]
+Rxmax = 25e-3; % Bending Stroke [rad]
+h = 22.5e-3;     % Height [m]
+
+%% Estimation of the force to test the full stroke
+Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
+
+%% Estimated maximum stroke [m]
+dxmax = h*tan(Rxmax);
+
+%% Stiffness
+ka = 94e6; % Axial Stiffness [N/m]
+ks = 13e6; % Shear Stiffness [N/m]
+kb = 5;    % Bending Stiffness [Nm/rad]
+kt = 260;  % Torsional Stiffness [Nm/rad]
+
+%% Maximum force
+Fa = 469;    % Axial Force before yield [N]
+Fs = 242;    % Shear Force before yield [N]
+Fb = 0.118;  % Bending Force before yield [Nm]
+Ft = 1.508; % Torsional Force before yield [Nm]
+
+%% Compute the corresponding stroke
+Xa = Fa/ka; % Axial Stroke before yield [m]
+Xs = Fs/ks; % Shear Stroke before yield [m]
+Xb = Fb/kb; % Bending Stroke before yield [rad]
+Xt = Ft/kt; % Torsional Stroke before yield [rad]
+
+%% Height between the joint's center and the force application point
+h = 22.5e-3; % [m]
+
+%% Estimated error due to shear
+epsilon_s = 100*abs(1-1/(1 + kb/(ks*h^2))); % Error in %
+
+%% Estimated error due to limited load cell stiffness
+kF = 50/0.05e-3; % Estimated load cell stiffness [N/m]
+epsilon_f = 100*abs(1-1/(1 + kb/(kF*h^2))); % Error in %

C2-test-bench-flexible-joints/test_joints_3_bending_stiff_meas.m

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+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./mat/'); % Path for data
+
+%% Colors for the figures
+colors = colororder;
+
+%% Force Sensor Calibration
+% Load measurement data
+load('calibration_force_sensor.mat', 't', 'F', 'Fc')
+
+% Remove any offset such that they are both measuring no force when not in contact.
+F  = F  - mean(F( t > 0.5 & t < 1.0)); % FC2231 [N]
+Fc = Fc - mean(Fc(t > 0.5 & t < 1.0)); % XFL212R [N]
+
+% Only get useful stroke
+F  = F( t > 2 & t < 2.2);
+Fc = Fc(t > 2 & t < 2.2);
+t  = t( t > 2 & t < 2.2);
+
+% Make a line fit
+fit_F = polyfit(Fc, F, 1);
+
+% Estimate the gain mismatch
+F_gain_mismatch = 100*(1 - fit_F(1)); % in %
+
+% Estimate non-linearity of the sensors
+F_non_linearity = 100*(F - (Fc*fit_F(1) + fit_F(2)))./Fc; % in %
+
+%% Measured two forces and linear fit
+figure;
+yyaxis left
+hold on;
+plot(Fc, F, '.', 'DisplayName', 'Raw Data');
+plot([0, 6], [0,6]*fit_F(1) + fit_F(2), '-', 'color', [colors(3,:), 0.5], 'DisplayName', 'Line Fit');
+hold off;
+xlabel('XFL212R [N]'); ylabel('FC2231 [N]');
+xlim([0,6]); ylim([0,6]);
+xticks([0:6])
+yticks([0:6])
+
+yyaxis right
+plot(Fc, movmean(F_non_linearity, 200), '-', 'DisplayName', 'Non linearity');
+ylim([-0.15, 0.15]);
+ylabel('Non Linearity [$\%$]')
+
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+%% Estimaetd load cell stiffness
+% Load measurement data
+load('force_sensor_stiffness_meas.mat', 't', 'F', 'd')
+
+% Remove offset
+F  = F - mean(F(t > 0.5 & t < 1.0));
+
+% Select important part of data
+F  = F( t > 4.55 & t < 7.24);
+d  = d( t > 4.55 & t < 7.24); d = d - d(1);
+t  = t( t > 4.55 & t < 7.24);
+
+% Linear fit
+fit_k = polyfit(F(F<10), d(F<10), 1);
+
+%% Displacement as a function of the measured force
+figure;
+hold on;
+plot(F, 1e6*d, 'k-', 'DisplayName', 'Raw Data');
+plot(F([1,end]), 1e6*(F([1,end])*fit_k(1) + fit_k(2)), '--', 'DisplayName', sprintf('Fit, $k_F \\approx %.2f N/\\mu m$', 1e-6/fit_k(1)));
+hold off;
+xlabel('Force [$N$]'); ylabel('Displacement [$\mu m$]');
+xlim([0,45]); ylim([0,60]);
+legend('location', 'southeast', 'FontSize', 8);
+
+%% 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');
+
+% 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');
+
+% 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(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');
+
+% 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
+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]
+Tx_at_theta_max = (fit_s(2) - fit_l(2))/(fit_l(1) - fit_s(1));
+
+%% Measured force and displacement as a function of time
+figure;
+yyaxis left
+hold on;
+plot(t, F);
+plot(0.8*cos(0:0.01:2*pi)+0.8, ...
+     2.8*sin(0:0.01:2*pi)-1, 'k--', 'HandleVisibility', 'off');
+text(1.8, -1.2, sprintf('Not in\ncontact'), 'horizontalalignment', 'left');
+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]');
+ylim([-4, 6])
+ylimr = get(gca,'Ylim');
+
+yyaxis right
+plot(t, 1e3*d);
+xlabel('Time [s]');
+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])
+
+% Regions where to plot the fitted data
+Tx_l_fit = [0, Tx_at_theta_max];
+Tx_s_fit = [Tx_at_theta_max, Tx(i_s_stop)];
+
+figure;
+hold on;
+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('$\\theta_{x,\\max} = %.1f$ mrad', 1e3*theta_max), 'horizontalalignment', 'left');
+hold off;
+xlabel('Torque $T_x$ [Nm]');
+ylabel('Angle $\theta_x$ [mrad]');
+xlim([0, 0.15]);
+ylim([-5,25]);
+leg = legend('location', 'southeast', 'FontSize', 8);
+leg.ItemTokenSize(1) = 15;
+
+%% 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(sprintf('meas_stiff_flex_%i_x.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]
+    Tx = h * F; % Applied torque in [Nm]
+    thetax = atan2(d, h); % Measured angle in [rad]
+
+    % 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');
+
+    % 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);
+
+figure;
+histogram([kRx, kRy], [4:0.2:5])
+xlabel('Bending stiffness [Nm/rad]')
+xticks([4:0.2:5])