286 lines
7.2 KiB
Mathematica
286 lines
7.2 KiB
Mathematica
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Path for functions, data and scripts
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addpath('./mat/'); % Path for data
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%% Colors for the figures
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colors = colororder;
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% Analysis of one measurement
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% In this section is shown how the data are analysis in order to measured:
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% - the bending stiffness
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% - the bending stroke
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% - the stiffness once the mechanical stops are in contact
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% The height from the flexible joint's center and the point of application force $h$ is defined below:
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h = 25e-3; % [m]
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%% Load Data
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load('meas_stiff_flex_1_x.mat', 't', 'F', 'd');
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%% Zero the force
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F = F - mean(F(t > 0.1 & t < 0.3));
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%% Start measurement at t = 0.2 s
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d = d(t > 0.2);
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F = F(t > 0.2);
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t = t(t > 0.2); t = t - t(1);
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% The obtained time domain measurements are shown in Figure ref:fig:test_joints_meas_bend_time.
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%% Time Domain plots
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figure;
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tiledlayout(2, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile;
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plot(t, F);
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ylabel('Force [N]'); set(gca, 'XTickLabel',[]);
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ax2 = nexttile;
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plot(t, 1e3*d);
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hold off;
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xlabel('Time [s]'); ylabel('Displacement [mm]');
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linkaxes([ax1,ax2],'x');
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xlim([0,5]);
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% #+name: fig:test_joints_meas_bend_time
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% #+caption: Typical time domain measurements
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% #+RESULTS:
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% [[file:figs/test_joints_meas_bend_time.png]]
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% The displacement as a function of the force is then shown in Figure ref:fig:test_joints_meas_F_d.
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figure;
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plot(F, 1e3*d);
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xlabel('Force [N]'); ylabel('Displacement [mm]');
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xlim([0,6]); ylim([0,1]);
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% #+name: fig:test_joints_meas_F_d
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% #+caption: Typical measurement of the diplacement as a function of the applied force
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% #+RESULTS:
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% [[file:figs/test_joints_meas_F_d.png]]
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% The bending stiffness can be estimated by computing the slope of the curve in Figure ref:fig:test_joints_meas_F_d.
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% The bending stroke and the stiffness when touching the mechanical stop can also be estimated from the same figure.
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%% Determine the linear region and region when touching the mechanical stop
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% Find when the force sensor touches the flexible joint
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i_l_start = find(F > 0.3, 1, 'first');
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% Reset the measured diplacement at that point
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d = d - d(i_l_start);
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% Find then the maximum force is applied
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[~, i_s_stop] = max(F);
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% Linear region stops ~ when 90% of the stroke is reached
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i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
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% "Stop" region start ~1N before maximum force is applied
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i_s_start = find(F > max(F)-1, 1, 'first');
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%% Define variables for the two regions
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F_l = F(i_l_start:i_l_stop);
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d_l = d(i_l_start:i_l_stop);
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F_s = F(i_s_start:i_s_stop);
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d_s = d(i_s_start:i_s_stop);
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%% Fit the best straight line for the two regions
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fit_l = polyfit(F_l, d_l, 1);
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fit_s = polyfit(F_s, d_s, 1);
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%% Reset displacement based on fit
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d = d - fit_l(2);
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fit_s(2) = fit_s(2) - fit_l(2);
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fit_l(2) = 0;
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% The raw data as well as the fit corresponding to the two stiffnesses are shown in Figure ref:fig:test_joints_meas_F_d_lin_fit.
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figure;
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hold on;
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plot(F(1:i_s_stop), 1e3*d(1:i_s_stop), '.k')
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plot(F_l, 1e3*(F_l*fit_l(1) + fit_l(2)))
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plot(F_s, 1e3*(F_s*fit_s(1) + fit_s(2)))
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hold off;
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xlabel('Force [N]'); ylabel('Displacement [mm]');
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xlim([0,6]);
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% #+name: fig:test_joints_meas_F_d_lin_fit
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% #+caption: Typical measurement of the diplacement as a function of the applied force with estimated linear fits
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% #+RESULTS:
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% [[file:figs/test_joints_meas_F_d_lin_fit.png]]
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% Then, the bending stroke is estimated as crossing point between the two fitted lines:
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d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
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% Bending stiffness and bending stroke of all the flexible joints
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% Now, let's estimate the bending stiffness and stroke for all the flexible joints.
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%% Initialize variables
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kRx = zeros(1,16);
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kSx = zeros(1,16);
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Rmx = zeros(1,16);
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for i = 1:16
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%% Load the data
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load(['meas_stiff_flex_' num2str(i) '_x.mat'], 't', 'F', 'd');
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%% Automatic Zero of the force
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F = F - mean(F(t > 0.1 & t < 0.3));
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%% Start measurement at t = 0.2 s
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d = d(t > 0.2);
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F = F(t > 0.2);
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t = t(t > 0.2); t = t - t(1);
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%% Estimate linear region and "stop" region
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i_l_start = find(F > 0.3, 1, 'first');
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d = d - d(i_l_start);
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[~, i_s_stop] = max(F);
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i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
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i_s_start = find(F > max(F)-1, 1, 'first');
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F_l = F(i_l_start:i_l_stop);
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d_l = d(i_l_start:i_l_stop);
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F_s = F(i_s_start:i_s_stop);
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d_s = d(i_s_start:i_s_stop);
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%% Straight line fit
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fit_l = polyfit(F_l, d_l, 1);
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fit_s = polyfit(F_s, d_s, 1);
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%% Reset displacement based on fit
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d = d - fit_l(2);
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fit_s(2) = fit_s(2) - fit_l(2);
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fit_l(2) = 0;
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%% Estimated Stroke
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d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
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%% Save stiffnesses and stroke
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kRx(i) = (h)^2/fit_l(1);
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kSx(i) = (h)^2/fit_s(1);
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Rmx(i) = atan2(d_max,h);
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end
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%% Initialize variables
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kRy = zeros(1,16);
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kSy = zeros(1,16);
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Rmy = zeros(1,16);
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for i = 1:16
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%% Load the data
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load(['meas_stiff_flex_' num2str(i) '_y.mat'], 't', 'F', 'd');
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%% Automatic Zero of the force
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F = F - mean(F(t > 0.1 & t < 0.3));
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%% Start measurement at t = 0.2 s
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d = d(t > 0.2);
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F = F(t > 0.2);
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t = t(t > 0.2); t = t - t(1);
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%% Estimate linear region and "stop" region
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i_l_start = find(F > 0.3, 1, 'first');
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d = d - d(i_l_start);
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[~, i_s_stop] = max(F);
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i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
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i_s_start = find(F > max(F)-1, 1, 'first');
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F_l = F(i_l_start:i_l_stop);
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d_l = d(i_l_start:i_l_stop);
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F_s = F(i_s_start:i_s_stop);
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d_s = d(i_s_start:i_s_stop);
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%% Straight line fit
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fit_l = polyfit(F_l, d_l, 1);
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fit_s = polyfit(F_s, d_s, 1);
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%% Reset displacement based on fit
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d = d - fit_l(2);
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fit_s(2) = fit_s(2) - fit_l(2);
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fit_l(2) = 0;
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%% Estimated Stroke
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d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
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%% Save stiffnesses and stroke
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kRy(i) = (h)^2/fit_l(1);
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kSy(i) = (h)^2/fit_s(1);
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Rmy(i) = atan2(d_max,h);
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end
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% Analysis
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% 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.
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figure;
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hold on;
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histogram(kRx, 'DisplayName', '$k_{R_x}$')
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histogram(kRy, 'DisplayName', '$k_{R_y}$')
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hold off;
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xlabel('Bending Stiffness [Nm/rad]')
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legend();
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% #+name: fig:test_joints_bend_stiff_hist
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% #+caption: Histogram of the measured bending stiffness
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% #+RESULTS:
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% [[file:figs/test_joints_bend_stiff_hist.png]]
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figure;
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hold on;
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histogram(1e3*Rmx, 'DisplayName', '$k_{R_x}$')
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histogram(1e3*Rmy, 'DisplayName', '$k_{R_y}$')
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hold off;
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xlabel('Bending Stroke [mrad]')
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legend();
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% #+name: fig:test_joints_bend_stroke_hist
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% #+caption: Histogram of the measured bending stroke
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% #+RESULTS:
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% [[file:figs/test_joints_bend_stroke_hist.png]]
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% The relation between the measured beam thickness and the measured bending stiffness is shown in Figure ref:fig:test_joints_thickness_stiffness.
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load('flex_meas_dim.mat', 'meas_flex');
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figure;
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hold on;
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plot((meas_flex(:,1)+meas_flex(:,2))/2, kRx, 'o', 'DisplayName', '$x$')
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plot((meas_flex(:,3)+meas_flex(:,4))/2, kRy, 'o', 'DisplayName', '$y$')
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hold off;
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xlabel('Flexible Beam Thickness [$\mu m$]');
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ylabel('Bending Stiffness [Nm/rad]');
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legend('location', 'southeast');
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