Analyze measurement of flexible joints

test-bench-flexible-joints.org

@@ -51,7 +51,8 @@ In this document, we present a test-bench that has been developed in order to me
 
 It is structured as follow:
 - Section [[sec:flexible_joints]]: the geometry of the flexible joints and the expected stiffness and stroke are presented
-- Section [[sec:test_bench_desc]]: the measurement bench is presented
+- Section [[sec:flex_dim_meas]]: each flexible joint is measured using a profile projector
+- Section [[sec:test_bench_desc]]: the stiffness measurement bench is presented
 - Section [[sec:error_budget]]: an error budget is performed in order to estimate the accuracy of the measured stiffness
 - Section [[sec:first_measurements]]: first measurements are performed
 - Section [[sec:bending_stiffness_meas]]: the bending stiffness of the flexible joints are measured
@@ -98,39 +99,47 @@ This has been done using a Finite Element Software and the obtained joint's char
 [[file:figs/flexible_joint_fem_geometry.png]]
 
 The obtained geometry are defined in the [[file:doc/flex_joints.pdf][drawings of the flexible joints]].
-The material is a special kind of stainless steel called "F16PH"..
+The material is a special kind of stainless steel called "F16PH".
 
-* Dimensions
+The flexible joints can be seen on Figure [[fig:received_flex]].
 
-** Measurements
+#+name: fig:received_flex
+#+caption: 15 of the 16 flexible joints
+#+attr_latex: :width \linewidth
+[[file:figs/IMG_20210302_173619.jpg]]
 
-Few notes:
+* Dimensional Measurements
-- dirt inside: 3,10,12,13
+:PROPERTIES:
-- strange surface quality: 15,16
+:header-args:matlab+: :tangle ./matlab/dim_meas.m
-- Strange shape: 5
+:END:
+<<sec:flex_dim_meas>>
+** Measurement Bench
+
+The axis corresponding to the flexible joints are defined in Figure [[fig:flexible_joint_axis]].
+
+#+name: fig:flexible_joint_axis
+#+caption: Define axis for the flexible joints
+#+attr_latex: :width 0.3\linewidth
+[[file:figs/flexible_joint_axis.png]]
+
+The dimensions of the flexible part in the Y-Z plane will contribute to the X-bending stiffness.
+Similarly, the dimensions of the flexible part in the X-Z plane will contribute to the Y-bending stiffness.
+
+The setup to measure the dimension of the "Y" flexible beam is shown in Figure [[fig:flexible_joint_y_flex_meas_setup]].
+
+#+name: fig:flexible_joint_y_flex_meas_setup
+#+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness
+#+attr_latex: :width 1.0\linewidth
+[[file:figs/flexible_joint_y_flex_meas_setup.png]]
+
+What we typically observe is shown in Figure [[fig:soft_measure_flex_size]].
+It is then possible to estimate to dimension of the flexible beam with an accuracy of $\approx 5\,\mu m$,
+
+#+name: fig:soft_measure_flex_size
+#+attr_latex: :width 1.0\linewidth
+#+caption: Image used to measure the flexible joint's dimensions
+[[file:figs/soft_measure_flex_size.jpg]]
 
-#+name: flex_dim
-#+caption: Table caption
-#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xcccc
-#+attr_latex: :center t :booktabs t :float t
-| Num |  X1 |  X2 |  Y1 |  Y2 |
-|-----+-----+-----+-----+-----|
-|   1 | 224 | 214 | 223 | 226 |
-|   2 | 237 | 224 | 229 | 231 |
-|   3 | 239 | 231 | 234 | 230 |
-|   4 | 229 | 232 | 233 | 227 |
-|   5 | 228 | 228 | 225 | 212 |
-|   6 | 224 | 220 | 220 | 221 |
-|   7 | 228 | 226 | 206 | 207 |
-|   8 | 224 | 223 | 230 | 224 |
-|   9 | 228 | 233 | 223 | 231 |
-|  10 | 235 | 231 | 228 | 230 |
-|  11 | 211 | 204 | 197 | 207 |
-|  12 | 225 | 226 | 227 | 226 |
-|  13 | 231 | 220 | 215 | 228 |
-|  14 | 224 | 221 | 216 | 224 |
-|  15 | 220 | 221 | 209 | 214 |
-|  16 | 230 | 229 | 213 | 210 |
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -150,12 +159,82 @@ addpath('./matlab/');
 addpath('./mat/');
 #+end_src
 
-** Analysis
+** Measurement Results
+# - Strange shape: 5
 
-#+begin_src matlab :var data=flex_dim
+The expected flexible beam thickness is $250\,\mu m$.
-mean(data(:,2:end))
+However, it is more important that the thickness of all beams are close to each other.
-std(data(:,2:end))
+
-std(data(:,2:end)')
+The dimension of the beams are been measured at each end to be able to estimate the mean of the beam thickness.
+
+All the measured dimensions are summarized in Table [[tab:flex_dim]].
+
+#+begin_src matlab :exports none
+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]];
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+data2orgtable(meas_flex, {'1','2','3','4','5','6','7','8','9','10','11','12','13','14','15','16'}, {'X1', 'X2', 'X3', 'X4'}, ' %.0f ');
+#+end_src
+
+#+name: tab:flex_dim
+#+caption: Measured Dimensions of the flexible beams in $\mu m$
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcccc
+#+attr_latex: :center t :booktabs t :float t
+#+RESULTS:
+|    |  X1 |  X2 |  X3 |  X4 |
+|----+-----+-----+-----+-----|
+|  1 | 223 | 226 | 224 | 214 |
+|  2 | 229 | 231 | 237 | 224 |
+|  3 | 234 | 230 | 239 | 231 |
+|  4 | 233 | 227 | 229 | 232 |
+|  5 | 225 | 212 | 228 | 228 |
+|  6 | 220 | 221 | 224 | 220 |
+|  7 | 206 | 207 | 228 | 226 |
+|  8 | 230 | 224 | 224 | 223 |
+|  9 | 223 | 231 | 228 | 233 |
+| 10 | 228 | 230 | 235 | 231 |
+| 11 | 197 | 207 | 211 | 204 |
+| 12 | 227 | 226 | 225 | 226 |
+| 13 | 215 | 228 | 231 | 220 |
+| 14 | 216 | 224 | 224 | 221 |
+| 15 | 209 | 214 | 220 | 221 |
+| 16 | 213 | 210 | 230 | 229 |
+
+An histogram of these measured dimensions is shown in Figure [[fig:beam_dim_histogram]].
+
+#+begin_src matlab :exports none
+figure;
+histogram([(meas_flex(:,1)+meas_flex(:,2))/2,(meas_flex(:,3)+meas_flex(:,4))/2], 7)
+xlabel("Beam's Thickness [$\mu m$]");
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/beam_dim_histogram.pdf', 'width', 'normal', 'height', 'normal');
+#+end_src
+
+#+name: fig:beam_dim_histogram
+#+caption:  Histogram for the (16x2) measured beams' thickness
+#+RESULTS:
+[[file:figs/beam_dim_histogram.png]]
+
+#+begin_src matlab :tangle no :exports none
+save('matlab/mat/flex_meas_dim.mat', 'meas_flex');
 #+end_src
 
 * Measurement Test Bench - Bending Stiffness
@@ -789,7 +868,38 @@ The Millimar 1318 probe has a stiffness of $\approx 0.04\,[N/mm]$.
 #+end_summary
 
 * Bending Stiffness Measurement
+:PROPERTIES:
+:header-args:matlab+: :tangle ./matlab/bending_stiff_meas.m
+:END:
 <<sec:bending_stiffness_meas>>
+** Introduction
+
+A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure [[fig:picture_bending_x_meas_side_overview]].
+A closer view on flexible joint is shown in Figure [[fig:picture_bending_x_meas_side_close]] and a zoom on the force sensor tip is shown in Figure [[fig:picture_bending_x_meas_side_zoom]].
+
+#+name: fig:picture_bending_x_meas_side_overview
+#+caption: Side view of the flexible joint stiffness bench. X-Bending stiffness is measured.
+#+attr_latex: :width \linewidth
+[[file:figs/picture_bending_x_meas_side_overview.jpg]]
+
+#+name: fig:picture_bending_x_meas_side_close
+#+caption: Zoom on the flexible joint - Side view
+#+attr_latex: :width \linewidth
+[[file:figs/picture_bending_x_meas_side_close.jpg]]
+
+
+#+name: fig:picture_bending_x_meas_side_zoom
+#+caption: Zoom on the tip of the force sensor
+#+attr_latex: :width 0.4\linewidth
+[[file:figs/picture_bending_x_meas_side_zoom.jpg]]
+
+The same bench used to measure the Y-bending stiffness of the flexible joint is shown in Figure [[fig:picture_bending_y_meas_side_close]].
+
+#+name: fig:picture_bending_y_meas_side_close
+#+caption: Stiffness measurement bench - Y-d bending measurement
+#+attr_latex: :width \linewidth
+[[file:figs/picture_bending_y_meas_side_close.jpg]]
+
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <<matlab-dir>>
@@ -808,21 +918,34 @@ addpath('./matlab/');
 addpath('./mat/');
 #+end_src
 
-** Results
+** Analysis of one measurement
+
+In this section is shown how the data are analysis in order to measured:
+- the bending stiffness
+- the bending stroke
+- the stiffness once the mechanical stops are in contact
+
+
+The height from the flexible joint's center and the point of application force $h$ is defined below:
 #+begin_src matlab
-load('meas_stiff_flex_12_x.mat', 't', 'd', 'F');
+h = 25e-3; % [m]
 #+end_src
 
 #+begin_src matlab
-d = d(t > 5.35 & t < 14.0);
+%% Load Data
-F = F(t > 5.35 & t < 14.0);
+load('meas_stiff_flex_1_x.mat', 't', 'F', 'd');
-t = t(t > 5.35 & t < 14.0);
 
-d = d - d(1);
+%% Zero the force
-F = F - F(1);
+F = F - mean(F(t > 0.1 & t < 0.3));
-t = t - t(1);
+
+%% 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);
 #+end_src
 
+The obtained time domain measurements are shown in Figure [[fig:flex_joint_meas_example_time_domain]].
+
 #+begin_src matlab :exports none
 %% Time Domain plots
 figure;
@@ -833,98 +956,154 @@ plot(t, F);
 ylabel('Force [N]'); set(gca, 'XTickLabel',[]);
 
 ax2 = nexttile;
-plot(t, d);
+plot(t, 1e3*d);
-xlabel('Time [s]'); ylabel('Displacement [m]');
-#+end_src
-
-#+begin_src matlab :exports none
-figure;
-hold on;
-plot(F, d, 'DisplayName', 'Raw data');
-% plot(F, F/(d\F), 'DisplayName', 'Linear fit');
 hold off;
-xlabel('Measured Force [N]');
+xlabel('Time [s]'); ylabel('Displacement [mm]');
-ylabel('Measured Displacement [m]');
-legend('location', 'southeast');
-#+end_src
-
-#+begin_src matlab
-ta = [0.05, 1.13];
-tb = [5.75, 7.2];
-
-d_l = [d(t > ta(1) & t < ta(2)); d(t > tb(1) & t < tb(2))];
-F_l = [F(t > ta(1) & t < ta(2)); F(t > tb(1) & t < tb(2))];
-
-d_l = d_l - d_l(1);
-F_l = F_l - F_l(1);
-#+end_src
-
-#+begin_src matlab :exports none
-figure;
-hold on;
-plot(F_l, d_l, '.', 'DisplayName', 'Raw data');
-plot(F_l, F_l/(d_l\F_l), 'DisplayName', 'Linear fit');
-hold off;
-xlabel('Measured Force [N]');
-ylabel('Measured Displacement [m]');
-legend('location', 'southeast');
-#+end_src
-
-#+begin_src matlab :results value replace :exports results :tangle no
-h = 25e-3;
-sprintf('Stiffness is %.3f [Nm/rad]', abs(h^2*(d_l\F_l)))
-#+end_src
-
-#+RESULTS:
-: Stiffness is 5.579 [Nm/rad]
-
-** Results - Y
-#+begin_src matlab
-load('meas_stiff_flex_12_y.mat', 't', 'd', 'F');
-#+end_src
-
-#+begin_src matlab
-%% Automatic Zero of the force
-F = F - mean(F(t > 0.9 & t < 1.1));
-
-%% Start measurement at t = 1.0 s
-d = d(t > 1.0);
-F = F(t > 1.0);
-t = t(t > 1.0); t = t - t(1);
-#+end_src
-
-#+begin_src matlab :exports none
-%% Time Domain plots
-figure;
-tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
-
-ax1 = nexttile;
-plot(t, F);
-ylabel('Force [N]'); set(gca, 'XTickLabel',[]);
-
-ax2 = nexttile;
-plot(t, d);
-hold off;
-xlabel('Time [s]'); ylabel('Displacement [m]');
 
 linkaxes([ax1,ax2],'x');
+xlim([0,5]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/flex_joint_meas_example_time_domain.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:flex_joint_meas_example_time_domain
+#+caption: Typical time domain measurements
+#+RESULTS:
+[[file:figs/flex_joint_meas_example_time_domain.png]]
+
+The displacement as a function of the force is then shown in Figure [[fig:flex_joint_meas_example_F_d]].
+
+#+begin_src matlab :exports none
+figure;
+plot(F, 1e3*d);
+xlabel('Force [N]'); ylabel('Displacement [mm]');
+xlim([0,6]); ylim([0,1]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/flex_joint_meas_example_F_d.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:flex_joint_meas_example_F_d
+#+caption: Typical measurement of the diplacement as a function of the applied force
+#+RESULTS:
+[[file:figs/flex_joint_meas_example_F_d.png]]
+
+The bending stiffness can be estimated by computing the slope of the curve in Figure [[fig:flex_joint_meas_example_F_d]].
+The bending stroke and the stiffness when touching the mechanical stop can also be estimated from the same figure.
+
+#+begin_src matlab
+%% Determine the linear region and region when touching the mechanical stop
+% 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
+d = d - d(i_l_start);
+% Find then the maximum force is applied
+[~, i_s_stop] = max(F);
+% Linear region stops ~ when 90% of the stroke is reached
+i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
+% "Stop" region start ~1N before maximum force is applied
+i_s_start = find(F > max(F)-1, 1, 'first');
+
+%% Define variables for the two regions
+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);
 #+end_src
 
 #+begin_src matlab
-t_l = [5.58, 6.75]; % Time of flexible joint's linear region
+%% Fit the best straight line for the two regions
-t_s = [6.9, 7.24]; % Time of stop's linear region
+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;
 #+end_src
 
+The raw data as well as the fit corresponding to the two stiffnesses are shown in Figure [[fig:flex_joint_meas_example_F_d_lin_fit]].
+
 #+begin_src matlab :exports none
-%% Linear Fit
+figure;
-h = 25e-3;
+hold on;
+plot(F(1:i_s_stop), 1e3*d(1:i_s_stop), '.k')
+plot(F_l, 1e3*(F_l*fit_l(1) + fit_l(2)))
+plot(F_s, 1e3*(F_s*fit_s(1) + fit_s(2)))
+hold off;
+xlabel('Force [N]'); ylabel('Displacement [mm]');
+xlim([0,6]);
+#+end_src
 
-F_l = F(t > t_l(1) & t < t_l(2));
+#+begin_src matlab :tangle no :exports results :results file replace
-d_l = d(t > t_l(1) & t < t_l(2));
+exportFig('figs/flex_joint_meas_example_F_d_lin_fit.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
 
-F_s = F(t > t_s(1) & t < t_s(2));
+#+name: fig:flex_joint_meas_example_F_d_lin_fit
-d_s = d(t > t_s(1) & t < t_s(2));
+#+caption: Typical measurement of the diplacement as a function of the applied force with estimated linear fits
+#+RESULTS:
+[[file:figs/flex_joint_meas_example_F_d_lin_fit.png]]
 
+Then, the bending stroke is estimated as crossing point between the two fitted lines:
+#+begin_src matlab
+d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
+#+end_src
+
+The obtained characteristics are summarized in Table [[tab:obtained_caracteristics_flex_1_x]].
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+data2orgtable([(h)^2/fit_l(1); (h)^2/fit_s(1); 1e3*atan2(d_max,h)], {'Bending Stiffness [Nm/rad]', 'Bending Stiffness @ stop [Nm/rad]', 'Bending Stroke [mrad]'}, {}, ' %.1f ');
+#+end_src
+
+#+name: tab:obtained_caracteristics_flex_1_x
+#+caption: Estimated characteristics of the flexible joint number 1 for the X-direction
+#+attr_latex: :environment tabularx :width 0.5\linewidth :align lc
+#+attr_latex: :center t :booktabs t :float t
+#+RESULTS:
+| Bending Stiffness [Nm/rad]        |   5.5 |
+| Bending Stiffness @ stop [Nm/rad] | 173.6 |
+| Bending Stroke [mrad]             |  18.9 |
+
+** Bending stiffness and bending stroke of all the flexible joints
+
+Now, let's estimate the bending stiffness and stroke for all the flexible joints.
+
+#+begin_src matlab :exports none
+%% Initialize variables
+kRx = zeros(1,16);
+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);
 
@@ -934,88 +1113,48 @@ fit_s(2) = fit_s(2) - fit_l(2);
     fit_l(2) = 0;
 
     %% Estimated Stroke
-F_max = fit_s(2)/(fit_l(1) - fit_s(1));
+    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
-d_max = fit_l(1)*F_max;
+
+    %% 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
-figure;
+%% Initialize variables
-hold on;
+kRy = zeros(1,16);
-% plot(F_l, d_l, 'k.', 'DisplayName', 'Raw Data');
+kSy = zeros(1,16);
-% plot(F_s, d_s, 'k.', 'HandleVisibility', 'off');
+Rmy = zeros(1,16);
-plot(F, d, 'k.', 'DisplayName', 'Raw data');
-set(gca,'ColorOrderIndex',1)
-plot(F_l, fit_l(1)*F_l + fit_l(2), '--', 'DisplayName', sprintf('$k_{R_x} = %.1f [Nm/rad]$', (h)^2/fit_l(1)));
-plot(F_s, fit_s(1)*F_s + fit_s(2), '--', 'DisplayName', sprintf('$k_s = %.1f [Nm/rad]$', (h)^2/fit_s(1)));
-plot([0.8*F_max, 1.2*F_max], [d_max, d_max], '--', 'DisplayName', sprintf('$R_{x,max} = %.1f [mrad]$', 1e3*atan2(d_max,h)));
-hold off;
-xlabel('Measured Force [N]');
-ylabel('Measured Displacement [m]');
-legend('location', 'southeast');
-ylim([-1e-4,inf])
-#+end_src
 
-#+begin_src matlab :results value replace :exports results :tangle no
+for i = 1:16
-sprintf('Bending Stiffness is %.1f [Nm/rad]', (h)^2/fit_l(1))
+    %% Load the data
-#+end_src
+    load(['meas_stiff_flex_' num2str(i) '_y.mat'], 't', 'F', 'd');
 
-#+RESULTS:
-: Bending Stiffness is 5.5 [Nm/rad]
-
-#+begin_src matlab :results value replace :exports results :tangle no
-sprintf('Bending Stroke is %.1f [mrad]', 1e3*atan2(d_max,h))
-#+end_src
-
-#+RESULTS:
-: Bending Stroke is 17.9 [mrad]
-
-** Results - X
-#+begin_src matlab
-load('meas_stiff_flex_12_x.mat', 't', 'd', 'F');
-#+end_src
-
-#+begin_src matlab
     %% Automatic Zero of the force
-F = F - mean(F(t > 0.9 & t < 1.1));
+    F = F - mean(F(t > 0.1 & t < 0.3));
 
-%% Start measurement at t = 1.0 s
+    %% Start measurement at t = 0.2 s
-d = d(t > 1.0);
+    d = d(t > 0.2);
-F = F(t > 1.0);
+    F = F(t > 0.2);
-t = t(t > 1.0); t = t - t(1);
+    t = t(t > 0.2); t = t - t(1);
-#+end_src
 
-#+begin_src matlab :exports none
+    %% Estimate linear region and "stop" region
-%% Time Domain plots
+    i_l_start = find(F > 0.3, 1, 'first');
-figure;
+    d = d - d(i_l_start);
-tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+    [~, 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');
 
-ax1 = nexttile;
+    F_l = F(i_l_start:i_l_stop);
-plot(t, F);
+    d_l = d(i_l_start:i_l_stop);
-ylabel('Force [N]'); set(gca, 'XTickLabel',[]);
 
-ax2 = nexttile;
+    F_s = F(i_s_start:i_s_stop);
-plot(t, d);
+    d_s = d(i_s_start:i_s_stop);
-hold off;
-xlabel('Time [s]'); ylabel('Displacement [m]');
-
-linkaxes([ax1,ax2],'x');
-#+end_src
-
-#+begin_src matlab
-t_l = [4.365, 5.47]; % Time of flexible joint's linear region
-t_s = [5.65, 6.09]; % Time of stop's linear region
-#+end_src
-
-#+begin_src matlab :exports none
-%% Linear Fit
-h = 25e-3;
-
-F_l = F(t > t_l(1) & t < t_l(2));
-d_l = d(t > t_l(1) & t < t_l(2));
-
-F_s = F(t > t_s(1) & t < t_s(2));
-d_s = d(t > t_s(1) & t < t_s(2));
 
+    %% Straight line fit
     fit_l = polyfit(F_l, d_l, 1);
     fit_s = polyfit(F_s, d_s, 1);
 
@@ -1025,124 +1164,143 @@ fit_s(2) = fit_s(2) - fit_l(2);
     fit_l(2) = 0;
 
     %% Estimated Stroke
-F_max = fit_s(2)/(fit_l(1) - fit_s(1));
+    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));
-d_max = fit_l(1)*F_max;
+
+    %% 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 [[tab:meas_flexible_joints_x_dir]] for the X direction and in Table [[tab:meas_flexible_joints_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:meas_flexible_joints_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:meas_flexible_joints_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 |
+
+** Analysis
+The dispersion of the measured bending stiffness is shown in Figure [[fig:bending_stiffness_histogram]] and of the bending stroke in Figure [[fig:bending_stroke_histogram]].
+
 #+begin_src matlab :exports none
 figure;
 hold on;
-plot(F, d, 'k.', 'DisplayName', 'Raw data');
+histogram(kRx, 'DisplayName', '$k_{R_x}$')
-set(gca,'ColorOrderIndex',1)
+histogram(kRy, 'DisplayName', '$k_{R_y}$')
-plot(F_l, fit_l(1)*F_l + fit_l(2), '-', 'DisplayName', sprintf('$k_{R_x} = %.1f [Nm/rad]$', (h)^2/fit_l(1)));
-plot(F_s, fit_s(1)*F_s + fit_s(2), '-', 'DisplayName', sprintf('$k_s = %.1f [Nm/rad]$', (h)^2/fit_s(1)));
-plot([0.8*F_max, 1.2*F_max], [d_max, d_max], '-', 'DisplayName', sprintf('$R_{x,max} = %.1f [mrad]$', 1e3*atan2(d_max,h)));
 hold off;
-xlabel('Measured Force [N]');
+xlabel('Bending Stiffness [Nm/rad]')
-ylabel('Measured Displacement [m]');
+legend();
-legend('location', 'southeast');
-ylim([-1e-4,inf])
 #+end_src
 
-#+begin_src matlab :results value replace :exports results :tangle no
+#+begin_src matlab :tangle no :exports results :results file replace
-sprintf('Bending Stiffness is %.1f [Nm/rad]', (h)^2/fit_l(1))
+exportFig('figs/bending_stiffness_histogram.pdf', 'width', 'wide', 'height', 'normal');
 #+end_src
 
+#+name: fig:bending_stiffness_histogram
+#+caption: Histogram of the measured bending stiffness
 #+RESULTS:
-: Bending Stiffness is 5.7 [Nm/rad]
+[[file:figs/bending_stiffness_histogram.png]]
-
-#+begin_src matlab :results value replace :exports results :tangle no
-sprintf('Bending Stroke is %.1f [mrad]', 1e3*atan2(d_max,h))
-#+end_src
-
-#+RESULTS:
-: Bending Stroke is 17.9 [mrad]
-
-** Results - XY
-#+begin_src matlab
-load('meas_stiff_flex_12_xy.mat', 't', 'd', 'F');
-#+end_src
-
-#+begin_src matlab
-%% Automatic Zero of the force
-F = F - mean(F(t > 0.9 & t < 1.1));
-
-%% Start measurement at t = 1.0 s
-d = d(t > 1.0);
-F = F(t > 1.0);
-t = t(t > 1.0); t = t - t(1);
-#+end_src
-
-#+begin_src matlab :exports none
-%% Time Domain plots
-figure;
-tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
-
-ax1 = nexttile;
-plot(t, F);
-ylabel('Force [N]'); set(gca, 'XTickLabel',[]);
-
-ax2 = nexttile;
-plot(t, d);
-hold off;
-xlabel('Time [s]'); ylabel('Displacement [m]');
-
-linkaxes([ax1,ax2],'x');
-#+end_src
-
-#+begin_src matlab
-t_l = [4.99, 6.5]; % Time of flexible joint's linear region
-t_s = [6.8, 7.10]; % Time of stop's linear region
-#+end_src
-
-#+begin_src matlab :exports none
-%% Linear Fit
-h = 25e-3;
-
-F_l = F(t > t_l(1) & t < t_l(2));
-d_l = d(t > t_l(1) & t < t_l(2));
-
-F_s = F(t > t_s(1) & t < t_s(2));
-d_s = d(t > t_s(1) & t < t_s(2));
-
-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
-F_max = fit_s(2)/(fit_l(1) - fit_s(1));
-d_max = fit_l(1)*F_max;
-#+end_src
 
 #+begin_src matlab :exports none
 figure;
 hold on;
-plot(F, d, 'k.', 'DisplayName', 'Raw data');
+histogram(1e3*Rmx, 'DisplayName', '$k_{R_x}$')
-set(gca,'ColorOrderIndex',1)
+histogram(1e3*Rmy, 'DisplayName', '$k_{R_y}$')
-plot(F_l, fit_l(1)*F_l + fit_l(2), '-', 'DisplayName', sprintf('$k_{R_x} = %.1f [Nm/rad]$', (h)^2/fit_l(1)));
-plot(F_s, fit_s(1)*F_s + fit_s(2), '-', 'DisplayName', sprintf('$k_s = %.1f [Nm/rad]$', (h)^2/fit_s(1)));
-plot([0.8*F_max, 1.2*F_max], [d_max, d_max], '-', 'DisplayName', sprintf('$R_{x,max} = %.1f [mrad]$', 1e3*atan2(d_max,h)));
 hold off;
-xlabel('Measured Force [N]');
+xlabel('Bending Stroke [mrad]')
-ylabel('Measured Displacement [m]');
+legend();
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/bending_stroke_histogram.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:bending_stroke_histogram
+#+caption: Histogram of the measured bending stroke
+#+RESULTS:
+[[file:figs/bending_stroke_histogram.png]]
+
+The relation between the measured beam thickness and the measured bending stiffness is shown in Figure [[fig:flex_thickness_vs_bending_stiff]].
+
+#+begin_src matlab :exports none
+load('flex_meas_dim.mat', 'meas_flex');
+figure;
+hold on;
+plot((meas_flex(:,1)+meas_flex(:,2))/2, kRx, 'o', 'DisplayName', '$x$')
+plot((meas_flex(:,3)+meas_flex(:,4))/2, kRy, 'o', 'DisplayName', '$y$')
+hold off;
+xlabel('Flexible Beam Thickness [$\mu m$]');
+ylabel('Bending Stiffness [Nm/rad]');
 legend('location', 'southeast');
-ylim([-1e-4,inf])
 #+end_src
 
-#+begin_src matlab :results value replace :exports results :tangle no
+#+begin_src matlab :tangle no :exports results :results file replace
-sprintf('Bending Stiffness is %.1f [Nm/rad]', (h)^2/fit_l(1))
+exportFig('figs/flex_thickness_vs_bending_stiff.pdf', 'width', 'wide', 'height', 'normal');
 #+end_src
 
+#+name: fig:flex_thickness_vs_bending_stiff
+#+caption: Measured bending stiffness as a function of the estimated flexible beam thickness
 #+RESULTS:
-: Bending Stiffness is 5.6 [Nm/rad]
+[[file:figs/flex_thickness_vs_bending_stiff.png]]
 
-#+begin_src matlab :results value replace :exports results :tangle no
+** Conclusion
-sprintf('Bending Stroke is %.1f [mrad]', 1e3*atan2(d_max,h))
+#+begin_important
-#+end_src
+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
 
-#+RESULTS:
-: Bending Stroke is 23.1 [mrad]