test-bench-nass-flexible-joints/test-bench-flexible-joints.org

#+TITLE: Flexible Joints - Test Bench
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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#+begin_export html
  <hr>
  <p>This report is also available as a <a href="./test-bench-flexible-joints.pdf">pdf</a>.</p>
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#+end_export

* Introduction                                                        :ignore:
In this document, we present a test-bench that has been developed in order to measure the bending stiffness of flexible joints.

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: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

* Flexible Joints
<<sec:flexible_joints>>

The flexible joints that are going to be measured in this document have been design to be used with a Nano-Hexapod (Figure [[fig:nano_hexapod]]).

#+name: fig:nano_hexapod
#+caption: CAD view of the Nano-Hexapod containing the flexible joints
#+attr_latex: :width 0.7\linewidth
[[file:figs/nano_hexapod.png]]

Ideally, these flexible joints would behave as perfect ball joints, that is to say:
- no bending and torsional stiffnesses
- infinite shear and axial stiffnesses
- un-limited bending and torsional stroke
- no friction, no backlash


The real characteristics of the flexible joints will influence the dynamics of the Nano-Hexapod.
Using a multi-body dynamical model of the nano-hexapod, the specifications in term of stiffness and stroke of the flexible joints have been determined and summarized in Table [[tab:flexible_joints_specs]].

#+name: tab:flexible_joints_specs
#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model
#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcc
#+attr_latex: :center t :booktabs t :float t
|                   | *Specification* | *FEM* |
|-------------------+-----------------+-------|
| Axial Stiffness   | > 100 [N/um]    |    94 |
| Shear Stiffness   | > 1 [N/um]      |    13 |
| Bending Stiffness | < 100 [Nm/rad]  |     5 |
| Torsion Stiffness | < 500 [Nm/rad]  |   260 |
| Bending Stroke    | > 1 [mrad]      |  24.5 |
| Torsion Stroke    | > 5 [urad]      |       |

Then, the classical geometry of a flexible ball joint shown in Figure [[fig:flexible_joint_fem_geometry]] has been optimized in order to meet the requirements.
This has been done using a Finite Element Software and the obtained joint's characteristics are summarized in Table [[tab:flexible_joints_specs]].

#+name: fig:flexible_joint_fem_geometry
#+caption: Flexible part of the Joint used for FEM - CAD view
#+attr_latex: :width 0.5\linewidth
[[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 flexible joints can be seen on Figure [[fig:received_flex]].

#+name: fig:received_flex
#+caption: 15 of the 16 flexible joints
#+attr_latex: :width \linewidth
[[file:figs/IMG_20210302_173619.jpg]]

* Dimensional Measurements
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/dim_meas.m
: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 "X" 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]]


** 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
#+end_src

** Measurement Results
# - Strange shape: 5

The expected flexible beam thickness is $250\,\mu m$.
However, it is more important that the thickness of all beams are close to each other.

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'}, {'Y1', 'Y2', 'X1', 'X2'}, ' %.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:
|    |  Y1 |  Y2 |  X1 |  X2 |
|----+-----+-----+-----+-----|
|  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
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/bench_dimensioning.m
:END:
<<sec:test_bench_desc>>
** Introduction                                                      :ignore:
The most important characteristic of the flexible joint that we want to measure is its bending stiffness $k_{R_x} \approx k_{R_y}$.

To do so, we have to apply a torque $T_x$ on the flexible joint and measure its angular deflection $\theta_x$.
The stiffness is then
\begin{equation}
k_{R_x} = \frac{T_x}{\theta_x}
\end{equation}

As it is quite difficult to apply a pure torque, a force will be applied instead.
The application point of the force should far enough from the flexible part such that the obtained bending is much larger than the displacement in shear.

The working principle of the bench is schematically shown in Figure [[fig:test_bench_principle]].
One part of the flexible joint is fixed. On the mobile part, a force $F_x$ is applied which is equivalent to a torque applied on the flexible joint center.
The induced rotation is measured with a displacement sensor $d_x$.

#+name: fig:test_bench_principle
#+caption: Test Bench - working principle
[[file:figs/test_bench_principle.png]]


This test-bench will be used to have a first approximation of the bending stiffnesss and stroke of the flexible joints.
Another test-bench, better engineered will be used to measure the flexible joint's characteristics with better accuracy.

** 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

** Flexible joint Geometry
The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]].
Its bending stiffness is foreseen to be $k_{R_y}\approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 25\,mrad$.

#+name: fig:flexible_joint_geometry
#+caption: Geometry of the flexible joint
[[file:figs/flexible_joint_geometry.png]]

The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$.

Let's define the parameters on Matlab.
#+begin_src matlab
kRx = 5; % Bending Stiffness [Nm/rad]
Rxmax = 25e-3; % Bending Stroke [rad]
h = 20e-3; % Height [m]
#+end_src

** Required external applied force

The bending $\theta_y$ of the flexible joint due to the force $F_x$ is:
\begin{equation}
  \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
\end{equation}

Therefore, the applied force to test the full range of the flexible joint is:
\begin{equation}
  F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
\end{equation}

#+begin_src matlab
Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
#+end_src

And we obtain:
#+begin_src matlab :results value raw replace :exports results
sprintf('\\begin{equation} F_{x,max} = %.1f\\, [N] \\end{equation}', Fxmax)
#+end_src

#+RESULTS:
\begin{equation} F_{x,max} = 6.2\, [N] \end{equation}

The measurement range of the force sensor should then be higher than $6.2\,N$.

** Required actuator stroke and sensors range

The flexible joint is designed to allow a bending motion of $\pm 25\,mrad$.
The corresponding stroke at the location of the force sensor is:
\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]

#+begin_src matlab
dxmax = h*tan(Rxmax);
#+end_src

#+begin_src matlab :results value raw replace :exports results
sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax)
#+end_src

#+RESULTS:
\begin{equation} d_{max} = 0.5\, [mm] \end{equation}

In order to test the full range of the flexible joint, the stroke of the translation stage used to move the force sensor should be higher than $0.5\,mm$.
Similarly, the measurement range of the displacement sensor should also be higher than $0.5\,mm$.

** Test Bench

A CAD view of the measurement bench is shown in Figure [[fig:test_bench_flex_overview]].

#+begin_note
Here are the different elements used in this bench:
- *Translation Stage*: [[file:doc/V-408-Datasheet.pdf][V-408]]
- *Load Cells*: [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]]
- *Encoder*: [[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][Renishaw Resolute 1nm]]
#+end_note

Both the measured force and displacement are acquired at the same time using a Speedgoat machine.

#+name: fig:test_bench_flex_overview
#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
#+attr_latex: :width 0.8\linewidth
[[file:figs/test_bench_flex_overview.png]]

A side view of the bench with the important quantities are shown in Figure [[fig:test_bench_flex_side]].

#+name: fig:test_bench_flex_side
#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
#+attr_latex: :width 0.25\linewidth
#+attr_html: :width 300px
[[file:figs/test_bench_flex_side.png]]

* Error budget
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/error_budget.m
:END:
<<sec:error_budget>>
** Introduction                                                      :ignore:

Many things can impact the accuracy of the measured bending stiffness such as:
- Errors in the force and displacement measurement
- Shear effects
- Deflection of the Force sensor
- Errors in the geometry of the bench

In this section, we wish to estimate the attainable accuracy with the current bench, and identified the limiting factors.


** 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

** Finite Element Model
From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables [[tab:axial_shear_characteristics]] and [[tab:bending_torsion_characteristics]].

#+begin_src matlab :exports none
%% 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]
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([1e-6*ka, Fa, 1e6*Xa; 1e-6*ks, Fs, 1e6*Xs], {'Axial', 'Shear'}, {'Stiffness [N/um]', 'Max Force [N]', 'Stroke [um]'}, ' %.0f ');
#+end_src

#+name: tab:axial_shear_characteristics
#+caption: Axial/Shear characteristics
#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|       | Stiffness [N/um] | Max Force [N] | Stroke [um] |
|-------+------------------+---------------+-------------|
| Axial |               94 |           469 |           5 |
| Shear |               13 |           242 |          19 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([kb, 1e3*Fb, 1e3*Xb; kt, 1e3*Ft, 1e3*Xt], {'Bending', 'Torsional'}, {'Stiffness [Nm/rad]', 'Max Torque [Nmm]', 'Stroke [mrad]'}, ' %.0f ');
#+end_src

#+name: tab:bending_torsion_characteristics
#+caption: Bending/Torsion characteristics
#+attr_latex: :environment tabularx :width 0.7\linewidth :align Xccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|           | Stiffness [Nm/rad] | Max Torque [Nmm] | Stroke [mrad] |
|-----------+--------------------+------------------+---------------|
| Bending   |                  5 |              118 |            24 |
| Torsional |                260 |             1508 |             6 |

** Setup

The setup is schematically represented in Figure [[fig:test_bench_flex_side_bis]].

The force is applied on top of the flexible joint with a distance $h$ with the joint's center.
The displacement of the flexible joint is also measured at the same height.

The height between the joint's center and the force application point is:
#+begin_src matlab
h = 25e-3; % Height [m]
#+end_src

#+name: fig:test_bench_flex_side_bis
#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
#+attr_latex: :width 0.25\linewidth
#+attr_html: :width 300px
[[file:figs/test_bench_flex_side.png]]

** Effect of Bending
The torque applied is:
\begin{equation}
  M_y = F_x \cdot h
\end{equation}

The flexible joint is experiencing a rotation $\theta_y$ due to the torque $M_y$:
\begin{equation}
  \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x \cdot h}{k_{R_y}}
\end{equation}

This rotation is then measured by the displacement sensor.
The measured displacement is:
\begin{equation}
  D_b = h \tan(\theta_y) = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) \label{eq:bending_stiffness_formula}
\end{equation}

** Computation of the bending stiffness
From equation eqref:eq:bending_stiffness_formula, we can compute the bending stiffness:
\begin{equation}
  k_{R_y} = \frac{F_x \cdot h}{\tan^{-1}\left( \frac{D_b}{h} \right)}
\end{equation}

For small displacement, we have
\begin{equation}
  \boxed{k_{R_y} \approx h^2 \frac{F_x}{d_x}}
\end{equation}

And therefore, to precisely measure $k_{R_y}$, we need to:
- precisely measure the motion $d_x$
- precisely measure the applied force $F_x$
- precisely now the height of the force application point $h$

** Estimation error due to force and displacement sensors accuracy
The maximum error on the measured displacement with the encoder is 40 nm.
This quite negligible compared to the measurement range of 0.5 mm.

The accuracy of the force sensor is around 1% and therefore, we should expect to have an accuracy on the measured stiffness of at most 1%.

** Estimation error due to Shear
The effect of Shear on the measured displacement is simply:
\begin{equation}
  D_s = \frac{F_x}{k_s}
\end{equation}

The measured displacement will be the effect of shear + effect of bending
\begin{equation}
  d_x = D_b + D_s = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_s} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_s} \right)
\end{equation}

The estimated bending stiffness $k_{\text{est}}$ will then be:
\begin{equation}
  k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}}
\end{equation}

#+begin_src matlab :results value replace :exports results :tangle no
sprintf('The measurement error due to Shear is %.1f %%', 100*abs(1-1/(1 + kb/(ks*h^2))))
#+end_src

#+RESULTS:
: The measurement error due to Shear is 0.1 %

** Estimation error due to force sensor compression
The measured displacement is not done directly at the joint's location.
The force sensor compression will then induce an error on the joint's stiffness.

The force sensor stiffness $k_F$ is estimated to be around:
#+begin_src matlab
kF = 50/0.05e-3; % [N/m]
#+end_src

#+begin_src matlab :results value replace :exports results
sprintf('k_F = %.1e [N/m]', kF)
#+end_src

#+RESULTS:
: k_F = 1.0e+06 [N/m]

The measured displacement will be the sum of the displacement induced by the bending and by the compression of the force sensor:
\begin{equation}
  d_x = D_b + \frac{F_x}{k_F} = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_F} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_F} \right)
\end{equation}

The estimated bending stiffness $k_{\text{est}}$ will then be:
\begin{equation}
  k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}}
\end{equation}

#+begin_src matlab :results value replace :exports results :tangle no
sprintf('The measurement error due to height estimation errors is %.1f %%', 100*abs(1-1/(1 + kb/(kF*h^2))))
#+end_src

#+RESULTS:
: The measurement error due to height estimation errors is 0.8 %

** Estimation error due to height estimation error
Let's consider an error in the estimation of the height from the application of the force to the joint's center:
\begin{equation}
h_{\text{est}} = h (1 + \epsilon)
\end{equation}

The computed bending stiffness will be:
\begin{equation}
  k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x}
\end{equation}

And the stiffness estimation error is:
\begin{equation}
\frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2
\end{equation}

#+begin_src matlab
h_err = 0.2e-3; % Height estimation error [m]
#+end_src

#+begin_src matlab :results value replace :exports results :tangle no
sprintf('The measurement error due to height estimation errors of %.1f [mm] is %.1f %%', 1e3*h_err, 100*abs(1-(1 + h_err/h)^2))
#+end_src

#+RESULTS:
: The measurement error due to height estimation errors of 0.2 [mm] is 1.6 %

** Conclusion
Based on the above analysis, we should expect no better than few percent of accuracy using the current test-bench.
This is well enough for a first estimation of the bending stiffness of the flexible joints.

Another measurement bench allowing better accuracy will be developed.

* First Measurements
<<sec:first_measurements>>
** Introduction                                                      :ignore:


- Section [[sec:test_meas_probe]]:
- Section [[sec:meas_probe_stiffness]]:

** Agreement between the probe and the encoder
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/probe_vs_encoder.m
:END:
<<sec:test_meas_probe>>
*** Introduction                                                    :ignore:

#+begin_note
- *Encoder*: [[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][Renishaw Resolute 1nm]]
- *Displacement Probe*: [[file:doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf][Millimar C1216 electronics]] and [[file:doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf][Millimar 1318 probe]]
#+end_note

*** Setup                                                           :ignore:
The measurement setup is made such that the probe measured the translation table displacement.
It should then measure the same displacement as the encoder.
Using this setup, we should be able to compare the probe and the encoder.

*** 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
#+end_src

*** Results                                                         :ignore:
Let's load the measurements.
#+begin_src matlab
load('meas_probe_against_encoder.mat', 't', 'd', 'dp', 'F')
#+end_src

#+begin_src matlab :exports none
%% Sampling time [s]
Ts = (t(end) - t(1))/(length(t)-1);

%% Remove first second
t = t(ceil(1/Ts):end);
d = -d(ceil(1/Ts):end);
dp = -dp(ceil(1/Ts):end);
F = F(ceil(1/Ts):end);
#+end_src

The time domain measured displacement by the probe and by the encoder is shown in Figure [[fig:comp_encoder_probe_time]].

#+begin_src matlab :exports none
%% Time Domain plots
figure;
hold on;
plot(t, d, 'DisplayName', 'Encoder');
plot(t, dp, 'DisplayName', 'Probe');
hold off;
xlabel('Time [s]'); ylabel('Displacement [m]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_encoder_probe_time.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:comp_encoder_probe_time
#+caption: Time domain measurement
#+RESULTS:
[[file:figs/comp_encoder_probe_time.png]]

If we zoom, we see that there is some delay between the encoder and the probe (Figure [[fig:comp_encoder_probe_time_zoom]]).

#+begin_src matlab :exports none
%% Zoom
figure;
hold on;
plot(t, d, 'DisplayName', 'Encoder');
plot(t, dp, 'DisplayName', 'Probe');
hold off;
xlabel('Time [s]'); ylabel('Displacement [m]');
xlim([7.7, 7.9])
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_encoder_probe_time_zoom.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:comp_encoder_probe_time_zoom
#+caption: Time domain measurement (Zoom)
#+RESULTS:
[[file:figs/comp_encoder_probe_time_zoom.png]]

This delay is estimated using the =finddelay= command.

#+begin_src matlab :results value replace :exports results :tangle no
sprintf('The time delay is approximately %.1f [ms]', 1e3*Ts*finddelay(d, dp))
#+end_src

#+RESULTS:
: The time delay is approximately 15.8 [ms]

The measured mismatch between the encoder and the probe with and without compensating for the time delay are shown in Figure [[fig:comp_encoder_probe_mismatch]].

#+begin_src matlab :exports none
figure;
hold on;
plot(t, d-dp, 'DisplayName', 'Raw Mismatch');
plot(t(1:end-finddelay(d, dp)), d(1:end-finddelay(d, dp))-dp(finddelay(d, dp)+1:end), 'DisplayName', 'Removed Delay');
hold off;
xlabel('Time [s]'); ylabel('Measurement Missmatch [m]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_encoder_probe_mismatch.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:comp_encoder_probe_mismatch
#+caption: Measurement mismatch, with and without delay compensation
#+RESULTS:
[[file:figs/comp_encoder_probe_mismatch.png]]

Finally, the displacement of the probe is shown as a function of the displacement of the encoder and a linear fit is made (Figure [[fig:comp_encoder_probe_linear_fit]]).

#+begin_src matlab :exports none
figure;
hold on;
plot(1e3*d, 1e3*dp, 'DisplayName', 'Raw data');
plot(1e3*d, 1e3*d*(d\dp), 'DisplayName', sprintf('Linear fit: $\\alpha = %.5f$', (d\dp)));
hold on;
xlabel('Encoder [mm]'); ylabel('Probe [mm]');
legend('location', 'southeast')
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_encoder_probe_linear_fit.pdf', 'width', 'normal', 'height', 'normal');
#+end_src

#+name: fig:comp_encoder_probe_linear_fit
#+caption: Measured displacement by the probe as a function of the measured displacement by the encoder
#+RESULTS:
[[file:figs/comp_encoder_probe_linear_fit.png]]

#+begin_important
From the measurement, it is shown that the probe is well calibrated.
However, there is some time delay of tens of milliseconds that could induce some measurement errors.
#+end_important

** Measurement of the Millimar 1318 probe stiffness
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/probe_stiffness.m
:END:
<<sec:meas_probe_stiffness>>

*** Introduction                                                    :ignore:

#+begin_note
- *Translation Stage*: [[file:doc/V-408-Datasheet.pdf][V-408]]
- *Load Cell*: [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]]
- *Encoder*: [[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][Renishaw Resolute 1nm]]
- *Displacement Probe*: [[file:doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf][Millimar C1216 electronics]] and [[file:doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf][Millimar 1318 probe]]
#+end_note

#+name: fig:setup_mahr_stiff_meas_side
#+caption: Setup - Side View
#+attr_latex: :width \linewidth
[[file:figs/setup_mahr_stiff_meas_side.jpg]]

#+name: fig:setup_mahr_stiff_meas_top
#+caption: Setup - Top View
#+attr_latex: :width \linewidth
[[file:figs/setup_mahr_stiff_meas_top.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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
#+end_src

*** Results                                                         :ignore:
Let's load the measurement results.
#+begin_src matlab
load('meas_stiff_probe.mat', 't', 'd', 'dp', 'F')
#+end_src

#+begin_src matlab :exports none
%% Sampling time [s]
Ts = (t(end) - t(1))/(length(t)-1);

%% Remove first second
t = t(ceil(1/Ts):end);
d = d(ceil(1/Ts):end);
dp = dp(ceil(1/Ts):end);
F = F(ceil(1/Ts):end);


%% Remove Offset
t = t - t(1);
F = F - mean(F(1:ceil(1/Ts)));
#+end_src

The time domain measured force and displacement are shown in Figure [[fig:mahr_time_domain]].

#+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);
xlabel('Time [s]'); ylabel('Displacement [m]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/mahr_time_domain.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:mahr_time_domain
#+caption: Time domain measurements
#+RESULTS:
[[file:figs/mahr_time_domain.png]]


Now we can estimate the stiffness with a linear fit.

#+begin_src matlab :results value replace :exports results :tangle no
sprintf('Stiffness is %.3f [N/mm]', abs(1e-3*(d\F)))
#+end_src

This is very close to the 0.04 [N/mm] written in the [[file:doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf][Millimar 1318 probe datasheet]].

And compare the linear fit with the raw measurement data (Figure [[fig:mahr_stiffness_f_d_plot]]).

#+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]');
ylabel('Measured Displacement [m]');
legend('location', 'southeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/mahr_stiffness_f_d_plot.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:mahr_stiffness_f_d_plot
#+caption: Measured displacement as a function of the measured force. Raw data and linear fit
#+RESULTS:
[[file:figs/mahr_stiffness_f_d_plot.png]]

#+begin_summary
The Millimar 1318 probe has a stiffness of $\approx 0.04\,[N/mm]$.
#+end_summary

** Force Sensor Calibration
*** Introduction                                                    :ignore:

#+begin_note
*Load Cells*:
- [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]]
- [[file:doc/FRE_DS_XFL212R_FR_A3.pdf][XFL212R]]
#+end_note

There are both specified to have $\pm 1 \%$ of non-linearity over the full range.

The XFL212R has a spherical interface while the FC2231 has a flat surface.
Therefore, we should have a nice point contact when using the two force sensors as shown in Figure [[fig:force_sensor_calibration_setup]].

#+name: fig:force_sensor_calibration_setup
#+caption: Zoom on the two force sensors in contact
#+attr_latex: :width 0.8\linewidth
[[file:figs/IMG_20210309_145333.jpg]]

The two force sensors are therefore measuring the exact same force, and we can compare the two measurements.

*** 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
#+end_src

*** Analysis                                                        :ignore:
Let's load the measured force of both sensors.
#+begin_src matlab
%% Load measurement data
load('calibration_force_sensor.mat', 't', 'F', 'Fc')
#+end_src

We remove any offset such that they are both measuring no force when not in contact.
#+begin_src matlab
%% Remove offset
F  = F  - mean(F( t > 0.5 & t < 1.0));
Fc = Fc - mean(Fc(t > 0.5 & t < 1.0));
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
plot(t, F, 'DisplayName', 'FC2231');
plot(t, Fc, 'DisplayName', 'XFL212R');
hold off;
xlabel('Time [s]'); ylabel('Measured Force [N]');
xlim([0,15]); ylim([0,55]);
legend('location', 'southeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/force_calibration_time.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:force_calibration_time
#+caption: Measured force using both sensors as a function of time
#+RESULTS:
[[file:figs/force_calibration_time.png]]

Let's select only the first part from the moment they are in contact until the maximum force is reached.

#+begin_src matlab
%% Only get the first part until maximum force
F  = F( t > 1.55 & t < 4.65);
Fc = Fc(t > 1.55 & t < 4.65);
#+end_src

Then, let's make a linear fit between the two measured forces.

#+begin_src matlab
%% Make a line fit
fit_F = polyfit(Fc, F, 1);
#+end_src

The two forces are plotted against each other as well as the linear fit in Figure [[fig:calibrated_force_dit]].

#+begin_src matlab :exports none
figure;
hold on;
plot(Fc, F, '-', 'DisplayName', 'Raw Data');
plot(Fc([1,end]), Fc([1,end])*fit_F(1) + fit_F(2), '--', 'DisplayName', 'Line Fit');
hold off;
xlabel('XFL212R [N]'); ylabel('FC2231 [N]');
xlim([0,50]); ylim([0,50]);
legend('location', 'southeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/calibrated_force_dit.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:calibrated_force_dit
#+caption: Measured two forces and linear fit
#+RESULTS:
[[file:figs/calibrated_force_dit.png]]

The measurement error between the two sensors is shown in Figure [[fig:force_meas_error]].
It is below 0.1N for the full measurement range.

#+begin_src matlab :exports none
figure;
hold on;
plot(Fc, F - (Fc*fit_F(1) + fit_F(2)), 'k-');
hold off;
xlim([0,50]); ylim([-0.12, 0.12]);
xlabel('Measured Force [N]');
ylabel('Error [N]')
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/force_meas_error.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:force_meas_error
#+caption: Error in Newtons
#+RESULTS:
[[file:figs/force_meas_error.png]]

The same error is shown in percentage in Figure [[fig:force_meas_error_percentage]].
The error is less than 1% when the measured force is above 5N.

#+begin_src matlab :exports none
figure;
plot(Fc, 100*(F - (Fc*fit_F(1) + fit_F(2)))./Fc, 'k-');
xlim([0,50]); ylim([-4, 1]);
xlabel('Measured Force [N]');
ylabel('Error [\%]')
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/force_meas_error_percentage.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:force_meas_error_percentage
#+caption: Error in percentage
#+RESULTS:
[[file:figs/force_meas_error_percentage.png]]

** Force Sensor Noise
*** Introduction                                                    :ignore:
The objective of this measurement is to estimate the noise of the force sensor [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]].
To do so, we don't apply any force to the sensor, and we measure its output for 100s.

*** 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
#+end_src

*** Analysis                                                        :ignore:
Let's load the measurement data.

#+begin_src matlab
%% Load measurement data
load('force_sensor_noise_meas.mat', 't', 'F');
Ts = t(2) - t(1);
#+end_src

The measured force is shown in Figure [[fig:force_noise_time]].

#+begin_src matlab :exports none
%% Take last 100s
F = F(t > t(end)-100);
t = t(t > t(end)-100);

%% Remove force offset and reset time
F = F - mean(F);
t = t - t(1);
#+end_src

#+begin_src matlab :exports none
figure;
plot(t, F)
xlabel('Time [s]');
ylabel('Force [N]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/force_noise_time.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:force_noise_time
#+caption: Measured force
#+RESULTS:
[[file:figs/force_noise_time.png]]

Let's now compute the Amplitude Spectral Density of the measured force.

#+begin_src matlab
%% Compute Spectral Density of Measured Force
% Hanning window
win = hanning(ceil(1/Ts));

% Power Spectral Density
[pxx, f] = pwelch(F, win, [], [], 1/Ts);
#+end_src

The results is shown in Figure [[fig:force_noise_asd]].

#+begin_src matlab :exports none
figure;
hold on;
plot(f, sqrt(pxx));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$N/\sqrt{Hz}$]');
xlim([1, 1/Ts/2]); ylim([4e-5, 1e-3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/force_noise_asd.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:force_noise_asd
#+caption: Amplitude Spectral Density of the meaured force
#+RESULTS:
[[file:figs/force_noise_asd.png]]


** Force Sensor Stiffness
*** Introduction                                                    :ignore:

The objective of this measurement is to estimate the stiffness of the force sensor [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]].

To do so, a very stiff element is fixed in front of the force sensor as shown in Figure [[fig:setup_meas_force_sensor_stiffness]].

Then, we apply a force on the stiff element through the force sensor.
We measure the deflection of the force sensor using an encoder.

Then, having the force and the deflection, we should be able to estimate the stiffness of the force sensor supposing the stiffness of the other elements are much larger.

#+name: fig:setup_meas_force_sensor_stiffness
#+caption: Bench used to measured the stiffness of the force sensor
#+attr_latex: :width 0.6\linewidth
[[file:figs/IMG_20210309_145242.jpg]]

From the documentation, the deflection of the sensor at the maximum load (50N) is 0.05mm, the stiffness is therefore foreseen to be around $1\,N/\mu m$.

*** 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
#+end_src

*** Analysis                                                        :ignore:

Let's load the measured force as well as the measured displacement.
#+begin_src matlab
%% Load measurement data
load('force_sensor_stiffness_meas.mat', 't', 'F', 'd')
#+end_src

Some pre-processing is applied on the data.
#+begin_src matlab
%% 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);
#+end_src

The linear fit is performed.
#+begin_src matlab
%% Linear fit
fit_k = polyfit(F, d, 1);
#+end_src

The displacement as a function of the force as well as the linear fit are shown in Figure [[fig:force_sensor_stiffness_fit]].
#+begin_src matlab :exports none
figure;
hold on;
plot(F, 1e6*d, '-', 'DisplayName', 'Raw Data');
plot(F([1,end]), 1e6*(F([1,end])*fit_k(1) + fit_k(2)), '--', 'DisplayName', 'Line Fit');
hold off;
xlabel('Force [$N$]'); ylabel('Displacement [$\mu m$]');
xlim([0,45]); ylim([0,60]);
legend('location', 'southeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/force_sensor_stiffness_fit.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:force_sensor_stiffness_fit
#+caption: Displacement as a function of the measured force
#+RESULTS:
[[file:figs/force_sensor_stiffness_fit.png]]

And we obtain the following stiffness:
#+begin_src matlab :results value replace :exports results
%% Force Sensor Stiffness
sprintf('k = %.2f [N/um]', 1e-6*1/fit_k(1));
#+end_src

#+RESULTS:
: k = 0.76 [N/um]

* 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>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
#+end_src

** 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
h = 25e-3; % [m]
#+end_src

#+begin_src matlab
%% Load Data
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
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;
tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile;
plot(t, F);
ylabel('Force [N]'); set(gca, 'XTickLabel',[]);

ax2 = nexttile;
plot(t, 1e3*d);
hold off;
xlabel('Time [s]'); ylabel('Displacement [mm]');

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
%% Fit the best straight line for the two regions
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
figure;
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

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/flex_joint_meas_example_F_d_lin_fit.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:flex_joint_meas_example_F_d_lin_fit
#+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);

    %% Reset displacement based on fit
    d = d - fit_l(2);
    fit_s(2) = fit_s(2) - fit_l(2);
    fit_l(2) = 0;

    %% Estimated Stroke
    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));

    %% Save stiffnesses and stroke
    kRx(i) = (h)^2/fit_l(1);
    kSx(i) = (h)^2/fit_s(1);
    Rmx(i) = atan2(d_max,h);

end
#+end_src

#+begin_src matlab :exports none
%% Initialize variables
kRy = zeros(1,16);
kSy = zeros(1,16);
Rmy = zeros(1,16);

for i = 1:16
    %% Load the data
    load(['meas_stiff_flex_' num2str(i) '_y.mat'], 't', 'F', 'd');

    %% Automatic Zero of the force
    F = F - mean(F(t > 0.1 & t < 0.3));

    %% Start measurement at t = 0.2 s
    d = d(t > 0.2);
    F = F(t > 0.2);
    t = t(t > 0.2); t = t - t(1);

    %% Estimate linear region and "stop" region
    i_l_start = find(F > 0.3, 1, 'first');
    d = d - d(i_l_start);
    [~, i_s_stop] = max(F);
    i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
    i_s_start = find(F > max(F)-1, 1, 'first');

    F_l = F(i_l_start:i_l_stop);
    d_l = d(i_l_start:i_l_stop);

    F_s = F(i_s_start:i_s_stop);
    d_s = d(i_s_start:i_s_stop);

    %% Straight line fit
    fit_l = polyfit(F_l, d_l, 1);
    fit_s = polyfit(F_s, d_s, 1);

    %% Reset displacement based on fit
    d = d - fit_l(2);
    fit_s(2) = fit_s(2) - fit_l(2);
    fit_l(2) = 0;

    %% Estimated Stroke
    d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));

    %% Save stiffnesses and stroke
    kRy(i) = (h)^2/fit_l(1);
    kSy(i) = (h)^2/fit_s(1);
    Rmy(i) = atan2(d_max,h);

end
#+end_src

The results are summarized in Table [[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;
histogram(kRx, 'DisplayName', '$k_{R_x}$')
histogram(kRy, 'DisplayName', '$k_{R_y}$')
hold off;
xlabel('Bending Stiffness [Nm/rad]')
legend();
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
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:
[[file:figs/bending_stiffness_histogram.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();
#+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');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
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:
[[file:figs/flex_thickness_vs_bending_stiff.png]]

** Conclusion
#+begin_important
The measured bending stiffness and bending stroke of the flexible joints are very close to the estimated one using a Finite Element Model.

The characteristics of all the flexible joints are also quite close to each other.
This should allow us to model them with unique parameters.
#+end_important