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

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

#+HTML_LINK_HOME: ../index.html
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  <p>This report is also available as a <a href="./test-bench-flexible-joints.pdf">pdf</a>.</p>
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* Notes                                                             :noexport:

Prefix for figures/section/tables =test_joints=

Compilation of the following reports:
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/test-bench-flexible-joints-old/index.org][test-bench-flexible-joints-adv]]
  Ideas of measuring shear stiffness, axial stiffness and torsion
  Not done in the end
- [X] [[/home/thomas/Cloud/work-projects/ID31-NASS/matlab/test-bench-flexible-joints-old/bending.org][bending measurement]]
  Some possible errors when measuring bending stiffness
- [X] [[file:~/Cloud/documents/internships/2021-martin-reichert/Bachelor thesis.pdf][Report of Martin]]
  Analytical estimation of the stiffness
  Estimation of possible errors in the estimation of the stiffness
  May not be useful here.
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/test-bench-nass-flexible-joints/test-bench-flexible-joints.org][test-bench-nass-flexible-joints]]
  Here only the bending stiffness is measured
- [ ] Also check start of this report: [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/nano_hexapod.org][nano_hexapod]]
  Model of the flexible joints
  Maybe add a section for the model of the flexible joints (of maybe this was done in the "detailed design" section?)
- [ ] Maybe say that some flexible joints where not machined properly (show picture with deformed machining and one with "chips" stuck inside)
- [ ] Explain why the encoder is here: in line with the measurement, no "abbe errors"

Goal:
- Characterization of flexible joints:
  - Remind the specifications
  - Most important: bending stiffness
- Dimensional measurement
- Presentation of test bench to measure stiffness
  - Possible measurement errors and how to prevent them
  - Calibration of force sensor
- Section 6 is most important: measurement results
- Conclusion


* Introduction                                                        :ignore:

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 ref:tab:test_joints_specs.

#+name: tab:test_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   | $> 100\,N/\mu m$       |    94 |
| Shear Stiffness   | $> 1\,N/\mu m$         |    13 |
| Bending Stiffness | $< 100\,Nm/\text{rad}$ |     5 |
| Torsion Stiffness | $< 500\,Nm/\text{rad}$ |   260 |
| Bending Stroke    | $> 1\,\text{mrad}$     |  24.5 |
| Torsion Stroke    | $> 5\,\mu\text{rad}$   |       |

Then, the classical geometry of a flexible ball joint shown in Figure ref:fig:test_joints_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 ref:tab:test_joints_specs.

#+name: fig:test_joints_fem_geometry
#+caption: Flexible part of the Joint used for FEM - CAD view
#+attr_latex: :width 0.5\linewidth
[[file:figs/test_joints_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 ref:fig:test_joints_received.

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

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 ref:sec:test_joints_flex_dim_meas: each flexible joint is measured using a profile projector
- Section ref:sec:test_joints_test_bench_desc: the stiffness measurement bench is presented
- Section ref:sec:test_joints_error_budget: an error budget is performed in order to estimate the accuracy of the measured stiffness
- Section ref:sec:test_joints_first_measurements: first measurements are performed
- Section ref:sec:test_joints_bending_stiffness_meas: the bending stiffness of the flexible joints are measured


#+name: tab:test_joints_section_matlab_code
#+caption: Report sections and corresponding Matlab files
#+attr_latex: :environment tabularx :width 0.6\linewidth :align lX
#+attr_latex: :center t :booktabs t
| *Sections*                  | *Matlab File*      |
|-----------------------------+--------------------|
| Section ref:sec:test_joints | =test_joints_1_.m= |
|                             |                    |
|                             |                    |

* Dimensional Measurements
:PROPERTIES:
:header-args:matlab+: :tangle matlab/test_joints_1_dim_meas.m
:END:
<<sec:test_joints_flex_dim_meas>>
** Measurement Bench

The axis corresponding to the flexible joints are defined in Figure ref:fig:test_joints_axis.

#+name: fig:test_joints_axis
#+caption: Define axis for the flexible joints
#+attr_latex: :width 0.3\linewidth
[[file:figs/test_joints_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 ref:fig:test_joints_y_flex_meas_setup.

#+name: fig:test_joints_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/test_joints_y_flex_meas_setup.png]]

What we typically observe is shown in Figure ref:fig:test_joints_soft_measure_size.
It is then possible to estimate to dimension of the flexible beam with an accuracy of $\approx 5\,\mu m$,

#+name: fig:test_joints_soft_measure_size
#+attr_latex: :width 1.0\linewidth
#+caption: Image used to measure the flexible joint's dimensions
[[file:figs/test_joints_soft_measure_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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+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 ref:tab:test_joints_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:test_joints_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 ref:fig:test_joints_size_hist.

#+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/test_joints_size_hist.pdf', 'width', 'normal', 'height', 'normal');
#+end_src

#+name: fig:test_joints_size_hist
#+caption:  Histogram for the (16x2) measured beams' thickness
#+RESULTS:
[[file:figs/test_joints_size_hist.png]]

#+begin_src matlab :exports none :tangle no
%% Save beam sizes
save('./matlab/mat/flex_meas_dim.mat', 'meas_flex');
#+end_src

#+begin_src matlab :exports none :eval no
%% Save beam sizes
save('./mat/flex_meas_dim.mat', 'meas_flex');
#+end_src

* Measurement Test Bench - Bending Stiffness
:PROPERTIES:
:header-args:matlab+: :tangle matlab/test_joints_2_bench_dimensioning.m
:END:
<<sec:test_joints_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 ref:fig:test_joints_bench_working_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_joints_bench_working_principle
#+caption: Test Bench - working principle
[[file:figs/test_joints_bench_working_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

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Flexible joint Geometry
The flexible joint used for the Nano-Hexapod is shown in Figure ref:fig:test_joints_bend_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:test_joints_bend_geometry
#+caption: Geometry of the flexible joint
[[file:figs/test_joints_bend_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 ref:fig:test_joints_bench_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_joints_bench_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_joints_bench_overview.png]]

A side view of the bench with the important quantities are shown in Figure ref:fig:test_joints_bench_side.

#+name: fig:test_joints_bench_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_joints_bench_side.png]]

* Error budget
:PROPERTIES:
:header-args:matlab+: :tangle matlab/test_joints_3_error_budget.m
:END:
<<sec:test_joints_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

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+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 ref:tab:test_joints_axial_shear_prop and ref:tab:test_joints_bending_torsion_prop.

#+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:test_joints_axial_shear_prop
#+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:test_joints_bending_torsion_prop
#+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 ref:fig:test_joints_bench_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

** 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
:PROPERTIES:
:header-args:matlab+: :tangle matlab/test_joints_4_first_meas.m
:END:
<<sec:test_joints_first_measurements>>
** Introduction                                                      :ignore:

- *Encoder*: [[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][Renishaw Resolute 1nm]]

** 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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Force Sensor Calibration

#+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 ref:fig:test_joints_force_sensor_calib.

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

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

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/test_joints_force_sensor_calib_time.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:test_joints_force_sensor_calib_time
#+caption: Measured force using both sensors as a function of time
#+RESULTS:
[[file:figs/test_joints_force_sensor_calib_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 ref:fig:test_joints_force_sensor_calib_fit.

#+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/test_joints_force_sensor_calib_fit.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

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

The measurement error between the two sensors is shown in Figure ref:fig:test_joints_force_sensor_calib_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/test_joints_force_sensor_calib_error.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

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

** Force Sensor Stiffness

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 ref:fig:test_joints_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:test_joints_meas_force_sensor_stiffness
#+caption: Bench used to measured the stiffness of the force sensor
#+attr_latex: :width 0.6\linewidth
[[file:figs/test_joints_meas_force_sensor_stiffness.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 $1\,N/\mu m$.

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 ref:fig:test_joints_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/test_joints_force_sensor_stiffness_fit.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:test_joints_force_sensor_stiffness_fit
#+caption: Displacement as a function of the measured force
#+RESULTS:
[[file:figs/test_joints_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/test_joints_5_bending_stiff_meas.m
:END:
<<sec:test_joints_bending_stiffness_meas>>
** Introduction

A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure ref:fig:test_joints_picture_bench_overview.
A closer view on flexible joint is shown in Figure ref:fig:test_joints_picture_bench_close and a zoom on the force sensor tip is shown in Figure ref:fig:test_joints_picture_bench_zoom.

#+name: fig:test_joints_picture_bench_overview
#+caption: Side view of the flexible joint stiffness bench. X-Bending stiffness is measured.
#+attr_latex: :width \linewidth
[[file:figs/test_joints_picture_bench_overview.jpg]]

#+name: fig:test_joints_picture_bench_close
#+caption: Zoom on the flexible joint - Side view
#+attr_latex: :width \linewidth
[[file:figs/test_joints_picture_bench_close.jpg]]


#+name: fig:test_joints_picture_bench_zoom
#+caption: Zoom on the tip of the force sensor
#+attr_latex: :width 0.4\linewidth
[[file:figs/test_joints_picture_bench_zoom.jpg]]

The same bench used to measure the Y-bending stiffness of the flexible joint by pivoting the flexible joint by 90 degrees.

** 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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+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 ref:fig:test_joints_meas_bend_time.

#+begin_src matlab :exports none
%% Time Domain plots
figure;
tiledlayout(2, 1, 'TileSpacing', 'Compact', '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/test_joints_meas_bend_time.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:test_joints_meas_bend_time
#+caption: Typical time domain measurements
#+RESULTS:
[[file:figs/test_joints_meas_bend_time.png]]

The displacement as a function of the force is then shown in Figure ref:fig:test_joints_meas_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/test_joints_meas_F_d.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:test_joints_meas_F_d
#+caption: Typical measurement of the diplacement as a function of the applied force
#+RESULTS:
[[file:figs/test_joints_meas_F_d.png]]

The bending stiffness can be estimated by computing the slope of the curve in Figure ref:fig:test_joints_meas_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 ref:fig:test_joints_meas_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/test_joints_meas_F_d_lin_fit.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:test_joints_meas_F_d_lin_fit
#+caption: Typical measurement of the diplacement as a function of the applied force with estimated linear fits
#+RESULTS:
[[file:figs/test_joints_meas_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:
- Bending Stiffness: 5.5Nm/rad
- Bending Stiffness at stop: 173.6Nm/rad
- Bending Stroke: 18.9mrad

** 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 ref:tab:test_joints_meas_results_x_dir for the X direction and in Table ref:tab:test_joints_meas_results_y_dir for the Y direction.

#+begin_src matlab :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:test_joints_meas_results_x_dir
#+caption: Measured characteristics of the flexible joints in the X direction
#+attr_latex: :environment tabularx :width 0.6\linewidth :align cccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|    | $R_{R_x}$ [Nm/rad] | $k_{R_x,s}$ [Nm/rad] | $R_{x,\text{max}}$ [mrad] |
|----+--------------------+----------------------+---------------------------|
|  1 |                5.5 |                173.6 |                      18.9 |
|  2 |                6.1 |                195.0 |                      17.6 |
|  3 |                6.1 |                191.3 |                      17.7 |
|  4 |                5.8 |                136.7 |                      18.3 |
|  5 |                5.7 |                 88.9 |                      22.0 |
|  6 |                5.7 |                183.9 |                      18.7 |
|  7 |                5.7 |                157.9 |                      17.9 |
|  8 |                5.8 |                166.1 |                      17.9 |
|  9 |                5.8 |                159.5 |                      18.2 |
| 10 |                6.0 |                143.6 |                      18.1 |
| 11 |                5.0 |                163.8 |                      17.7 |
| 12 |                6.1 |                111.9 |                      17.0 |
| 13 |                6.0 |                142.0 |                      17.4 |
| 14 |                5.8 |                130.1 |                      17.9 |
| 15 |                5.7 |                170.7 |                      18.6 |
| 16 |                6.0 |                148.7 |                      17.5 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([kRy; kSy; 1e3*Rmy]', {'1','2','3','4','5','6','7','8','9','10','11','12','13','14','15','16'}, {'$R_{R_y}$ [Nm/rad]', '$k_{R_y,s}$ [Nm/rad]', '$R_{y,\text{may}}$ [mrad]'}, ' %.1f ');
#+end_src

#+name: tab:test_joints_meas_results_y_dir
#+caption: Measured characteristics of the flexible joints in the Y direction
#+attr_latex: :environment tabularx :width 0.6\linewidth :align cccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|    | $R_{R_y}$ [Nm/rad] | $k_{R_y,s}$ [Nm/rad] | $R_{y,\text{may}}$ [mrad] |
|----+--------------------+----------------------+---------------------------|
|  1 |                5.7 |                323.5 |                      17.9 |
|  2 |                5.9 |                306.0 |                      17.2 |
|  3 |                6.0 |                224.4 |                      16.8 |
|  4 |                5.7 |                247.3 |                      17.8 |
|  5 |                5.8 |                250.9 |                      13.0 |
|  6 |                5.8 |                244.5 |                      17.8 |
|  7 |                5.3 |                214.8 |                      18.1 |
|  8 |                5.8 |                217.2 |                      17.6 |
|  9 |                5.7 |                225.0 |                      17.6 |
| 10 |                6.0 |                254.7 |                      17.3 |
| 11 |                4.9 |                261.1 |                      18.4 |
| 12 |                5.9 |                161.5 |                      16.7 |
| 13 |                6.1 |                227.6 |                      16.8 |
| 14 |                5.9 |                221.3 |                      17.8 |
| 15 |                5.4 |                241.5 |                      17.8 |
| 16 |                5.3 |                291.1 |                      17.7 |

** Analysis
The dispersion of the measured bending stiffness is shown in Figure ref:fig:test_joints_bend_stiff_hist and of the bending stroke in Figure ref:fig:test_joints_bend_stroke_hist.

#+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/test_joints_bend_stiff_hist.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:test_joints_bend_stiff_hist
#+caption: Histogram of the measured bending stiffness
#+RESULTS:
[[file:figs/test_joints_bend_stiff_hist.png]]

#+begin_src matlab :exports none
figure;
hold on;
histogram(1e3*Rmx, 'DisplayName', '$k_{R_x}$')
histogram(1e3*Rmy, 'DisplayName', '$k_{R_y}$')
hold off;
xlabel('Bending Stroke [mrad]')
legend();
#+end_src

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

#+name: fig:test_joints_bend_stroke_hist
#+caption: Histogram of the measured bending stroke
#+RESULTS:
[[file:figs/test_joints_bend_stroke_hist.png]]

The relation between the measured beam thickness and the measured bending stiffness is shown in Figure ref:fig:test_joints_thickness_stiffness.

#+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/test_joints_thickness_stiffness.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:test_joints_thickness_stiffness
#+caption: Measured bending stiffness as a function of the estimated flexible beam thickness
#+RESULTS:
[[file:figs/test_joints_thickness_stiffness.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

* Conclusion
<<sec:test_joints_conclusion>>

* Bibliography                                                        :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

* Helping Functions                                                 :noexport:
** Initialize Path
#+NAME: m-init-path
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./matlab/mat/'); % Path for data
addpath('./matlab/');     % Path for scripts
#+END_SRC

#+NAME: m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./mat/'); % Path for data
#+END_SRC

** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;
#+END_SRC