#+TITLE: Finite Element Model with Simscape
:DRAWER:
#+STARTUP: overview

#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :mkdirp yes

#+PROPERTY: header-args:shell  :eval no-export

#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
:END:

* Introduction                                                        :ignore:

In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis.

- Section [[sec:APA300ML]]:
  A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape.
  The static and dynamical properties of the APA300ML are then estimated using the Simscape model.
- Section [[sec:flexor_ID16]]:
  A first geometry of a Flexible joint is modelled and its characteristics are identified from the Stiffness matrix as well as from the Simscape model.
- Section [[sec:flexor_025]]:
  An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model.
- Section [[sec:integral_force_feedback]]:
- Section [[sec:strut_fem]]:
  A super element of a complete strut is exported.

* APA300ML
:PROPERTIES:
:header-args:matlab+: :tangle matlab/APA300ML.m
:END:
<<sec:APA300ML>>
** Introduction                                                      :ignore:

In this section, the Amplified Piezoelectric Actuator APA300ML ([[file:doc/APA300ML.pdf][doc]]) is modeled using a Finite Element Software.
Then a /super element/ is exported and imported in Simscape where its dynamic is studied.

A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure [[fig:apa300ml_ansys]].
The remote point used are also shown in this figure.

#+name: fig:apa300ml_ansys
#+caption: Ansys FEM of the APA300ML
[[file:figs/apa300ml_ansys.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/');
  addpath('matlab/APA300ML/');
#+end_src

#+begin_src matlab :eval no
  addpath('APA300ML/');
#+end_src

#+begin_src matlab
  open('APA300ML.slx');
#+end_src

** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
We first extract the stiffness and mass matrices.
#+begin_src matlab
  K = readmatrix('mat_K.CSV');
  M = readmatrix('mat_M.CSV');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src

#+caption: First 10x10 elements of the Stiffness matrix
#+RESULTS:
| 200000000.0 |    30000.0 |  -20000.0 |     -70.0 | 300000.0 |    40.0 |  10000000.0 |    10000.0 |   -6000.0 |     30.0 |
|     30000.0 | 30000000.0 |    2000.0 | -200000.0 |     60.0 |   -10.0 |      4000.0 |  2000000.0 |    -500.0 |   9000.0 |
|    -20000.0 |     2000.0 | 7000000.0 |     -10.0 |    -30.0 |    10.0 |      6000.0 |      900.0 | -500000.0 |        3 |
|       -70.0 |  -200000.0 |     -10.0 |    1000.0 |     -0.1 |    0.08 |       -20.0 |    -9000.0 |         3 |    -30.0 |
|    300000.0 |       60.0 |     -30.0 |      -0.1 |    900.0 |     0.1 |     30000.0 |       20.0 |     -10.0 |     0.06 |
|        40.0 |      -10.0 |      10.0 |      0.08 |      0.1 | 10000.0 |        20.0 |          9 |        -5 |     0.03 |
|  10000000.0 |     4000.0 |    6000.0 |     -20.0 |  30000.0 |    20.0 | 200000000.0 |    10000.0 |    9000.0 |     50.0 |
|     10000.0 |  2000000.0 |     900.0 |   -9000.0 |     20.0 |       9 |     10000.0 | 30000000.0 |    -500.0 | 200000.0 |
|     -6000.0 |     -500.0 | -500000.0 |         3 |    -10.0 |      -5 |      9000.0 |     -500.0 | 7000000.0 |       -2 |
|        30.0 |     9000.0 |         3 |     -30.0 |     0.06 |    0.03 |        50.0 |   200000.0 |        -2 |   1000.0 |


#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src

#+caption: First 10x10 elements of the Mass matrix
#+RESULTS:
|    0.01 |  -2e-06 |  1e-06 |  6e-09 |  5e-05 | -5e-09 | -0.0005 |  -7e-07 |  6e-07 | -3e-09 |
|  -2e-06 |    0.01 |  8e-07 | -2e-05 | -8e-09 |  2e-09 |  -9e-07 | -0.0002 |  1e-08 | -9e-07 |
|   1e-06 |   8e-07 |  0.009 |  5e-10 |  1e-09 | -1e-09 |  -5e-07 |   3e-08 |  6e-05 |  1e-10 |
|   6e-09 |  -2e-05 |  5e-10 |  3e-07 |  2e-11 | -3e-12 |   3e-09 |   9e-07 | -4e-10 |  3e-09 |
|   5e-05 |  -8e-09 |  1e-09 |  2e-11 |  6e-07 | -4e-11 |  -1e-06 |  -2e-09 |  1e-09 | -8e-12 |
|  -5e-09 |   2e-09 | -1e-09 | -3e-12 | -4e-11 |  1e-07 |  -2e-09 |  -1e-09 | -4e-10 | -5e-12 |
| -0.0005 |  -9e-07 | -5e-07 |  3e-09 | -1e-06 | -2e-09 |    0.01 |   1e-07 | -3e-07 | -2e-08 |
|  -7e-07 | -0.0002 |  3e-08 |  9e-07 | -2e-09 | -1e-09 |   1e-07 |    0.01 | -4e-07 |  2e-05 |
|   6e-07 |   1e-08 |  6e-05 | -4e-10 |  1e-09 | -4e-10 |  -3e-07 |  -4e-07 |  0.009 | -2e-10 |
|  -3e-09 |  -9e-07 |  1e-10 |  3e-09 | -8e-12 | -5e-12 |  -2e-08 |   2e-05 | -2e-10 |  3e-07 |


Then, we extract the coordinates of the interface nodes.
#+begin_src matlab
  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number |   x [m] | y [m] |  z [m] |
|--------+-------------+---------+-------+--------|
|    1.0 |    697783.0 |     0.0 |   0.0 | -0.015 |
|    2.0 |    697784.0 |     0.0 |   0.0 |  0.015 |
|    3.0 |    697785.0 | -0.0325 |   0.0 |    0.0 |
|    4.0 |    697786.0 | -0.0125 |   0.0 |    0.0 |
|    5.0 |    697787.0 | -0.0075 |   0.0 |    0.0 |
|    6.0 |    697788.0 |  0.0125 |   0.0 |    0.0 |
|    7.0 |    697789.0 |  0.0325 |   0.0 |    0.0 |

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+caption: Some extracted parameters of the FEM
#+RESULTS:
| Total number of Nodes     |   7 |
| Number of interface Nodes |   7 |
| Number of Modes           | 120 |
| Size of M and K matrices  | 162 |

Using =K=, =M= and =int_xyz=, we can now use the =Reduced Order Flexible Solid= simscape block.

** Piezoelectric parameters
In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
#+begin_src matlab
  d33 = 300e-12; % Strain constant [m/V]
  n   = 80;      % Number of layers per stack
  eT  = 1.6e-8;  % Permittivity under constant stress [F/m]
  sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
  ka  = 235e6;   % Stack stiffness [N/m]
  C   = 5e-6;    % Stack capactiance [F]
#+end_src

The ratio of the developed force to applied voltage is:
#+name: eq:piezo_voltage_to_force
\begin{equation}
  F_a = g_a V_a, \quad g_a = d_{33} n k_a
\end{equation}
where:
- $F_a$: developed force in [N]
- $n$: number of layers of the actuator stack
- $d_{33}$: strain constant in [m/V]
- $k_a$: actuator stack stiffness in [N/m]
- $V_a$: applied voltage in [V]

If we take the numerical values, we obtain:
#+begin_src matlab :results replace value
  d33*n*ka % [N/V]
#+end_src

#+RESULTS:
: 5.64

From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
#+name: eq:piezo_strain_to_voltage
\begin{equation}
  V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
\end{equation}
where:
- $V_s$: measured voltage in [V]
- $d_{33}$: strain constant in [m/V]
- $\epsilon^T$: permittivity under constant stress in [F/m]
- $s^D$: elastic compliance under constant electric displacement in [m^2/N]
- $n$: number of layers of the sensor stack
- $\Delta h$: relative displacement in [m]

If we take the numerical values, we obtain:
#+begin_src matlab :results replace value
  1e-6*d33/(eT*sD*n) % [V/um]
#+end_src

#+RESULTS:
: 23.438

** Simscape Model
The flexible element is imported using the =Reduced Order Flexible Solid= simscape block.

Let's say we use two stacks as a force sensor and one stack as an actuator:
- A =Relative Motion Sensor= block is added between the nodes A and C
- An =Internal Force= block is added between the remote points E and B

The interface nodes are shown in Figure [[fig:apa300ml_ansys]].

One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G).

** Identification of the APA Characteristics
*** Stiffness
#+begin_src matlab :exports none
  m = 0.001;
#+end_src

The transfer function from vertical external force to the relative vertical displacement is identified.

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/z'],  1, 'openoutput'); io_i = io_i + 1;

  G = linearize(mdl, io);
#+end_src

The inverse of its DC gain is the axial stiffness of the APA:
#+begin_src matlab :results replace value
  1e-6/dcgain(G) % [N/um]
#+end_src

#+RESULTS:
: 1.753

The specified stiffness in the datasheet is $k = 1.8\, [N/\mu m]$.

*** Resonance Frequency
The resonance frequency is specified to be between 650Hz and 840Hz.
This is also the case for the FEM model (Figure [[fig:apa300ml_resonance]]).

#+begin_src matlab :exports none
  freqs = logspace(2, 4, 5000);

  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude');
  hold off;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/apa300ml_resonance.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:apa300ml_resonance
#+caption: First resonance is around 800Hz
#+RESULTS:
[[file:figs/apa300ml_resonance.png]]

*** Amplification factor
The amplification factor is the ratio of the vertical displacement to the stack displacement.

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;

  G = linearize(mdl, io);
#+end_src

The ratio of the two displacement is computed from the FEM model.
#+begin_src matlab :results replace value
  abs(dcgain(G(1,1))./dcgain(G(2,1)))
#+end_src

#+RESULTS:
: 5.0749

This is actually correct and approximately corresponds to the ratio of the piezo height and length:
#+begin_src matlab :results replace value
  75/15
#+end_src

#+RESULTS:
: 5

*** Stroke

Estimation of the actuator stroke:
\[ \Delta H = A n \Delta L \]
with:
- $\Delta H$ Axial Stroke of the APA
- $A$ Amplification factor (5 for the APA300ML)
- $n$ Number of stack used
- $\Delta L$ Stroke of the stack (0.1% of its length)

#+begin_src matlab :results replace value
  1e6 * 5 * 3 * 20e-3 * 0.1e-2
#+end_src

#+RESULTS:
: 300

This is exactly the specified stroke in the data-sheet.

** Identification of the Dynamics from actuator to replace displacement
We first set the mass to be approximately zero.
#+begin_src matlab :exports none
  m = 0.01;
#+end_src

The dynamics is identified from the applied force to the measured relative displacement.
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;

  Gh = -linearize(mdl, io);
#+end_src

The same dynamics is identified for a payload mass of 10Kg.
#+begin_src matlab
  m = 10;
#+end_src

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;

  Ghm = -linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 4, 5000);

  figure;
  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

  ax1 = nexttile([2,1]);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-');
  plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
  hold off;

  ax2 = nexttile;
  hold on;
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gh, freqs, 'Hz')))), '-', ...
       'DisplayName', '$m = 0kg$');
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))), '-', ...
       'DisplayName', '$m = 10kg$');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  yticks(-360:90:360);
  ylim([-360 0]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  hold off;
  linkaxes([ax1,ax2],'x');
  xlim([freqs(1), freqs(end)]);
  legend('location', 'southwest');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/apa300ml_plant_dynamics.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:apa300ml_plant_dynamics
#+caption: Transfer function from forces applied by the stack to the axial displacement of the APA
#+RESULTS:
[[file:figs/apa300ml_plant_dynamics.png]]

The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure [[fig:apa300ml_dvf_root_locus]].
#+begin_src matlab :exports none
  figure;

  gains = logspace(0, 5, 500);

  hold on;
  plot(real(pole(Ghm)),  imag(pole(G)), 'kx');
  plot(real(tzero(Ghm)),  imag(tzero(G)), 'ko');
  for k = 1:length(gains)
      cl_poles = pole(feedback(Ghm, gains(k)*s));
      plot(real(cl_poles), imag(cl_poles), 'k.');
  end
  hold off;
  axis square;
  xlim([-500, 10]); ylim([0, 510]);

  xlabel('Real Part'); ylabel('Imaginary Part');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/apa300ml_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:apa300ml_dvf_root_locus
#+caption: Root Locus for Direct Velocity Feedback
#+RESULTS:
[[file:figs/apa300ml_dvf_root_locus.png]]

** Identification of the Dynamics from actuator to force sensor
Let's use 2 stacks as a force sensor and 1 stack as force actuator.

The transfer function from actuator voltage to sensor voltage is identified and shown in Figure [[fig:apa300ml_iff_plant]].
#+begin_src matlab :exports none
  m = 10;
#+end_src

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;

  Giff = -linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 4, 5000);

  figure;
  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

  ax1 = nexttile([2,1]);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), '-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
  hold off;

  ax2 = nexttile;
  hold on;
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Giff, freqs, 'Hz')))), '-');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  yticks(-360:90:360);
  ylim([-180 180]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  hold off;
  linkaxes([ax1,ax2],'x');
  xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/apa300ml_iff_plant.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:apa300ml_iff_plant
#+caption: Transfer function from actuator to force sensor
#+RESULTS:
[[file:figs/apa300ml_iff_plant.png]]

For root locus corresponding to IFF is shown in Figure [[fig:apa300ml_iff_root_locus]].
#+begin_src matlab :exports none
  figure;

  gains = logspace(0, 5, 500);

  hold on;
  plot(real(pole(Giff)),  imag(pole(Giff)), 'kx');
  plot(real(tzero(Giff)),  imag(tzero(Giff)), 'ko');
  for k = 1:length(gains)
      cl_poles = pole(feedback(Giff, gains(k)/s));
      plot(real(cl_poles), imag(cl_poles), 'k.');
  end
  hold off;
  axis square;
  xlim([-500, 10]); ylim([0, 510]);

  xlabel('Real Part'); ylabel('Imaginary Part');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/apa300ml_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:apa300ml_iff_root_locus
#+caption: Root Locus for IFF
#+RESULTS:
[[file:figs/apa300ml_iff_root_locus.png]]

** Identification for a simpler model
The goal in this section is to identify the parameters of a simple APA model from the FEM.
This can be useful is a lower order model is to be used for simulations.

The presented model is based on cite:souleille18_concep_activ_mount_space_applic.

The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
The parameters are shown in the table below.

#+name: fig:souleille18_model_piezo
#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
[[file:./figs/souleille18_model_piezo.png]]

#+caption:Parameters used for the model of the APA 100M
|       | Meaning                                                        |
|-------+----------------------------------------------------------------|
| $k_e$ | Stiffness used to adjust the pole of the isolator              |
| $k_1$ | Stiffness of the metallic suspension when the stack is removed |
| $k_a$ | Stiffness of the actuator                                      |
| $c_1$ | Added viscous damping                                          |

The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model.

\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]

If we can fix $k_a$, we can determine $k_e$ and $k_1$ with:
\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]

#+begin_src matlab :exports none
  m = 10;
#+end_src

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
  io(io_i) = linio([mdl, '/w'],  1, 'openinput');  io_i = io_i + 1; % Base Motion [m]
  io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force [N]
  io(io_i) = linio([mdl, '/z'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
  io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
  io(io_i) = linio([mdl, '/d'],  1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m]

  G = linearize(mdl, io);

  G.InputName = {'Fd', 'w', 'Fa'};
  G.OutputName = {'y', 'Fs', 'd'};
#+end_src

From the identified dynamics, compute $\alpha$ and $\beta$
#+begin_src matlab
  alpha = abs(dcgain(G('y', 'Fa')));
  beta  = abs(dcgain(G('y', 'Fd')));
#+end_src

$k_a$ is estimated using the following formula:
#+begin_src matlab
  ka = 0.8/abs(dcgain(G('y', 'Fa')));
#+end_src
The factor can be adjusted to better match the curves.

Then $k_e$ and $k_1$ are computed.
#+begin_src matlab
  ke = ka/(beta/alpha - 1);
  k1 = 1/beta - ke*ka/(ke + ka);
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f ');
#+end_src

#+RESULTS:
|    | Value [N/um] |
|----+--------------|
| ka |         40.5 |
| ke |          1.5 |
| k1 |          0.4 |

The damping in the system is adjusted to match the FEM model if necessary.
#+begin_src matlab
  c1 = 1e2;
#+end_src

The analytical model of the simpler system is defined below:
#+begin_src matlab
  Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
       [ 1 ,              k1 + c1*s + ke*ka/(ke + ka)  , ke/(ke + ka) ;
        -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 ,       -ke/(ke + ka)*(m*s^2 + c1*s + k1)];

  Ga.InputName = {'Fd', 'w', 'Fa'};
  Ga.OutputName = {'y', 'Fs'};
#+end_src

And the DC gain is adjusted for the force sensor:
#+begin_src matlab
  F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
#+end_src

The dynamics of the FEM model and the simpler model are compared in Figure [[fig:apa300ml_comp_simpler_model]].

#+begin_src matlab :exports none
  freqs = logspace(0, 5, 1000);

  figure;
  tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');

  ax1 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Ga('y', 'w'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('$x_1/w$ [m/m]');
  ylim([1e-6, 1e2]);

  ax2 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('$x_1/f$ [m/N]');
  ylim([1e-14, 1e-6]);

  ax3 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('$x_1/F$ [m/N]');
  ylim([1e-14, 1e-4]);

  ax4 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'w'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('$F_s/w$ [m/m]');
  ylim([1e2, 1e8]);

  ax5 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('$F_s/f$ [m/N]');
  ylim([1e-4, 1e1]);

  ax6 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('$F_s/F$ [m/N]');
  ylim([1e-7, 1e2]);

  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/apa300ml_comp_simpler_model.pdf', 'width', 'full', 'height', 'full');
#+end_src

#+name: fig:apa300ml_comp_simpler_model
#+caption: Comparison of the Dynamics between the FEM model and the simplified one
#+RESULTS:
[[file:figs/apa300ml_comp_simpler_model.png]]

The simplified model has also been implemented in Simscape.

The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure [[fig:apa300ml_comp_simpler_simscape]].
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML_simplified';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
  io(io_i) = linio([mdl, '/w'],  1, 'openinput');  io_i = io_i + 1; % Base Motion [m]
  io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force [N]
  io(io_i) = linio([mdl, '/y'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
  io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]

  Gs = linearize(mdl, io);

  Gs.InputName = {'Fd', 'w', 'Fa'};
  Gs.OutputName = {'y', 'Fs'};
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 5, 1000);

  figure;
  tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');

  ax1 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Gs('y', 'w'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('$x_1/w$ [m/m]');
  ylim([1e-6, 1e2]);

  ax2 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fa'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('$x_1/f$ [m/N]');
  ylim([1e-14, 1e-6]);

  ax3 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fd'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('$x_1/F$ [m/N]');
  ylim([1e-14, 1e-4]);

  ax4 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'w'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('$F_s/w$ [m/m]');
  ylim([1e2, 1e8]);

  ax5 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fa'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('$F_s/f$ [m/N]');
  ylim([1e-4, 1e1]);

  ax6 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fd'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('$F_s/F$ [m/N]');
  ylim([1e-7, 1e2]);

  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/apa300ml_comp_simpler_simscape.pdf', 'width', 'full', 'height', 'full');
#+end_src

#+name: fig:apa300ml_comp_simpler_simscape
#+caption: Comparison of the Dynamics between the FEM model and the simplified simscape model
#+RESULTS:
[[file:figs/apa300ml_comp_simpler_simscape.png]]

* First Flexible Joint Geometry
:PROPERTIES:
:header-args:matlab+: :tangle matlab/flexor_ID16.m
:END:
<<sec:flexor_ID16>>
** Introduction                                                      :ignore:
The studied flexor is shown in Figure [[fig:flexor_id16_screenshot]].

The stiffness and mass matrices representing the dynamics of the flexor are exported from a FEM.
It is then imported into Simscape.

A simplified model of the flexor is then developped.

#+name: fig:flexor_id16_screenshot
#+caption: Flexor studied
[[file:figs/flexor_id16_screenshot.png]]

** 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/');
  addpath('matlab/flexor_ID16/');
#+end_src

#+begin_src matlab :eval no
  addpath('flexor_ID16/');
#+end_src

#+begin_src matlab
  open('flexor_ID16.slx');
#+end_src

** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
We first extract the stiffness and mass matrices.
#+begin_src matlab
  K = extractMatrix('mat_K_6modes_2MDoF.matrix');
  M = extractMatrix('mat_M_6modes_2MDoF.matrix');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(K(1:10, 1:10), {}, {}, ' %.2e ');
#+end_src

#+caption: First 10x10 elements of the Stiffness matrix
#+RESULTS:
|  11200000.0 |       195.0 |       2220.0 |    -0.719 |   -265.0 |      1.59 | -11200000.0 |      -213.0 |      -2220.0 |    0.147 |
|       195.0 |  11400000.0 |       1290.0 |    -148.0 |   -0.188 |      2.41 |      -212.0 | -11400000.0 |      -1290.0 |    148.0 |
|      2220.0 |      1290.0 |  119000000.0 |      1.31 |     1.49 |      1.79 |     -2220.0 |     -1290.0 | -119000000.0 |    -1.31 |
|      -0.719 |      -148.0 |         1.31 |      33.0 | 0.000488 | -0.000977 |       0.141 |       148.0 |        -1.31 |    -33.0 |
|      -265.0 |      -0.188 |         1.49 |  0.000488 |     33.0 |   0.00293 |       266.0 |       0.154 |        -1.49 |  0.00026 |
|        1.59 |        2.41 |         1.79 | -0.000977 |  0.00293 |     236.0 |       -1.32 |       -2.55 |        -1.79 | 0.000379 |
| -11200000.0 |      -212.0 |      -2220.0 |     0.141 |    266.0 |     -1.32 |  11400000.0 |     24600.0 |       1640.0 |    120.0 |
|      -213.0 | -11400000.0 |      -1290.0 |     148.0 |    0.154 |     -2.55 |     24600.0 |  11400000.0 |       1290.0 |    -72.0 |
|     -2220.0 |     -1290.0 | -119000000.0 |     -1.31 |    -1.49 |     -1.79 |      1640.0 |      1290.0 |  119000000.0 |     1.32 |
|       0.147 |       148.0 |        -1.31 |     -33.0 |  0.00026 |  0.000379 |       120.0 |       -72.0 |         1.32 |     34.7 |

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src

#+caption: First 10x10 elements of the Mass matrix
#+RESULTS:
|   0.02 |   1e-09 | -4e-08 |  -1e-10 | 0.0002 | -3e-11 |  0.004 |  5e-08 |  7e-08 |  1e-10 |
|  1e-09 |    0.02 | -3e-07 | -0.0002 | -1e-10 | -2e-09 |  2e-08 |  0.004 |  3e-07 |  1e-05 |
| -4e-08 |  -3e-07 |   0.02 |   7e-10 | -2e-09 |  1e-09 |  3e-07 |  7e-08 |  0.003 |  1e-09 |
| -1e-10 | -0.0002 |  7e-10 |   4e-06 | -1e-12 | -6e-13 |  2e-10 | -7e-06 | -8e-10 | -1e-09 |
| 0.0002 |  -1e-10 | -2e-09 |  -1e-12 |  3e-06 |  2e-13 |  9e-06 |  4e-11 |  2e-09 | -3e-13 |
| -3e-11 |  -2e-09 |  1e-09 |  -6e-13 |  2e-13 |  4e-07 |  8e-11 |  9e-10 | -1e-09 |  2e-12 |
|  0.004 |   2e-08 |  3e-07 |   2e-10 |  9e-06 |  8e-11 |   0.02 | -7e-08 | -3e-07 | -2e-10 |
|  5e-08 |   0.004 |  7e-08 |  -7e-06 |  4e-11 |  9e-10 | -7e-08 |   0.01 | -4e-08 | 0.0002 |
|  7e-08 |   3e-07 |  0.003 |  -8e-10 |  2e-09 | -1e-09 | -3e-07 | -4e-08 |   0.02 | -1e-09 |
|  1e-10 |   1e-05 |  1e-09 |  -1e-09 | -3e-13 |  2e-12 | -2e-10 | 0.0002 | -1e-09 |  2e-06 |

Then, we extract the coordinates of the interface nodes.
#+begin_src matlab
  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+-------+-------+-------|
|    1.0 |    181278.0 |   0.0 |   0.0 |   0.0 |
|    2.0 |    181279.0 |   0.0 |   0.0 |  -0.0 |

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |  2 |
| Number of interface Nodes |  2 |
| Number of Modes           |  6 |
| Size of M and K matrices  | 18 |

Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.

** Identification of the parameters using Simscape and looking at the Stiffness Matrix
The flexor is now imported into Simscape and its parameters are estimated using an identification.

#+begin_src matlab :exports none
  m = 1;
#+end_src

The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'flexor_ID16';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;

  G = linearize(mdl, io);
#+end_src

And we find the same parameters as the one estimated from the Stiffness matrix.

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Identification*'}, ' %0.f ');
#+end_src

#+RESULTS:
| *Caracteristic*               | *Value* | *Identification* |
|-------------------------------+---------+------------------|
| Axial Stiffness Dz [N/um]     |     119 |              119 |
| Bending Stiffness Rx [Nm/rad] |      33 |               33 |
| Bending Stiffness Ry [Nm/rad] |      33 |               33 |
| Torsion Stiffness Rz [Nm/rad] |     236 |              236 |

** Simpler Model
Let's now model the flexible joint with a "perfect" Bushing joint as shown in Figure [[fig:flexible_joint_simscape]].

#+name: fig:flexible_joint_simscape
#+caption: Bushing Joint used to model the flexible joint
[[file:figs/flexible_joint_simscape.png]]

The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
#+begin_src matlab
  Kx = K(1,1); % [N/m]
  Ky = K(2,2); % [N/m]
  Kz = K(3,3); % [N/m]
  Krx = K(4,4); % [Nm/rad]
  Kry = K(5,5); % [Nm/rad]
  Krz =  K(6,6); % [Nm/rad]
#+end_src

The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'flexor_ID16_simplified';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;

  Gs = linearize(mdl, io);
#+end_src

The two obtained dynamics are compared in Figure

#+begin_src matlab :exports none
  freqs = logspace(0, 5, 1000);

  figure;
  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');

  ax1 = nexttile;
  hold on;
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), '-', 'DisplayName', '$D_x/F_x$');
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(Gs(1,1), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', '$D_y/F_y$');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(Gs(2,2), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', '$D_z/F_z$');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(Gs(3,3), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
  hold off;
  legend('location', 'southwest');

  ax2 = nexttile;
  hold on;
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(G(4,4), freqs, 'Hz'))), '-', 'DisplayName', '$R_x/M_x$');
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(Gs(4,4), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(G(5,5), freqs, 'Hz'))), '-', 'DisplayName', '$R_y/M_y$');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(Gs(5,5), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(G(6,6), freqs, 'Hz'))), '-', 'DisplayName', '$R_z/M_z$');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(Gs(6,6), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [rad/Nm]');
  hold off;
  legend('location', 'southwest');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/flexor_ID16_compare_bushing_joint.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:flexor_ID16_compare_bushing_joint
#+caption: Comparison of the Joint compliance between the FEM model and the simpler model
#+RESULTS:
[[file:figs/flexor_ID16_compare_bushing_joint.png]]

* Optimized Flexible Joint
:PROPERTIES:
:header-args:matlab+: :tangle matlab/flexor_025.m
:END:
<<sec:flexor_025>>
** Introduction                                                      :ignore:

The joint geometry has been optimized using Ansys to have lower bending stiffness while keeping a large axial stiffness.

The obtained geometry is shown in Figure [[fig:optimal_flexor]].

#+name: fig:optimal_flexor
#+caption: Flexor studied
[[file:figs/flexor_025_MDoF.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/');
  addpath('matlab/flexor_025/');
#+end_src

#+begin_src matlab :eval no
  addpath('flexor_025/');
#+end_src

#+begin_src matlab
  open('flexor_025.slx');
#+end_src

** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
We first extract the stiffness and mass matrices.
#+begin_src matlab
  K = readmatrix('mat_K.CSV');
  M = readmatrix('mat_M.CSV');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(K(1:10, 1:10), {}, {}, ' %.2e ');
#+end_src

#+caption: First 10x10 elements of the Stiffness matrix
#+RESULTS:
|  12700000.0 |       -18.5 |       -26.8 |  0.00162 |    -4.63 |      64.0 | -12700000.0 |        18.3 |        26.7 |   0.00234 |
|       -18.5 |  12700000.0 |      -499.0 |   -132.0 |  0.00414 |    -0.495 |        18.4 | -12700000.0 |       499.0 |     132.0 |
|       -26.8 |      -499.0 |  94000000.0 |   -470.0 |  0.00771 |    -0.855 |        26.8 |       498.0 | -94000000.0 |     470.0 |
|     0.00162 |      -132.0 |      -470.0 |     4.83 | 2.61e-07 |  0.000123 |    -0.00163 |       132.0 |       470.0 |     -4.83 |
|       -4.63 |     0.00414 |     0.00771 | 2.61e-07 |     4.83 |  4.43e-05 |        4.63 |    -0.00413 |    -0.00772 |  -4.3e-07 |
|        64.0 |      -0.495 |      -0.855 | 0.000123 | 4.43e-05 |     260.0 |       -64.0 |       0.495 |       0.855 | -0.000124 |
| -12700000.0 |        18.4 |        26.8 | -0.00163 |     4.63 |     -64.0 |  12700000.0 |       -18.2 |       -26.7 |  -0.00234 |
|        18.3 | -12700000.0 |       498.0 |    132.0 | -0.00413 |     0.495 |       -18.2 |  12700000.0 |      -498.0 |    -132.0 |
|        26.7 |       499.0 | -94000000.0 |    470.0 | -0.00772 |     0.855 |       -26.7 |      -498.0 |  94000000.0 |    -470.0 |
|     0.00234 |       132.0 |       470.0 |    -4.83 | -4.3e-07 | -0.000124 |    -0.00234 |      -132.0 |      -470.0 |      4.83 |


#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src

#+caption: First 10x10 elements of the Mass matrix
#+RESULTS:
|  0.006 |  8e-09 | -2e-08 | -1e-10 |  3e-05 |  3e-08 |  0.003 | -3e-09 |  9e-09 |  2e-12 |
|  8e-09 |   0.02 |  1e-07 | -3e-05 |  1e-11 |  6e-10 |  1e-08 |  0.003 | -5e-08 |  3e-09 |
| -2e-08 |  1e-07 |   0.01 | -6e-08 | -6e-11 | -8e-12 | -1e-07 |  1e-08 |  0.003 | -1e-08 |
| -1e-10 | -3e-05 | -6e-08 |  1e-06 |  7e-14 |  6e-13 |  1e-10 |  1e-06 | -1e-08 |  3e-10 |
|  3e-05 |  1e-11 | -6e-11 |  7e-14 |  2e-07 |  1e-10 |  3e-08 | -7e-12 |  6e-11 | -6e-16 |
|  3e-08 |  6e-10 | -8e-12 |  6e-13 |  1e-10 |  5e-07 |  1e-08 | -5e-10 | -1e-11 |  1e-13 |
|  0.003 |  1e-08 | -1e-07 |  1e-10 |  3e-08 |  1e-08 |   0.02 | -2e-08 |  1e-07 | -4e-12 |
| -3e-09 |  0.003 |  1e-08 |  1e-06 | -7e-12 | -5e-10 | -2e-08 |  0.006 | -8e-08 |  3e-05 |
|  9e-09 | -5e-08 |  0.003 | -1e-08 |  6e-11 | -1e-11 |  1e-07 | -8e-08 |   0.01 | -6e-08 |
|  2e-12 |  3e-09 | -1e-08 |  3e-10 | -6e-16 |  1e-13 | -4e-12 |  3e-05 | -6e-08 |  2e-07 |


Then, we extract the coordinates of the interface nodes.
#+begin_src matlab
  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |  2 |
| Number of interface Nodes |  2 |
| Number of Modes           |  6 |
| Size of M and K matrices  | 18 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+-------+-------+-------|
|    1.0 |    528875.0 |   0.0 |   0.0 |   0.0 |
|    2.0 |    528876.0 |   0.0 |   0.0 |  -0.0 |

Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.

** Identification of the parameters using Simscape
The flexor is now imported into Simscape and its parameters are estimated using an identification.

#+begin_src matlab :exports none
  m = 1;
#+end_src

The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'flexor_025';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;

  G = linearize(mdl, io);
#+end_src

And we find the same parameters as the one estimated from the Stiffness matrix.

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Identification*'}, ' %.1f ');
#+end_src

#+RESULTS:
| *Caracteristic*               | *Value* | *Identification* |
|-------------------------------+---------+------------------|
| Axial Stiffness Dz [N/um]     |    94.0 |             93.9 |
| Bending Stiffness Rx [Nm/rad] |     4.8 |              4.8 |
| Bending Stiffness Ry [Nm/rad] |     4.8 |              4.8 |
| Torsion Stiffness Rz [Nm/rad] |   260.2 |            260.2 |

** Simpler Model
Let's now model the flexible joint with a "perfect" Bushing joint as shown in Figure [[fig:flexible_joint_simscape]].

#+name: fig:flexible_joint_simscape
#+caption: Bushing Joint used to model the flexible joint
[[file:figs/flexible_joint_simscape.png]]

The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
#+begin_src matlab
  Kx = K(1,1); % [N/m]
  Ky = K(2,2); % [N/m]
  Kz = K(3,3); % [N/m]
  Krx = K(4,4); % [Nm/rad]
  Kry = K(5,5); % [Nm/rad]
  Krz =  K(6,6); % [Nm/rad]
#+end_src

The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'flexor_025_simplified';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;

  Gs = linearize(mdl, io);
#+end_src

The two obtained dynamics are compared in Figure

#+begin_src matlab :exports none
  freqs = logspace(0, 5, 1000);

  figure;
  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');

  ax1 = nexttile;
  hold on;
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), '-', 'DisplayName', '$D_x/F_x$');
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(Gs(1,1), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', '$D_y/F_y$');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(Gs(2,2), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', '$D_z/F_z$');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(Gs(3,3), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
  hold off;
  legend('location', 'southwest');

  ax2 = nexttile;
  hold on;
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(G(4,4), freqs, 'Hz'))), '-', 'DisplayName', '$R_x/M_x$');
  set(gca,'ColorOrderIndex',1)
  plot(freqs, abs(squeeze(freqresp(Gs(4,4), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(G(5,5), freqs, 'Hz'))), '-', 'DisplayName', '$R_y/M_y$');
  set(gca,'ColorOrderIndex',2)
  plot(freqs, abs(squeeze(freqresp(Gs(5,5), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(G(6,6), freqs, 'Hz'))), '-', 'DisplayName', '$R_z/M_z$');
  set(gca,'ColorOrderIndex',3)
  plot(freqs, abs(squeeze(freqresp(Gs(6,6), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [rad/Nm]');
  hold off;
  legend('location', 'southwest');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/flexor_ID16_compare_bushing_joint.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:flexor_ID16_compare_bushing_joint
#+caption: Comparison of the Joint compliance between the FEM model and the simpler model
#+RESULTS:
[[file:figs/flexor_ID16_compare_bushing_joint.png]]

* Force Sensor and Integral Force Feedback with an Amplified Piezoelectric Actuator
<<sec:integral_force_feedback>>
** Introduction                                                      :ignore:
In this section, we try to replicate the results obtained in cite:souleille18_concep_activ_mount_space_applic.

** 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
  addpath('./data/piezo_amplified_IFF/');
#+end_src

#+begin_src matlab :exports none
  open('piezo_amplified_IFF');
#+end_src

** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
We first extract the stiffness and mass matrices.
#+begin_src matlab
  K = extractMatrix('piezo_amplified_IFF_K.txt');
  M = extractMatrix('piezo_amplified_IFF_M.txt');
#+end_src

Then, we extract the coordinates of the interface nodes.
#+begin_src matlab
  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_IFF.txt');
#+end_src

** IFF Plant
The transfer function from the force actuator to the force sensor is identified and shown in Figure [[fig:piezo_amplified_iff_plant]].

#+begin_src matlab
  Kiff = tf(0);
#+end_src

#+begin_src matlab
  m = 0;
#+end_src

#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'piezo_amplified_IFF';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Kiff'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/G'],    1, 'openoutput'); io_i = io_i + 1;

  Gf = linearize(mdl, io);
#+end_src

#+begin_src matlab
  m = 10;
#+end_src

#+begin_src matlab
  Gfm = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(1, 5, 1000);

  figure;

  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gf, freqs, 'Hz'))), '-');
  plot(freqs, abs(squeeze(freqresp(Gfm, freqs, 'Hz'))), '-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
  hold off;

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf, freqs, 'Hz')))), '-', ...
       'DisplayName', '$m = 0kg$');
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gfm, freqs, 'Hz')))), '-', ...
       'DisplayName', '$m = 10kg$');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  yticks(-360:90:360);
  ylim([-390 30]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  hold off;
  linkaxes([ax1,ax2],'x');
  xlim([freqs(1), freqs(end)]);
  legend('location', 'southwest');
#+end_src

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

#+name: fig:piezo_amplified_iff_plant
#+caption: IFF Plant
#+RESULTS:
[[file:figs/piezo_amplified_iff_plant.png]]

** IFF controller
The controller is defined and the loop gain is shown in Figure [[fig:piezo_amplified_iff_loop_gain]].
#+begin_src matlab
  Kiff = -1e12/s;
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(1, 5, 1000);

  figure;

  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gf*Kiff, freqs, 'Hz'))), '-');
  plot(freqs, abs(squeeze(freqresp(Gfm*Kiff, freqs, 'Hz'))), '-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
  hold off;

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf*Kiff, freqs, 'Hz')))), '-', ...
       'DisplayName', '$m = 0kg$');
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gfm*Kiff, freqs, 'Hz')))), '-', ...
       'DisplayName', '$m = 10kg$');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  yticks(-360:90:360);
  ylim([-180 180]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  hold off;
  linkaxes([ax1,ax2],'x');
  xlim([freqs(1), freqs(end)]);
  legend('location', 'southwest');
#+end_src

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

#+name: fig:piezo_amplified_iff_loop_gain
#+caption: IFF Loop Gain
#+RESULTS:
[[file:figs/piezo_amplified_iff_loop_gain.png]]

** Closed Loop System
#+begin_src matlab
  m = 10;
#+end_src

#+begin_src matlab
  Kiff = -1e12/s;
#+end_src

#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'piezo_amplified_IFF';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Dw'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/d'],  1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/G'],  1, 'output'); io_i = io_i + 1;

  Giff = linearize(mdl, io);
  Giff.InputName  = {'w', 'f', 'F'};
  Giff.OutputName  = {'x1', 'Fs'};
#+end_src

#+begin_src matlab
  Kiff = tf(0);
#+end_src

#+begin_src matlab
  G = linearize(mdl, io);
  G.InputName  = {'w', 'f', 'F'};
  G.OutputName  = {'x1', 'Fs'};
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 3, 1);
  title('$\displaystyle \frac{x_1}{w}$')
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');

  ax2 = subplot(2, 3, 2);
  title('$\displaystyle \frac{x_1}{f}$')
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');

  ax3 = subplot(2, 3, 3);
  title('$\displaystyle \frac{x_1}{F}$')
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');

  ax4 = subplot(2, 3, 4);
  title('$\displaystyle \frac{F_s}{w}$')
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');

  ax5 = subplot(2, 3, 5);
  title('$\displaystyle \frac{F_s}{f}$')
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');

  ax6 = subplot(2, 3, 6);
  title('$\displaystyle \frac{F_s}{F}$')
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');

  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/piezo_amplified_iff_comp.pdf', 'width', 1500, 'height', 'full');
#+end_src

#+name: fig:piezo_amplified_iff_comp
#+caption: OL and CL transfer functions
#+RESULTS:
[[file:figs/piezo_amplified_iff_comp.png]]


#+name: fig:souleille18_results
#+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic
[[file:figs/souleille18_results.png]]

* Complete Strut with Encoder
<<sec:strut_fem>>
** Introduction

#+name: fig:strut_encoder_points3
#+caption: Interface points
[[file:data/strut_encoder/points3.jpg]]

Flexible joints have 0.25mm width.

** 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
  addpath('./data/strut_encoder/');
#+end_src

#+begin_src matlab
  open('strut_encoder.slx');
#+end_src

** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
We first extract the stiffness and mass matrices.
#+begin_src matlab
  K = readmatrix('mat_K.CSV');
  M = readmatrix('mat_M.CSV');
#+end_src

Then, we extract the coordinates of the interface nodes.
#+begin_src matlab
  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
#+end_src

#+begin_src matlab
  save('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

** Output parameters
#+begin_src matlab
  load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |  8 |
| Number of interface Nodes |  8 |
| Number of Modes           |  6 |
| Size of M and K matrices  | 54 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number |   x [m] |  y [m] |    z [m] |
|--------+-------------+---------+--------+----------|
|    1.0 |    504411.0 |     0.0 |    0.0 |   0.0405 |
|    2.0 |    504412.0 |     0.0 |    0.0 |  -0.0405 |
|    3.0 |    504413.0 | -0.0325 |    0.0 |      0.0 |
|    4.0 |    504414.0 | -0.0125 |    0.0 |      0.0 |
|    5.0 |    504415.0 | -0.0075 |    0.0 |      0.0 |
|    6.0 |    504416.0 |  0.0325 |    0.0 |      0.0 |
|    7.0 |    504417.0 |   0.004 | 0.0145 | -0.00175 |
|    8.0 |    504418.0 |   0.004 | 0.0166 | -0.00175 |

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src

#+caption: First 10x10 elements of the Stiffness matrix
#+RESULTS:
|  2000000.0 |  1000000.0 | -3000000.0 |  -400.0 |  300.0 | 200.0 |      -30.0 |     2000.0 |   -10000.0 |     0.3 |
|  1000000.0 |  4000000.0 | -8000000.0 |  -900.0 |  400.0 | -50.0 |    -6000.0 |    10000.0 |   -20000.0 |       3 |
| -3000000.0 | -8000000.0 | 20000000.0 |  2000.0 | -900.0 | 200.0 |   -10000.0 |    20000.0 |  -300000.0 |       7 |
|     -400.0 |     -900.0 |     2000.0 |       5 |   -0.1 |  0.05 |          1 |         -3 |          6 | -0.0007 |
|      300.0 |      400.0 |     -900.0 |    -0.1 |      5 |  0.04 |       -0.1 |        0.5 |         -3 |  0.0001 |
|      200.0 |      -50.0 |      200.0 |    0.05 |   0.04 | 300.0 |          4 |      -0.01 |         -1 |   3e-05 |
|      -30.0 |    -6000.0 |   -10000.0 |       1 |   -0.1 |     4 |  3000000.0 | -1000000.0 | -2000000.0 |  -300.0 |
|     2000.0 |    10000.0 |    20000.0 |      -3 |    0.5 | -0.01 | -1000000.0 |  6000000.0 |  7000000.0 |  1000.0 |
|   -10000.0 |   -20000.0 |  -300000.0 |       6 |     -3 |    -1 | -2000000.0 |  7000000.0 | 20000000.0 |  2000.0 |
|        0.3 |          3 |          7 | -0.0007 | 0.0001 | 3e-05 |     -300.0 |     1000.0 |     2000.0 |       5 |


#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src

#+caption: First 10x10 elements of the Mass matrix
#+RESULTS:
|   0.04 |  -0.005 |   0.007 |   2e-06 | 0.0001 | -5e-07 | -1e-05 |  -9e-07 |  8e-05 | -5e-10 |
| -0.005 |    0.03 |    0.02 | -0.0001 |  1e-06 | -3e-07 |  3e-05 | -0.0001 |  8e-05 | -3e-08 |
|  0.007 |    0.02 |    0.08 |  -6e-06 | -5e-06 | -7e-07 |  4e-05 | -0.0001 | 0.0005 | -3e-08 |
|  2e-06 | -0.0001 |  -6e-06 |   2e-06 | -4e-10 |  2e-11 | -8e-09 |   3e-08 | -2e-08 |  6e-12 |
| 0.0001 |   1e-06 |  -5e-06 |  -4e-10 |  3e-06 |  2e-10 | -3e-09 |   3e-09 | -7e-09 |  6e-13 |
| -5e-07 |  -3e-07 |  -7e-07 |   2e-11 |  2e-10 |  5e-07 | -2e-08 |   5e-09 | -5e-09 |  1e-12 |
| -1e-05 |   3e-05 |   4e-05 |  -8e-09 | -3e-09 | -2e-08 |   0.04 |   0.004 |  0.003 |  1e-06 |
| -9e-07 | -0.0001 | -0.0001 |   3e-08 |  3e-09 |  5e-09 |  0.004 |    0.02 |  -0.02 | 0.0001 |
|  8e-05 |   8e-05 |  0.0005 |  -2e-08 | -7e-09 | -5e-09 |  0.003 |   -0.02 |   0.08 | -5e-06 |
| -5e-10 |  -3e-08 |  -3e-08 |   6e-12 |  6e-13 |  1e-12 |  1e-06 |  0.0001 | -5e-06 |  2e-06 |

Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.




** Piezoelectric parameters
Parameters for the APA300ML:

#+begin_src matlab
  d33 = 3e-10; % Strain constant [m/V]
  n = 80; % Number of layers per stack
  eT = 1.6e-8; % Permittivity under constant stress [F/m]
  sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
  ka = 235e6; % Stack stiffness [N/m]
  C = 5e-6; % Stack capactiance [F]
#+end_src

#+begin_src matlab
  na = 2; % Number of stacks used as actuator
  ns = 1; % Number of stacks used as force sensor
#+end_src

** Identification of the Dynamics
#+begin_src matlab :exports none
  m = 0.01;
#+end_src

The dynamics is identified from the applied force to the measured relative displacement.
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'strut_encoder';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/L'], 1, 'openoutput'); io_i = io_i + 1;

  Gh = -linearize(mdl, io);
#+end_src

The same dynamics is identified for a payload mass of 10Kg.
#+begin_src matlab
  m = 10;
#+end_src

#+begin_src matlab :exports none
  Ghm = -linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 4, 5000);

  figure;
  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');

  % ax1 = subplot(2,1,1);
  ax1 = nexttile;
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-');
  plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
  axis padded 'auto x'
  hold off;

  ax2 = nexttile;
  % ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gh, freqs, 'Hz'))), '-', ...
       'DisplayName', '$m = 0kg$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-', ...
       'DisplayName', '$m = 10kg$');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  yticks(-360:90:360);
  % ylim([-360 0]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  hold off;
  axis padded 'auto x'

  % linkaxes([ax1,ax2],'x');
  % xlim([freqs(1), freqs(end)]);
  % legend('location', 'southwest');
#+end_src

* To Order                                                          :noexport:
** TODO Identification of the stiffness properties of the APA300ML
*** APA Alone
#+begin_src matlab :exports none
  m = 10;
#+end_src

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML_characterisation';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
  io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]

  G = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([1e-6./dcgain(G(1,1)), 1e-6./dcgain(G(2,2)), 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristics*', '*Value*'}, ' %.1f ');
#+end_src

#+RESULTS:
| *Caracteristics* | *Value* |
|------------------+---------|
| Kx [N/um]        |     0.8 |
| Ky [N/um]        |     1.6 |
| Kz [N/um]        |     1.8 |
| Rx [Nm/rad]      |    71.4 |
| Ry [Nm/rad]      |   148.2 |
| Rz [Nm/rad]      |  4241.8 |

*** See how the global stiffness is changing with the flexible joints
#+begin_src matlab
  flex = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML_characterisation_with_joints';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
  io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]

  Gf = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([1e-6./dcgain(Gf(1,1)), 1e-6./dcgain(Gf(2,2)), 1e-6./dcgain(Gf(3,3)), 1./dcgain(Gf(4,4)), 1./dcgain(Gf(5,5)), 1./dcgain(Gf(6,6))]', {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %.1f ');
#+end_src

#+RESULTS:
| *Caracteristic* | *Value* |
|-----------------+---------|
| Kx [N/um]       |     0.0 |
| Ky [N/um]       |     0.0 |
| Kz [N/um]       |     1.8 |
| Rx [Nm/rad]     |   722.9 |
| Ry [Nm/rad]     |   129.6 |
| Rz [Nm/rad]     |   115.3 |


#+begin_src matlab
  freqs = logspace(-2, 5, 1000);

  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
  plot(freqs, abs(squeeze(freqresp(Gf(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
  hold off;
  legend('location', 'northeast');
#+end_src

#+begin_src matlab
  freqs = logspace(-2, 5, 1000);

  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
  plot(freqs, abs(squeeze(freqresp(Gf(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
  hold off;
  legend('location', 'northeast');
#+end_src

** TODO Effect of APA300ML in the flexibility of the leg
#+begin_src matlab :exports none
  m = 10;
#+end_src

#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'APA300ML_flex_joints';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
  io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]
#+end_src

#+begin_src matlab :exports none
  G = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
  Gf = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1e-6./dcgain(G(1,1)), 1e-6./dcgain(G(2,2)), 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', [1e-6./dcgain(Gf(1,1)), 1e-6./dcgain(Gf(2,2)), 1e-6./dcgain(Gf(3,3)), 1./dcgain(Gf(4,4)), 1./dcgain(Gf(5,5)), 1./dcgain(Gf(6,6))]'], {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristic*', '*Rigid APA*', '*Flexible APA*'}, ' %.3f ');
#+end_src


#+RESULTS:
| *Caracteristic* | *Rigid APA* | *Flexible APA* |
|-----------------+-------------+----------------|
| Kx [N/um]       |       0.018 |          0.019 |
| Ky [N/um]       |       0.018 |          0.018 |
| Kz [N/um]       |        60.0 |          2.647 |
| Rx [Nm/rad]     |      16.705 |        557.682 |
| Ry [Nm/rad]     |      16.535 |        185.939 |
| Rz [Nm/rad]     |       118.0 |        114.803 |

* Bibliography                                                        :ignore:
bibliographystyle:unsrt
bibliography:ref.bib