#+TITLE: Compliance Measurement of the Micro Station
:DRAWER:
#+STARTUP: overview

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

#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP:   ../index.html

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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:shell  :eval no-export
:END:

* Setup
** Position of inertial sensors on top of the micro-hexapod
Orientation is relative to the frame determined by the X-ray
| *Num* | *Position* | *Orientation* | *Sensibility* | *Channels* |
|-------+------------+---------------+---------------+------------|
|     1 | [0, +A, 0] | [x, y, z]     | 1V/g          |        1-3 |
|     2 | [-B, 0, 0] | [x, y, z]     | 1V/g          |        4-6 |
|     3 | [0, -A, 0] | [x, y, z]     | 0.1V/g        |        7-9 |
|     4 | [+B, 0, 0] | [x, y, z]     | 1V/g          |      10-12 |

Instrumented Hammer:
- Channel 13
- Sensibility: 230 uV/N

| Acc Number | Dir | Channel Number |
|------------+-----+----------------|
|          1 | x   |              1 |
|          1 | y   |              2 |
|          1 | z   |              3 |
|          2 | x   |              4 |
|          2 | y   |              5 |
|          2 | z   |              6 |
|          3 | x   |              7 |
|          3 | y   |              8 |
|          3 | z   |              9 |
|          4 | x   |             10 |
|          4 | y   |             11 |
|          4 | z   |             12 |
|     Hammer |     |             13 |

From the acceleration measurement of the 4 accelerometers, we can compute the translations and rotations:
|       | *Formula*                |
|-------+--------------------------|
| $D_x$ | (1x + 2x + 3x + 4x)/4    |
| $D_y$ | (1y + 2y + 3y + 4y)/4    |
| $D_z$ | (1z + 2z + 3z + 4z)/4    |
| $R_x$ | (1z - 3z)/A              |
| $R_y$ | (2z - 4z)/B              |
| $R_z$ | (3x - 1x)/A, (4y - 2y)/B |


|       | *Formula*             |
|-------+-----------------------|
| $D_x$ | (1 + 4 + 7 + 10)/4    |
| $D_y$ | (2 + 5 + 8 + 11)/4    |
| $D_z$ | (3 + 6 + 9 + 12)/4    |
| $R_x$ | (1 - 9)/A             |
| $R_y$ | (6 - 12)/B            |
| $R_z$ | (7 - 1)/A, (11 - 5)/B |

** Hammer blow position/orientation

| *Num* | *Direction* | *Position* |
|-------+-------------+------------|
|     1 | -Y          | [0, +A, 0] |
|     2 | -Z          | [0, +A, 0] |
|     3 | X           | [-B, 0, 0] |
|     4 | -Z          | [-B, 0, 0] |
|     5 | Y           | [0, -A, 0] |
|     6 | -Z          | [0, -A, 0] |
|     7 | -X          | [+B, 0, 0] |
|     8 | -Z          | [+B, 0, 0] |
|     9 | -X          | [0, -A, 0] |
|    10 | -X          | [0, +A, 0] |

From hammer blows to pure forces / torques:
|       | *Formula*    | Alternative |
|-------+--------------+-------------|
| $F_x$ | +3           |          -7 |
| $F_y$ | -1           |          +5 |
| $F_z$ | -(2 + 6)/2   |  -(4 + 8)/2 |
| $M_x$ | A/2*(2 - 6)  |             |
| $M_y$ | B/2*(8 - 4)  |             |
| $M_z$ | A/2*(10 - 9) |             |

* Results
** 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

** Load Data
#+begin_src matlab
  m1 = load('data/Measurement1.mat');
  m2 = load('data/Measurement2.mat');
  m3 = load('data/Measurement3.mat');
  m4 = load('data/Measurement4.mat');
  m5 = load('data/Measurement5.mat');
  m6 = load('data/Measurement6.mat');
  m7 = load('data/Measurement7.mat');
  m8 = load('data/Measurement8.mat');
  m9 = load('data/Measurement9.mat');
  m10 = load('data/Measurement10.mat');
#+end_src

** Compute Transfer Functions
#+begin_src matlab
  freqs = m3.FFT1_H1_1_13_X_Val;
  w = 2*pi*freqs';

  A = 0.14;
  B = 0.14;
#+end_src

#+begin_src matlab
  G = zeros(6,6,length(freqs));

  % Fx
  G(1,1,:) = (m3.FFT1_H1_1_13_Y_ReIm + m3.FFT1_H1_4_13_Y_ReIm + m3.FFT1_H1_7_13_Y_ReIm + m3.FFT1_H1_10_13_Y_ReIm)./4;
  G(2,1,:) = (m3.FFT1_H1_2_13_Y_ReIm + m3.FFT1_H1_5_13_Y_ReIm + m3.FFT1_H1_8_13_Y_ReIm + m3.FFT1_H1_11_13_Y_ReIm)./4;
  G(3,1,:) = (m3.FFT1_H1_3_13_Y_ReIm + m3.FFT1_H1_6_13_Y_ReIm + m3.FFT1_H1_9_13_Y_ReIm + m3.FFT1_H1_12_13_Y_ReIm)./4;
  G(4,1,:) = (m3.FFT1_H1_1_13_Y_ReIm - m3.FFT1_H1_9_13_Y_ReIm )./A;
  G(5,1,:) = (m3.FFT1_H1_6_13_Y_ReIm - m3.FFT1_H1_12_13_Y_ReIm)./B;
  G(6,1,:) = (m3.FFT1_H1_7_13_Y_ReIm - m3.FFT1_H1_1_13_Y_ReIm )./A;

  % Fy
  G(1,2,:) = -(m1.FFT1_H1_2_13_Y_ReIm + m1.FFT1_H1_5_13_Y_ReIm + m1.FFT1_H1_8_13_Y_ReIm + m1.FFT1_H1_11_13_Y_ReIm)./4;
  G(2,2,:) = -(m1.FFT1_H1_2_13_Y_ReIm + m1.FFT1_H1_5_13_Y_ReIm + m1.FFT1_H1_8_13_Y_ReIm + m1.FFT1_H1_11_13_Y_ReIm)./4;
  G(3,2,:) = -(m1.FFT1_H1_3_13_Y_ReIm + m1.FFT1_H1_6_13_Y_ReIm + m1.FFT1_H1_9_13_Y_ReIm + m1.FFT1_H1_12_13_Y_ReIm)./4;
  G(4,2,:) = -(m1.FFT1_H1_1_13_Y_ReIm - m1.FFT1_H1_9_13_Y_ReIm )./A;
  G(5,2,:) = -(m1.FFT1_H1_6_13_Y_ReIm - m1.FFT1_H1_12_13_Y_ReIm)./B;
  G(6,2,:) = -(m1.FFT1_H1_7_13_Y_ReIm - m1.FFT1_H1_1_13_Y_ReIm )./A;


  % Fz
  G(1,3,:) = -1/2./(1./(m2.FFT1_H1_1_13_Y_ReIm + m2.FFT1_H1_4_13_Y_ReIm + m2.FFT1_H1_7_13_Y_ReIm + m2.FFT1_H1_10_13_Y_ReIm) + ...
                    1./(m6.FFT1_H1_1_13_Y_ReIm + m6.FFT1_H1_4_13_Y_ReIm + m6.FFT1_H1_7_13_Y_ReIm + m6.FFT1_H1_10_13_Y_ReIm));
  G(2,3,:) = -1/2./(1./(m2.FFT1_H1_2_13_Y_ReIm + m2.FFT1_H1_5_13_Y_ReIm + m2.FFT1_H1_8_13_Y_ReIm + m2.FFT1_H1_11_13_Y_ReIm) + ...
                    1./(m6.FFT1_H1_2_13_Y_ReIm + m6.FFT1_H1_5_13_Y_ReIm + m6.FFT1_H1_8_13_Y_ReIm + m6.FFT1_H1_11_13_Y_ReIm));
  G(3,3,:) = -1/2./(1./(m2.FFT1_H1_3_13_Y_ReIm + m2.FFT1_H1_6_13_Y_ReIm + m2.FFT1_H1_9_13_Y_ReIm + m2.FFT1_H1_12_13_Y_ReIm) + ...
                    1./(m6.FFT1_H1_3_13_Y_ReIm + m6.FFT1_H1_6_13_Y_ReIm + m6.FFT1_H1_9_13_Y_ReIm + m6.FFT1_H1_12_13_Y_ReIm));
  G(4,3,:) = -2/A./(1./(m2.FFT1_H1_1_13_Y_ReIm - m2.FFT1_H1_9_13_Y_ReIm) + ...
                    1./(m6.FFT1_H1_1_13_Y_ReIm - m6.FFT1_H1_9_13_Y_ReIm));
  G(5,3,:) = -2/B./(1./(m2.FFT1_H1_6_13_Y_ReIm - m2.FFT1_H1_12_13_Y_ReIm) + ...
                    1./(m6.FFT1_H1_6_13_Y_ReIm - m6.FFT1_H1_12_13_Y_ReIm));
  G(6,3,:) = -2/A./(1./(m2.FFT1_H1_7_13_Y_ReIm - m2.FFT1_H1_1_13_Y_ReIm) + ...
                    1./(m6.FFT1_H1_7_13_Y_ReIm - m6.FFT1_H1_1_13_Y_ReIm));

  % Mx
  G(1,4,:) = 1/A/2./(1./(m2.FFT1_H1_1_13_Y_ReIm + m2.FFT1_H1_4_13_Y_ReIm + m2.FFT1_H1_7_13_Y_ReIm + m2.FFT1_H1_10_13_Y_ReIm) - ...
                   1./(m6.FFT1_H1_1_13_Y_ReIm + m6.FFT1_H1_4_13_Y_ReIm + m6.FFT1_H1_7_13_Y_ReIm + m6.FFT1_H1_10_13_Y_ReIm));
  G(2,4,:) = 1/A/2./(1./(m2.FFT1_H1_2_13_Y_ReIm + m2.FFT1_H1_5_13_Y_ReIm + m2.FFT1_H1_8_13_Y_ReIm + m2.FFT1_H1_11_13_Y_ReIm) - ...
                   1./(m6.FFT1_H1_2_13_Y_ReIm + m6.FFT1_H1_5_13_Y_ReIm + m6.FFT1_H1_8_13_Y_ReIm + m6.FFT1_H1_11_13_Y_ReIm));
  G(3,4,:) = 1/A/2./(1./(m2.FFT1_H1_3_13_Y_ReIm + m2.FFT1_H1_6_13_Y_ReIm + m2.FFT1_H1_9_13_Y_ReIm + m2.FFT1_H1_12_13_Y_ReIm) - ...
                   1./(m6.FFT1_H1_3_13_Y_ReIm + m6.FFT1_H1_6_13_Y_ReIm + m6.FFT1_H1_9_13_Y_ReIm + m6.FFT1_H1_12_13_Y_ReIm));
  G(4,4,:) = 1/A^2*2./(1./(m2.FFT1_H1_1_13_Y_ReIm - m2.FFT1_H1_9_13_Y_ReIm) - ...
                 1./(m6.FFT1_H1_1_13_Y_ReIm - m6.FFT1_H1_9_13_Y_ReIm));
  G(5,4,:) = 2/A/B./(1./(m2.FFT1_H1_6_13_Y_ReIm - m2.FFT1_H1_12_13_Y_ReIm) - ...
                 1./(m6.FFT1_H1_6_13_Y_ReIm - m6.FFT1_H1_12_13_Y_ReIm));
  G(6,4,:) = 1/A^2*2./(1./(m2.FFT1_H1_7_13_Y_ReIm - m2.FFT1_H1_1_13_Y_ReIm) - ...
                 1./(m6.FFT1_H1_7_13_Y_ReIm - m6.FFT1_H1_1_13_Y_ReIm));

  % My
  G(1,5,:) = 1/B/2./(1./(m8.FFT1_H1_1_13_Y_ReIm + m8.FFT1_H1_4_13_Y_ReIm + m8.FFT1_H1_7_13_Y_ReIm + m8.FFT1_H1_10_13_Y_ReIm) - ...
                   1./(m4.FFT1_H1_1_13_Y_ReIm + m4.FFT1_H1_4_13_Y_ReIm + m4.FFT1_H1_7_13_Y_ReIm + m4.FFT1_H1_10_13_Y_ReIm));
  G(2,5,:) = 1/B/2./(1./(m8.FFT1_H1_2_13_Y_ReIm + m8.FFT1_H1_5_13_Y_ReIm + m8.FFT1_H1_8_13_Y_ReIm + m8.FFT1_H1_11_13_Y_ReIm) - ...
                   1./(m4.FFT1_H1_2_13_Y_ReIm + m4.FFT1_H1_5_13_Y_ReIm + m4.FFT1_H1_8_13_Y_ReIm + m4.FFT1_H1_11_13_Y_ReIm));
  G(3,5,:) = 1/B/2./(1./(m8.FFT1_H1_3_13_Y_ReIm + m8.FFT1_H1_6_13_Y_ReIm + m8.FFT1_H1_9_13_Y_ReIm + m8.FFT1_H1_12_13_Y_ReIm) - ...
                   1./(m4.FFT1_H1_3_13_Y_ReIm + m4.FFT1_H1_6_13_Y_ReIm + m4.FFT1_H1_9_13_Y_ReIm + m4.FFT1_H1_12_13_Y_ReIm));
  G(4,5,:) = 2/B/A./(1./(m8.FFT1_H1_1_13_Y_ReIm - m8.FFT1_H1_9_13_Y_ReIm) - ...
                     1./(m4.FFT1_H1_1_13_Y_ReIm - m4.FFT1_H1_9_13_Y_ReIm));
  G(5,5,:) = 1/B^2*2./(1./(m8.FFT1_H1_6_13_Y_ReIm - m8.FFT1_H1_12_13_Y_ReIm) - ...
                 1./(m4.FFT1_H1_6_13_Y_ReIm - m4.FFT1_H1_12_13_Y_ReIm));
  G(6,5,:) = 2/B/A./(1./(m8.FFT1_H1_7_13_Y_ReIm - m8.FFT1_H1_1_13_Y_ReIm) - ...
                     1./(m4.FFT1_H1_7_13_Y_ReIm - m4.FFT1_H1_1_13_Y_ReIm));

  % Mz
  G(1,6,:) = 1/A/2./(1./(m10.FFT1_H1_1_13_Y_ReIm + m10.FFT1_H1_4_13_Y_ReIm + m10.FFT1_H1_7_13_Y_ReIm + m10.FFT1_H1_10_13_Y_ReIm) - ...
                   1./(m9.FFT1_H1_1_13_Y_ReIm + m9.FFT1_H1_4_13_Y_ReIm + m9.FFT1_H1_7_13_Y_ReIm + m9.FFT1_H1_10_13_Y_ReIm));
  G(2,6,:) = 1/A/2./(1./(m10.FFT1_H1_2_13_Y_ReIm + m10.FFT1_H1_5_13_Y_ReIm + m10.FFT1_H1_8_13_Y_ReIm + m10.FFT1_H1_11_13_Y_ReIm) - ...
                   1./(m9.FFT1_H1_2_13_Y_ReIm + m9.FFT1_H1_5_13_Y_ReIm + m9.FFT1_H1_8_13_Y_ReIm + m9.FFT1_H1_11_13_Y_ReIm));
  G(3,6,:) = 1/A/2./(1./(m10.FFT1_H1_3_13_Y_ReIm + m10.FFT1_H1_6_13_Y_ReIm + m10.FFT1_H1_9_13_Y_ReIm + m10.FFT1_H1_12_13_Y_ReIm) - ...
                   1./(m9.FFT1_H1_3_13_Y_ReIm + m9.FFT1_H1_6_13_Y_ReIm + m9.FFT1_H1_9_13_Y_ReIm + m9.FFT1_H1_12_13_Y_ReIm));
  G(4,6,:) = 1/A^2*2./(1./(m10.FFT1_H1_1_13_Y_ReIm - m10.FFT1_H1_9_13_Y_ReIm) - ...
                 1./(m9.FFT1_H1_1_13_Y_ReIm - m9.FFT1_H1_9_13_Y_ReIm));
  G(5,6,:) = 2*A/B./(1./(m10.FFT1_H1_6_13_Y_ReIm - m10.FFT1_H1_12_13_Y_ReIm) - ...
                     1./(m9.FFT1_H1_6_13_Y_ReIm - m9.FFT1_H1_12_13_Y_ReIm));
  G(6,6,:) = 1/A^2*2./(1./(m10.FFT1_H1_7_13_Y_ReIm - m10.FFT1_H1_1_13_Y_ReIm) - ...
                 1./(m9.FFT1_H1_7_13_Y_ReIm - m9.FFT1_H1_1_13_Y_ReIm));
#+end_src

** Diagonal Dynamics
#+begin_src matlab
  figure;
  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(G(1,1,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(2,2,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(3,3,:))./(-w.^2)), '.')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
  ylim([1e-9, 2e-6]);

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(G(1,1,:))./(-w.^2)), '.', 'DisplayName', '$D_x/F_x$')
  plot(freqs, 180/pi*angle(squeeze(G(2,2,:))./(-w.^2)), '.', 'DisplayName', '$D_y/F_y$')
  plot(freqs, 180/pi*angle(squeeze(G(3,3,:))./(-w.^2)), '.', 'DisplayName', '$D_z/F_z$')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  xlabel('Freqency [Hz]'); ylabel('Phase [deg]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2],'x');
  xlim([30, 300]);
#+end_src

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

#+name: fig:compliance_diagonal_translations
#+caption: Dynamics from Forces to Translations
#+RESULTS:
[[file:figs/compliance_diagonal_translations.png]]

#+begin_src matlab
  figure;
  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(G(4,4,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(5,5,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(6,6,:))./(-w.^2)), '.')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Magnitude [rad/Nm]'); set(gca, 'XTickLabel',[]);
  ylim([1e-7, 2e-4]);

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(G(4,4,:))./(-w.^2)), '.', 'DisplayName', '$R_x/M_x$')
  plot(freqs, 180/pi*angle(squeeze(G(5,5,:))./(-w.^2)), '.', 'DisplayName', '$R_y/M_y$')
  plot(freqs, 180/pi*angle(squeeze(G(6,6,:))./(-w.^2)), '.', 'DisplayName', '$R_z/M_z$')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  xlabel('Freqency [Hz]'); ylabel('Phase [deg]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2],'x');
  xlim([30, 300]);
#+end_src

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

#+name: fig:compliance_diagonal_rotations
#+caption: Dynamics from Torques to Rotations
#+RESULTS:
[[file:figs/compliance_diagonal_rotations.png]]

** Equivalent Stiffness and Mass Estimation

#+begin_src matlab
  K = [1e7, 1e7, 2e8, 5e7, 3e7, 2e7];
  f_res = [125, 135, 390, 335, 335, 160];
#+end_src

#+begin_src matlab
  M = [20, 20, 20, 11, 7, 20];
  f_res_est = sqrt(K./M)./(2*pi);
#+end_src

Here is the inertia / stiffness to the granite that can represent the micro-station compliance dynamics:
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([K'], {'x', 'y', 'z', 'Rx', 'Ry', 'Rz'}, {'Stiffness', 'Inertia'}, ' %.1g ');
#+end_src


#+RESULTS:
| Stiffness |     Inertia |
|-----------+-------------|
| x         |  10000000.0 |
| y         |  10000000.0 |
| z         | 200000000.0 |
| Rx        |  50000000.0 |
| Ry        |  30000000.0 |
| Rz        |  20000000.0 |

** Compare with Model
#+begin_src matlab
  load('./mat/model.mat', 'Gm');
#+end_src

#+begin_src matlab
  figure;
  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(G(1,1,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(2,2,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(3,3,:))./(-w.^2)), '.')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(Gm(1,1,:), freqs, 'Hz'))), '-')
  plot(freqs, abs(squeeze(freqresp(Gm(2,2,:), freqs, 'Hz'))), '-')
  plot(freqs, abs(squeeze(freqresp(Gm(3,3,:), freqs, 'Hz'))), '-')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
  ylim([1e-9, 2e-6]);

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(G(1,1,:))./(-w.^2)), '.', 'DisplayName', '$D_x/F_x$')
  plot(freqs, 180/pi*angle(squeeze(G(2,2,:))./(-w.^2)), '.', 'DisplayName', '$D_y/F_y$')
  plot(freqs, 180/pi*angle(squeeze(G(3,3,:))./(-w.^2)), '.', 'DisplayName', '$D_z/F_z$')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(1,1,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(2,2,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(3,3,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  xlabel('Freqency [Hz]'); ylabel('Phase [deg]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2],'x');
  xlim([30, 300]);
#+end_src

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

#+name: fig:compliance_diagonal_translations_comp_model
#+caption: Dynamics from Forces to Translations
#+RESULTS:
[[file:figs/compliance_diagonal_translations_comp_model.png]]

#+begin_src matlab
  figure;
  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(G(4,4,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(5,5,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(6,6,:))./(-w.^2)), '.')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(Gm(4,4,:), freqs, 'Hz'))), '-')
  plot(freqs, abs(squeeze(freqresp(Gm(5,5,:), freqs, 'Hz'))), '-')
  plot(freqs, abs(squeeze(freqresp(Gm(6,6,:), freqs, 'Hz'))), '-')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Magnitude [rad/Nm]'); set(gca, 'XTickLabel',[]);
  % ylim([1e-9, 2e-6]);

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(G(4,4,:))./(-w.^2)), '.', 'DisplayName', '$R_x/M_x$')
  plot(freqs, 180/pi*angle(squeeze(G(5,5,:))./(-w.^2)), '.', 'DisplayName', '$R_y/M_y$')
  plot(freqs, 180/pi*angle(squeeze(G(6,6,:))./(-w.^2)), '.', 'DisplayName', '$R_z/M_z$')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(4,4,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(5,5,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(6,6,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  xlabel('Freqency [Hz]'); ylabel('Phase [deg]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2],'x');
  xlim([30, 300]);
#+end_src

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

#+name: fig:compliance_diagonal_rotations_comp_model
#+caption: Dynamics from Torques to Rotations
#+RESULTS:
[[file:figs/compliance_diagonal_rotations_comp_model.png]]

|           | Stiffness | Unit     |
|-----------+-----------+----------|
| $K_x$     |       1e7 | [N/m]    |
| $K_y$     |       1e7 | [N/m]    |
| $K_z$     |       2e8 | [N/m]    |
| $K_{R_x}$ |       5e7 | [Nm/rad] |
| $K_{R_y}$ |       3e7 | [Nm/rad] |
| $K_{R_z}$ |       2e7 | [Nm/rad] |

** Coupling Dynamics
#+begin_src matlab
  figure;
  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(G(1,1,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(2,1,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(3,1,:))./(-w.^2)), '.')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(Gm(1,1,:), freqs, 'Hz'))), '-')
  plot(freqs, abs(squeeze(freqresp(Gm(2,1,:), freqs, 'Hz'))), '-')
  plot(freqs, abs(squeeze(freqresp(Gm(3,1,:), freqs, 'Hz'))), '-')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
  ylim([1e-9, 2e-6]);

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(G(1,1,:))./(-w.^2)), '.', 'DisplayName', '$D_x/F_x$')
  plot(freqs, 180/pi*angle(squeeze(G(2,1,:))./(-w.^2)), '.', 'DisplayName', '$D_y/F_x$')
  plot(freqs, 180/pi*angle(squeeze(G(3,1,:))./(-w.^2)), '.', 'DisplayName', '$D_z/F_x$')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(1,1,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(2,1,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(3,1,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  xlabel('Freqency [Hz]'); ylabel('Phase [deg]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2],'x');
  xlim([30, 300]);
#+end_src

#+begin_src matlab
  figure;
  ax1 = subplot(2,1,1);
  hold on;
  plot(freqs, abs(squeeze(G(5,1,:))./(-w.^2)), '.')
  plot(freqs, abs(squeeze(G(4,2,:))./(-w.^2)), '.')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(Gm(5,1,:), freqs, 'Hz'))), '-')
  plot(freqs, abs(squeeze(freqresp(Gm(4,2,:), freqs, 'Hz'))), '-')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
  ylim([1e-9, 2e-6]);

  ax2 = subplot(2,1,2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(G(5,1,:))./(-w.^2)), '.', 'DisplayName', '$R_y/F_x$')
  plot(freqs, 180/pi*angle(squeeze(G(4,2,:))./(-w.^2)), '.', 'DisplayName', '$R_x/F_y$')
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(5,1,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gm(4,2,:), freqs, 'Hz'))), '-', 'HandleVisibility', 'off')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  xlabel('Freqency [Hz]'); ylabel('Phase [deg]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2],'x');
  xlim([30, 300]);
#+end_src