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#+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
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/readtheorg.js"></script>
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+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-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', '$R_z/M_z$')
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]]
| | Stiffness | Unit |
|-----------+-----------+----------|
| $K_x$ | 1e7 | [N/m] |
| $K_y$ | 1e7 | [N/m] |
| $K_z$ | 2e8 | [N/m] |
| $K_{R_x}$ | ? | [Nm/rad] |
| $K_{R_y}$ | 1.8e7 | [Nm/rad] |
| $K_{R_z}$ | 1e7 | [Nm/rad] |
** 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
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