#+TITLE:Effect on the control system of each stages on the vibration of the 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="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.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:

* Introduction                                                        :ignore:
This file is organized as follow:
- Section [[sec:effect_control_all]]:
  - One geophone on the marble and one at the sample location
  - Each stage is turned on one by one
- Section [[sec:effect_control_one]]:
  - One geophone on the marble and one at the sample location
  - Each stage is turned on one at a time
- Section [[sec:effect_symetrie_driver]]:
  - We check if the Symetrie driver induces some vibrations when placed on the marble

* Effect of all the control systems on the Sample vibrations
  :PROPERTIES:
  :header-args:matlab+: :tangle matlab/effect_control_all.m
  :header-args:matlab+: :comments org :mkdirp yes
  :END:
  <<sec:effect_control_all>>

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/effect_control_all.m -nt data/effect_control_all.zip ]; then
    cp matlab/effect_control_all.m effect_control_all.m;
    zip data/effect_control_all \
        mat/data_003.mat \
        mat/data_004.mat \
        mat/data_005.mat \
        mat/data_006.mat \
        mat/data_007.mat \
        mat/data_008.mat \
        effect_control_all.m;
    rm effect_control_all.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/effect_control_all.zip][here]].
#+end_note

** Experimental Setup
We here measure the signals of two L22 geophones:
- One is located on top of the Sample platform
- One is located on the marble

The signals are amplified with voltage amplifiers with the following settings:
- gain of 60dB
- AC/DC option set on AC
- Low pass filter set at 1kHz

The signal from the top geophone does not go trought the slip-ring.

First, all the control systems are turned ON, then, they are turned one by one.
Each measurement are done during 50s.

#+name: tab:control_system_on_off
#+caption: Summary of the measurements and the states of the control systems
| Ty   | Ry   | Slip Ring | Spindle | Hexapod | Meas. file     |
|------+------+-----------+---------+---------+----------------|
| *ON* | *ON* | *ON*      | *ON*    | *ON*    | =meas_003.mat= |
| OFF  | *ON* | *ON*      | *ON*    | *ON*    | =meas_004.mat= |
| OFF  | OFF  | *ON*      | *ON*    | *ON*    | =meas_005.mat= |
| OFF  | OFF  | OFF       | *ON*    | *ON*    | =meas_006.mat= |
| OFF  | OFF  | OFF       | OFF     | *ON*    | =meas_007.mat= |
| OFF  | OFF  | OFF       | OFF     | OFF     | =meas_008.mat= |

Each of the =mat= file contains one array =data= with 3 columns:
| Column number | Description       |
|---------------+-------------------|
|             1 | Geophone - Marble |
|             2 | Geophone - Sample |
|             3 | Time              |

** 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>>
  addpath('../src');
#+end_src

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

** Load data
We load the data of the z axis of two geophones.
#+begin_src matlab
  d3 = load('mat/data_003.mat', 'data'); d3 = d3.data;
  d4 = load('mat/data_004.mat', 'data'); d4 = d4.data;
  d5 = load('mat/data_005.mat', 'data'); d5 = d5.data;
  d6 = load('mat/data_006.mat', 'data'); d6 = d6.data;
  d7 = load('mat/data_007.mat', 'data'); d7 = d7.data;
  d8 = load('mat/data_008.mat', 'data'); d8 = d8.data;
#+end_src

** Analysis - Time Domain
First, we can look at the time domain data and compare all the measurements:
- comparison for the geophone at the sample location (figure [[fig:time_domain_sample]])
- comparison for the geophone on the granite (figure [[fig:time_domain_marble]])

#+begin_src matlab
  figure;
  hold on;
  plot(d3(:, 3), d3(:, 2), 'DisplayName', 'Hexa, Rz, SR, Ry, Ty');
  plot(d4(:, 3), d4(:, 2), 'DisplayName', 'Hexa, Rz, SR, Ry');
  plot(d5(:, 3), d5(:, 2), 'DisplayName', 'Hexa, Rz, SR');
  plot(d6(:, 3), d6(:, 2), 'DisplayName', 'Hexa, Rz');
  plot(d7(:, 3), d7(:, 2), 'DisplayName', 'Hexa');
  plot(d8(:, 3), d8(:, 2), 'DisplayName', 'All OFF');
  hold off;
  xlabel('Time [s]'); ylabel('Voltage [V]');
  xlim([0, 50]);
  legend('Location', 'bestoutside');
#+end_src

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

#+name: fig:time_domain_sample
#+caption: Comparison of the time domain data when turning off the control system of the stages - Geophone at the sample location
#+RESULTS:
[[file:figs/time_domain_sample.png]]

#+begin_src matlab :results none
  figure;
  hold on;
  plot(d3(:, 3), d3(:, 1), 'DisplayName', 'Hexa, Rz, SR, Ry, Ty');
  plot(d4(:, 3), d4(:, 1), 'DisplayName', 'Hexa, Rz, SR, Ry');
  plot(d5(:, 3), d5(:, 1), 'DisplayName', 'Hexa, Rz, SR');
  plot(d6(:, 3), d6(:, 1), 'DisplayName', 'Hexa, Rz');
  plot(d7(:, 3), d7(:, 1), 'DisplayName', 'Hexa');
  plot(d8(:, 3), d8(:, 1), 'DisplayName', 'All OFF');
  hold off;
  xlabel('Time [s]'); ylabel('Voltage [V]');
  xlim([0, 50]);
  legend('Location', 'bestoutside');
#+end_src

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

#+name: fig:time_domain_marble
#+caption: Comparison of the time domain data when turning off the control system of the stages - Geophone on the marble
#+RESULTS:
[[file:figs/time_domain_marble.png]]

** Analysis - Frequency Domain
#+begin_src matlab :exports none
  dt = d3(2, 3) - d3(1, 3);

  Fs = 1/dt;
  win = hanning(ceil(10*Fs));
#+end_src

*** Vibrations at the sample location
First, we compute the Power Spectral Density of the signals coming from the Geophone located at the sample location.
#+begin_src matlab :results none
  [px3, f] = pwelch(d3(:, 2), win, [], [], Fs);
  [px4, ~] = pwelch(d4(:, 2), win, [], [], Fs);
  [px5, ~] = pwelch(d5(:, 2), win, [], [], Fs);
  [px6, ~] = pwelch(d6(:, 2), win, [], [], Fs);
  [px7, ~] = pwelch(d7(:, 2), win, [], [], Fs);
  [px8, ~] = pwelch(d8(:, 2), win, [], [], Fs);
#+end_src

And we compare all the signals (figures [[fig:psd_sample_comp]] and [[fig:psd_sample_comp_high_freq]]).
#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(px3), 'DisplayName', 'Hexa, Rz, SR, Ry, Ty');
  plot(f, sqrt(px4), 'DisplayName', 'Hexa, Rz, SR, Ry');
  plot(f, sqrt(px5), 'DisplayName', 'Hexa, Rz, SR');
  plot(f, sqrt(px6), 'DisplayName', 'Hexa, Rz');
  plot(f, sqrt(px7), 'DisplayName', 'Hexa');
  plot(f, sqrt(px8), 'DisplayName', 'All OFF');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude Spectral Density $\left[\frac{V}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);
  legend('Location', 'southwest');
#+end_src

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

#+name: fig:psd_sample_comp
#+caption: Amplitude Spectral Density of the signal coming from the top geophone
#+RESULTS:
[[file:figs/psd_sample_comp.png]]

#+begin_src matlab :results none :tangle no :exports none
  xlim([80, 500]);
#+end_src

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

#+name: fig:psd_sample_comp_high_freq
#+caption: Amplitude Spectral Density of the signal coming from the top geophone (zoom at high frequencies)
#+RESULTS:
[[file:figs/psd_sample_comp_high_freq.png]]

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(px3)))), 'DisplayName', 'Hexa, Rz, SR, Ry, Ty');
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(px4)))), 'DisplayName', 'Hexa, Rz, SR, Ry');
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(px5)))), 'DisplayName', 'Hexa, Rz, SR');
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(px6)))), 'DisplayName', 'Hexa, Rz');
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(px7)))), 'DisplayName', 'Hexa');
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(px8)))), 'DisplayName', 'All OFF');
  hold off;
  xlabel('Frequency [Hz]');
  ylabel('CAS [V]');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  legend('location', 'southwest');
  xlim([0.1, 300]);
#+end_src

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

#+name: fig:cas_sample_comp
#+caption:
#+RESULTS:
[[file:figs/cas_sample_comp.png]]

*** Vibrations on the marble
Now we plot the same curves for the geophone located on the marble.
#+begin_src matlab :results none
  [px3, f] = pwelch(d3(:, 1), win, [], [], Fs);
  [px4, ~] = pwelch(d4(:, 1), win, [], [], Fs);
  [px5, ~] = pwelch(d5(:, 1), win, [], [], Fs);
  [px6, ~] = pwelch(d6(:, 1), win, [], [], Fs);
  [px7, ~] = pwelch(d7(:, 1), win, [], [], Fs);
  [px8, ~] = pwelch(d8(:, 1), win, [], [], Fs);
#+end_src

And we compare the Amplitude Spectral Densities (figures [[fig:psd_marble_comp]] and [[fig:psd_marble_comp_high_freq]])
#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(px3), 'DisplayName', 'Hexa, Rz, SR, Ry, Ty');
  plot(f, sqrt(px4), 'DisplayName', 'Hexa, Rz, SR, Ry');
  plot(f, sqrt(px5), 'DisplayName', 'Hexa, Rz, SR');
  plot(f, sqrt(px6), 'DisplayName', 'Hexa, Rz');
  plot(f, sqrt(px7), 'DisplayName', 'Hexa');
  plot(f, sqrt(px8), 'DisplayName', 'All OFF');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude Spectral Density $\left[\frac{V}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);
  legend('Location', 'northeast');
#+end_src

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

#+name: fig:psd_marble_comp
#+caption: Amplitude Spectral Density of the signal coming from the top geophone
#+RESULTS:
[[file:figs/psd_marble_comp.png]]

#+begin_src matlab :results none :tangle no :exports none
  legend('Location', 'southwest');
  xlim([80, 500]);
#+end_src

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

#+name: fig:psd_marble_comp_high_freq
#+caption: Amplitude Spectral Density of the signal coming from the top geophone (zoom at high frequencies)
#+RESULTS:
[[file:figs/psd_marble_comp_high_freq.png]]

** Conclusion
#+begin_important
  - The control system of the Ty stage induces a lot of vibrations of the marble above 100Hz
  - The hexapod control system add vibrations of the sample only above 200Hz
  - When the Slip-Ring is ON, white noise appears at high frequencies. This is studied [[file:../slip-ring-electrical-noise/index.org][here]]
#+end_important

* Effect of all the control systems on the Sample vibrations - One stage at a time
  :PROPERTIES:
  :header-args:matlab+: :tangle matlab/effect_control_one.m
  :header-args:matlab+: :comments org :mkdirp yes
  :END:
  <<sec:effect_control_one>>

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/effect_control_one.m -nt data/effect_control_one.zip ]; then
    cp matlab/effect_control_one.m effect_control_one.m;
    zip data/effect_control_one \
        mat/data_013.mat \
        mat/data_014.mat \
        mat/data_015.mat \
        mat/data_016.mat \
        mat/data_017.mat \
        mat/data_018.mat \
        effect_control_one.m
    rm effect_control_one.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/effect_control_one.zip][here]].
#+end_note

** Experimental Setup
We here measure the signals of two geophones:
- One is located on top of the Sample platform
- One is located on the marble

The signal from the top geophone does go trought the slip-ring.

All the control systems are turned OFF, then, they are turned on one at a time.

Each measurement are done during 100s.

The settings of the voltage amplifier are shown on figure [[fig:amplifier_settings]]:
- gain of 60dB
- AC/DC option set on DC
- Low pass filter set at 1kHz

A first order low pass filter with a cut-off frequency of 1kHz is added before the voltage amplifier.

#+name: tab:control_system_on_off
#+caption: Summary of the measurements and the states of the control systems
| Ty   | Ry   | Slip Ring | Spindle | Hexapod | Meas. file     |
|------+------+-----------+---------+---------+----------------|
| OFF  | OFF  | OFF       | OFF     | OFF     | =meas_013.mat= |
| *ON* | OFF  | OFF       | OFF     | OFF     | =meas_014.mat= |
| OFF  | *ON* | OFF       | OFF     | OFF     | =meas_015.mat= |
| OFF  | OFF  | *ON*      | OFF     | OFF     | =meas_016.mat= |
| OFF  | OFF  | OFF       | *ON*    | OFF     | =meas_017.mat= |
| OFF  | OFF  | OFF       | OFF     | *ON*    | =meas_018.mat= |

Each of the =mat= file contains one array =data= with 3 columns:
| Column number | Description       |
|---------------+-------------------|
|             1 | Geophone - Marble |
|             2 | Geophone - Sample |
|             3 | Time              |

#+name: fig:amplifier_settings
#+caption: Voltage amplifier settings for the measurement
#+attr_html: :width 500px
[[file:./img/IMG_20190507_101459.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>>
  addpath('../src');
#+end_src

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

** Load data
We load the data of the z axis of two geophones.
#+begin_src matlab :results none
  d_of = load('mat/data_013.mat', 'data'); d_of = d_of.data;
  d_ty = load('mat/data_014.mat', 'data'); d_ty = d_ty.data;
  d_ry = load('mat/data_015.mat', 'data'); d_ry = d_ry.data;
  d_sr = load('mat/data_016.mat', 'data'); d_sr = d_sr.data;
  d_rz = load('mat/data_017.mat', 'data'); d_rz = d_rz.data;
  d_he = load('mat/data_018.mat', 'data'); d_he = d_he.data;
#+end_src

** Voltage to Velocity
We convert the measured voltage to velocity using the function =voltageToVelocityL22= (accessible [[file:~/Cloud/thesis/meas/srcindex.org][here]]).

#+begin_src matlab
  gain = 60; % [dB]

  d_of(:, 1) = voltageToVelocityL22(d_of(:, 1), d_of(:, 3), gain);
  d_ty(:, 1) = voltageToVelocityL22(d_ty(:, 1), d_ty(:, 3), gain);
  d_ry(:, 1) = voltageToVelocityL22(d_ry(:, 1), d_ry(:, 3), gain);
  d_sr(:, 1) = voltageToVelocityL22(d_sr(:, 1), d_sr(:, 3), gain);
  d_rz(:, 1) = voltageToVelocityL22(d_rz(:, 1), d_rz(:, 3), gain);
  d_he(:, 1) = voltageToVelocityL22(d_he(:, 1), d_he(:, 3), gain);

  d_of(:, 2) = voltageToVelocityL22(d_of(:, 2), d_of(:, 3), gain);
  d_ty(:, 2) = voltageToVelocityL22(d_ty(:, 2), d_ty(:, 3), gain);
  d_ry(:, 2) = voltageToVelocityL22(d_ry(:, 2), d_ry(:, 3), gain);
  d_sr(:, 2) = voltageToVelocityL22(d_sr(:, 2), d_sr(:, 3), gain);
  d_rz(:, 2) = voltageToVelocityL22(d_rz(:, 2), d_rz(:, 3), gain);
  d_he(:, 2) = voltageToVelocityL22(d_he(:, 2), d_he(:, 3), gain);
#+end_src

** Analysis - Time Domain
First, we can look at the time domain data and compare all the measurements:
- comparison for the geophone at the sample location (figure [[fig:time_domain_sample_lpf]])
- comparison for the geophone on the granite (figure [[fig:time_domain_marble_lpf]])
- relative displacement of the sample with respect to the marble (figure [[fig:time_domain_marble_lpf]])

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(d_of(:, 3), d_of(:, 2), 'DisplayName', 'All OFF');
  plot(d_ty(:, 3), d_ty(:, 2), 'DisplayName', 'Ty ON');
  plot(d_ry(:, 3), d_ry(:, 2), 'DisplayName', 'Ry ON');
  plot(d_sr(:, 3), d_sr(:, 2), 'DisplayName', 'S-R ON');
  plot(d_rz(:, 3), d_rz(:, 2), 'DisplayName', 'Rz ON');
  plot(d_he(:, 3), d_he(:, 2), 'DisplayName', 'Hexa ON');
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  xlim([0, 50]);
  legend('Location', 'bestoutside');
#+end_src

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

#+name: fig:time_domain_sample_lpf
#+caption: Comparison of the time domain data when turning off the control system of the stages - Geophone at the sample location
#+RESULTS:
[[file:figs/time_domain_sample_lpf.png]]

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(d_of(:, 3), d_of(:, 1), 'DisplayName', 'All OFF');
  plot(d_ty(:, 3), d_ty(:, 1), 'DisplayName', 'Ty ON');
  plot(d_ry(:, 3), d_ry(:, 1), 'DisplayName', 'Ry ON');
  plot(d_sr(:, 3), d_sr(:, 1), 'DisplayName', 'S-R ON');
  plot(d_rz(:, 3), d_rz(:, 1), 'DisplayName', 'Rz ON');
  plot(d_he(:, 3), d_he(:, 1), 'DisplayName', 'Hexa ON');
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  xlim([0, 50]);
  legend('Location', 'bestoutside');
#+end_src

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

#+name: fig:time_domain_marble_lpf
#+caption: Comparison of the time domain data when turning off the control system of the stages - Geophone on the marble
#+RESULTS:
[[file:figs/time_domain_marble_lpf.png]]

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(d_of(:, 3), 1e6*lsim(1/(1+s/(2*pi*0.5)), d_of(:, 2)-d_of(:, 1), d_of(:, 3)), 'DisplayName', 'All OFF');
  plot(d_ty(:, 3), 1e6*lsim(1/(1+s/(2*pi*0.5)), d_ty(:, 2)-d_ty(:, 1), d_ty(:, 3)), 'DisplayName', 'Ty ON');
  plot(d_ry(:, 3), 1e6*lsim(1/(1+s/(2*pi*0.5)), d_ry(:, 2)-d_ry(:, 1), d_ry(:, 3)), 'DisplayName', 'Ry ON');
  plot(d_sr(:, 3), 1e6*lsim(1/(1+s/(2*pi*0.5)), d_sr(:, 2)-d_sr(:, 1), d_sr(:, 3)), 'DisplayName', 'S-R ON');
  plot(d_rz(:, 3), 1e6*lsim(1/(1+s/(2*pi*0.5)), d_rz(:, 2)-d_rz(:, 1), d_rz(:, 3)), 'DisplayName', 'Rz ON');
  plot(d_he(:, 3), 1e6*lsim(1/(1+s/(2*pi*0.5)), d_he(:, 2)-d_he(:, 1), d_he(:, 3)), 'DisplayName', 'Hexa ON');
  hold off;
  xlabel('Time [s]'); ylabel('Relative Displacement [$\mu m$]');
  xlim([0, 50]);
  legend('Location', 'bestoutside');
#+end_src

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

#+name: fig:time_domain_relative_disp
#+caption: Relative displacement of the sample with respect to the marble
#+RESULTS:
[[file:figs/time_domain_relative_disp.png]]

** Analysis - Frequency Domain
#+begin_src matlab :results none
  dt = d_of(2, 3) - d_of(1, 3);

  Fs = 1/dt;
  win = hanning(ceil(10*Fs));
#+end_src

*** Vibrations at the sample location
First, we compute the Power Spectral Density of the signals coming from the Geophone located at the sample location.
#+begin_src matlab :results none
  [px_of, f] = pwelch(d_of(:, 2), win, [], [], Fs);
  [px_ty, ~] = pwelch(d_ty(:, 2), win, [], [], Fs);
  [px_ry, ~] = pwelch(d_ry(:, 2), win, [], [], Fs);
  [px_sr, ~] = pwelch(d_sr(:, 2), win, [], [], Fs);
  [px_rz, ~] = pwelch(d_rz(:, 2), win, [], [], Fs);
  [px_he, ~] = pwelch(d_he(:, 2), win, [], [], Fs);
#+end_src

And we compare all the signals (figures [[fig:psd_sample_comp_lpf]] and [[fig:psd_sample_comp_high_freq_lpf]]).
#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(px_of), 'DisplayName', 'All OFF');
  plot(f, sqrt(px_ty), 'DisplayName', 'Ty ON');
  plot(f, sqrt(px_ry), 'DisplayName', 'Ry ON');
  plot(f, sqrt(px_sr), 'DisplayName', 'S-R ON');
  plot(f, sqrt(px_rz), 'DisplayName', 'Rz ON');
  plot(f, sqrt(px_he), 'DisplayName', 'Hexa ON');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude Spectral Density $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);
  legend('Location', 'southwest');
#+end_src

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

#+name: fig:psd_sample_comp_lpf
#+caption: Amplitude Spectral Density of the sample velocity
#+RESULTS:
[[file:figs/psd_sample_comp_lpf.png]]


#+begin_src matlab :results none :tangle no :exports none
  xlim([80, 500]);
#+end_src

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

#+name: fig:psd_sample_comp_high_freq_lpf
#+caption: Amplitude Spectral Density of the sample velocity (zoom at high frequencies)
#+RESULTS:
[[file:figs/psd_sample_comp_high_freq_lpf.png]]

*** Vibrations on the marble
Now we plot the same curves for the geophone located on the marble.
#+begin_src matlab :exports none
  [px_of, f] = pwelch(d_of(:, 1), win, [], [], Fs);
  [px_ty, ~] = pwelch(d_ty(:, 1), win, [], [], Fs);
  [px_ry, ~] = pwelch(d_ry(:, 1), win, [], [], Fs);
  [px_sr, ~] = pwelch(d_sr(:, 1), win, [], [], Fs);
  [px_rz, ~] = pwelch(d_rz(:, 1), win, [], [], Fs);
  [px_he, ~] = pwelch(d_he(:, 1), win, [], [], Fs);
#+end_src

And we compare the Amplitude Spectral Densities (figures [[fig:psd_marble_comp_lpf]] and [[fig:psd_marble_lpf_high_freq]])
#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f, sqrt(px_of), 'DisplayName', 'All OFF');
  plot(f, sqrt(px_ty), 'DisplayName', 'Ty ON');
  plot(f, sqrt(px_ry), 'DisplayName', 'Ry ON');
  plot(f, sqrt(px_sr), 'DisplayName', 'S-R ON');
  plot(f, sqrt(px_rz), 'DisplayName', 'Rz ON');
  plot(f, sqrt(px_he), 'DisplayName', 'Hexa ON');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude Spectral Density $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);
  legend('Location', 'northeast');
#+end_src

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

#+name: fig:psd_marble_comp_lpf
#+caption: Amplitude Spectral Density of the marble velocity
#+RESULTS:
[[file:figs/psd_marble_comp_lpf.png]]

#+begin_src matlab :results none :tangle no :exports none
  legend('Location', 'southwest');
  xlim([80, 500]);
#+end_src

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

#+name: fig:psd_marble_lpf_high_freq
#+caption: Amplitude Spectral Density of the marble velocity (zoom at high frequencies)
#+RESULTS:
[[file:figs/psd_marble_lpf_high_freq.png]]

** Conclusion
#+begin_important
  - The Ty stage induces vibrations of the marble and at the sample location above 100Hz
  - The hexapod stage induces vibrations at the sample position above 220Hz
#+end_important

* Effect of the Symetrie Driver
  :PROPERTIES:
  :header-args:matlab+: :tangle matlab/effect_symetrie_driver.m
  :header-args:matlab+: :comments org :mkdirp yes
  :END:
  <<sec:effect_symetrie_driver>>

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/effect_symetrie_driver.m -nt data/effect_symetrie_driver.zip ]; then
    cp matlab/effect_symetrie_driver.m effect_symetrie_driver.m;
    zip data/effect_symetrie_driver \
        mat/data_018.mat \
        mat/data_019.mat \
        effect_symetrie_driver.m
    rm effect_symetrie_driver.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/effect_symetrie_driver.zip][here]].
#+end_note

** Experimental Setup
We here measure the signals of two geophones:
- One is located on top of the Sample platform
- One is located on the marble

The signal from the top geophone does go trought the slip-ring.

All the control systems are turned OFF except the Hexapod one.

Each measurement are done during 100s.

The settings of the voltage amplifier are:
- gain of 60dB
- AC/DC option set on DC
- Low pass filter set at 1kHz

A first order low pass filter with a cut-off frequency of 1kHz is added before the voltage amplifier.

The measurements are:
- =meas_018.mat=: Hexapod's driver on the granite
- =meas_019.mat=: Hexapod's driver on the ground

Each of the =mat= file contains one array =data= with 3 columns:
| Column number | Description       |
|---------------+-------------------|
|             1 | Geophone - Marble |
|             2 | Geophone - Sample |
|             3 | Time              |

** 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
We load the data of the z axis of two geophones.
#+begin_src matlab :results none
  d_18 = load('mat/data_018.mat', 'data'); d_18 = d_18.data;
  d_19 = load('mat/data_019.mat', 'data'); d_19 = d_19.data;
#+end_src

** Analysis - Time Domain
#+begin_src matlab :results none
  figure;
  hold on;
  plot(d_19(:, 3), d_19(:, 1), 'DisplayName', 'Driver - Ground');
  plot(d_18(:, 3), d_18(:, 1), 'DisplayName', 'Driver - Granite');
  hold off;
  xlabel('Time [s]'); ylabel('Voltage [V]');
  xlim([0, 50]);
  legend('Location', 'bestoutside');
#+end_src

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

#+name: fig:time_domain_hexa_driver
#+caption: Comparison of the time domain data when turning off the control system of the stages - Geophone at the sample location
#+RESULTS:
[[file:figs/time_domain_hexa_driver.png]]

** Analysis - Frequency Domain
#+begin_src matlab :results none
  dt = d_18(2, 3) - d_18(1, 3);

  Fs = 1/dt;
  win = hanning(ceil(10*Fs));
#+end_src

*** Vibrations at the sample location
First, we compute the Power Spectral Density of the signals coming from the Geophone located at the sample location.
#+begin_src matlab :results none
  [px_18, f] = pwelch(d_18(:, 1), win, [], [], Fs);
  [px_19, ~] = pwelch(d_19(:, 1), win, [], [], Fs);
#+end_src

#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(px_19), 'DisplayName', 'Driver - Ground');
  plot(f, sqrt(px_18), 'DisplayName', 'Driver - Granite');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude Spectral Density $\left[\frac{V}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);
  legend('Location', 'southwest');
#+end_src

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

#+name: fig:psd_hexa_driver
#+caption: Amplitude Spectral Density of the signal coming from the top geophone
#+RESULTS:
[[file:figs/psd_hexa_driver.png]]

#+begin_src matlab :results none :tangle no :exports none
  xlim([80, 500]);
#+end_src

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

#+name: fig:psd_hexa_driver_high_freq
#+caption: Amplitude Spectral Density of the signal coming from the top geophone (zoom at high frequencies)
#+RESULTS:
[[file:figs/psd_hexa_driver_high_freq.png]]

** Conclusion
#+begin_important
  Even tough the Hexapod's driver vibrates quite a lot, it does not generate significant vibrations of the granite when either placed on the granite or on the ground.
#+end_important