nass-micro-station-measurements/2018-10-15 - Marc/index.org

#+TITLE: Measurement Analysis
: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:shell  :eval no-export
:END:

* 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

* Measurement Description
#+name: fig:setup_picture
#+attr_html: :width 500px
#+caption: Picture of the setup for the measurement
[[file:./figs/setup_picture.png]]

The sensor used are 3 L-4C geophones ([[file:../actuators-sensors/index.org::*L-4C][Documentation]]).

Each motor are turn off and then on.

The goal is to see what noise is injected in the system due to the regulation loop of each stage.

* Importation of the data
First, load all the measurement files:
#+begin_src matlab :exports code :results silent
  meas = {};
  meas{1} = load('./mat/Measurement1.mat');
  meas{2} = load('./mat/Measurement2.mat');
  meas{3} = load('./mat/Measurement3.mat');
  meas{4} = load('./mat/Measurement4.mat');
  meas{5} = load('./mat/Measurement5.mat');
#+end_src

Change the track name for measurements 3 and 4.
#+begin_src matlab :exports code :results silent
  meas{3}.Track1_Name = 'Input 1: Hexa Z';
  meas{4}.Track1_Name = 'Input 1: Hexa Z';
#+end_src

For the measurements 1 to 4, the measurement channels are shown table [[tab:meas_14]].

#+begin_src matlab :exports results :results table :post addhdr(*this*)
  table_string = sprintf(' | Channel 1 | Channel 2 | Channel 3 \n');
  for i = 1:4
    table_string = [table_string, sprintf('Meas. %i | %s | %s | %s \n', i, meas{i}.Track1_Name, meas{i}.Track2_Name, meas{i}.Track3_Name)];
  end
  ans = table_string
#+end_src

#+NAME: tab:meas_14
#+CAPTION: Channels for measurements 1 to 4
#+RESULTS:
|         | Channel 1        | Channel 2        | Channel 3     |
|---------+------------------+------------------+---------------|
| Meas. 1 | Input 1: tilt1 Z | Input 2: tilt2 Z | Input 3: Ty Y |
| Meas. 2 | Input 1: tilt1 Z | Input 2: tilt2 Z | Input 3: Ty Y |
| Meas. 3 | Input 1: Hexa Z  | Input 2: tilt2 Z | Input 3: Ty Y |
| Meas. 4 | Input 1: Hexa Z  | Input 2: tilt2 Z | Input 3: Ty Y |


For the measurement 5, the channels are shown table [[tab:meas_5]].
#+begin_src matlab :exports results :results table :post addhdr(*this*)
  table_string = sprintf(' | Channel 1 | Channel 2 | Channel 3 | Channel 4 \n');
  i = 5
  table_string = [table_string, sprintf('Meas. %i | %s | %s | %s | %s \n', i, meas{i}.Track1_Name, meas{i}.Track2_Name, meas{i}.Track3_Name, meas{i}.Track4_Name)];
  ans = table_string
#+end_src

#+NAME: tab:meas_5
#+CAPTION: Channels for measurement 5
#+RESULTS:
|         | Channel 1        | Channel 2         | Channel 3        | Channel 4         |
|---------+------------------+-------------------+------------------+-------------------|
| Meas. 5 | Input 1: Floor Z | Input 2: Marble Z | Input 3: Floor Y | Input 4: Marble Y |

* Variables for analysis
We define the sampling frequency and the time vectors for the plots.

#+begin_src matlab :exports code :results silent
  Fs = 256; % [Hz]
  dt = 1/(Fs);
  t1  = dt*[0:length(meas{1}.Track1)-1];
  t2  = dt*[0:length(meas{2}.Track1)-1];
  t3  = dt*[0:length(meas{3}.Track1)-1];
  t4  = dt*[0:length(meas{4}.Track1)-1];
  t5  = dt*[0:length(meas{5}.Track1)-1];
#+end_src

For the frequency analysis, we define the frequency limits for the plot.
#+begin_src matlab :exports code :results silent
  fmin = 1; % [Hz]
  fmax = 100; % [Hz]
#+end_src

Then we define the windows that will be used to average the results.
#+begin_src matlab :exports code :results silent
  psd_window = hanning(2*fmin/dt);
#+end_src

* Coherence between the two vertical geophones on the Tilt Stage
We first compute the coherence between the two geophones located on the tilt stage. The result is shown on figure [[fig:coherence_vertical_tilt_sensors]].
#+begin_src matlab :results none
  [coh, f] = mscohere(meas{1}.Track1(:), meas{1}.Track2(:), psd_window, [], [], Fs);
#+end_src

#+begin_src matlab :results none :exports none
  figure;
  plot(f, coh);
  set(gca, 'xscale', 'log');
  ylim([0, 1]);
  xlabel('Frequency [Hz]'); ylabel('Coherence');
#+end_src

#+NAME: fig:coherence_vertical_tilt_sensors
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/coherence_vertical_tilt_sensors.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:coherence_vertical_tilt_sensors
#+CAPTION: Coherence between the two vertical sensors positionned on the Tilt Stage
#+RESULTS: fig:coherence_vertical_tilt_sensors
[[file:figs/coherence_vertical_tilt_sensors.png]]

We then compute the transfer function from one sensor to the other (figure [[fig:tf_vertical_tilt_sensors]]).
#+begin_src matlab :results none
  [tf23, f] = tfestimate(meas{1}.Track1(:), meas{1}.Track2(:), psd_window, [], [], Fs);
#+end_src

#+begin_src matlab :results none :exports none
  figure;
  ax1 = subaxis(2,1,1);
  plot(f, abs(tf23));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Magnitude [V/(m/s)]');

  ax2 = subaxis(2,1,2);
  plot(f, 180/pi*angle(tf23));
  set(gca,'xscale','log');
  yticks(-180:90:180);
  ylim([-180 180]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  linkaxes([ax1,ax2],'x');
#+end_src

#+NAME: fig:tf_vertical_tilt_sensors
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/tf_vertical_tilt_sensors.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:tf_vertical_tilt_sensors
#+CAPTION: Transfer function from one vertical geophone on the tilt stage to the other vertical geophone on the tilt stage
#+RESULTS: fig:tf_vertical_tilt_sensors
[[file:figs/tf_vertical_tilt_sensors.png]]

Even though the coherence is not very good, we observe no resonance between the two sensors.

* Data Post Processing
When using two geophone sensors on the same tilt stage (measurements 1 and 2), we post-process the data to obtain the z displacement and the rotation of the tilt stage:

#+begin_src matlab :results silent
  meas1_z    = (meas{1}.Track1+meas{1}.Track2)/2;
  meas1_tilt = (meas{1}.Track1-meas{1}.Track2)/2;

  meas{1}.Track1 = meas1_z;
  meas{1}.Track1_Y_Magnitude = 'Meter / second';
  meas{1}.Track1_Name = 'Ry Z';
  meas{1}.Track2 = meas1_tilt;
  meas{1}.Track2_Y_Magnitude = 'Rad / second';
  meas{1}.Track2_Name = 'Ry Tilt';

  meas2_z    = (meas{2}.Track1+meas{2}.Track2)/2;
  meas2_tilt = (meas{2}.Track1-meas{2}.Track2)/2;
  meas{2}.Track1 = meas2_z;
  meas{2}.Track1_Y_Magnitude = 'Meter / second';
  meas{2}.Track1_Name = 'Ry Z';
  meas{2}.Track2 = meas2_tilt;
  meas{2}.Track2_Y_Magnitude = 'Rad / second';
  meas{2}.Track2_Name = 'Ry Tilt';
#+end_src

* Normalization
Parameters of the geophone are defined below.
The transfer function from geophone velocity to measured voltage is shown on figure [[fig:L4C_bode_plot]].

Measurements will be normalized by the inverse of this transfer function in order to go from voltage measurement to velocity measurement.

#+begin_src matlab :results none
  L4C_w0 = 2*pi; % [rad/s]
  L4C_ksi = 0.28;
  L4C_G0 = 276.8; % [V/(m/s)]
  L4C_G = L4C_G0*(s/L4C_w0)^2/((s/L4C_w0)^2 + 2*L4C_ksi*(s/L4C_w0) + 1);
#+end_src

#+begin_src matlab :results none :exports none
  freqs = logspace(-2, 2, 1000);

  figure;
  ax1 = subaxis(2,1,1);
  plot(freqs, abs(squeeze(freqresp(L4C_G, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Magnitude [V/(m/s)]');

  ax2 = subaxis(2,1,2);
  plot(freqs, 180/pi*angle(squeeze(freqresp(L4C_G, freqs, 'Hz'))));
  set(gca,'xscale','log');
  yticks(-180:90:180);
  ylim([-180 180]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  linkaxes([ax1,ax2],'x');
#+end_src

#+NAME: fig:L4C_bode_plot
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/L4C_bode_plot.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:L4C_bode_plot
#+CAPTION: Bode plot of the L4C Geophone
#+RESULTS: fig:L4C_bode_plot
[[file:figs/L4C_bode_plot.png]]

#+begin_src matlab :results none :exports code
  meas{1}.Track1 = (meas{1}.Track1)./276.8;
  meas{1}.Track2 = (meas{1}.Track2)./276.8;
  meas{1}.Track3 = (meas{1}.Track3)./276.8;

  meas{2}.Track1 = (meas{2}.Track1)./276.8;
  meas{2}.Track2 = (meas{2}.Track2)./276.8;
  meas{2}.Track3 = (meas{2}.Track3)./276.8;

  meas{3}.Track1 = (meas{3}.Track1)./276.8;
  meas{3}.Track2 = (meas{3}.Track2)./276.8;
  meas{3}.Track3 = (meas{3}.Track3)./276.8;

  meas{4}.Track1 = (meas{4}.Track1)./276.8;
  meas{4}.Track2 = (meas{4}.Track2)./276.8;
  meas{4}.Track3 = (meas{4}.Track3)./276.8;

  meas{5}.Track1 = (meas{5}.Track1)./276.8;
  meas{5}.Track2 = (meas{5}.Track2)./276.8;
  meas{5}.Track3 = (meas{5}.Track3)./276.8;
  meas{5}.Track4 = (meas{5}.Track4)./276.8;
#+end_src

#+begin_src matlab :results none
  meas{1}.Track1_norm = lsim(inv(L4C_G), meas{1}.Track1, t1);
  meas{1}.Track2_norm = lsim(inv(L4C_G), meas{1}.Track2, t1);
  meas{1}.Track3_norm = lsim(inv(L4C_G), meas{1}.Track3, t1);

  meas{2}.Track1_norm = lsim(inv(L4C_G), meas{2}.Track1, t2);
  meas{2}.Track2_norm = lsim(inv(L4C_G), meas{2}.Track2, t2);
  meas{2}.Track3_norm = lsim(inv(L4C_G), meas{2}.Track3, t2);

  meas{3}.Track1_norm = lsim(inv(L4C_G), meas{3}.Track1, t3);
  meas{3}.Track2_norm = lsim(inv(L4C_G), meas{3}.Track2, t3);
  meas{3}.Track3_norm = lsim(inv(L4C_G), meas{3}.Track3, t3);

  meas{4}.Track1_norm = lsim(inv(L4C_G), meas{4}.Track1, t4);
  meas{4}.Track2_norm = lsim(inv(L4C_G), meas{4}.Track2, t4);
  meas{4}.Track3_norm = lsim(inv(L4C_G), meas{4}.Track3, t4);

  meas{5}.Track1_norm = lsim(inv(L4C_G), meas{5}.Track1, t5);
  meas{5}.Track2_norm = lsim(inv(L4C_G), meas{5}.Track2, t5);
  meas{5}.Track3_norm = lsim(inv(L4C_G), meas{5}.Track3, t5);
  meas{5}.Track4_norm = lsim(inv(L4C_G), meas{5}.Track4, t5);
#+end_src

* Measurement 1 - Effect of Ty stage
The configuration for this measurement is shown table [[tab:conf_meas1]].

#+CAPTION: Stages configuration - Measurement 1
#+NAME: tab:conf_meas1
| Time     | 0-309 | 309-end |
|----------+-------+---------|
| Ty       | OFF   | *ON*    |
| Ry       | OFF   | OFF     |
| SlipRing | OFF   | OFF     |
| Spindle  | OFF   | OFF     |
| Hexa     | OFF   | OFF     |

We then plot the measurements in time domain (figure [[fig:meas1]]).

#+begin_important
  We observe strange behavior when the Ty stage is turned on.
  How can we explain that?
#+end_important

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(t1(ceil(300/dt):ceil(340/dt)), meas{1}.Track1(ceil(300/dt):ceil(340/dt)));
  plot(t1(ceil(300/dt):ceil(340/dt)), meas{1}.Track2(ceil(300/dt):ceil(340/dt)));
  plot(t1(ceil(300/dt):ceil(340/dt)), meas{1}.Track3(ceil(300/dt):ceil(340/dt)));
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  legend({meas{1}.Track1_Name, meas{1}.Track2_Name, meas{1}.Track3_Name}, 'Location', 'northeast')
#+end_src

#+NAME: fig:meas1
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas1.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas1
#+CAPTION: Time domain - measurement 1
#+RESULTS: fig:meas1
[[file:figs/meas1.png]]

To understand what is going on, instead of looking at the velocity, we can look at the displacement by integrating the data. The displacement is computed by integrating the velocity using =cumtrapz= function.

Then we plot the position with respect to time (figure [[fig:meas1_disp]]).
#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(t1, cumtrapz(t1, meas{1}.Track3));
  hold off;
  xlim([300, 340]);
  xlabel('Time [s]'); ylabel('Displacement [m]');
#+end_src

#+NAME: fig:meas1_disp
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas1_disp.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas1_disp
#+CAPTION: Y displacement of the Ty stage
#+RESULTS: fig:meas1_disp
[[file:figs/meas1_disp.png]]

We when compute the power spectral density of each measurement before and after turning on the stage.
#+begin_src matlab :exports code :results silent
  [pxx111, f11] = pwelch(meas{1}.Track1(1:ceil(300/dt)),   psd_window, [], [], Fs);
  [pxx112, f12] = pwelch(meas{1}.Track1(ceil(350/dt):end), psd_window, [], [], Fs);

  [pxx121, ~] = pwelch(meas{1}.Track2(1:ceil(300/dt)),   psd_window, [], [], Fs);
  [pxx122, ~] = pwelch(meas{1}.Track2(ceil(350/dt):end), psd_window, [], [], Fs);

  [pxx131, ~] = pwelch(meas{1}.Track3(1:ceil(300/dt)),   psd_window, [], [], Fs);
  [pxx132, ~] = pwelch(meas{1}.Track3(ceil(350/dt):end), psd_window, [], [], Fs);
#+end_src

We finally plot the power spectral density of each track (figures [[fig:meas1_ry_z_psd]], [[fig:meas1_ry_tilt_psd]], [[fig:meas1_ty_y_psd]]).

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f11, sqrt(pxx111)./abs(squeeze(freqresp(L4C_G, f11, 'Hz'))));
  plot(f12, sqrt(pxx112)./abs(squeeze(freqresp(L4C_G, f12, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{1}.Track1_Name));
  legend({'0-300', '350-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas1_ry_z_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas1_ry_z_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas1_ry_z_psd
#+CAPTION: PSD of the Z velocity of Ry stage - measurement 1
#+RESULTS: fig:meas1_ry_z_psd
[[file:figs/meas1_ry_z_psd.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f11, sqrt(pxx121)./abs(squeeze(freqresp(L4C_G, f11, 'Hz'))));
  plot(f12, sqrt(pxx122)./abs(squeeze(freqresp(L4C_G, f12, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$rad/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{1}.Track2_Name));
  legend({'0-300', '350-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas1_ry_tilt_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas1_ry_tilt_psd.png" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas1_ry_tilt_psd
#+CAPTION: PSD of the Rotation of Ry Stage - measurement 1
#+RESULTS: fig:meas1_ry_tilt_psd
[[file:figs/meas1_ry_tilt_psd.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f11, sqrt(pxx131)./abs(squeeze(freqresp(L4C_G, f11, 'Hz'))));
  plot(f12, sqrt(pxx132)./abs(squeeze(freqresp(L4C_G, f12, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{1}.Track3_Name));
  legend({'0-300', '350-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas1_ty_y_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas1_ty_y_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas1_ty_y_psd
#+CAPTION: PSD of the Ty velocity in the Y direction - measurement 1
#+RESULTS: fig:meas1_ty_y_psd
[[file:figs/meas1_ty_y_psd.png]]

#+begin_important
Turning on the Y-translation stage increases the velocity of the Ty stage in the Y direction and the rotation motion of the tilt stage:
- at 20Hz
- at 40Hz
- between 80Hz and 90Hz

It does not seems to have any effect on the Z motion of the tilt stage.
#+end_important

* Measurement 2 - Effect of Ry stage
The tilt stage is turned ON at around 326 seconds (table [[tab:conf_meas2]]).

#+CAPTION: Stages configuration - Measurement 2
#+NAME: tab:conf_meas2
| Time     | 0-326 | 326-end |
|----------+-------+---------|
| Ty       | OFF   | OFF     |
| Ry       | OFF   | *ON*    |
| SlipRing | OFF   | OFF     |
| Spindle  | OFF   | OFF     |
| Hexa     | OFF   | OFF     |

We plot the time domain (figure [[fig:meas2]]) and we don't observe anything special in the time domain.

#+begin_src matlab :exports results :results silent
  figure;
  hold on;
  plot(t2(ceil(300/dt):ceil(350/dt)), meas{2}.Track1(ceil(300/dt):ceil(350/dt)));
  plot(t2(ceil(300/dt):ceil(350/dt)), meas{2}.Track3(ceil(300/dt):ceil(350/dt)));
  plot(t2(ceil(300/dt):ceil(350/dt)), meas{2}.Track2(ceil(300/dt):ceil(350/dt)));
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  legend({meas{2}.Track1_Name, meas{2}.Track2_Name, meas{2}.Track3_Name}, 'Location', 'northeast')
  xlim([300, 350]);
#+end_src

#+NAME: fig:meas2
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas2.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas2
#+CAPTION: Time domain - measurement 2
#+RESULTS: fig:meas2
[[file:figs/meas2.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(t2, cumtrapz(t2, meas{2}.Track1));
  plot(t2, cumtrapz(t2, meas{2}.Track2));
  plot(t2, cumtrapz(t2, meas{2}.Track3));
  hold off;
  xlim([300, 350]);
  xlabel('Time [s]'); ylabel('Displacement [m]');
  legend({meas{2}.Track1_Name, meas{2}.Track2_Name, meas{2}.Track3_Name}, 'Location', 'northeast')
#+end_src

#+NAME: fig:meas2_disp
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas2_disp.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas2_disp
#+CAPTION: Time domain - measurement 2
#+RESULTS: fig:meas2_disp
[[file:figs/meas2_disp.png]]

We compute the PSD of each track and we plot them (figures [[fig:meas2_ry_z_psd]], [[fig:meas2_ry_tilt_psd]] and [[fig:meas2_ty_y_psd]] ).
#+begin_src matlab :exports code :results silent
  [pxx211, f21] = pwelch(meas{2}.Track1(1:ceil(326/dt)),   psd_window, [], [], Fs);
  [pxx212, f22] = pwelch(meas{2}.Track1(ceil(326/dt):end), psd_window, [], [], Fs);

  [pxx221, ~] = pwelch(meas{2}.Track2(1:ceil(326/dt)),   psd_window, [], [], Fs);
  [pxx222, ~] = pwelch(meas{2}.Track2(ceil(326/dt):end), psd_window, [], [], Fs);

  [pxx231, ~] = pwelch(meas{2}.Track3(1:ceil(326/dt)),   psd_window, [], [], Fs);
  [pxx232, ~] = pwelch(meas{2}.Track3(ceil(326/dt):end), psd_window, [], [], Fs);
#+end_src

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f21, sqrt(pxx211)./abs(squeeze(freqresp(L4C_G, f21, 'Hz'))));
  plot(f22, sqrt(pxx212)./abs(squeeze(freqresp(L4C_G, f22, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{2}.Track1_Name));
  legend({'0-326', '326-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas2_ry_z_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas2_ry_z_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas2_ry_z_psd
#+CAPTION: PSD of the Z velocity of Ry Stage - measurement 2
#+RESULTS: fig:meas2_ry_z_psd
[[file:figs/meas2_ry_z_psd.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f21, sqrt(pxx221)./abs(squeeze(freqresp(L4C_G, f21, 'Hz'))));
  plot(f22, sqrt(pxx222)./abs(squeeze(freqresp(L4C_G, f22, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$rad/s/\sqrt(Hz)$]');
  title(sprintf('%s', meas{2}.Track2_Name));
  legend({'0-326', '326-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas2_ry_tilt_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas2_ry_tilt_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas2_ry_tilt_psd
#+CAPTION: PSD of the Rotation motion of Ry Stage - measurement 2
#+RESULTS: fig:meas2_ry_tilt_psd
[[file:figs/meas2_ry_tilt_psd.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f21, sqrt(pxx231)./abs(squeeze(freqresp(L4C_G, f21, 'Hz'))));
  plot(f22, sqrt(pxx232)./abs(squeeze(freqresp(L4C_G, f22, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{2}.Track3_Name));
  legend({'0-326', '326-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas2_ty_y_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas2_ty_y_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas2_ty_y_psd
#+CAPTION: PSD of the Ty velocity in the Y direction - measurement 2
#+RESULTS: fig:meas2_ty_y_psd
[[file:figs/meas2_ty_y_psd.png]]

#+begin_important
  We observe no noticeable difference when the Tilt-stage is turned ON expect a small decrease of the Z motion of the tilt stage around 10Hz.
#+end_important

* Measurement 3 - Effect of the Hexapod
The hexapod is turned off after 406 seconds (table [[tab:conf_meas3]]).

#+CAPTION: Stages configuration - Measurement 3
#+NAME: tab:conf_meas3
| Time     | 0-406 | 406-end |
|----------+-------+---------|
| Ty       | OFF   | OFF     |
| Ry       | *ON*  | *ON*    |
| SlipRing | OFF   | OFF     |
| Spindle  | OFF   | OFF     |
| Hexa     | *ON*  | OFF     |

The time domain result is shown figure [[fig:meas3]].

#+begin_src matlab :exports results :results silent
  figure;
  hold on;
  plot(t3(ceil(380/dt):ceil(420/dt)), meas{3}.Track1(ceil(380/dt):ceil(420/dt)));
  plot(t3(ceil(380/dt):ceil(420/dt)), meas{3}.Track2(ceil(380/dt):ceil(420/dt)));
  plot(t3(ceil(380/dt):ceil(420/dt)), meas{3}.Track3(ceil(380/dt):ceil(420/dt)));
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  legend({meas{3}.Track1_Name, meas{3}.Track2_Name, meas{3}.Track3_Name}, 'Location', 'northeast')
#+end_src

#+NAME: fig:meas3
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas3.pdf" :var figsize="wide-noral" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas3
#+CAPTION: Time domain - measurement 3
#+RESULTS: fig:meas3
[[file:figs/meas3.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(t3, cumtrapz(t3, meas{3}.Track1));
  plot(t3, cumtrapz(t3, meas{3}.Track2));
  plot(t3, cumtrapz(t3, meas{3}.Track3));
  hold off;
  xlim([350, 450]);
  xlabel('Time [s]'); ylabel('Displacement [m]');
  legend({meas{3}.Track1_Name, meas{3}.Track2_Name, meas{3}.Track3_Name}, 'Location', 'northeast')
#+end_src

#+NAME: fig:meas3_disp
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas3_disp.pdf" :var figsize="wide-noral" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas3_disp
#+CAPTION: Time domain - measurement 3
#+RESULTS: fig:meas3_disp
[[file:figs/meas3_disp.png]]

We then compute the PSD of each track before and after turning off the hexapod and plot the results in the figures [[fig:meas3_hexa_z_psd]], [[fig:meas3_ry_z_psd]] and [[fig:meas3_ty_y_psd]].
#+begin_src matlab :exports code :results silent
  [pxx311, f31] = pwelch(meas{3}.Track1(1:ceil(400/dt)),   psd_window, [], [], Fs);
  [pxx312, f32] = pwelch(meas{3}.Track1(ceil(420/dt):end), psd_window, [], [], Fs);

  [pxx321, ~] = pwelch(meas{3}.Track2(1:ceil(400/dt)),   psd_window, [], [], Fs);
  [pxx322, ~] = pwelch(meas{3}.Track2(ceil(420/dt):end), psd_window, [], [], Fs);

  [pxx331, ~] = pwelch(meas{3}.Track3(1:ceil(400/dt)),   psd_window, [], [], Fs);
  [pxx332, ~] = pwelch(meas{3}.Track3(ceil(420/dt):end), psd_window, [], [], Fs);
#+end_src

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f31, sqrt(pxx311)./abs(squeeze(freqresp(L4C_G, f31, 'Hz'))));
  plot(f32, sqrt(pxx312)./abs(squeeze(freqresp(L4C_G, f32, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{3}.Track1_Name));
  legend({'0-400', '420-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas3_hexa_z_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas3_hexa_z_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas3_hexa_z_psd
#+CAPTION: PSD of the Z velocity of the Hexapod - measurement 3
#+RESULTS: fig:meas3_hexa_z_psd
[[file:figs/meas3_hexa_z_psd.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f31, sqrt(pxx321)./abs(squeeze(freqresp(L4C_G, f31, 'Hz'))));
  plot(f32, sqrt(pxx322)./abs(squeeze(freqresp(L4C_G, f32, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{3}.Track2_Name));
  legend({'0-400', '420-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas3_ry_z_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas3_ry_z_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas3_ry_z_psd
#+CAPTION: PSD of the Z velocity of the Ry stage - measurement 3
#+RESULTS: fig:meas3_ry_z_psd
[[file:figs/meas3_ry_z_psd.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f31, sqrt(pxx331)./abs(squeeze(freqresp(L4C_G, f31, 'Hz'))));
  plot(f32, sqrt(pxx332)./abs(squeeze(freqresp(L4C_G, f32, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{3}.Track3_Name));
  legend({'0-400', '420-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas3_ty_y_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas3_ty_y_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas3_ty_y_psd
#+CAPTION: PSD of the Ty velocity in the Y direction - measurement 3
#+RESULTS: fig:meas3_ty_y_psd
[[file:figs/meas3_ty_y_psd.png]]

#+begin_important
Turning ON induces some motion on the hexapod in the z direction (figure [[fig:meas3_hexa_z_psd]]), on the tilt stage in the z direction (figure [[fig:meas3_ry_z_psd]]) and on the y motion of the Ty stage (figure [[fig:meas3_ty_y_psd]]):
- at 17Hz
- at 34Hz
#+end_important

* Measurement 4 - Effect of the Splip-Ring and Spindle
The slip ring is turned on at 300s, then the spindle is turned on at 620s (table [[tab:conf_meas4]]). The time domain signals are shown figure [[fig:meas4]].

#+CAPTION: Stages configuration - Measurement 4
#+NAME: tab:conf_meas4
| Time     | 0-300 | 300-620 | 620-end |
|----------+-------+---------+---------|
| Ty       | OFF   | OFF     | OFF     |
| Ry       | OFF   | OFF     | OFF     |
| SlipRing | OFF   | *ON*    | *ON*    |
| Spindle  | OFF   | OFF     | *ON*    |
| Hexa     | OFF   | OFF     | OFF     |

#+begin_src matlab :exports results :results silent
  figure;
  hold on;
  plot(t4, meas{4}.Track1);
  plot(t4, meas{4}.Track2);
  plot(t4, meas{4}.Track3);
  hold off;
  xlim([t4(1), t4(end)]);
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  legend({meas{4}.Track1_Name, meas{4}.Track2_Name, meas{4}.Track3_Name}, 'Location', 'southwest')
#+end_src

#+NAME: fig:meas4
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas4.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas4
#+CAPTION: Time domain - measurement 4
#+RESULTS: fig:meas4
[[file:figs/meas4.png]]

#+begin_src matlab :exports none :results silent
  figure;
  subaxis(1, 2, 1);
  hold on;
  plot(t4, cumtrapz(t4, meas{4}.Track1));
  plot(t4, cumtrapz(t4, meas{4}.Track2));
  plot(t4, cumtrapz(t4, meas{4}.Track3));
  hold off;
  xlim([250, 350]);
  xlabel('Time [s]'); ylabel('Displacement [m]');
  legend({meas{4}.Track1_Name, meas{4}.Track2_Name, meas{4}.Track3_Name}, 'Location', 'northwest')

  subaxis(1, 2, 2);
  hold on;
  plot(t4, cumtrapz(t4, meas{4}.Track1));
  plot(t4, cumtrapz(t4, meas{4}.Track2));
  plot(t4, cumtrapz(t4, meas{4}.Track3));
  hold off;
  xlim([600, 650]);
  xlabel('Time [s]'); ylabel('Displacement [m]');
#+end_src

If we integrate this signal, we obtain Figure [[fig:meas4_int]].

#+NAME: fig:meas4_int
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas4_int.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas4_int
#+CAPTION: Time domain - measurement 4
#+RESULTS: fig:meas4_int
[[file:figs/meas4_int.png]]

The PSD of each track are computed using the code below.
#+begin_src matlab :exports none :results silent
  [pxx411, f41] = pwelch(meas{4}.Track1(1:ceil(280/dt)),            psd_window, [], [], Fs);
  [pxx412, f42] = pwelch(meas{4}.Track1(ceil(280/dt):ceil(600/dt)), psd_window, [], [], Fs);
  [pxx413, f43] = pwelch(meas{4}.Track1(ceil(640/dt):end),          psd_window, [], [], Fs);

  [pxx421, ~] = pwelch(meas{4}.Track2(1:ceil(280/dt)),            psd_window, [], [], Fs);
  [pxx422, ~] = pwelch(meas{4}.Track2(ceil(280/dt):ceil(600/dt)), psd_window, [], [], Fs);
  [pxx423, ~] = pwelch(meas{4}.Track2(ceil(640/dt):end),          psd_window, [], [], Fs);

  [pxx431, ~] = pwelch(meas{4}.Track3(1:ceil(280/dt)),            psd_window, [], [], Fs);
  [pxx432, ~] = pwelch(meas{4}.Track3(ceil(280/dt):ceil(600/dt)), psd_window, [], [], Fs);
  [pxx433, ~] = pwelch(meas{4}.Track3(ceil(640/dt):end),          psd_window, [], [], Fs);

  f41 = f41(2:end);
  f42 = f42(2:end);
  f43 = f43(2:end);

  pxx411 = pxx411(2:end);
  pxx412 = pxx412(2:end);
  pxx413 = pxx413(2:end);

  pxx421 = pxx421(2:end);
  pxx422 = pxx422(2:end);
  pxx423 = pxx423(2:end);

  pxx431 = pxx431(2:end);
  pxx432 = pxx432(2:end);
  pxx433 = pxx433(2:end);
#+end_src

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f41, sqrt(pxx411)./abs(squeeze(freqresp(L4C_G, f41, 'Hz'))));
  plot(f42, sqrt(pxx412)./abs(squeeze(freqresp(L4C_G, f42, 'Hz'))));
  plot(f43, sqrt(pxx413)./abs(squeeze(freqresp(L4C_G, f43, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{4}.Track1_Name));
  legend({'0-280', '320-600', '640-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas4_hexa_z_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas4_hexa_z_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas4_hexa_z_psd
#+CAPTION: PSD of the Z velocity of the Hexapod - measurement 4
#+RESULTS: fig:meas4_hexa_z_psd
[[file:figs/meas4_hexa_z_psd.png]]

We plot the PSD of the displacement.
#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f41, sqrt(pxx411)./abs(squeeze(freqresp(L4C_G, f41, 'Hz')))./(2*pi*f41));
  plot(f42, sqrt(pxx412)./abs(squeeze(freqresp(L4C_G, f42, 'Hz')))./(2*pi*f42));
  plot(f43, sqrt(pxx413)./abs(squeeze(freqresp(L4C_G, f43, 'Hz')))./(2*pi*f43));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/\sqrt{Hz}$]');
  title(sprintf('%s', meas{4}.Track1_Name));
  legend({'0-280', '320-600', '640-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas4_hexa_z_psd_int
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas4_hexa_z_psd_int.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas4_hexa_z_psd_int
#+CAPTION: PSD_INT of the Z velocity of the Hexapod - measurement 4
#+RESULTS: fig:meas4_hexa_z_psd_int
[[file:figs/meas4_hexa_z_psd_int.png]]

And we compute the Cumulative amplitude spectrum.
#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f41, sqrt(cumsum(pxx431./abs(squeeze(freqresp(L4C_G, f41, 'Hz'))).^2./(2*pi*f41).*(f41 - [0; f41(1:end-1)]))));
  plot(f42, sqrt(cumsum(pxx432./abs(squeeze(freqresp(L4C_G, f42, 'Hz'))).^2./(2*pi*f42).*(f42 - [0; f42(1:end-1)]))));
  plot(f43, sqrt(cumsum(pxx433./abs(squeeze(freqresp(L4C_G, f43, 'Hz'))).^2./(2*pi*f43).*(f43 - [0; f43(1:end-1)]))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS [$m$ rms]');
  title(sprintf('%s', meas{4}.Track1_Name));
  legend({'0-280', '320-600', '640-end'}, 'Location', 'southwest');
  hold off;
#+end_src



#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f41, sqrt(pxx421)./abs(squeeze(freqresp(L4C_G, f41, 'Hz'))));
  plot(f42, sqrt(pxx422)./abs(squeeze(freqresp(L4C_G, f42, 'Hz'))));
  plot(f43, sqrt(pxx423)./abs(squeeze(freqresp(L4C_G, f43, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{4}.Track2_Name));
  legend({'0-280', '320-600', '640-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas4_ry_z_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas4_ry_z_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas4_ry_z_psd
#+CAPTION: PSD of the Ry rotation in the Y direction - measurement 4
#+RESULTS: fig:meas4_ry_z_psd
[[file:figs/meas4_ry_z_psd.png]]


#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f41, sqrt(pxx431)./abs(squeeze(freqresp(L4C_G, f41, 'Hz'))));
  plot(f42, sqrt(pxx432)./abs(squeeze(freqresp(L4C_G, f42, 'Hz'))));
  plot(f43, sqrt(pxx433)./abs(squeeze(freqresp(L4C_G, f43, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  title(sprintf('%s', meas{4}.Track3_Name));
  legend({'0-280', '320-600', '640-end'}, 'Location', 'southwest');
  hold off;
#+end_src

#+NAME: fig:meas4_ty_y_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas4_ty_y_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas4_ty_y_psd
#+CAPTION: PSD of the Ty velocity in the Y direction - measurement 4
#+RESULTS: fig:meas4_ty_y_psd
[[file:figs/meas4_ty_y_psd.png]]


#+begin_important
Turning ON the splipring seems to not add motions on the stages measured.
It even seems to lower the motion of the Ty stage (figure [[fig:meas4_ty_y_psd]]): does that make any sense?

Turning ON the spindle induces motions:
- at 5Hz on each motion measured
- at 22.5Hz on the Z motion of the Hexapod. Can this is due to some 50Hz?
- at 62Hz on each motion measured
#+end_important

* Measurement 5 - Transmission from ground to marble

This measurement just consists of measurement of Y-Z motion of the ground and the marble.

The time domain signals are shown on figure [[fig:meas5]].

#+begin_src matlab :exports results :results silent
  figure;
  hold on;
  plot(t5, meas{5}.Track1);
  plot(t5, meas{5}.Track2);
  plot(t5, meas{5}.Track3);
  plot(t5, meas{5}.Track4);
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  legend({meas{5}.Track1_Name, meas{5}.Track2_Name, meas{5}.Track3_Name, meas{5}.Track4_Name}, 'Location', 'northeast')
#+end_src

#+NAME: fig:meas5
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas5.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas5
#+CAPTION: Time domain - measurement 5
#+RESULTS: fig:meas5
[[file:figs/meas5.png]]

We compute the PSD of each track and we plot the PSD of the Z motion for the ground and marble on figure [[fig:meas5_z_psd]] and for the Y motion on figure [[fig:meas5_y_psd]].

#+begin_src matlab :exports code :results silent
  [pxx51, f51] = pwelch(meas{5}.Track1(:), psd_window, [], [], Fs);
  [pxx52, f52] = pwelch(meas{5}.Track2(:), psd_window, [], [], Fs);
  [pxx53, f53] = pwelch(meas{5}.Track3(:), psd_window, [], [], Fs);
  [pxx54, f54] = pwelch(meas{5}.Track4(:), psd_window, [], [], Fs);
#+end_src

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f51, sqrt(pxx51)./abs(squeeze(freqresp(L4C_G, f51, 'Hz'))));
  plot(f52, sqrt(pxx52)./abs(squeeze(freqresp(L4C_G, f52, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  legend({meas{5}.Track1_Name, meas{5}.Track2_Name}, 'Location', 'northwest');
  hold off;
#+end_src

#+NAME: fig:meas5_z_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas5_z_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas5_z_psd
#+CAPTION: PSD of the ground and marble in the Z direction
#+RESULTS: fig:meas5_z_psd
[[file:figs/meas5_z_psd.png]]

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(f53, sqrt(pxx53)./abs(squeeze(freqresp(L4C_G, f53, 'Hz'))));
  plot(f54, sqrt(pxx54)./abs(squeeze(freqresp(L4C_G, f54, 'Hz'))));
  xlim([fmin, fmax]);
  xticks([1, 10, 100]);
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  legend({meas{5}.Track3_Name, meas{5}.Track4_Name}, 'Location', 'northwest');
  hold off;
#+end_src

#+NAME: fig:meas5_y_psd
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas5_y_psd.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas5_y_psd
#+CAPTION: PSD of the ground and marble in the Y direction
#+RESULTS: fig:meas5_y_psd
[[file:figs/meas5_y_psd.png]]

Then, instead of looking at the Power Spectral Density, we can try to estimate the transfer function from a ground motion to the motion of the marble.
The transfer functions are shown on figure [[fig:meas5_tf]] and the coherence on figure [[fig:meas5_coh]].

#+begin_src matlab :exports code :results silent
  [tfz, fz] = tfestimate(meas{5}.Track1(:), meas{5}.Track2(:), psd_window, [], [], Fs);
  [tfy, fy] = tfestimate(meas{5}.Track3(:), meas{5}.Track4(:), psd_window, [], [], Fs);
#+end_src

#+begin_src matlab :exports none :results silent
  figure;
  ax1 = subaxis(2,1,1);
  hold on;
  plot(fz, abs(tfz));
  plot(fy, abs(tfy));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Magnitude');
  hold off;
  ax2 = subaxis(2,1,2);
  hold on;
  plot(fz, 180/pi*angle(tfz));
  plot(fy, 180/pi*angle(tfy));
  set(gca,'xscale','log');
  yticks(-180:90:180);
  ylim([-180 180]);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  hold off;
  linkaxes([ax1,ax2],'x');
  xlim([fmin, fmax]);
  legend({'Z direction', 'Y direction'}, 'Location', 'southwest')
#+end_src

#+NAME: fig:meas5_tf
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas5_tf.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas5_tf
#+CAPTION: Transfer function estimation - measurement 5
#+RESULTS: fig:meas5_tf
[[file:figs/meas5_tf.png]]

#+begin_src matlab :exports code :results silent
  [cohz, fz] = mscohere(meas{5}.Track1(:), meas{5}.Track2(:), psd_window, [], [], Fs);
  [cohy, fy] = mscohere(meas{5}.Track3(:), meas{5}.Track4(:), psd_window, [], [], Fs);
#+end_src

#+begin_src matlab :exports none :results silent
  figure;
  hold on;
  plot(fz, cohz);
  plot(fy, cohy);
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Coherence');
  xlabel('Frequency [Hz]');
  xlim([fmin, fmax]);
  legend({'Z direction', 'Y direction'}, 'Location', 'southwest')
#+end_src

#+NAME: fig:meas5_coh
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/meas5_coh.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+LABEL: fig:meas5_coh
#+CAPTION: Coherence - measurement 5
#+RESULTS: fig:meas5_coh
[[file:figs/meas5_coh.png]]

#+begin_important
The marble seems to have a resonance at around 20Hz on the Y direction.
But the coherence is not good above 20Hz, so it is difficult to estimate resonances.
#+end_important