Update Noise measurement analysis

This includes normalization with transfer function of the geophone

2018-10-15 - Marc/index.org

@@ -23,25 +23,22 @@
 
 [[../index.org][Back to main page]].
 
-* Measurement Description
+#+begin_src matlab :exports none :results silent
+  <<matlab-init>>
+#+end_src
 
+* Measurement Description
 #+name: fig:setup_picture
 #+caption: Picture of the setup for the measurement
-#+attr_latex: :scale 1
 [[file:./figs/setup_picture.png]]
 
-Sensors:
-- 3 L-4C ([[file:../actuators-sensors/index.org::*L-4C][Documentation]]) [[file:~/MEGA/These/Measurements/actuators-sensors/index.org::tab:L4C][table]]
+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
-#+begin_src matlab :exports none :results silent
-  <<matlab-init>>
-#+end_src
-
 First, load all the measurement files:
 #+begin_src matlab :exports code :results silent
   meas = {};
@@ -94,10 +91,97 @@ For the measurement 5, the channels are shown table [[tab:meas_5]].
 |---------+------------------+-------------------+------------------+-------------------|
 | 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';
@@ -115,39 +199,63 @@ When using two geophone sensors on the same tilt stage (measurements 1 and 2), w
   meas{2}.Track2_Name = 'Ry Tilt';
 #+end_src
 
-* Variables for analysis
-We define the sampling frequency and the time vectors for the plots.
+* 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]].
 
-#+begin_src matlab :exports code :results silent
-  Fs = 256; % [Hz]
-  dt = 1/(Fs);
-  t1  = dt*[1:length(meas{1}.Track1)];
-  t2  = dt*[1:length(meas{2}.Track1)];
-  t3  = dt*[1:length(meas{3}.Track1)];
-  t4  = dt*[1:length(meas{4}.Track1)];
-  t5  = dt*[1:length(meas{5}.Track1)];
+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
 
-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]
+#+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
 
-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);
+#+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]]
+
+Time domain data are just normalized using the sensibility of the sensor ($276.8 V/m/s$).
+
 * 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      |
+| 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]]).
 
@@ -159,9 +267,9 @@ We then plot the measurements in time domain (figure [[fig:meas1]]).
 #+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)));
+  plot(t1(ceil(300/dt):ceil(340/dt)), (1/276.8).*meas{1}.Track1(ceil(300/dt):ceil(340/dt)));
+  plot(t1(ceil(300/dt):ceil(340/dt)), (1/276.8).*meas{1}.Track2(ceil(300/dt):ceil(340/dt)));
+  plot(t1(ceil(300/dt):ceil(340/dt)), (1/276.8).*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')
@@ -181,7 +289,7 @@ We then plot the measurements in time domain (figure [[fig:meas1]]).
 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.
 #+begin_src matlab :exports code :results silent
   tdisp = t1(ceil(300/dt):ceil(340/dt));
-  xdisp = cumtrapz(tdisp, meas{1}.Track3(ceil(300/dt):ceil(340/dt)));
+  xdisp = cumtrapz(tdisp, (1/276.8).*meas{1}.Track3(ceil(300/dt):ceil(340/dt)));
 #+end_src
 
 Then we plot the position with respect to time (figure [[fig:meas1_disp]]).
@@ -206,14 +314,14 @@ Then we plot the position with respect to time (figure [[fig:meas1_disp]]).
 
 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, f111] = pwelch(meas{1}.Track1(1:ceil(300/dt)),   psd_window, [], [], Fs);
-  [pxx112, f112] = pwelch(meas{1}.Track1(ceil(350/dt):end), psd_window, [], [], Fs);
+  [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, f121] = pwelch(meas{1}.Track2(1:ceil(300/dt)),   psd_window, [], [], Fs);
-  [pxx122, f122] = pwelch(meas{1}.Track2(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, f131] = pwelch(meas{1}.Track3(1:ceil(300/dt)),   psd_window, [], [], Fs);
-  [pxx132, f132] = pwelch(meas{1}.Track3(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]]).
@@ -221,8 +329,8 @@ We finally plot the power spectral density of each track (figures [[fig:meas1_ry
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f111, sqrt(pxx111));
-  plot(f112, sqrt(pxx112));
+  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');
@@ -246,8 +354,8 @@ We finally plot the power spectral density of each track (figures [[fig:meas1_ry
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f121, sqrt(pxx121));
-  plot(f122, sqrt(pxx122));
+  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');
@@ -271,8 +379,8 @@ We finally plot the power spectral density of each track (figures [[fig:meas1_ry
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f131, sqrt(pxx131));
-  plot(f132, sqrt(pxx132));
+  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');
@@ -303,26 +411,30 @@ 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 |
-|------+-------+---------|
-| Tilt |   OFF | ON      |
+| 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)));
+  plot(t2(ceil(300/dt):ceil(350/dt)), (1/276.8).*meas{2}.Track1(ceil(300/dt):ceil(350/dt)));
+  plot(t2(ceil(300/dt):ceil(350/dt)), (1/276.8).*meas{2}.Track3(ceil(300/dt):ceil(350/dt)));
+  plot(t2(ceil(300/dt):ceil(350/dt)), (1/276.8).*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
@@ -338,21 +450,21 @@ We plot the time domain (figure [[fig:meas2]]) and we don't observe anything spe
 
 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, f211] = pwelch(meas{2}.Track1(1:ceil(326/dt)),   psd_window, [], [], Fs);
-  [pxx212, f212] = pwelch(meas{2}.Track1(ceil(326/dt):end), psd_window, [], [], Fs);
+  [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, f221] = pwelch(meas{2}.Track2(1:ceil(326/dt)),   psd_window, [], [], Fs);
-  [pxx222, f222] = pwelch(meas{2}.Track2(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, f231] = pwelch(meas{2}.Track3(1:ceil(326/dt)),   psd_window, [], [], Fs);
-  [pxx232, f232] = pwelch(meas{2}.Track3(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(f211, sqrt(pxx211));
-  plot(f212, sqrt(pxx212));
+  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');
@@ -376,8 +488,8 @@ We compute the PSD of each track and we plot them (figures [[fig:meas2_ry_z_psd]
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f221, sqrt(pxx221));
-  plot(f222, sqrt(pxx222));
+  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');
@@ -401,8 +513,8 @@ We compute the PSD of each track and we plot them (figures [[fig:meas2_ry_z_psd]
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f231, sqrt(pxx231));
-  plot(f232, sqrt(pxx232));
+  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');
@@ -428,24 +540,26 @@ We compute the PSD of each track and we plot them (figures [[fig:meas2_ry_z_psd]
 #+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 |
-|------+-------+---------|
-| Tilt | ON    | ON      |
-| Hexa | ON    | OFF     |
+| 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)));
+  plot(t3(ceil(380/dt):ceil(420/dt)), (1/276.8).*meas{3}.Track1(ceil(380/dt):ceil(420/dt)));
+  plot(t3(ceil(380/dt):ceil(420/dt)), (1/276.8).*meas{3}.Track2(ceil(380/dt):ceil(420/dt)));
+  plot(t3(ceil(380/dt):ceil(420/dt)), (1/276.8).*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')
@@ -464,21 +578,21 @@ The time domain result is shown figure [[fig:meas3]].
 
 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, f311] = pwelch(meas{3}.Track1(1:ceil(400/dt)),   psd_window, [], [], Fs);
-  [pxx312, f312] = pwelch(meas{3}.Track1(ceil(420/dt):end), psd_window, [], [], Fs);
+  [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, f321] = pwelch(meas{3}.Track2(1:ceil(400/dt)),   psd_window, [], [], Fs);
-  [pxx322, f322] = pwelch(meas{3}.Track2(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, f331] = pwelch(meas{3}.Track3(1:ceil(400/dt)),   psd_window, [], [], Fs);
-  [pxx332, f332] = pwelch(meas{3}.Track3(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(f311, sqrt(pxx311));
-  plot(f312, sqrt(pxx312));
+  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');
@@ -502,8 +616,8 @@ We then compute the PSD of each track before and after turning off the hexapod a
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f321, sqrt(pxx321));
-  plot(f322, sqrt(pxx322));
+  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');
@@ -527,8 +641,8 @@ We then compute the PSD of each track before and after turning off the hexapod a
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f331, sqrt(pxx331));
-  plot(f332, sqrt(pxx332));
+  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');
@@ -556,23 +670,24 @@ Turning ON induces some motion on the hexapod in the z direction (figure [[fig:m
 #+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 |
 |----------+-------+---------+---------|
-| SlipRing | OFF   | ON      | ON      |
+| Ty       | OFF   | OFF     | OFF     |
+| Ry       | OFF   | OFF     | OFF     |
+| SlipRing | OFF   | *ON*    | *ON*    |
+| Spindle  | OFF   | OFF     | *ON*    |
 | Hexa     | OFF   | OFF     | OFF     |
-| Spindle  | OFF   | OFF     | ON      |
 
 #+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);
+  plot(t4, (1/276.8).*meas{4}.Track1);
+  plot(t4, (1/276.8).*meas{4}.Track2);
+  plot(t4, (1/276.8).*meas{4}.Track3);
   hold off;
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
   legend({meas{4}.Track1_Name, meas{4}.Track2_Name, meas{4}.Track3_Name}, 'Location', 'southwest')
@@ -591,25 +706,25 @@ The slip ring is turned on at 300s, then the spindle is turned on at 620s (table
 
 The PSD of each track are computed using the code below.
 #+begin_src matlab :exports none :results silent
-  [pxx411, f411] = pwelch(meas{4}.Track1(1:ceil(280/dt)),            psd_window, [], [], Fs);
-  [pxx412, f412] = pwelch(meas{4}.Track1(ceil(280/dt):ceil(600/dt)), psd_window, [], [], Fs);
-  [pxx413, f413] = pwelch(meas{4}.Track1(ceil(640/dt):end),          psd_window, [], [], Fs);
+  [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, f421] = pwelch(meas{4}.Track2(1:ceil(280/dt)),            psd_window, [], [], Fs);
-  [pxx422, f422] = pwelch(meas{4}.Track2(ceil(280/dt):ceil(600/dt)), psd_window, [], [], Fs);
-  [pxx423, f423] = pwelch(meas{4}.Track2(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, f431] = pwelch(meas{4}.Track3(1:ceil(280/dt)),            psd_window, [], [], Fs);
-  [pxx432, f432] = pwelch(meas{4}.Track3(ceil(280/dt):ceil(600/dt)), psd_window, [], [], Fs);
-  [pxx433, f433] = pwelch(meas{4}.Track3(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);
 #+end_src
 
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f411, sqrt(pxx411));
-  plot(f412, sqrt(pxx412));
-  plot(f413, sqrt(pxx413));
+  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');
@@ -633,9 +748,9 @@ The PSD of each track are computed using the code below.
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f421, sqrt(pxx421));
-  plot(f422, sqrt(pxx422));
-  plot(f423, sqrt(pxx423));
+  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');
@@ -660,9 +775,9 @@ The PSD of each track are computed using the code below.
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f431, sqrt(pxx431));
-  plot(f432, sqrt(pxx432));
-  plot(f433, sqrt(pxx433));
+  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');
@@ -703,10 +818,10 @@ 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);
+  plot(t5, (1/276.8).*meas{5}.Track1);
+  plot(t5, (1/276.8).*meas{5}.Track2);
+  plot(t5, (1/276.8).*meas{5}.Track3);
+  plot(t5, (1/276.8).*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')
@@ -735,8 +850,8 @@ We compute the PSD of each track and we plot the PSD of the Z motion for the gro
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f51, sqrt(pxx51));
-  plot(f52, sqrt(pxx52));
+  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');
@@ -759,8 +874,8 @@ We compute the PSD of each track and we plot the PSD of the Z motion for the gro
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(f53, sqrt(pxx53));
-  plot(f54, sqrt(pxx54));
+  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');