#+TITLE: SVD Control
:DRAWER:
#+STARTUP: overview

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

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

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#+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:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :mkdirp yes

#+PROPERTY: header-args:shell  :eval no-export

#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:

* Gravimeter - Simscape Model
** 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

** Simulink
#+begin_src matlab
  open('gravimeter.slx')
#+end_src

#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'gravimeter';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;

  G = linearize(mdl, io);
  G.InputName  = {'F1', 'F2', 'F3'};
  G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
#+end_src

The plant as 6 states as expected (2 translations + 1 rotation)

#+begin_src matlab :results output replace
  size(G)
#+end_src

#+RESULTS:
: State-space model with 4 outputs, 3 inputs, and 6 states.

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

  figure;
  for in_i = 1:3
      for out_i = 1:4
          subplot(4, 3, 3*(out_i-1)+in_i);
          plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
          set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      end
  end
#+end_src

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

#+name: fig:open_loop_tf
#+caption: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers
#+RESULTS:
[[file:figs/open_loop_tf.png]]

** Matlab Code                                                     :noexport:
#+begin_src matlab
  clc;
  % close all

  g = 100000;

  w0 = 2*pi*.5; % MinusK BM1 tablle
  l = 0.5; %[m]
  la = 1; %[m]
  h = 1.7; %[m]
  ha = 1.7;% %[m]
  m = 400; %[kg]
  k = 15e3;%[N/m]
  kv = k;
  kh = 15e3;
  I = 115;%[kg m^2]
          % c = 0.06;
          % l = 0.4719; %[m]
          % la = .477; %[m]
          % h = 1.8973; %[m]
          % ha = 1.9060;% %[m]
          % m = 98.1421; %[kg]
          % k = 14512;%[N/m]
          % I = 28.5372;%[kg m^2]
  cv = 0.03;
  ch = 0.03;

  %% System definition
  [Fr, x1, z1, x2, z2, wx, wz, x12, z12, PHIwx, PHIwz,xsum,zsum,xdelta,zdelta,rot]...
      = modelGeneration(m,I,k,h,ha,l,la,cv,ch,kv,kh);

  %% Bode options
  P = bodeoptions;
  P.FreqUnits = 'Hz';
  P.MagUnits = 'abs';
  P.MagScale = 'log';
  P.Grid = 'on';
  P.PhaseWrapping = 'on';
  P.Xlim = [1e-1,1e2];
  %P.PhaseVisible = 'off';
  w = 2*pi*logspace(-1,2,1000);

  %% curves points
  % slide 4
  F_sl4 = [2e-1 4e-1 7e-1 1 2 3 5];
  Amp_sl4 = [ 1 2 4 2.5 1 7e-1 7e-1];
  F_sl4_phase = [2e-1 4e-1 7e-1 1 ];
  Phase_sl4 = (180/pi).*[0 0 -0.5 -1.7];

  %slide 6
  F_sl6 = [2e-1 4e-1 1 2 3 5];
  Amp_sl6 = [1 1 6e-1 2e-1 3e-1 3e-1];
  F_sl6_phase = [2e-1 4e-1 1 ];
  Phase_sl6 = (180/pi).*[0 0 0 ];

  %slide 9
  F_sl9 = [2.5e-1 4e-1 6e-1 1 1.7 2.2 3 5 10];
  Amp_sl9 = [3 6 1 5e-1 1 2 7e-1 2.5e-1 7e-2];
  Phase_sl9 = (180/pi)*[0 -1 -pi 0 -1 -1.5 -pi -pi -pi];

  % slide 14
  F_sl14 = [ 2e-1 4e-1 6e-1 8e-1 1 2 3 5 10];
  Amp_sl14 = [9e-1 1.5 1.2 0.35 .3 1.2 .3 .1 5e-2];
  F_sl14_phase = [ 2e-1 4e-1 6e-1 8e-1 ];
  Phase_sl14 = (180/pi).*[0 0 -1.7 -2];

  %rotation
  F_rot = [1e-1 2e-1 4e-1 5e-1 7e-1 1 2 3 6.5 10 20];
  Amp_rot = [7e-8 2.2e-7 3e-7 1e-7 2e-8 9e-9 3e-8 9e-9 1e-9 4e-10 8e-11];

  %% Plots
  % %slide 3
  % figure
  % loglog(Fr,abs(x2).^.5,Fr,abs(x1).^.5,Fr,abs(xsum).^.5,Fr,abs(xdelta).^.5)
  % xlabel('Frequency [Hz]');ylabel('Acceleration [m/s^2/rtHz]')
  % legend('Top sensor','Bottom sensor','Half sum','Half difference');
  % title('Horizontal')
  % xlim([7e-2 1e1]);

  %slide 4
  figure
  subplot 211
  loglog(Fr, abs(x12)./abs(x1));hold on;
  loglog(F_sl4,Amp_sl4,'*');
  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
  title('X direction Top/bottom sensor');
  xlim([7e-2 1e1]);
  subplot 212
  semilogx(Fr, (180/pi).*angle(x12./abs(x1)));hold on;
  loglog(F_sl4_phase,Phase_sl4,'*');
  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
  xlim([7e-2 1e1]);

  %slide 6
  figure
  subplot 211
  loglog(Fr, abs(z12)./abs(z1));hold on;
  loglog(F_sl6,Amp_sl6,'*');
  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
  title('Z direction Top/bottom sensor');
  xlim([7e-2 1e1]);
  subplot 212
  semilogx(Fr, (180/pi).*angle(z12./abs(z1)));hold on;
  loglog(F_sl6_phase,Phase_sl6,'*');
  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
  xlim([7e-2 1e1]);ylim([-180 180]);

  % %slide 6
  % figure
  % loglog(Fr,abs(z2).^.5,Fr,abs(z1).^.5,Fr,abs(zsum).^.5,Fr,abs(zdelta).^.5)
  % xlabel('Frequency [Hz]');ylabel('Acceleration [m/s^2/rtHz]')
  % legend('Top sensor','Bottom sensor','Half sum','Half difference');
  % title('Vertical')
  % xlim([7e-2 1e1]);

  %slide 9
  figure
  subplot 211
  loglog(Fr, abs(PHIwx)./abs(wx));hold on;
  loglog(F_sl9,Amp_sl9,'*');
  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
  title('X direction bottom/ground sensor');
  xlim([7e-2 1e1]);
  ylim([0.01 10]);
  subplot 212
  semilogx(Fr, (180/pi).*angle(PHIwx./abs(wx)));hold on;
  loglog(F_sl9,Phase_sl9,'*');
  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
  xlim([7e-2 1e1]);

  % %slide 8
  % figure
  % loglog(Fr,abs(wx).^.5,Fr,abs(x1).^.5,'-.',Fr,abs(x2).^.5,'.');
  % grid on;xlabel('Frequency [Hz]');
  % ylabel('ASD [m/s^2/rtHz]');
  % xlim([7e-2 1e1]);
  % legend('Ground','Bottom sensor','Top sensor');
  % title('Horizontal');
  %
  % %slide 13
  % figure
  % loglog(Fr,abs(wz).^.5,Fr,abs(z1).^.5,'-.',Fr,abs(z2).^.5,'.');
  % grid on;xlabel('Frequency [Hz]');
  % ylabel('ASD [m/s^2/rtHz]');
  % xlim([7e-2 1e1]);
  % legend('Ground','Bottom sensor','Top sensor');
  % title('Vertical');

  %slide 14
  figure
  subplot 211
  loglog(Fr, abs(PHIwz)./abs(wz));hold on;
  loglog(F_sl14,Amp_sl14,'*');
  xlabel('Frequency [Hz]');ylabel('Amplitude [-]');
  title('Z direction bottom/ground sensor');
  xlim([7e-2 1e1]);
  ylim([0.01 10]);
  subplot 212
  semilogx(Fr, (180/pi).*angle(PHIwz./abs(wz)));hold on;
  loglog(F_sl14_phase,Phase_sl14,'*');
  xlabel('Frequency [Hz]');ylabel('Phase [deg]');
  xlim([7e-2 1e1]);

  %rotation
  figure
  loglog(Fr,abs(rot).^.5./((2*pi*Fr').^2),F_rot,Amp_rot,'*');
  xlabel('Frequency [Hz]');ylabel('Rotation [rad/rtHz]')
  xlim([7e-2 1e1]);
#+end_src

** Model Generation                                                :noexport:
#+begin_src matlab
  function [Fr, x1, z1, x2, z2, wx, wz, x12, z12, PHIwx, PHIwz,xsum,zsum,xdelta,zdelta,rot] = modelGeneration(m,I,k,h,ha,l,la,dampv,damph,kv,kh)
      %% generation of the state space model
      M = [m 0 0
           0 m 0
           0 0 I];

      %Jacobian of the bottom sensor
      Js1 = [1 0 h/2
             0 1 -l/2];
      %Jacobian of the top sensor
      Js2 = [1 0 -h/2
             0 1 0];

      %Jacobian of the actuators
      Ja = [1 0 ha/2 %Left horizontal actuator
                     %1 0 h/2 %Right horizontal actuator
            0 1 -la/2 %Left vertical actuator
            0 1 la/2]; %Right vertical actuator
      Jah = [1 0 ha/2];
      Jav = [0 1 -la/2 %Left vertical actuator
             0 1 la/2]; %Right vertical actuator
      Jta = Ja';
      Jtah = Jah';
      Jtav = Jav';
      K = kv*Jtav*Jav + kh*Jtah*Jah;
      C = dampv*kv*Jtav*Jav+damph*kh*Jtah*Jah;

      E = [1 0 0
           0 1 0
           0 0 1]; %projecting ground motion in the directions of the legs

      AA = [zeros(3) eye(3)
            -M\K -M\C];

      BB = [zeros(3,6)
            M\Jta M\(k*Jta*E)];

      CC = [[Js1;Js2] zeros(4,3);
            zeros(2,6)
            (Js1+Js2)./2 zeros(2,3)
            (Js1-Js2)./2 zeros(2,3)
            (Js1-Js2)./(2*h) zeros(2,3)];

      DD = [zeros(4,6)
            zeros(2,3) eye(2,3)
            zeros(6,6)];

      system_dec = ss(AA,BB,CC,DD);
      %input = three actuators and three ground motions
      %output = the bottom sensor; the top sensor; the ground motion; the half
      %sum; the half difference; the rotation

      %% Injecting ground motion in the system to have the output
      Fr = logspace(-2,3,1e3);
      w=2*pi*Fr*1i;
      %fit of the ground motion data in m/s^2/rtHz
      Fr_ground_x = [0.07 0.1 0.15 0.3 0.7 0.8 0.9 1.2 5 10];
      n_ground_x1 = [4e-7 4e-7 2e-6 1e-6 5e-7 5e-7 5e-7 1e-6 1e-5 3.5e-5];
      Fr_ground_v = [0.07 0.08 0.1 0.11 0.12 0.15 0.25 0.6 0.8 1 1.2 1.6 2 6 10];
      n_ground_v1 = [7e-7 7e-7 7e-7 1e-6 1.2e-6 1.5e-6 1e-6 9e-7 7e-7 7e-7 7e-7 1e-6 2e-6 1e-5 3e-5];

      n_ground_x = interp1(Fr_ground_x,n_ground_x1,Fr,'linear');
      n_ground_v = interp1(Fr_ground_v,n_ground_v1,Fr,'linear');
      % figure
      % loglog(Fr,abs(n_ground_v),Fr_ground_v,n_ground_v1,'*');
      % xlabel('Frequency [Hz]');ylabel('ASD [m/s^2 /rtHz]');
      % return

      %converting into PSD
      n_ground_x = (n_ground_x).^2;
      n_ground_v = (n_ground_v).^2;

      %Injecting ground motion in the system and getting the outputs
      system_dec_f = (freqresp(system_dec,abs(w)));
      PHI = zeros(size(Fr,2),12,12);
      for p = 1:size(Fr,2)
          Sw=zeros(6,6);
          Iact = zeros(3,3);
          Sw(4,4) = n_ground_x(p);
          Sw(5,5) = n_ground_v(p);
          Sw(6,6) = n_ground_v(p);
          Sw(1:3,1:3) = Iact;
          PHI(p,:,:) = (system_dec_f(:,:,p))*Sw(:,:)*(system_dec_f(:,:,p))';
      end
      x1 = PHI(:,1,1);
      z1 = PHI(:,2,2);
      x2 = PHI(:,3,3);
      z2 = PHI(:,4,4);
      wx = PHI(:,5,5);
      wz = PHI(:,6,6);
      x12 = PHI(:,1,3);
      z12 = PHI(:,2,4);
      PHIwx = PHI(:,1,5);
      PHIwz = PHI(:,2,6);
      xsum = PHI(:,7,7);
      zsum = PHI(:,8,8);
      xdelta = PHI(:,9,9);
      zdelta = PHI(:,10,10);
      rot = PHI(:,11,11);
#+end_src

* Stewart Platform - Simscape Model
** 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

** Jacobian
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
#+begin_src matlab
  open('stewart_platform/drone_platform_jacobian.slx');
#+end_src

#+begin_src matlab
  sim('drone_platform_jacobian');
#+end_src

#+begin_src matlab
  Aa = [a1.Data(1,:);
        a2.Data(1,:);
        a3.Data(1,:);
        a4.Data(1,:);
        a5.Data(1,:);
        a6.Data(1,:)]';

  Ab = [b1.Data(1,:);
        b2.Data(1,:);
        b3.Data(1,:);
        b4.Data(1,:);
        b5.Data(1,:);
        b6.Data(1,:)]';

  As = (Ab - Aa)./vecnorm(Ab - Aa);

  l = vecnorm(Ab - Aa)';

  J = [As' , cross(Ab, As)'];

  save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src

** Simscape Model
#+begin_src matlab
  open('stewart_platform/drone_platform.slx');
#+end_src

Definition of spring parameters
#+begin_src matlab
  kx = 50; % [N/m]
  ky = 50;
  kz = 50;

  cx = 0.025; % [Nm/rad]
  cy = 0.025;
  cz = 0.025;
#+end_src

We load the Jacobian.
#+begin_src matlab
  load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src

** Identification of the plant
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'drone_platform';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/u'],               1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;

  G = linearize(mdl, io);
  G.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
                  'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
#+end_src

There are 24 states (6dof for the bottom platform + 6dof for the top platform).
#+begin_src matlab :results output replace
  size(G)
#+end_src

#+RESULTS:
: State-space model with 6 outputs, 12 inputs, and 24 states.

#+begin_src matlab
  % G = G*blkdiag(inv(J), eye(6));
  % G.InputName  = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
  %                 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
#+end_src

Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
#+begin_src matlab
  Gx = G*blkdiag(eye(6), inv(J'));
  Gx.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
                   'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

  Gl = J*G;
  Gl.OutputName  = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
#+end_src

** Obtained Dynamics
#+begin_src matlab :exports none
  freqs = logspace(-1, 2, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
  plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
  plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-360:90:360]);

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

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

#+name: fig:stewart_platform_translations
#+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
#+RESULTS:
[[file:figs/stewart_platform_translations.png]]

#+begin_src matlab :exports none
  freqs = logspace(-1, 2, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
  plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
  plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-360:90:360]);

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

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

#+name: fig:stewart_platform_rotations
#+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
#+RESULTS:
[[file:figs/stewart_platform_rotations.png]]

#+begin_src matlab :exports none
  freqs = logspace(-1, 2, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', i)), freqs, 'Hz'))));
  end
  for i = 1:5
    for j = i+1:6
      plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', j)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', i)), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-360:90:360]);

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

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

#+name: fig:stewart_platform_legs
#+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
#+RESULTS:
[[file:figs/stewart_platform_legs.png]]

#+begin_src matlab :exports none
  freqs = logspace(-1, 2, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
  plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
  plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - Translations');  xlabel('Frequency [Hz]');
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
  plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
  plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - Rotations');  xlabel('Frequency [Hz]');
  legend('location', 'northeast');

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

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

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

** Real Approximation of $G$ at the decoupling frequency
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
#+begin_src matlab
  wc = 2*pi*20; % Decoupling frequency [rad/s]
  Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
#+end_src

#+begin_src matlab
  H1 = evalfr(Gc, j*wc);
#+end_src

The real approximation is computed as follows:
#+begin_src matlab
  D = pinv(real(H1'*H1));
  H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
#+end_src

** Verification of the decoupling using the "Gershgorin Radii"
First, the Singular Value Decomposition of $H_1$ is performed:
\[ H_1 = U \Sigma V^H \]

#+begin_src matlab
  [U,S,V] = svd(H1);
#+end_src

Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
\[ G_d(s) = U^T G_c(s) V \]

It is done over the following frequencies.
#+begin_src matlab
  freqs = logspace(-1,2,1000); % [Hz]
#+end_src

#+begin_src matlab
  for i = 1:length(freqs)
      H = abs(evalfr(Gc, j*2*pi*freqs(i)));
      for j = 1:size(H,2)
          g_r1(i,j) =  (sum(H(j,:)) - H(j,j))/H(j,j);
      end
  end
#+end_src

#+begin_src matlab
  Gd = U'*Gc*V;
 
  for i  = 1:length(freqs)
      H_dec = abs(evalfr(Gd, j*2*pi*freqs(i)));
      for j = 1:size(H,2)
          g_r2(i,j) =  (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
      end
  end
#+end_src

#+begin_src matlab :exports results
  figure;
  hold on;
  plot(freqs, g_r1(:,1), 'DisplayName', '$a_x$')
  plot(freqs, g_r1(:,2), 'DisplayName', '$a_y$')
  plot(freqs, g_r1(:,3), 'DisplayName', '$a_z$')
  plot(freqs, g_r1(:,4), 'DisplayName', '$a_{R_x}$')
  plot(freqs, g_r1(:,5), 'DisplayName', '$a_{R_y}$')
  plot(freqs, g_r1(:,6), 'DisplayName', '$a_{R_z}$')
  plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  hold off;
  xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
#+end_src

#+begin_src matlab :exports results
  figure;
  hold on;
  plot(freqs, g_r2(:,1), 'DisplayName', '$a_x$')
  plot(freqs, g_r2(:,2), 'DisplayName', '$a_y$')
  plot(freqs, g_r2(:,3), 'DisplayName', '$a_z$')
  plot(freqs, g_r2(:,4), 'DisplayName', '$a_{R_x}$')
  plot(freqs, g_r2(:,5), 'DisplayName', '$a_{R_y}$')
  plot(freqs, g_r2(:,6), 'DisplayName', '$a_{R_z}$')
  plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
#+end_src

** Decoupled Plant
Let's see the bode plot of the decoupled plant $G_d(s)$.
\[ G_d(s) = U^T G_c(s) V \]

#+begin_src matlab :exports results
  freqs = logspace(-1, 2, 1000);

  figure;
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gd(i, i), freqs, 'Hz'))), ...
         'DisplayName', sprintf('$G(%i, %i)$', i, i));
  end
  for i = 1:5
    for j = i+1:6
      plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
           'HandleVisibility', 'off');
    end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude'); xlabel('Frequency [Hz]');
  legend('location', 'southeast');
#+end_src

** Diagonal Controller
The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.

#+begin_src matlab
  wc = 2*pi*0.1; % Crossover Frequency [rad/s]
  C_g = 50; % DC Gain

  K = eye(6)*C_g/(s+wc);
#+end_src

** Centralized Control
The control diagram for the centralized control is shown below.

The controller $K_c$ is "working" in an cartesian frame.
The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.

#+begin_src latex :file centralized_control.pdf
  \begin{tikzpicture}
    \node[block={2cm}{1.5cm}] (G) {$G$};
    \node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
    \node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$};

    % Inputs of the controllers
    \coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
    \coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);

    % Connections and labels
    \draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
    \draw[->] (G.east) -- ++(2.0, 0)  node[above left]{$a$};
    \draw[->] ($(G.east)+(1.4, 0)$)node[branch]{} |- (K.east);
    \draw[->] (K.west) -- (J.east) node[above right]{$\mathcal{F}$};
    \draw[->] (J.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
  \end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/centralized_control.png]]

#+begin_src matlab
  G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
#+end_src

** SVD Control
The SVD control architecture is shown below.
The matrices $U$ and $V$ are used to decoupled the plant $G$.

#+begin_src latex :file svd_control.pdf
  \begin{tikzpicture}
    \node[block={2cm}{1.5cm}] (G) {$G$};
    \node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$};
    \node[block, below=0.6 of G] (K) {$K_{\text{SVD}}$};
    \node[block, below left= 0.6 and 0 of G] (V) {$V^{-T}$};

    % Inputs of the controllers
    \coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
    \coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);

    % Connections and labels
    \draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
    \draw[->] (G.east) -- ++(2.5, 0) node[above left]{$a$};
    \draw[->] ($(G.east)+(2.0, 0)$) node[branch]{} |- (U.east);
    \draw[->] (U.west) -- (K.east);
    \draw[->] (K.west) -- (V.east);
    \draw[->] (V.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
  \end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/svd_control.png]]

SVD Control
#+begin_src matlab
  G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
#+end_src

** Results
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].

#+begin_src matlab :exports results
  freqs = logspace(-3, 3, 1000);

  figure

  ax1 = subplot(2, 3, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(    'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
  plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
  plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - $D_x/D_{w,x}$');  xlabel('Frequency [Hz]');
  legend('location', 'southwest');

  ax2 = subplot(2, 3, 2);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(    'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - $D_y/D_{w,y}$');  xlabel('Frequency [Hz]');

  ax3 = subplot(2, 3, 3);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(    'Az', 'Dwz')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - $D_z/D_{w,z}$');  xlabel('Frequency [Hz]');

  ax4 = subplot(2, 3, 4);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(    'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - $R_x/R_{w,x}$');  xlabel('Frequency [Hz]');

  ax5 = subplot(2, 3, 5);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(    'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - $R_y/R_{w,y}$');  xlabel('Frequency [Hz]');

  ax6 = subplot(2, 3, 6);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(    'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Transmissibility - $R_z/R_{w,z}$');  xlabel('Frequency [Hz]');

  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
  xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'width', 1600, 'height', 'full');
#+end_src

#+name: fig:stewart_platform_simscape_cl_transmissibility
#+caption: Obtained Transmissibility
#+RESULTS:
[[file:figs/stewart_platform_simscape_cl_transmissibility.png]]

* Stewart Platform - Analytical Model
** 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

#+begin_src matlab
  %% Bode plot options
  opts = bodeoptions('cstprefs');
  opts.FreqUnits = 'Hz';
  opts.MagUnits = 'abs';
  opts.MagScale = 'log';
  opts.PhaseWrapping = 'on';
  opts.xlim = [1 1000];
#+end_src

** Characteristics
#+begin_src matlab
  L  = 0.055;
  Zc = 0;
  m  = 0.2;
  k  = 1e3;
  c  = 2*0.1*sqrt(k*m);

  Rx = 0.04;
  Rz = 0.04;
  Ix = m*Rx^2;
  Iy = m*Rx^2;
  Iz = m*Rz^2;
#+end_src

** Mass Matrix
#+begin_src matlab
  M = m*[1 0 0 0 Zc 0;
         0 1 0 -Zc 0 0;
         0 0 1 0 0 0;
         0 -Zc 0 Rx^2+Zc^2 0 0;
         Zc 0 0 0 Rx^2+Zc^2 0;
         0 0 0 0 0 Rz^2];
#+end_src

** Jacobian Matrix
#+begin_src matlab
  Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
                 sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
                 sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
                 0 0 L L -L -L;
                 -L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
                 L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
#+end_src

** Stifnness matrix and Damping matrix
#+begin_src matlab
  kv = k/3; % [N/m]
  kh = 0.5*k/3; % [N/m]

  K = diag([3*kh,3*kh,3*kv,3*kv*Rx^2/2,3*kv*Rx^2/2,3*kh*Rx^2]); % Stiffness Matrix

  C = c*K/100000; % Damping Matrix
#+end_src

** State Space System
#+begin_src matlab
  A  = [zeros(6) eye(6); -M\K -M\C];
  Bw = [zeros(6); -eye(6)];
  Bu = [zeros(6); M\Bj];
  Co = [-M\K -M\C];
  D  = [zeros(6) M\Bj];
 
  ST = ss(A,[Bw Bu],Co,D);
#+end_src

- OUT 1-6: 6 dof
- IN 1-6 : ground displacement in the directions of the legs
- IN 7-12: forces in the actuators.
#+begin_src matlab
  ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
                  'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
  ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
                  'u1';'u2';'u3';'u4';'u5';'u6'};
  ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
#+end_src

** Transmissibility
#+begin_src matlab
  TR=ST*[eye(6); zeros(6)];
#+end_src

#+begin_src matlab
  figure
  subplot(231)
  bodemag(TR(1,1),opts);
  subplot(232)
  bodemag(TR(2,2),opts);
  subplot(233)
  bodemag(TR(3,3),opts);
  subplot(234)
  bodemag(TR(4,4),opts);
  subplot(235)
  bodemag(TR(5,5),opts);
  subplot(236)
  bodemag(TR(6,6),opts);
#+end_src

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

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

** Real approximation of $G(j\omega)$ at decoupling frequency
#+begin_src matlab
  sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs

  dec_fr = 20;
  H1 = evalfr(sys1,j*2*pi*dec_fr);
  H2 = H1;
  D = pinv(real(H2'*H2));
  H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
  [U,S,V] = svd(H1);

  wf = logspace(-1,2,1000);
  for i  = 1:length(wf)
      H = abs(evalfr(sys1,j*2*pi*wf(i)));
      H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
      for j = 1:size(H,2)
          g_r1(i,j) =  (sum(H(j,:))-H(j,j))/H(j,j);
          g_r2(i,j) =  (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
          %     keyboard
      end
      g_lim(i) = 0.5;
  end
#+end_src

** Coupled and Decoupled Plant "Gershgorin Radii"
#+begin_src matlab
  figure;
  title('Coupled plant')
  loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
  legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
  xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
#+end_src

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

#+name: fig:gershorin_raddii_coupled_analytical
#+caption: Gershorin Raddi for the coupled plant
#+RESULTS:
[[file:figs/gershorin_raddii_coupled_analytical.png]]

#+begin_src matlab
  figure;
  title('Decoupled plant (10 Hz)')
  loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
  legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
  xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
#+end_src

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

#+name: fig:gershorin_raddii_decoupled_analytical
#+caption: Gershorin Raddi for the decoupled plant
#+RESULTS:
[[file:figs/gershorin_raddii_decoupled_analytical.png]]

** Decoupled Plant
#+begin_src matlab
  figure;
  bodemag(U'*sys1*V,opts)
#+end_src

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

#+name: fig:stewart_platform_analytical_decoupled_plant
#+caption: Decoupled Plant
#+RESULTS:
[[file:figs/stewart_platform_analytical_decoupled_plant.png]]

** Controller
#+begin_src matlab
  fc = 2*pi*0.1; % Crossover Frequency [rad/s]
  c_gain = 50; %

  cont = eye(6)*c_gain/(s+fc);
#+end_src

** Closed Loop System
#+begin_src matlab
  FEEDIN  = [7:12]; % Input of controller
  FEEDOUT = [1:6]; % Output of controller
#+end_src

Centralized Control
#+begin_src matlab
  STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
  TRcen = STcen*[eye(6); zeros(6)];
#+end_src

SVD Control
#+begin_src matlab
  STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
  TRsvd = STsvd*[eye(6); zeros(6)];
#+end_src

** Results
#+begin_src matlab
  figure
  subplot(231)
  bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
  legend('OL','Centralized','SVD')
  subplot(232)
  bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
  legend('OL','Centralized','SVD')
  subplot(233)
  bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
  legend('OL','Centralized','SVD')
  subplot(234)
  bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
  legend('OL','Centralized','SVD')
  subplot(235)
  bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
  legend('OL','Centralized','SVD')
  subplot(236)
  bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
  legend('OL','Centralized','SVD')
#+end_src

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

#+name: fig:stewart_platform_analytical_svd_cen_comp
#+caption: Comparison of the obtained transmissibility for the centralized control and the SVD control
#+RESULTS:
[[file:figs/stewart_platform_analytical_svd_cen_comp.png]]
