nass-simscape/org/control_virtual_mass.org

#+TITLE: Decentralize control to add virtual mass
#+SETUPFILE: ./setup/org-setup-file.org

* 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 :tangle no
  simulinkproject('../');
#+end_src

#+begin_src matlab
  load('mat/conf_simulink.mat');
  open('nass_model.slx')
#+end_src

* Initialization
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

  initializeSimscapeConfiguration();
  initializeDisturbances('enable', false);
  initializeLoggingConfiguration('log', 'none');

  initializeController('type', 'hac-dvf');
#+end_src

The nano-hexapod has the following leg's stiffness and damping.
#+begin_src matlab
  initializeNanoHexapod('k', 1e5, 'c', 2e2);
#+end_src

We set the stiffness of the payload fixation:
#+begin_src matlab
  Kp = 1e8; % [N/m]
#+end_src

#+begin_src matlab :exports none
  K = tf(zeros(6));
  Kdvf = tf(zeros(6));
#+end_src

* Identification
We identify the system for the following payload masses:
#+begin_src matlab
  Ms = [1, 10, 50];
#+end_src

Identification of the transfer function from $\tau$ to $d\mathcal{L}$.
#+begin_src matlab :exports none
  Gm = {zeros(length(Ms), 1)};

  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],    1, 'openinput');               io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm');  io_i = io_i + 1;

  for i = 1:length(Ms)
    initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
    initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));

    %% Run the linearization
    G = linearize(mdl, io);
    G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
    G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
    Gm(i) = {G};
  end
#+end_src

Identification of the Primary plant without virtual add of mass
#+begin_src matlab :exports none
  G_x = {zeros(length(Ms), 1)};
  G_l = {zeros(length(Ms), 1)};

  load('mat/stages.mat', 'nano_hexapod');

  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],     1, 'input');            io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror

  for i = 1:length(Ms)
      initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
      initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));

      %% Run the linearization
      G = linearize(mdl, io);
      G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

      Gx = -G*inv(nano_hexapod.J');
      Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
      G_x(i) = {Gx};

      Gl = -nano_hexapod.J*G;
      Gl.OutputName  = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
      G_l(i) = {Gl};
  end
#+end_src

* Adding Virtual Mass in the Leg's Space
** Plant
#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz'))));
  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:length(Ms)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz')))), ...
         'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
  legend('location', 'northeast');

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

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

#+name: fig:virtual_mass_plant_L
#+caption: Transfer function from $\tau_i$ to $d\mathcal{L}_i$ for three payload masses
#+RESULTS:
[[file:figs/virtual_mass_plant_L.png]]

** Controller Design
#+begin_src matlab
  Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6);
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ms)
      isstable(feedback(Gm{i}*Kdvf, eye(6), -1))
  end
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(-1, 4, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(1,1), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:length(Ms)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(1,1), freqs, 'Hz')))), ...
         'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-360:90:360]);
  legend('location', 'northeast');

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

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

#+name: fig:virtual_mass_loop_gain_L
#+caption: Loop Gain for the addition of virtual mass in the leg's space
#+RESULTS:
[[file:figs/virtual_mass_loop_gain_L.png]]

** Identification of the Primary Plant
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],     1, 'input');            io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror

  load('mat/stages.mat', 'nano_hexapod');

  GmL_x = {zeros(length(Ms), 1)};
  GmL_l = {zeros(length(Ms), 1)};

  for i = 1:length(Ms)
      initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
      initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));

      %% Run the linearization
      G = linearize(mdl, io);
     G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

      Gx = -G*inv(nano_hexapod.J');
      Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
      GmL_x(i) = {Gx};

      Gl = -nano_hexapod.J*G;
      Gl.OutputName  = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
      GmL_l(i) = {Gl};
  end
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmL_x{i}(1, 1), freqs, 'Hz'))), '--');
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmL_x{i}(2, 2), freqs, 'Hz'))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')

  ax2 = subplot(2, 2, 2);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmL_x{i}(3, 3), freqs, 'Hz'))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('$\mathcal{X}_z/\mathcal{F}_z$')

  ax3 = subplot(2, 2, 3);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz')))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz')))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(1, 1), freqs, 'Hz')))), '--');
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(2, 2), freqs, 'Hz')))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(3, 3), freqs, 'Hz')))), '--', ...
           'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

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

#+name: fig:virtual_mass_L_primary_plant_X
#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the leg's space
#+RESULTS:
[[file:figs/virtual_mass_L_primary_plant_X.png]]

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmL_l{i}(1, 1), freqs, 'Hz'))), '--');
  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:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_l{i}(1, 1), freqs, 'Hz')))), '--', ...
           'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
  legend('location', 'southwest');

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

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

#+name: fig:virtual_mass_L_primary_plant_L
#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the leg's space
#+RESULTS:
[[file:figs/virtual_mass_L_primary_plant_L.png]]

* Adding Virtual Mass in the Task Space
** Plant
Let's look at the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$:
\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \]

#+begin_src matlab :exports none
  load('mat/stages.mat', 'nano_hexapod');

  GmX = {zeros(length(Ms), 1)};
  for i = 1:length(Ms)
      GmX(i) = {inv(nano_hexapod.J) * Gm{i} * inv(nano_hexapod.J')};
  end
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmX{i}(1, 1), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmX{i}(2, 2), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 2, 3);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(1, 1), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(2, 2), freqs, 'Hz')))), ...
           'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
  legend('location', 'northeast');

  ax3 = subplot(2, 2, 2);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmX{i}(3, 3), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(3, 3), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
  legend('location', 'northeast');

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

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

#+name: fig:virtual_mass_plant_X
#+caption: Dynamics from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ used for virtual mass addition in the task space
#+RESULTS:
[[file:figs/virtual_mass_plant_X.png]]

** Controller Design
#+begin_src matlab
  KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0]));
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ms)
      isstable(feedback(GmX{i}*KmX, eye(6), -1))
  end
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:length(Ms)
      LmX = GmX{i}*KmX;
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(LmX(1, 1), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(LmX(2, 2), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);

  ax3 = subplot(2, 2, 3);
  hold on;
  for i = 1:length(Ms)
      LmX = GmX{i}*KmX;
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(1, 1), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(2, 2), freqs, 'Hz')))), ...
           'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-360:90:360]);
  legend('location', 'southwest');

  ax2 = subplot(2, 2, 2);
  hold on;
  for i = 1:length(Ms)
      LmX = GmX{i}*KmX;
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(LmX(3, 3), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:length(Ms)
      LmX = GmX{i}*KmX;
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(3, 3), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-360:90:360]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

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

#+name: fig:virtual_mass_loop_gain_X
#+caption: Loop gain for virtual mass addition in the task space
#+RESULTS:
[[file:figs/virtual_mass_loop_gain_X.png]]

#+begin_src matlab
  Kdvf = inv(nano_hexapod.J')*KmX*inv(nano_hexapod.J);
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ms)
      isstable(feedback(Gm{i}*Kdvf, eye(6), -1))
  end
#+end_src

** Identification of the Primary Plant
#+begin_src matlab :exports none
  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],     1, 'input');            io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror

  load('mat/stages.mat', 'nano_hexapod');

  GmX_x = {zeros(length(Ms), 1)};
  GmX_l = {zeros(length(Ms), 1)};

  for i = 1:length(Ms)
      initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
      initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));

      %% Run the linearization
      G = linearize(mdl, io);
      G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

      Gx = -G*inv(nano_hexapod.J');
      Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
      GmX_x(i) = {Gx};

      Gl = -nano_hexapod.J*G;
      Gl.OutputName  = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
      GmX_l(i) = {Gl};
  end
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmX_x{i}(1, 1), freqs, 'Hz'))), '--');
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmX_x{i}(2, 2), freqs, 'Hz'))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')

  ax2 = subplot(2, 2, 2);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmX_x{i}(3, 3), freqs, 'Hz'))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('$\mathcal{X}_z/\mathcal{F}_z$')

  ax3 = subplot(2, 2, 3);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz')))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz')))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(1, 1), freqs, 'Hz')))), '--');
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(2, 2), freqs, 'Hz')))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(3, 3), freqs, 'Hz')))), '--', ...
           'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
  legend('location', 'southwest');

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

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

#+name: fig:virtual_mass_X_primary_plant_X
#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the task space
#+RESULTS:
[[file:figs/virtual_mass_X_primary_plant_X.png]]

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz'))));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(GmX_l{i}(1, 1), freqs, 'Hz'))), '--');
  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:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))), ...
           'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_l{i}(1, 1), freqs, 'Hz')))), '--', ...
           'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
  legend('location', 'southwest');

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

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

#+name: fig:virtual_mass_X_primary_plant_L
#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the task space
#+RESULTS:
[[file:figs/virtual_mass_X_primary_plant_L.png]]