#+TITLE: Determination of the optimal nano-hexapod's stiffness
:DRAWER:
#+STARTUP: overview

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

#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>

#+HTML_MATHJAX: align: center tagside: right font: TeX

#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle ../matlab/optimal_stiffness.m
#+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/thesis/latex/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 raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
:END:

* Introduction                                                        :ignore:
As shown before, many parameters other than the nano-hexapod itself do influence the plant dynamics:
- The micro-station compliance (studied [[file:uncertainty_support.org][here]])
- The payload mass and dynamical properties (studied [[file:uncertainty_payload.org][here]] and [[file:uncertainty_experiment.org][here]])
- The experimental conditions, mainly the spindle rotation speed (studied [[file:uncertainty_experiment.org][here]])

As seen before, the stiffness of the nano-hexapod greatly influence the effect of such parameters.

We wish here to see if we can determine an optimal stiffness of the nano-hexapod such that:
- Section [[sec:spindle_rotation_speed]]: the change of its dynamics due to the spindle rotation speed is acceptable
- Section [[sec:micro_station_compliance]]: the support compliance dynamics is not much present in the nano-hexapod dynamics
- Section [[sec:payload_impedance]]: the change of payload impedance has acceptable effect on the plant dynamics

The overall goal is to design a nano-hexapod that will allow the highest possible control bandwidth.

* Spindle Rotation Speed
<<sec:spindle_rotation_speed>>

** Introduction                                                      :ignore:


** 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
We initialize all the stages with the default parameters.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
#+end_src

The worst case scenario is a rotation speed of 60rpm with a payload mass of 10Kg.
#+begin_src matlab
  initializeSample('mass', 10);
#+end_src

We don't include gravity nor disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics.
#+begin_src matlab
  initializeSimscapeConfiguration('gravity', true);
  initializeDisturbances('enable', false);
  initializeLoggingConfiguration('log', 'none');
  initializeController();
#+end_src

#+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, 'openinput');              io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'],  3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'],  3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En');   io_i = io_i + 1;
#+end_src

** Identification when rotating at maximum speed
#+begin_src matlab
  Rz_rpm = linspace(0, 60, 6);
#+end_src

#+begin_src matlab
  Ks = logspace(3,9,7); % [N/m]
#+end_src

#+begin_src matlab :exports none
  Gk_wz_iff = {zeros(length(Ks), length(Rz_rpm))};
  Gk_wz_dvf = {zeros(length(Ks), length(Rz_rpm))};
  Gk_wz_err = {zeros(length(Ks), length(Rz_rpm))};
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ks)
    for j = 1:length(Rz_rpm)
      initializeReferences('Rz_type', 'rotating-not-filtered', ...
                          'Rz_period', 60/Rz_rpm(j));
      initializeNanoHexapod('k', Ks(i));

      %% Run the linearization
      G = linearize(mdl, io);
      G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                      'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                      'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

      Gk_wz_iff(i,j) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
      Gk_wz_dvf(i,j) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};

      Jinvt = tf(inv(nano_hexapod.J)');
      Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
      Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      Gk_wz_err(i,j) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'},                {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt};
    end
  end
#+end_src

#+begin_src matlab :exports none
  save('mat/optimal_stiffness_Gk_wz.mat', 'Ks', 'Rz_rpm', ...
       'Gk_wz_iff', 'Gk_wz_dvf', 'Gk_wz_err');
#+end_src

** Change of dynamics
#+begin_src matlab :exports none
  load('mat/optimal_stiffness_Gk_wz.mat');
#+end_src

Change of dynamics for IFF
#+begin_src matlab :exports none
  freqs = logspace(-2, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
      for j = 1:length(Rz_rpm)
          set(gca,'ColorOrderIndex',i);
          plot(freqs, abs(squeeze(freqresp(Gk_wz_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:length(Ks)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_iff{i,1}('Fnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
      for j = 2:length(Rz_rpm)
          set(gca,'ColorOrderIndex',i);
          plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
               'HandleVisibility', 'off');
      end
  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');
  xlim([freqs(1), freqs(end)]);
#+end_src

Change of dynamics for DVF
#+begin_src matlab :exports none
  freqs = logspace(-2, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
      for j = 1:length(Rz_rpm)
          set(gca,'ColorOrderIndex',i);
          plot(freqs, abs(squeeze(freqresp(Gk_wz_dvf{i,j}('Dnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
      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:length(Ks)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_dvf{i,1}('Dnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
      for j = 2:length(Rz_rpm)
          set(gca,'ColorOrderIndex',i);
          plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_dvf{i,j}('Dnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
               'HandleVisibility', 'off');
      end
  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');
  xlim([freqs(1), freqs(end)]);
#+end_src

Change of dynamics from $F_x$ to $D_x$.
#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
      for j = 1:length(Rz_rpm)
          set(gca,'ColorOrderIndex',i);
          plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-');
      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:length(Ks)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_err{i,1}('Ex', 'Fx'), freqs, 'Hz')))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
      for j = 2:length(Rz_rpm)
          set(gca,'ColorOrderIndex',i);
          plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), '-', ...
               'HandleVisibility', 'off');
      end
  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');
  xlim([freqs(1), freqs(end)]);
#+end_src

Change of coupling from $F_x$ to $D_y$ with the rotating speed.
#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  hold on;
  for i = 1:length(Ks)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i,1}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
      for j = 1:length(Rz_rpm)
          set(gca,'ColorOrderIndex',i);
          plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i,j}('Ey', 'Fx'), freqs, 'Hz'))), '--', ...
           'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);
  legend('location', 'northeast');
#+end_src

** Conclusion                                                        :ignore:
#+begin_important
  The leg stiffness should be at higher than $k_i = 10^4\ [N/m]$ such that the main resonance frequency does not shift too much when rotating.
  For the coupling, it is more difficult to conclude about the minimum required leg stiffness.
#+end_important

#+begin_notes
  Note that we can use very soft nano-hexapod if we limit the spindle rotating speed.
#+end_notes

* Micro-Station Compliance Effect
<<sec:micro_station_compliance>>

** Introduction                                                      :ignore:
- take the 6dof compliance of the micro-station
- simple model + uncertainty

** 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

** Identification of the micro-station compliance
We initialize all the stages with the default parameters.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod('type', 'compliance');
#+end_src

We put nothing on top of the micro-hexapod.
#+begin_src matlab
  initializeAxisc('type', 'none');
  initializeMirror('type', 'none');
  initializeNanoHexapod('type', 'none');
  initializeSample('type', 'none');
#+end_src

#+begin_src matlab :exports none
  initializeReferences();
  initializeDisturbances();
  initializeController();
  initializeSimscapeConfiguration();
  initializeLoggingConfiguration();
#+end_src

And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.
#+begin_src matlab :exports noone
  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Compliance/Fm'], 1, 'openinput');       io_i = io_i + 1; % Direct Forces/Torques applied on the micro-hexapod top platform
  io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Compliance/Dm'], 1, 'output'); io_i = io_i + 1; % Absolute displacement of the top platform

  %% Run the linearization
  Gm = linearize(mdl, io, 0);
  Gm.InputName  = {'Fmx', 'Fmy', 'Fmz', 'Mmx', 'Mmy', 'Mmz'};
  Gm.OutputName = {'Dx', 'Dy', 'Dz', 'Drx', 'Dry', 'Drz'};
#+end_src

#+begin_src matlab :exports none
  labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'};

  freqs = logspace(1, 3, 1000);

  figure;

  hold on;
  plot(freqs, abs(squeeze(freqresp(Gm(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', labels{1});
  for i = 2:6
    plot(freqs, abs(squeeze(freqresp(Gm(1, i), freqs, 'Hz'))), 'DisplayName', labels{i});
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]');
  ylabel('Compliance');
  legend('location', 'northwest');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/opt_stiff_micro_station_compliance.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:opt_stiff_micro_station_compliance
#+caption: Identified Compliance of the Micro-Station ([[./figs/opt_stiff_micro_station_compliance.png][png]], [[./figs/opt_stiff_micro_station_compliance.pdf][pdf]])
[[file:figs/opt_stiff_micro_station_compliance.png]]

** Identification of the dynamics with a rigid micro-station
#+begin_src matlab :exports none
  initializeReferences();
  initializeDisturbances();
  initializeController();
  initializeSimscapeConfiguration();
  initializeLoggingConfiguration();
  initializeSimscapeConfiguration('gravity', false);
#+end_src

#+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, 'openinput');              io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'],  3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'],  3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En');   io_i = io_i + 1;
#+end_src

#+begin_src matlab
  Ks = logspace(3,9,7); % [N/m]
#+end_src

#+begin_src matlab
  initializeSample('type', 'rigid', 'mass', 10);
#+end_src

#+begin_src matlab
  initializeGround('type', 'rigid');
  initializeGranite('type', 'rigid');
  initializeTy('type', 'rigid');
  initializeRy('type', 'rigid');
  initializeRz('type', 'rigid');
  initializeMicroHexapod('type', 'rigid');

  initializeAxisc('type', 'rigid');
  initializeMirror('type', 'rigid');
#+end_src

#+begin_src matlab :exports none
  Gmr_iff = {zeros(length(Ks))};
  Gmr_dvf = {zeros(length(Ks))};
  Gmr_err = {zeros(length(Ks))};
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ks)
    initializeNanoHexapod('k', Ks(i));

    G = linearize(mdl, io);
    G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
    G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                    'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                    'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

    Gmr_iff(i) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
    Gmr_dvf(i) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};

    Jinvt = tf(inv(nano_hexapod.J)');
    Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
    Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
    Gmr_err(i) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'},                {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt};
  end
#+end_src

** Identification of the dynamics with a flexible micro-station
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();

  initializeAxisc();
  initializeMirror();
#+end_src

#+begin_src matlab :exports none
  Gmf_iff = {zeros(length(Ks))};
  Gmf_dvf = {zeros(length(Ks))};
  Gmf_err = {zeros(length(Ks))};
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ks)
    initializeNanoHexapod('k', Ks(i));

    G = linearize(mdl, io);
    G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
    G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                    'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                    'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

    Gmf_iff(i) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
    Gmf_dvf(i) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};

    Jinvt = tf(inv(nano_hexapod.J)');
    Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
    Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
    Gmf_err(i) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'},                {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt};
  end
#+end_src

#+begin_src matlab :exports none
  save('mat/optimal_stiffness_micro_station_compliance.mat', 'Ks', ...
       'Gmr_iff',    'Gmr_dvf',    'Gmr_err', ...
       'Gmf_iff',    'Gmf_dvf',    'Gmf_err');
#+end_src

** Obtained Dynamics
#+begin_src matlab :exports none
  load('mat/optimal_stiffness_micro_station_compliance.mat');
#+end_src

The IFF plant only changes when the stiffness is $10^7 [N/m]$ or higher.
#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmr_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmf_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:length(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmr_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
         'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmf_iff{i}('Fnlm1', 'Fnl1'), 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', 'northwest');

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

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmr_dvf{i}('Dnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmf_dvf{i}('Dnlm1', 'Fnl1'), freqs, 'Hz'))), '--');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:length(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmr_dvf{i}('Dnlm1', 'Fnl1'), freqs, 'Hz')))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmf_dvf{i}('Dnlm1', 'Fnl1'), 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]);

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

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), 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(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz')))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), 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]);

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

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmr_err{i}('Ez', 'Fz'), freqs, 'Hz'))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gmf_err{i}('Ez', 'Fz'), 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(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmr_err{i}('Ez', 'Fz'), freqs, 'Hz')))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmf_err{i}('Ez', 'Fz'), 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]);

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

** Conclusion                                                        :ignore:
* Payload "Impedance" Effect
<<sec:payload_impedance>>

** Introduction                                                      :ignore:

** 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
We initialize all the stages with the default parameters.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
#+end_src

We don't include disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics.
#+begin_src matlab
  initializeSimscapeConfiguration('gravity', true);
  initializeDisturbances('enable', false);
#+end_src

We set the controller type to Open-Loop, and we do not need to log any signal.
#+begin_src matlab
  initializeController();
  initializeLoggingConfiguration('log', 'none');
  initializeReferences();
#+end_src

#+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, 'openinput');              io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'],  3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'],  3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En');   io_i = io_i + 1;
#+end_src

** Identification of the dynamics while change the payload dynamics
- Change of mass: from 1kg to 50kg
- Change of resonance frequency: from 50Hz to 500Hz
- The damping ratio of the payload is fixed to $\xi = 0.2$

We identify the dynamics for the following payload masses =Ms= and nano-hexapod leg's stiffnesses =Ks=:
#+begin_src matlab
  Ms = [1, 20, 50]; % [Kg]
  Ks = logspace(3,9,7); % [N/m]
#+end_src

#+begin_src matlab :exports none
  Gm_iff = {zeros(length(Ks), length(Ms))};
  Gm_dvf = {zeros(length(Ks), length(Ms))};
  Gm_err = {zeros(length(Ks), length(Ms))};
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ks)
    for j = 1:length(Ms)
      initializeNanoHexapod('k', Ks(i));
      initializeSample('mass', Ms(j), 'freq', 100*ones(6,1));

      G = linearize(mdl, io);
      G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                      'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                      'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

      Gm_iff(i,j) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
      Gm_dvf(i,j) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};

      Jinvt = tf(inv(nano_hexapod.J)');
      Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
      Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      Gm_err(i,j) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'},                {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt};
    end
  end
#+end_src

We then identify the dynamics for the following payload resonance frequencies =Fs=:
#+begin_src matlab
  Fs = [50, 200, 500]; % [Hz]
#+end_src

#+begin_src matlab :exports none
  Gf_iff = {zeros(length(Ks), length(Fs))};
  Gf_dvf = {zeros(length(Ks), length(Fs))};
  Gf_err = {zeros(length(Ks), length(Fs))};
#+end_src

#+begin_src matlab :exports none
  for i = 1:length(Ks)
    for j = 1:length(Fs)
      initializeNanoHexapod('k', Ks(i));
      initializeSample('mass', 20, 'freq', Fs(j)*ones(6,1));

      G = linearize(mdl, io);
      G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                      'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                      'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

      Gf_iff(i,j) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
      Gf_dvf(i,j) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};

      Jinvt = tf(inv(nano_hexapod.J)');
      Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
      Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
      Gf_err(i,j) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'},                {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt};
    end
  end
#+end_src

#+begin_src matlab :exports none
  save('mat/optimal_stiffness_Gm_Gf.mat', 'Ks', 'Ms', 'Fs', ...
       'Gm_iff',    'Gm_dvf',    'Gm_err', ...
       'Gf_iff',    'Gf_dvf',    'Gf_err');
#+end_src

** TODO Change of optimal gain for decentralized control
For each payload, compute the optimal gain for the IFF control.
The optimal value corresponds to critical damping to *all* the 6 modes of the nano-hexapod.

#+begin_src matlab
  load('mat/optimal_stiffness_Gm_Gf.mat');
#+end_src

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    for j = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(Gm_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
    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:length(Ks)
    for j = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      if j == 1
        plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
            'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i)));
      else
        plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
             'HandleVisibility', 'off');
      end
    end
  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');
  xlim([freqs(1), freqs(end)]);
#+end_src

Optimal gains:
#+begin_src matlab
  opt_gains = [20 60 200 600 2000 6000 20000];
#+end_src

Change of poles with mass
#+begin_src matlab :exports none
  i = 7;

  figure;
  hold on;
  for j = 1:length(Ms)
    set(gca,'ColorOrderIndex',j);
    cl_poles = pole(feedback(Gm_iff{i,j}, (-gains(k)/s)*eye(6)));
    plot(real(cl_poles), imag(cl_poles), '.');
  end
#+end_src

#+begin_src matlab :exports none
  i = 4;

  gains = logspace(1, 3, 500);

  figure;
  hold on;
  for j = 1:length(Ms)
    for k = 1:length(gains)
        set(gca,'ColorOrderIndex',j);
        cl_poles = pole(feedback(Gm_iff{i,j}, (-gains(k)/s)*eye(6)));
        poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
        plot(gains(k)*ones(size(poles_damp)), poles_damp, '.');
    end
  end
  xlabel('Control Gain');
  ylabel('Damping of the Poles');
  set(gca, 'XScale', 'log');
  ylim([0,pi/2]);
#+end_src

#+begin_src matlab :exports none
  i = 7;

  gains = logspace(3, 5, 500);

  figure;
  hold on;
  for j = 1:length(Ms)
    for k = 1:length(gains)
        set(gca,'ColorOrderIndex',j);
        cl_poles = pole(feedback(Gm_iff{i,j}, (-gains(k)/s)*eye(6)));
        poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
        plot(gains(k)*ones(size(poles_damp)), poles_damp, '.');
    end
  end
  xlabel('Control Gain');
  ylabel('Damping of the Poles');
  set(gca, 'XScale', 'log');
  ylim([0,pi/2]);
#+end_src

Change of payload resonance frequency
#+begin_src matlab :exports none
  i = 1;

  gains = logspace(0, 2, 100);

  figure;
  hold on;
  for j = 1:length(Fs)
    for k = 1:length(gains)
        set(gca,'ColorOrderIndex',j);
        cl_poles = pole(feedback(Gf_iff{i,j}, (-gains(k)/s)*eye(6)));
        poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
        plot(gains(k)*ones(size(poles_damp)), poles_damp, '.');
    end
  end
  xlabel('Control Gain');
  ylabel('Damping of the Poles');
  set(gca, 'XScale', 'log');
  ylim([0,pi/2]);
#+end_src

#+begin_src matlab :exports none
  i = 7;

  gains = logspace(3, 5, 100);

  figure;
  hold on;
  for j = 1:length(Fs)
    for k = 1:length(gains)
        set(gca,'ColorOrderIndex',j);
        cl_poles = pole(feedback(Gf_iff{i,j}, (-gains(k)/s)*eye(6)));
        poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
        plot(gains(k)*ones(size(poles_damp)), poles_damp, '.');
    end
  end
  xlabel('Control Gain');
  ylabel('Damping of the Poles');
  set(gca, 'XScale', 'log');
  ylim([0,pi/2]);
#+end_src


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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    for j = 1:length(Fs)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(Gf_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
    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:length(Ks)
    for j = 1:length(Fs)
      set(gca,'ColorOrderIndex',i);
      if j == 1
        plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
            'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i)));
      else
        plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz')))), '-', ...
             'HandleVisibility', 'off');
      end
    end
  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');
  xlim([freqs(1), freqs(end)]);
#+end_src

** Change of dynamics for the primary controller
For each stiffness, plot the total spread of dynamics.

#+begin_src matlab
  load('mat/optimal_stiffness_Gm_Gf.mat');
#+end_src

*** Frequency variation
Same payload mass, but different stiffness resulting in different resonance frequency.

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    for j = 1:length(Fs)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-');
    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:length(Ks)
    for j = 1:length(Fs)
      set(gca,'ColorOrderIndex',i);
      if j == 1
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
            'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i)));
      else
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
             'HandleVisibility', 'off');
      end
    end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

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

X direction
#+begin_src matlab :exports none
  i = 1;

  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title(sprintf('$k = %.0e$ [N/m]', Ks(i)))

  ax2 = subplot(2, 2, 3);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
         'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

  linkaxes([ax1,ax2],'x');
  xlim([freqs(1), freqs(end)]);

  i = 7;

  ax1 = subplot(2, 2, 2);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title(sprintf('$k = %.0e$ [N/m]', Ks(i)))

  ax2 = subplot(2, 2, 4);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
         'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

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

Z direction:
We can see two mass lines for the soft nano-hexapod:
- The first mass line corresponds to $\frac{1}{(m_n + m_p)s^2}$ where $m_p = 20\ [kg]$ is the mass of the payload and $m_n = 15\ [Kg]$ is the mass of the nano-hexapod top platform and attached mirror
- The second mass line corresponds to $\frac{1}{m_n s^2}$

#+begin_src matlab :exports none
  i = 1;

  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title(sprintf('$k = %.0e$ [N/m]', Ks(i)))

  ax2 = subplot(2, 2, 3);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ...
         'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

  linkaxes([ax1,ax2],'x');
  xlim([freqs(1), freqs(end)]);

  i = 7;

  ax1 = subplot(2, 2, 2);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title(sprintf('$k = %.0e$ [N/m]', Ks(i)))

  ax2 = subplot(2, 2, 4);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ...
         'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

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

*** Mass variation
All mixed, X direction
#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    for j = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-');
    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:length(Ks)
    for j = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      if j == 1
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
            'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i)));
      else
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
             'HandleVisibility', 'off');
      end
    end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

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

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Ks)
    for j = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-');
    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:length(Ks)
    for j = 1:length(Ms)
      set(gca,'ColorOrderIndex',i);
      if j == 1
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ...
            'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i)));
      else
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ...
             'HandleVisibility', 'off');
      end
    end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

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

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

  figure;

  i = 1;

  ax1 = subplot(2, 2, 1);
  hold on;
  for j = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title(sprintf('$k = %.0e$ [N/m]', Ks(i)))

  ax2 = subplot(2, 2, 3);
  hold on;
  for j = 1:length(Ms)
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
  xlim([freqs(1), freqs(end)]);

  i = 7;

  ax1 = subplot(2, 2, 2);
  hold on;
  for j = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title(sprintf('$k = %.0e$ [N/m]', Ks(i)))

  ax2 = subplot(2, 2, 4);
  hold on;
  for j = 1:length(Ms)
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ...
          'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(j)));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northeast');

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

*** Total variation
Total change of dynamics due to change of the payload:
- mass from 1kg to 50kg
- main resonance from 50Hz to 500Hz

For a soft nano-hexapod
#+begin_src matlab :exports none
  i = 1;

  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  end
  for j = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  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 j = 1:length(Fs)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  for j = 1:length(Ms)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);

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

For a stiff nano-hexapod
#+begin_src matlab :exports none
  i = 7;

  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  end
  for j = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  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 j = 1:length(Fs)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  for j = 1:length(Ms)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);

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

** Conclusion                                                        :ignore:
* Total Change of dynamics
#+begin_src matlab :exports none
  load('mat/optimal_stiffness_Gm_Gf.mat');
  load('mat/optimal_stiffness_micro_station_compliance.mat');
  load('mat/optimal_stiffness_Gk_wz.mat');
#+end_src

- =Gk_wz_err= - Change of spindle rotation speed
- =Gf_err= - Change of payload resonance
- =Gm_err= - Change of payload mass
- =Gmr_err= - Rigid Micro-Station
- =Gmf_err= - Flexible Micro-Station

Soft nano-hexapod
#+begin_src matlab :exports none
  i = 1;

  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  % =Gf_err= - Change of payload resonance
  plot(freqs, abs(squeeze(freqresp(Gf_err{i,1}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
       'DisplayName', 'Payload Freq');
  for j = 2:length(Fs)
    set(gca,'ColorOrderIndex',1);
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
  end
  % =Gm_err= - Change of payload mass
  set(gca,'ColorOrderIndex',2);
  plot(freqs, abs(squeeze(freqresp(Gm_err{i,1}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
       'DisplayName', 'Payload Mass');
  for j = 2:length(Ms)
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
  end
  % =Gm_err= - Change of payload mass
  set(gca,'ColorOrderIndex',3);
  plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i,1}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
       'DisplayName', 'Rotationg Speed');
  for j = 2:length(Rz_rpm)
    set(gca,'ColorOrderIndex',3);
    plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
  end
  set(gca,'ColorOrderIndex',4);
  % =Gmr_err= - Rigid Micro-Station
  plot(freqs, abs(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
       'DisplayName', 'Rigid $\mu$-station');
  % =Gmf_err= - Flexible Micro-Station
  plot(freqs, abs(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '-', ...
       'DisplayName', 'Flexible $\mu$-station');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southwest');

  ax2 = subplot(2, 1, 2);
  hold on;
  for j = 1:length(Rz_rpm)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  for j = 1:length(Fs)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  % =Gm_err= - Change of payload mass
  for j = 1:length(Ms)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  % =Gmr_err= - Rigid Micro-Station
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  % =Gmf_err= - Flexible Micro-Station
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);

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

#+begin_src matlab :exports none
  i = 1;

  freqs = logspace(-1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  % =Gf_err= - Change of payload resonance
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  end
  % =Gm_err= - Change of payload mass
  for j = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  end
  % Spindle Rotation Speed
  for j = 1:length(Rz_rpm)
    plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  end
  % =Gmr_err= - Rigid Micro-Station
  plot(freqs, abs(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  % =Gmf_err= - Flexible Micro-Station
  plot(freqs, abs(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), freqs, 'Hz'))), 'k-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  % =Gf_err= - Change of payload resonance
  for j = 1:length(Fs)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  % =Gm_err= - Change of payload mass
  for j = 1:length(Ms)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  % Spindle Rotation Speed
  for j = 1:length(Rz_rpm)
    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gk_wz_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  end
  % =Gmr_err= - Rigid Micro-Station
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  % =Gmf_err= - Flexible Micro-Station
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), freqs, 'Hz')))), 'k-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);

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

Comparison with initial TF
#+begin_src matlab :exports none
  i = 1;

  G0 = abs(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz')));

  freqs = logspace(-1, 3, 1000);

  figure;

  hold on;
  % =Gf_err= - Change of payload resonance
  for j = 1:length(Fs)
    plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))./G0, 'k-');
  end
  % =Gm_err= - Change of payload mass
  for j = 1:length(Ms)
    plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))./G0, 'k-');
  end
  % Spindle Rotation Speed
  for j = 1:length(Rz_rpm)
    plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i,j}('Ex', 'Fx'), freqs, 'Hz')))./G0, 'k-');
  end
  % =Gmf_err= - Flexible Micro-Station
  plot(freqs, abs(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), freqs, 'Hz')))./G0, 'k-');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);
  ylim([1e-2, 1e2])
#+end_src

- [ ] Make a gif with the stiffness varying