#+TITLE: Identification
#+SETUPFILE: ./setup/org-setup-file.org

* Introduction                                                       :ignore:
The goal here is to tune the Simscape model of the station in order to have a good dynamical representation of the real system.

In order to do so, we reproduce the Modal Analysis done on the station using the Simscape model.

We can then compare the measured Frequency Response Functions with the identified dynamics of the model.

Finally, this should help to tune the parameters of the model such that the dynamics is closer to the measured FRF.

* Some notes about the Simscape Model
The Simscape Model of the micro-station consists of several solid bodies:
- Bottom Granite
- Top Granite
- Translation Stage
- Tilt Stage
- Spindle
- Hexapod

Each solid body has some characteristics: Center of Mass, mass, moment of inertia, etc...
These parameters are automatically computed from the geometry and from the density of the materials.

Then, the solid bodies are connected with springs and dampers.
Some of the springs and dampers values can be estimated from the joints/stages specifications, however, we here prefer to tune these values based on the measurements.

* Compare with measurements at the CoM of each element
** 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

** Prepare the Simulation
We load the configuration.
#+begin_src matlab
  load('mat/conf_simulink.mat');
#+end_src

We set a small =StopTime=.
#+begin_src matlab
  set_param(conf_simulink, 'StopTime', '0.5');
#+end_src

We initialize all the stages.
#+begin_src matlab
  initializeGround(      'type', 'rigid');
  initializeGranite(     'type', 'modal-analysis');
  initializeTy(          'type', 'modal-analysis');
  initializeRy(          'type', 'modal-analysis');
  initializeRz(          'type', 'modal-analysis');
  initializeMicroHexapod('type', 'modal-analysis');
  initializeAxisc(       'type', 'flexible');

  initializeMirror(      'type', 'none');
  initializeNanoHexapod( 'type', 'none');
  initializeSample(      'type', 'none');

  initializeController(  'type', 'open-loop');

  initializeLoggingConfiguration('log', 'none');

  initializeReferences();
  initializeDisturbances('enable', false);
#+end_src

** Estimate the position of the CoM of each solid and compare with the one took for the Measurement Analysis
Thanks to the [[https://fr.mathworks.com/help/physmod/sm/ref/inertiasensor.html][Inertia Sensor]] simscape block, it is possible to estimate the position of the Center of Mass of a solid body with respect to a defined frame.

#+begin_src matlab
  sim('nass_model')
#+end_src

The results are shown in the table [[tab:com_simscape]].

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  stages_com = 1e3*[granite_bot_com.Data(end, :) ;
                    granite_top_com.Data(end, :) ;
                    ty_com.Data(end, :) ;
                    ry_com.Data(end, :) ;
                    rz_com.Data(end, :) ;
                    hexa_com.Data(end, :) ]';

  data2orgtable(stages_com, {'X [mm]', 'Y [mm]', 'Z [mm]'}, {'granite bot', 'granite top', 'ty', 'ry', 'rz', 'hexa'}, ' %.1f ');
#+end_src

#+name: tab:com_simscape
#+caption: Center of Mass of each solid body as defined in Simscape
#+RESULTS:
|        | granite bot | granite top |     ty |     ry |     rz |   hexa |
|--------+-------------+-------------+--------+--------+--------+--------|
| X [mm] |        52.4 |        51.7 |    0.9 |   -0.1 |    0.0 |   -0.0 |
| Y [mm] |       190.4 |       263.2 |    0.7 |    5.2 |   -0.0 |    0.1 |
| Z [mm] |     -1200.0 |      -777.1 | -598.9 | -627.7 | -643.2 | -317.1 |

We can compare the obtained center of mass (table [[tab:com_simscape]]) with the one used for the Modal Analysis shown in table [[tab:com_solidworks]].

#+name: tab:com_solidworks
#+caption: Estimated Center of Mass of each solid body using Solidworks
|        | granite bot | granite top |   ty |   ry |   rz | hexa |
|--------+-------------+-------------+------+------+------+------|
| X [mm] |          45 |          52 |    0 |    0 |    0 |   -4 |
| Y [mm] |         144 |         258 |   14 |   -5 |    0 |    6 |
| Z [mm] |       -1251 |        -778 | -600 | -628 | -580 | -319 |

The results are quite similar.
The differences can be explained by some differences in the chosen density of the materials or by the fact that not exactly all the same elements have been chosen for each stage.

For instance, on simscape, the fixed part of the translation stage counts for the top granite solid body.
However, in SolidWorks, this has probably not be included with the top granite.

** Create a frame at the CoM of each solid body
Now we use one =inertiasensor= block connected on each solid body that measured the center of mass of this solid with respect to the same connected frame.

We do that in order to position an accelerometer on the Simscape model at this particular point.

#+begin_src matlab
  open('identification/matlab/sim_micro_station_com_estimation.slx')
#+end_src

#+begin_src matlab
  sim('sim_micro_station_com_estimation')
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  stages_com = 1e3*[granite_bot_com.Data(end, :) ;
                    granite_top_com.Data(end, :) ;
                    ty_com.Data(end, :) ;
                    ry_com.Data(end, :) ;
                    rz_com.Data(end, :) ;
                    hexa_com.Data(end, :) ]';

  data2orgtable(stages_com, {'X [mm]', 'Y [mm]', 'Z [mm]'}, {'granite bot', 'granite top', 'ty', 'ry', 'rz', 'hexa'}, ' %.1f ');
#+end_src

#+RESULTS:
|        | granite bot | granite top |    ty |     ry |    rz |  hexa |
|--------+-------------+-------------+-------+--------+-------+-------|
| X [mm] |         0.0 |        51.7 |   0.9 |   -0.1 |   0.0 |  -0.0 |
| Y [mm] |         0.0 |       753.2 |   0.7 |    5.2 |  -0.0 |   0.1 |
| Z [mm] |      -250.0 |        22.9 | -17.1 | -146.5 | -23.2 | -47.1 |

We now same this for further use:
#+begin_src matlab
  granite_bot_com = granite_bot_com.Data(end, :)';
  granite_top_com = granite_top_com.Data(end, :)';
  ty_com = ty_com.Data(end, :)';
  ry_com = ry_com.Data(end, :)';
  rz_com = rz_com.Data(end, :)';
  hexa_com = hexa_com.Data(end, :)';

  save('./mat/solids_com.mat', 'granite_bot_com', 'granite_top_com', 'ty_com', 'ry_com', 'rz_com', 'hexa_com');
#+end_src

Then, we use the obtained results to add a =rigidTransform= block in order to create a new frame at the center of mass of each solid body.

** Identification of the dynamics of the Simscape Model
We now use a new Simscape Model where 6DoF inertial sensors are located at the Center of Mass of each solid body.

#+begin_src matlab
  % load('mat/solids_com.mat', 'granite_bot_com', 'granite_top_com', 'ty_com', 'ry_com', 'rz_com', 'hexa_com');
#+end_src

#+begin_src matlab
  open('nass_model.slx')
#+end_src

We use the =linearize= function in order to estimate the dynamics from forces applied on the Translation stage at the same position used for the real modal analysis to the inertial sensors.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Micro-Station/Translation Stage/Modal Analysis/F_hammer'],          1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station/Granite/Modal Analysis/accelerometer'],               1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station/Translation Stage/Modal Analysis/accelerometer'],     1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station/Tilt Stage/Modal Analysis/accelerometer'],            1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station/Spindle/Modal Analysis/accelerometer'],               1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Modal Analysis/accelerometer'],         1, 'openoutput'); io_i = io_i + 1;
#+end_src

#+begin_src matlab
  % Run the linearization
  G_ms = linearize(mdl, io, 0);

  %% Input/Output definition
  G_ms.InputName  = {'Fx', 'Fy', 'Fz'};
  G_ms.OutputName = {'gtop_x', 'gtop_y', 'gtop_z', 'gtop_rx', 'gtop_ry', 'gtop_rz', ...
                     'ty_x', 'ty_y', 'ty_z', 'ty_rx', 'ty_ry', 'ty_rz', ...
                     'ry_x', 'ry_y', 'ry_z', 'ry_rx', 'ry_ry', 'ry_rz', ...
                     'rz_x', 'rz_y', 'rz_z', 'rz_rx', 'rz_ry', 'rz_rz', ...
                     'hexa_x', 'hexa_y', 'hexa_z', 'hexa_rx', 'hexa_ry', 'hexa_rz'};
#+end_src

The output of =G_ms= is the acceleration of each solid body.
In order to obtain a displacement, we divide the obtained transfer function by $1/s^{2}$;
#+begin_src matlab
  G_ms = G_ms/s^2;
#+end_src

** Compare with measurements
We now load the Frequency Response Functions measurements during the Modal Analysis (accessible [[file:../../meas/modal-analysis/index.org][here]]).

#+begin_src matlab
  load('../meas/modal-analysis/mat/frf_coh_matrices.mat', 'freqs');
  load('../meas/modal-analysis/mat/frf_com.mat', 'FRFs_CoM');
#+end_src

We then compare the measurements with the identified transfer functions using the Simscape Model.

#+begin_src matlab :exports none
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'gbot', 'gtop', 'ty', 'ry', 'rz', 'hexa'}

  n_stg = 3;
  n_dir = 6; % x, y, z, Rx, Ry, Rz
  n_exc = 2; % x, y, z

  f = logspace(0, 3, 1000);

  figure;
  hold on;
  plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg-1) + n_dir, n_exc, :)))./((2*pi*freqs).^2)');
  plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_exc}]), f, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]');
  hold off;
  xlim([1, 200]);
#+end_src

#+begin_src matlab :exports none
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'gtop', 'ty', 'ry', 'rz', 'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1:2
    for n_dir = 1:3
      subplot(3, 2, (n_dir-1)*2 + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      if n_dir == 3
        xlabel('Frequency [Hz]');
      end
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end
#+end_src

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

#+NAME: fig:identification_comp_bot_stages
#+CAPTION: caption ([[./figs/identification_comp_bot_stages.png][png]], [[./figs/identification_comp_bot_stages.pdf][pdf]])
[[file:figs/identification_comp_bot_stages.png]]


#+begin_src matlab :exports none
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'ry', 'rz', 'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1:2
    for n_dir = 1:3
      subplot(3, 2, (n_dir-1)*2 + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg+2) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      if n_dir == 3
        xlabel('Frequency [Hz]');
      end
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end
#+end_src

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

#+NAME: fig:identification_comp_mid_stages
#+CAPTION: caption ([[./figs/identification_comp_mid_stages.png][png]], [[./figs/identification_comp_mid_stages.pdf][pdf]])
[[file:figs/identification_comp_mid_stages.png]]


#+begin_src matlab :exports none
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1
    for n_dir = 1:3
      subplot(3, 1, (n_dir-1) + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg+4) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      if n_dir == 3
        xlabel('Frequency [Hz]');
      end
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end
#+end_src

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

#+NAME: fig:identification_comp_top_stages
#+CAPTION: caption ([[./figs/identification_comp_top_stages.png][png]], [[./figs/identification_comp_top_stages.pdf][pdf]])
[[file:figs/identification_comp_top_stages.png]]


* Obtained Compliance of the Micro-Station
** 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
  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('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
  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.

The obtained compliance is shown in Figure [[fig:compliance_micro_station]].
#+begin_src matlab
  %% 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
  save('../meas/micro-station-compliance/mat/model.mat', 'Gm');
#+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;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gm(i, 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/compliance_micro_station.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+name: fig:compliance_micro_station
#+caption: Obtained compliance of the Micro-Station ([[./figs/compliance_micro_station.png][png]], [[./figs/compliance_micro_station.pdf][pdf]])
[[file:figs/compliance_micro_station.png]]

* Conclusion
#+begin_important
  For such a complex system, we believe that the Simscape Model represents the dynamics of the system with enough fidelity.
#+end_important