stewart-simscape/identification.org

#+TITLE: Identification of the Stewart Platform using Simscape
:DRAWER:
#+HTML_LINK_HOME: ./index.html
#+HTML_LINK_UP: ./index.html

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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :tangle matlab/identification.m
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs
:END:

* Identification
** 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 :results none :exports none
  simulinkproject('./');
#+end_src

** Script
#+begin_src matlab :results none :exports none
  open stewart
#+end_src

The hexapod structure and Sample structure are initialized.
#+begin_src matlab :results none
  stewart = initializeGeneralConfiguration();
  stewart = computeGeometricalProperties(stewart);
  stewart = initializeMechanicalElements(stewart);
  save('./mat/stewart.mat', 'stewart');

  initializeSample();
#+end_src

#+begin_src matlab :results none
  G = identifyPlant();
#+end_src

#+begin_src matlab :results none
  freqs = logspace(2, 4, 1000);
#+end_src

* Cartesian Plot
From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_cart(1, 1), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart(2, 1), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart(3, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude');
#+end_src

#+begin_src matlab :results none
  figure;
  bode(G.G_cart, freqs);
#+end_src

* From a force to force sensor
#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
  legend('location', 'southeast');
#+end_src

#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
  plot(freqs, abs(squeeze(freqresp(G.G_forc(2, 1), freqs, 'Hz'))), 'k--', 'DisplayName', '$F_{m_j}/F_{i}$');
  plot(freqs, abs(squeeze(freqresp(G.G_forc(3, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G.G_forc(4, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G.G_forc(5, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G.G_forc(6, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
  legend('location', 'southeast');
#+end_src

* From a force applied in the leg to the displacement of the leg
#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$D_{i}/F_{i}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src

#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$D_{i}/F_{i}$');
  plot(freqs, abs(squeeze(freqresp(G.G_legs(2, 1), freqs, 'Hz'))), 'k--', 'DisplayName', '$D_{j}/F_{i}$');
  plot(freqs, abs(squeeze(freqresp(G.G_legs(3, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G.G_legs(4, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G.G_legs(5, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G.G_legs(6, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
  legend('location', 'northeast');
#+end_src

* Transmissibility
#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_tran(2, 2), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_tran(3, 3), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/m]');
#+end_src

#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_tran(4, 4), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_tran(5, 5), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_tran(6, 6), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [$\frac{rad/s}{rad/s}$]');
#+end_src

#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 2), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 3), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/m]');
#+end_src

* Compliance
From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.

#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_comp(1, 1), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_comp(2, 2), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_comp(3, 3), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src

* Inertial
From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.

#+begin_src matlab :results none
  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_iner(1, 1), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_iner(2, 2), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_iner(3, 3), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src

* identifyPlant
  :PROPERTIES:
  :HEADER-ARGS:matlab+: :exports code
  :HEADER-ARGS:matlab+: :comments yes
  :HEADER-ARGS:matlab+: :eval no
  :HEADER-ARGS:matlab+: :tangle src/identifyPlant.m
  :END:

#+begin_src matlab
  function [sys] = identifyPlant(opts_param)
#+end_src

We use this code block to pass optional parameters.
#+begin_src matlab
  %% Default values for opts
  opts = struct();

  %% Populate opts with input parameters
  if exist('opts_param','var')
      for opt = fieldnames(opts_param)'
          opts.(opt{1}) = opts_param.(opt{1});
      end
  end
#+end_src

We defined the options for the =linearize= command.
Here, we just identify the system at time $t = 0$.
#+begin_src matlab
  options = linearizeOptions;
  options.SampleTime = 0;
#+end_src

We define the name of the Simulink File used to identification.
#+begin_src matlab
  mdl = 'stewart';
#+end_src

Then we defined the input/output of the transfer function we want to identify.
#+begin_src matlab
  %% Inputs
  io(1) = linio([mdl, '/F'],  1, 'input'); % Cartesian forces
  io(2) = linio([mdl, '/Fl'], 1, 'input'); % Leg forces
  io(3) = linio([mdl, '/Fd'], 1, 'input'); % Direct forces
  io(4) = linio([mdl, '/Dw'], 1, 'input'); % Base motion

  %% Outputs
  io(5) = linio([mdl, '/Dm'],  1, 'output'); % Relative Motion
  io(6) = linio([mdl, '/Dlm'], 1, 'output'); % Displacement of each leg
  io(7) = linio([mdl, '/Flm'], 1, 'output'); % Force sensor in each leg
  io(8) = linio([mdl, '/Xm'],  1, 'output'); % Absolute motion of platform
#+end_src

The linearization is run.
#+begin_src matlab
  G = linearize(mdl, io, 0);
#+end_src

We defined all the Input/Output names of the identified transfer function.
#+begin_src matlab
  G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz', ...
                  'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
                  'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz', ...
                  'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'};
  G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
                  'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m', ...
                  'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m', ...
                  'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'};
#+end_src

We split the transfer function into sub transfer functions and we compute their minimum realization.
#+begin_src matlab
  sys.G_cart = minreal(G({'Dxm',  'Dym',  'Dzm',  'Rxm',  'Rym',  'Rzm'},  {'Fx',  'Fy',  'Fz',  'Mx',  'My',  'Mz'}));
  sys.G_forc = minreal(G({'F1m',  'F2m',  'F3m',  'F4m',  'F5m',  'F6m'},  {'F1',  'F2',  'F3',  'F4',  'F5',  'F6'}));
  sys.G_legs = minreal(G({'D1m',  'D2m',  'D3m',  'D4m',  'D5m',  'D6m'},  {'F1',  'F2',  'F3',  'F4',  'F5',  'F6'}));
  sys.G_tran = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));
  sys.G_comp = minreal(G({'Dxm',  'Dym',  'Dzm',  'Rxm',  'Rym',  'Rzm'},  {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
  sys.G_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
  % sys.G_all  = minreal(G);
#+end_src

#+begin_src matlab
  end
#+end_src