#+TITLE: Stewart Platform with Flexible Elements
:DRAWER:
#+STARTUP: overview

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

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

#+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: <script src="./js/jquery.min.js"></script>
#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script src="./js/readtheorg.js"></script>

#+PROPERTY: header-args:matlab  :session *MATLAB*
#+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

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

* Simscape Model
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :results none :exports none
  simulinkproject('../');
#+end_src

** Flexible APA
#+begin_src matlab
  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(apa.n_i); length(apa.int_i); size(apa.M,1) - 6*length(apa.int_i); size(apa.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |  7 |
| Number of interface Nodes |  7 |
| Number of Modes           |  6 |
| Size of M and K matrices  | 48 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(apa.int_i)]', apa.int_i, apa.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number |   x [m] |  y [m] | z [m] |
|--------+-------------+---------+--------+-------|
|    1.0 |     53917.0 |     0.0 | -0.015 |   0.0 |
|    2.0 |     53918.0 |     0.0 |  0.015 |   0.0 |
|    3.0 |     53919.0 | -0.0325 |    0.0 |   0.0 |
|    4.0 |     53920.0 | -0.0125 |    0.0 |   0.0 |
|    5.0 |     53921.0 | -0.0075 |    0.0 |   0.0 |
|    6.0 |     53922.0 |  0.0125 |    0.0 |   0.0 |
|    7.0 |     53923.0 |  0.0325 |    0.0 |   0.0 |

** Flexible Joint
#+begin_src matlab
  flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(flex_joint.n_i); length(flex_joint.int_i); size(flex_joint.M,1) - 6*length(flex_joint.int_i); size(flex_joint.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |  2 |
| Number of interface Nodes |  2 |
| Number of Modes           |  6 |
| Size of M and K matrices  | 18 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(flex_joint.int_i)]', flex_joint.int_i, flex_joint.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+-------+-------+-------|
|    1.0 |    181278.0 |   0.0 |   0.0 |   0.0 |
|    2.0 |    181279.0 |   0.0 |   0.0 |  -0.0 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([1e-6*flex_joint.K(3,3), flex_joint.K(4,4), flex_joint.K(5,5), flex_joint.K(6,6)]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %0.f ');
#+end_src

#+RESULTS:
| *Caracteristic*            | *Value* |
|----------------------------+---------|
| Axial Stiffness [N/um]     |     119 |
| Bending Stiffness [Nm/rad] |      33 |
| Bending Stiffness [Nm/rad] |      33 |
| Torsion Stiffness [Nm/rad] |     236 |

** Identification
And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
#+end_src

#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');
#+end_src

#+begin_src matlab
  disturbances = initializeDisturbances();
  references = initializeReferences(stewart);
#+end_src

** No Flexible Elements
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  % stewart = initializeStrutDynamics(stewart, 'K', 1.8e6*ones(6,1));
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235*ones(6,1), 'Kf_F', 33*ones(6,1), 'Kt_F', 235*ones(6,1));
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

** Flexible joints
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  %% Run the linearization
  Gj = linearize(mdl, io, options);
  Gj.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

** Flexible APA
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235, 'Kf_F', 33*ones(6,1), 'Kt_F', 235);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  %% Run the linearization
  Ga = -linearize(mdl, io, options);
  Ga.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Ga.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

** Flexible Joints and APA
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  Gf = -linearize(mdl, io, options);
  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(44*Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(44*Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  h = zeros(4, 1);
  h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
  h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
  h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
  h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
  legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
  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');
#+end_src

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

#+name: fig:flexible_elements_effect_dvf
#+caption: Change of the DVF plant dynamics with the added flexible elements
#+RESULTS:
[[file:figs/flexible_elements_effect_dvf.png]]

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(1e9*Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(1e9*Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  h = zeros(4, 1);
  h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
  h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
  h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
  h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
  legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
  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');
#+end_src

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

#+name: fig:flexible_elements_effect_iff
#+caption: Change of the IFF plant dynamics with the added flexible elements
#+RESULTS:
[[file:figs/flexible_elements_effect_iff.png]]