stewart-simscape/org/dynamics-study.org

#+TITLE: Stewart Platform - Dynamics Study
: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/thesis/latex/}{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:

* Some tests
** 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
  simulinkproject('../');
#+end_src

** Simscape Model
#+begin_src matlab
  open('stewart_platform_dynamics.slx')
#+end_src

** test
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
#+end_src

Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src


#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/J-T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/X'],   1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

#+begin_src matlab
  G_cart = minreal(G*inv(stewart.J'));
  G_cart.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src

#+begin_src matlab
  figure; bode(G, G_cart)
#+end_src

#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fext'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/X'],    1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  Gd = linearize(mdl, io, options);
  Gd.InputName  = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
  Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

#+begin_src matlab
  freqs = logspace(0, 3, 1000);

  figure;
  bode(Gd, G)
#+end_src

** Compare external forces and forces applied by the actuators
Initialization of the Stewart platform.
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
#+end_src

Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

Estimation of the transfer function from $\mathcal{\bm{F}}_{d}$ to $\mathcal{\bm{X}}$:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fext'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/X'],    1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  Gd = linearize(mdl, io, options);
  Gd.InputName  = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
  Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

Comparison of the two transfer function matrices.
#+begin_src matlab
  freqs = logspace(0, 4, 1000);

  figure;
  bode(Gd, G, freqs)
#+end_src

#+begin_important
Seems quite similar.
#+end_important

** Comparison of the static transfer function and the Compliance matrix
Initialization of the Stewart platform.
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
#+end_src

Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

Let's first look at the low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$.
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(real(freqresp(G, 0.1)), {}, {}, ' %.1e ');
#+end_src

#+RESULTS:
|  2.0e-06 | -9.1e-19 | -5.3e-12 |  7.3e-18 |  1.7e-05 |  1.3e-18 |
| -1.7e-18 |  2.0e-06 |  8.6e-19 | -1.7e-05 | -1.5e-17 |  6.7e-12 |
|  3.6e-13 |  3.2e-19 |  5.0e-07 | -2.5e-18 |  8.1e-12 | -1.5e-19 |
|  1.0e-17 | -1.7e-05 | -5.0e-18 |  1.9e-04 |  9.1e-17 | -3.5e-11 |
|  1.7e-05 | -6.9e-19 | -5.3e-11 |  6.9e-18 |  1.9e-04 |  4.8e-18 |
| -3.5e-18 | -4.5e-12 |  1.5e-18 |  7.1e-11 | -3.4e-17 |  4.6e-05 |

And now at the Compliance matrix.
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(stewart.C, {}, {}, ' %.1e ');
#+end_src

#+RESULTS:
|  2.0e-06 |  2.9e-22 |  2.8e-22 | -3.2e-21 |  1.7e-05 |  1.5e-37 |
| -2.1e-22 |  2.0e-06 | -1.8e-23 | -1.7e-05 | -2.3e-21 |  1.1e-22 |
|  3.1e-22 | -1.6e-23 |  5.0e-07 |  1.7e-22 |  2.2e-21 | -8.1e-39 |
|  2.1e-21 | -1.7e-05 |  2.0e-22 |  1.9e-04 |  2.3e-20 | -8.7e-21 |
|  1.7e-05 |  2.5e-21 |  2.0e-21 | -2.8e-20 |  1.9e-04 |  1.3e-36 |
|  3.7e-23 |  3.1e-22 | -6.0e-39 | -1.0e-20 |  3.1e-22 |  4.6e-05 |

#+begin_important
The low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$ corresponds to the compliance matrix of the Stewart platform.
#+end_important

** Transfer function from forces applied in the legs to the displacement of the legs
Initialization of the Stewart platform.
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
#+end_src

Estimation of the transfer function from $\bm{\tau}$ to $\bm{L}$:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/J-T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/L'], 1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};
#+end_src

#+begin_src matlab
  freqs = logspace(1, 3, 1000);
  figure; bode(G, 2*pi*freqs)
#+end_src

#+begin_src matlab
  bodeFig({G(1,1), G(1,2)}, freqs, struct('phase', true));
#+end_src