stewart-simscape/org/dynamics-study.org

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#+TITLE: Stewart Platform - Dynamics Study
:DRAWER:
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#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
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#+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
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#+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
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#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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: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
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simulinkproject('../');
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#+end_src
** Simscape Model
#+begin_src matlab
open('stewart_platform_dynamics.slx')
#+end_src
** test
#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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#+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
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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#+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
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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#+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
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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#+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