404 lines
17 KiB
Org Mode
404 lines
17 KiB
Org Mode
#+TITLE: Cubic configuration for the Stewart Platform
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:DRAWER:
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#+HTML_LINK_HOME: ./index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :tangle matlab/cubic_configuration.m
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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:END:
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* Introduction :ignore:
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The discovery of the Cubic configuration is done in cite:geng94_six_degree_of_freed_activ.
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Further analysis is conducted in
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The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
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The cubic (or orthogonal) configuration of the Stewart platform is now widely used (cite:preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm).
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According to cite:preumont07_six_axis_singl_stage_activ, the cubic configuration provides a uniform stiffness in all directions and *minimizes the crosscoupling* from actuator to sensor of different legs (being orthogonal to each other).
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To generate and study the Cubic configuration, =generateCubicConfiguration= is used (description in section [[sec:generateCubicConfiguration]]).
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The goal is to study the benefits of using a cubic configuration:
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- Equal stiffness in all the degrees of freedom?
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- No coupling between the actuators?
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- Is the center of the cube an important point?
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* Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :results none :exports none
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simulinkproject('./');
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#+end_src
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* TODO Configuration Analysis - Stiffness Matrix
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** Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
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We create a cubic Stewart platform (figure [[fig:3d-cubic-stewart-aligned]]) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
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The Jacobian matrix is estimated at the location of the center of the cube.
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#+name: fig:3d-cubic-stewart-aligned
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#+caption: Centered cubic configuration
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[[file:./figs/3d-cubic-stewart-aligned.png]]
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#+begin_src matlab :results silent
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opts = struct(...
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'H_tot', 100, ... % Total height of the Hexapod [mm]
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'L', 200/sqrt(3), ... % Size of the Cube [mm]
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'H', 60, ... % Height between base joints and platform joints [mm]
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'H0', 200/2-60/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
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);
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stewart = initializeCubicConfiguration(opts);
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opts = struct(...
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'Jd_pos', [0, 0, -50], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
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'Jf_pos', [0, 0, -50] ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
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);
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stewart = computeGeometricalProperties(stewart, opts);
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save('./mat/stewart.mat', 'stewart');
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#+end_src
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#+begin_src matlab :results none :exports code
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K = stewart.Jf'*stewart.Jf;
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#+end_src
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#+begin_src matlab :results value table :exports results
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data = K;
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data2orgtable(data, {}, {}, ' %.2g ');
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#+end_src
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#+RESULTS:
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| 2 | 1.9e-18 | -2.3e-17 | 1.8e-18 | 5.5e-17 | -1.5e-17 |
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| 1.9e-18 | 2 | 6.8e-18 | -6.1e-17 | -1.6e-18 | 4.8e-18 |
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| -2.3e-17 | 6.8e-18 | 2 | -6.7e-18 | 4.9e-18 | 5.3e-19 |
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| 1.8e-18 | -6.1e-17 | -6.7e-18 | 0.0067 | -2.3e-20 | -6.1e-20 |
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| 5.5e-17 | -1.6e-18 | 4.9e-18 | -2.3e-20 | 0.0067 | 1e-18 |
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| -1.5e-17 | 4.8e-18 | 5.3e-19 | -6.1e-20 | 1e-18 | 0.027 |
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** Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
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We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure [[fig:3d-cubic-stewart-aligned]]).
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The Jacobian matrix is not estimated at the location of the center of the cube.
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#+begin_src matlab :results silent
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opts = struct(...
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'H_tot', 100, ... % Total height of the Hexapod [mm]
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'L', 200/sqrt(3), ... % Size of the Cube [mm]
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'H', 60, ... % Height between base joints and platform joints [mm]
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'H0', 200/2-60/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
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);
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stewart = initializeCubicConfiguration(opts);
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opts = struct(...
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'Jd_pos', [0, 0, 0], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
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'Jf_pos', [0, 0, 0] ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
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);
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stewart = computeGeometricalProperties(stewart, opts);
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#+end_src
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#+begin_src matlab :results none :exports code
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K = stewart.Jf'*stewart.Jf;
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#+end_src
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#+begin_src matlab :results value table :exports results
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data = K;
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data2orgtable(data', {}, {}, ' %.2g ');
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#+end_src
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#+RESULTS:
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| 2 | 1.9e-18 | -2.3e-17 | 1.5e-18 | -0.1 | -1.5e-17 |
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| 1.9e-18 | 2 | 6.8e-18 | 0.1 | -1.6e-18 | 4.8e-18 |
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| -2.3e-17 | 6.8e-18 | 2 | -5.1e-19 | -5.5e-18 | 5.3e-19 |
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| 1.5e-18 | 0.1 | -5.1e-19 | 0.012 | -3e-19 | 3.1e-19 |
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| -0.1 | -1.6e-18 | -5.5e-18 | -3e-19 | 0.012 | 1.9e-18 |
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| -1.5e-17 | 4.8e-18 | 5.3e-19 | 3.1e-19 | 1.9e-18 | 0.027 |
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** Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
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Here, the "center" of the Stewart platform is not at the cube center (figure [[fig:3d-cubic-stewart-misaligned]]).
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The Jacobian is estimated at the cube center.
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#+name: fig:3d-cubic-stewart-misaligned
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#+caption: Not centered cubic configuration
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[[file:./figs/3d-cubic-stewart-misaligned.png]]
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The center of the cube is at $z = 110$.
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The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$.
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The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$.
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The center of the cube from the top platform is at $z = 110 - 175 = -65$.
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#+begin_src matlab :results silent
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opts = struct(...
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'H_tot', 100, ... % Total height of the Hexapod [mm]
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'L', 220/sqrt(3), ... % Size of the Cube [mm]
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'H', 60, ... % Height between base joints and platform joints [mm]
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'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm]
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);
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stewart = initializeCubicConfiguration(opts);
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opts = struct(...
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'Jd_pos', [0, 0, -65], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
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'Jf_pos', [0, 0, -65] ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
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);
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stewart = computeGeometricalProperties(stewart, opts);
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#+end_src
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#+begin_src matlab :results none :exports code
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K = stewart.Jf'*stewart.Jf;
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#+end_src
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#+begin_src matlab :results value table :exports results
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data = K;
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data2orgtable(data', {}, {}, ' %.2g ');
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#+end_src
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#+RESULTS:
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| 2 | -1.8e-17 | 2.6e-17 | 3.3e-18 | 0.04 | 1.7e-19 |
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| -1.8e-17 | 2 | 1.9e-16 | -0.04 | 2.2e-19 | -5.3e-19 |
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| 2.6e-17 | 1.9e-16 | 2 | -8.9e-18 | 6.5e-19 | -5.8e-19 |
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| 3.3e-18 | -0.04 | -8.9e-18 | 0.0089 | -9.3e-20 | 9.8e-20 |
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| 0.04 | 2.2e-19 | 6.5e-19 | -9.3e-20 | 0.0089 | -2.4e-18 |
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| 1.7e-19 | -5.3e-19 | -5.8e-19 | 9.8e-20 | -2.4e-18 | 0.032 |
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We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.
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** Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center
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Here, the "center" of the Stewart platform is not at the cube center.
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The Jacobian is estimated at the center of the Stewart platform.
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The center of the cube is at $z = 110$.
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The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$.
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The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$.
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The center of the cube from the top platform is at $z = 110 - 175 = -65$.
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#+begin_src matlab :results silent
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opts = struct(...
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'H_tot', 100, ... % Total height of the Hexapod [mm]
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'L', 220/sqrt(3), ... % Size of the Cube [mm]
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'H', 60, ... % Height between base joints and platform joints [mm]
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'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm]
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);
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stewart = initializeCubicConfiguration(opts);
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opts = struct(...
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'Jd_pos', [0, 0, -60], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
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'Jf_pos', [0, 0, -60] ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
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);
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stewart = computeGeometricalProperties(stewart, opts);
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#+end_src
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#+begin_src matlab :results none :exports code
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K = stewart.Jf'*stewart.Jf;
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#+end_src
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#+begin_src matlab :results value table :exports results
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data = K;
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data2orgtable(data', {}, {}, ' %.2g ');
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#+end_src
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#+RESULTS:
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| 2 | -1.8e-17 | 2.6e-17 | -5.7e-19 | 0.03 | 1.7e-19 |
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| -1.8e-17 | 2 | 1.9e-16 | -0.03 | 2.2e-19 | -5.3e-19 |
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| 2.6e-17 | 1.9e-16 | 2 | -1.5e-17 | 6.5e-19 | -5.8e-19 |
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| -5.7e-19 | -0.03 | -1.5e-17 | 0.0085 | 4.9e-20 | 1.7e-19 |
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| 0.03 | 2.2e-19 | 6.5e-19 | 4.9e-20 | 0.0085 | -1.1e-18 |
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| 1.7e-19 | -5.3e-19 | -5.8e-19 | 1.7e-19 | -1.1e-18 | 0.032 |
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We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.
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** Conclusion
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#+begin_important
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- The cubic configuration permits to have $k_x = k_y = k_z$ and $k_{\theta\x} = k_{\theta_y}$
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- The stiffness matrix $K$ is diagonal for the cubic configuration if the Stewart platform and the cube are centered *and* the Jacobian is estimated at the cube center
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#+end_important
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* TODO Cubic size analysis
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We here study the effect of the size of the cube used for the Stewart configuration.
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We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform.
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We only vary the size of the cube.
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#+begin_src matlab :results silent
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H_cubes = 250:20:350;
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stewarts = {zeros(length(H_cubes), 1)};
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#+end_src
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#+begin_src matlab :results silent
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for i = 1:length(H_cubes)
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H_cube = H_cubes(i);
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H_tot = 100;
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H = 80;
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opts = struct(...
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'H_tot', H_tot, ... % Total height of the Hexapod [mm]
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'L', H_cube/sqrt(3), ... % Size of the Cube [mm]
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'H', H, ... % Height between base joints and platform joints [mm]
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'H0', H_cube/2-H/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
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);
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stewart = initializeCubicConfiguration(opts);
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opts = struct(...
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'Jd_pos', [0, 0, H_cube/2-opts.H0-opts.H_tot], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
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'Jf_pos', [0, 0, H_cube/2-opts.H0-opts.H_tot] ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
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);
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stewart = computeGeometricalProperties(stewart, opts);
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stewarts(i) = {stewart};
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end
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#+end_src
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The Stiffness matrix is computed for all generated Stewart platforms.
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#+begin_src matlab :results none :exports code
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Ks = zeros(6, 6, length(H_cube));
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for i = 1:length(H_cubes)
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Ks(:, :, i) = stewarts{i}.Jd'*stewarts{i}.Jd;
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end
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#+end_src
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The only elements of $K$ that vary are $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$.
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Finally, we plot $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$
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#+begin_src matlab :results none :exports code
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figure;
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hold on;
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plot(H_cubes, squeeze(Ks(4, 4, :)), 'DisplayName', '$k_{\theta_x}$');
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plot(H_cubes, squeeze(Ks(6, 6, :)), 'DisplayName', '$k_{\theta_z}$');
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hold off;
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legend('location', 'northwest');
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xlabel('Cube Size [mm]'); ylabel('Rotational stiffnes [normalized]');
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#+end_src
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#+NAME: fig:stiffness_cube_size
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#+HEADER: :tangle no :exports results :results raw :noweb yes
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#+begin_src matlab :var filepath="figs/stiffness_cube_size.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+NAME: fig:stiffness_cube_size
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#+CAPTION: $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ function of the size of the cube
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#+RESULTS: fig:stiffness_cube_size
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[[file:figs/stiffness_cube_size.png]]
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We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linearly with the cube size.
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#+begin_important
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In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
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In that case, the legs will the further separated. Size of the cube is then limited by allowed space.
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#+end_important
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* Functions
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<<sec:functions>>
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** =generateCubicConfiguration=: Generate a Cubic Configuration
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:PROPERTIES:
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:header-args:matlab+: :tangle src/generateCubicConfiguration.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:generateCubicConfiguration>>
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This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here]].
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*** Function description
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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#+begin_src matlab
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function [stewart] = generateCubicConfiguration(stewart, args)
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% generateCubicConfiguration - Generate a Cubic Configuration
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%
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% Syntax: [stewart] = generateCubicConfiguration(stewart, args)
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%
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% Inputs:
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% - stewart - A structure with the following fields
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% - H [1x1] - Total height of the platform [m]
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% - args - Can have the following fields:
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% - Hc [1x1] - Height of the "useful" part of the cube [m]
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% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
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% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
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% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
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%
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% Outputs:
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% - stewart - updated Stewart structure with the added fields:
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% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
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% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
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#+end_src
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*** Documentation
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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#+name: fig:cubic-configuration-definition
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#+caption: Cubic Configuration
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[[file:figs/cubic-configuration-definition.png]]
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*** Optional Parameters
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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#+begin_src matlab
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arguments
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stewart
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args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
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args.FOc (1,1) double {mustBeNumeric} = 50e-3
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args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
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args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
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end
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#+end_src
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*** Position of the Cube
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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We define the useful points of the cube with respect to the Cube's center.
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${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is
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located at the center of the cube and aligned with {F} and {M}.
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#+begin_src matlab
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sx = [ 2; -1; -1];
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sy = [ 0; 1; -1];
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sz = [ 1; 1; 1];
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R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
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L = args.Hc*sqrt(3);
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Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];
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CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
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CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
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#+end_src
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*** Compute the pose
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).
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#+begin_src matlab
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CSi = (CCm - CCf)./vecnorm(CCm - CCf);
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#+end_src
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We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
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#+begin_src matlab
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stewart.Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
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stewart.Mb = CCf + [0; 0; args.FOc-stewart.H] + ((stewart.H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
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#+end_src
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* Bibliography :ignore:
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bibliographystyle:unsrtnat
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bibliography:ref.bib
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