Update study: cubic configuration, renew the function for generation

2019-03-25 18:12:43 +01:00 · 2019-03-25 18:12:43 +01:00 · 7fb03f7b90
commit 7fb03f7b90
parent 2914d01e8f
29 changed files with 3773 additions and 817 deletions
--- a/.gitignore
+++ b/.gitignore
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--- a/cubic-configuration.html
+++ b/cubic-configuration.html
--- a/cubic-configuration.org
+++ b/cubic-configuration.org
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+#+TITLE: Cubic configuration for the Stewart Platform
+:DRAWER:
+#+STARTUP: overview
+
+#+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 type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
+
+#+LATEX_CLASS: cleanreport
+#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
+#+LaTeX_HEADER: \usepackage{svg}
+#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
+#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
+#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
+
+#+PROPERTY: header-args:matlab  :session *MATLAB*
+#+PROPERTY: header-args:matlab+ :comments org
+#+PROPERTY: header-args:matlab+ :exports both
+#+PROPERTY: header-args:matlab+ :eval no-export
+#+PROPERTY: header-args:matlab+ :output-dir figs
+#+PROPERTY: header-args:matlab+ :mkdirp yes
+:END:
+
+#+begin_src matlab :results none :exports none :noweb yes
+  <<matlab-init>>
+  addpath('src');
+  addpath('library');
+#+end_src
+
+The discovery of the Cubic configuration is done in citenum:geng94_six_degree_of_freed_activ.
+The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
+
+To generate and study the Cubic configuration, =initializeCubicConfiguration= is used (description in section [[sec:initializeCubicConfiguration]]).
+
+* Questions we wish to answer with this analysis
+The goal is to study the benefits of using a cubic configuration:
+- Equal stiffness in all the degrees of freedom?
+- No coupling between the actuators?
+- Is the center of the cube an important point?
+
+* Configuration Analysis - Stiffness Matrix
+** Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
+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).
+The Jacobian matrix is estimated at the location of the center of the cube.
+
+#+name: fig:3d-cubic-stewart-aligned
+#+caption: Centered cubic configuration
+[[file:./figs/3d-cubic-stewart-aligned.png]]
+
+#+begin_src matlab :results silent
+  opts = struct(...
+      'H_tot', 100, ... % Total height of the Hexapod [mm]
+      'L',     200/sqrt(3), ... % Size of the Cube [mm]
+      'H',     60, ... % Height between base joints and platform joints [mm]
+      'H0',    200/2-60/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
+      );
+  stewart = initializeCubicConfiguration(opts);
+  opts = struct(...
+      'Jd_pos', [0, 0, -50], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
+      'Jf_pos', [0, 0, -50]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
+      );
+  stewart = computeGeometricalProperties(stewart, opts);
+
+  save('./mat/stewart.mat', 'stewart');
+#+end_src
+
+#+begin_src matlab :results none :exports code
+  K = stewart.Jd'*stewart.Jd;
+#+end_src
+
+#+begin_src matlab :results value table :exports results
+  data = K;
+  data2orgtable(data, {}, {}, ' %.2g ');
+#+end_src
+
+#+RESULTS:
+|        2 |  1.9e-18 | -2.3e-17 |  1.8e-18 |  5.5e-17 | -1.5e-17 |
+|  1.9e-18 |        2 |  6.8e-18 | -6.1e-17 | -1.6e-18 |  4.8e-18 |
+| -2.3e-17 |  6.8e-18 |        2 | -6.7e-18 |  4.9e-18 |  5.3e-19 |
+|  1.8e-18 | -6.1e-17 | -6.7e-18 |   0.0067 | -2.3e-20 | -6.1e-20 |
+|  5.5e-17 | -1.6e-18 |  4.9e-18 | -2.3e-20 |   0.0067 |    1e-18 |
+| -1.5e-17 |  4.8e-18 |  5.3e-19 | -6.1e-20 |    1e-18 |    0.027 |
+
+** Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
+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]]).
+The Jacobian matrix is not estimated at the location of the center of the cube.
+
+#+begin_src matlab :results silent
+  opts = struct(...
+      'H_tot', 100, ... % Total height of the Hexapod [mm]
+      'L',     200/sqrt(3), ... % Size of the Cube [mm]
+      'H',     60, ... % Height between base joints and platform joints [mm]
+      'H0',    200/2-60/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
+      );
+  stewart = initializeCubicConfiguration(opts);
+  opts = struct(...
+      'Jd_pos', [0, 0, 0], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
+      'Jf_pos', [0, 0, 0]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
+      );
+  stewart = computeGeometricalProperties(stewart, opts);
+#+end_src
+
+#+begin_src matlab :results none :exports code
+  K = stewart.Jd'*stewart.Jd;
+#+end_src
+
+#+begin_src matlab :results value table :exports results
+  data = K;
+  data2orgtable(data', {}, {}, ' %.2g ');
+#+end_src
+
+#+RESULTS:
+|        2 |  1.9e-18 | -2.3e-17 |  1.5e-18 |     -0.1 | -1.5e-17 |
+|  1.9e-18 |        2 |  6.8e-18 |      0.1 | -1.6e-18 |  4.8e-18 |
+| -2.3e-17 |  6.8e-18 |        2 | -5.1e-19 | -5.5e-18 |  5.3e-19 |
+|  1.5e-18 |      0.1 | -5.1e-19 |    0.012 |   -3e-19 |  3.1e-19 |
+|     -0.1 | -1.6e-18 | -5.5e-18 |   -3e-19 |    0.012 |  1.9e-18 |
+| -1.5e-17 |  4.8e-18 |  5.3e-19 |  3.1e-19 |  1.9e-18 |    0.027 |
+
+** Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
+Here, the "center" of the Stewart platform is not at the cube center (figure [[fig:3d-cubic-stewart-misaligned]]).
+The Jacobian is estimated at the cube center.
+
+#+name: fig:3d-cubic-stewart-misaligned
+#+caption: Not centered cubic configuration
+[[file:./figs/3d-cubic-stewart-misaligned.png]]
+
+The center of the cube is at $z = 110$.
+The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$.
+The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$.
+The center of the cube from the top platform is at $z = 110 - 175 = -65$.
+
+#+begin_src matlab :results silent
+  opts = struct(...
+      'H_tot', 100,         ... % Total height of the Hexapod [mm]
+      'L',     220/sqrt(3), ... % Size of the Cube [mm]
+      'H',     60,          ... % Height between base joints and platform joints [mm]
+      'H0',    75           ... % Height between the corner of the cube and the plane containing the base joints [mm]
+      );
+  stewart = initializeCubicConfiguration(opts);
+  opts = struct(...
+      'Jd_pos', [0, 0, -65], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
+      'Jf_pos', [0, 0, -65]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
+      );
+  stewart = computeGeometricalProperties(stewart, opts);
+#+end_src
+
+#+begin_src matlab :results none :exports code
+  K = stewart.Jd'*stewart.Jd;
+#+end_src
+
+#+begin_src matlab :results value table :exports results
+  data = K;
+  data2orgtable(data', {}, {}, ' %.2g ');
+#+end_src
+
+#+RESULTS:
+|        2 | -1.8e-17 |  2.6e-17 |  3.3e-18 |     0.04 |  1.7e-19 |
+| -1.8e-17 |        2 |  1.9e-16 |    -0.04 |  2.2e-19 | -5.3e-19 |
+|  2.6e-17 |  1.9e-16 |        2 | -8.9e-18 |  6.5e-19 | -5.8e-19 |
+|  3.3e-18 |    -0.04 | -8.9e-18 |   0.0089 | -9.3e-20 |  9.8e-20 |
+|     0.04 |  2.2e-19 |  6.5e-19 | -9.3e-20 |   0.0089 | -2.4e-18 |
+|  1.7e-19 | -5.3e-19 | -5.8e-19 |  9.8e-20 | -2.4e-18 |    0.032 |
+
+We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.
+
+** Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center
+Here, the "center" of the Stewart platform is not at the cube center.
+The Jacobian is estimated at the center of the Stewart platform.
+
+The center of the cube is at $z = 110$.
+The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$.
+The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$.
+The center of the cube from the top platform is at $z = 110 - 175 = -65$.
+
+#+begin_src matlab :results silent
+  opts = struct(...
+      'H_tot', 100, ... % Total height of the Hexapod [mm]
+      'L',     220/sqrt(3), ... % Size of the Cube [mm]
+      'H',     60, ... % Height between base joints and platform joints [mm]
+      'H0',    75 ... % Height between the corner of the cube and the plane containing the base joints [mm]
+      );
+  stewart = initializeCubicConfiguration(opts);
+  opts = struct(...
+      'Jd_pos', [0, 0, -60], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
+      'Jf_pos', [0, 0, -60]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
+      );
+  stewart = computeGeometricalProperties(stewart, opts);
+#+end_src
+
+#+begin_src matlab :results none :exports code
+  K = stewart.Jd'*stewart.Jd;
+#+end_src
+
+#+begin_src matlab :results value table :exports results
+  data = K;
+  data2orgtable(data', {}, {}, ' %.2g ');
+#+end_src
+
+#+RESULTS:
+|        2 | -1.8e-17 |  2.6e-17 | -5.7e-19 |     0.03 |  1.7e-19 |
+| -1.8e-17 |        2 |  1.9e-16 |    -0.03 |  2.2e-19 | -5.3e-19 |
+|  2.6e-17 |  1.9e-16 |        2 | -1.5e-17 |  6.5e-19 | -5.8e-19 |
+| -5.7e-19 |    -0.03 | -1.5e-17 |   0.0085 |  4.9e-20 |  1.7e-19 |
+|     0.03 |  2.2e-19 |  6.5e-19 |  4.9e-20 |   0.0085 | -1.1e-18 |
+|  1.7e-19 | -5.3e-19 | -5.8e-19 |  1.7e-19 | -1.1e-18 |    0.032 |
+
+We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.
+
+** Conclusion
+#+begin_important
+  - The cubic configuration permits to have $k_x = k_y = k_z$ and $k_{\theta\x} = k_{\theta_y}$
+  - 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
+#+end_important
+
+* Cubic size analysis
+We here study the effect of the size of the cube used for the Stewart configuration.
+
+We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform.
+
+We only vary the size of the cube.
+
+#+begin_src matlab :results silent
+  H_cubes = 250:20:350;
+  stewarts = {zeros(length(H_cubes), 1)};
+#+end_src
+
+#+begin_src matlab :results silent
+  for i = 1:length(H_cubes)
+    H_cube = H_cubes(i);
+    H_tot = 100;
+    H = 80;
+
+    opts = struct(...
+        'H_tot', H_tot, ... % Total height of the Hexapod [mm]
+        'L',     H_cube/sqrt(3), ... % Size of the Cube [mm]
+        'H',     H, ... % Height between base joints and platform joints [mm]
+        'H0',    H_cube/2-H/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
+        );
+    stewart = initializeCubicConfiguration(opts);
+
+    opts = struct(...
+        '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]
+        '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]
+        );
+    stewart = computeGeometricalProperties(stewart, opts);
+    stewarts(i) = {stewart};
+  end
+#+end_src
+
+
+The Stiffness matrix is computed for all generated Stewart platforms.
+#+begin_src matlab :results none :exports code
+  Ks = zeros(6, 6, length(H_cube));
+  for i = 1:length(H_cubes)
+    Ks(:, :, i) = stewarts{i}.Jd'*stewarts{i}.Jd;
+  end
+#+end_src
+
+The only elements of $K$ that vary are $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$.
+
+Finally, we plot $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$
+#+begin_src matlab :results none :exports code
+  figure;
+  hold on;
+  plot(H_cubes, squeeze(Ks(4, 4, :)), 'DisplayName', '$k_{\theta_x}$');
+  plot(H_cubes, squeeze(Ks(6, 6, :)), 'DisplayName', '$k_{\theta_z}$');
+  hold off;
+  legend('location', 'northwest');
+  xlabel('Cube Size [mm]'); ylabel('Rotational stiffnes [normalized]');
+#+end_src
+
+#+NAME: fig:stiffness_cube_size
+#+HEADER: :tangle no :exports results :results raw :noweb yes
+#+begin_src matlab :var filepath="figs/stiffness_cube_size.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:stiffness_cube_size
+#+CAPTION: $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ function of the size of the cube
+#+RESULTS: fig:stiffness_cube_size
+[[file:figs/stiffness_cube_size.png]]
+
+
+We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linearly with the cube size.
+
+#+begin_important
+  In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
+  In that case, the legs will the further separated. Size of the cube is then limited by allowed space.
+#+end_important
+
+* initializeCubicConfiguration
+  :PROPERTIES:
+  :HEADER-ARGS:matlab+: :exports code
+  :HEADER-ARGS:matlab+: :comments no
+  :HEADER-ARGS:matlab+: :eval no
+  :HEADER-ARGS:matlab+: :tangle src/initializeCubicConfiguration.m
+  :END:
+  <<sec:initializeCubicConfiguration>>
+
+** Function description
+#+begin_src matlab
+  function [stewart] = initializeCubicConfiguration(opts_param)
+#+end_src
+
+** Optional Parameters
+Default values for opts.
+#+begin_src matlab
+  opts = struct(...
+      'H_tot', 90,  ... % Total height of the Hexapod [mm]
+      'L',     110, ... % Size of the Cube [mm]
+      'H',     40,  ... % Height between base joints and platform joints [mm]
+      'H0',    75   ... % Height between the corner of the cube and the plane containing the base joints [mm]
+      );
+#+end_src
+
+Populate opts with input parameters
+#+begin_src matlab
+  if exist('opts_param','var')
+      for opt = fieldnames(opts_param)'
+          opts.(opt{1}) = opts_param.(opt{1});
+      end
+  end
+#+end_src
+
+** Cube Creation
+#+begin_src matlab :results none
+  points = [0, 0, 0; ...
+            0, 0, 1; ...
+            0, 1, 0; ...
+            0, 1, 1; ...
+            1, 0, 0; ...
+            1, 0, 1; ...
+            1, 1, 0; ...
+            1, 1, 1];
+  points = opts.L*points;
+#+end_src
+
+We create the rotation matrix to rotate the cube
+#+begin_src matlab :results none
+  sx = cross([1, 1, 1], [1 0 0]);
+  sx = sx/norm(sx);
+
+  sy = -cross(sx, [1, 1, 1]);
+  sy = sy/norm(sy);
+
+  sz = [1, 1, 1];
+  sz = sz/norm(sz);
+
+  R = [sx', sy', sz']';
+#+end_src
+
+We use to rotation matrix to rotate the cube
+#+begin_src matlab :results none
+  cube = zeros(size(points));
+  for i = 1:size(points, 1)
+    cube(i, :) = R * points(i, :)';
+  end
+#+end_src
+
+** Vectors of each leg
+#+begin_src matlab :results none
+  leg_indices = [3, 4; ...
+                 2, 4; ...
+                 2, 6; ...
+                 5, 6; ...
+                 5, 7; ...
+                 3, 7];
+#+end_src
+
+Vectors are:
+#+begin_src matlab :results none
+  legs = zeros(6, 3);
+  legs_start = zeros(6, 3);
+
+  for i = 1:6
+    legs(i, :) = cube(leg_indices(i, 2), :) - cube(leg_indices(i, 1), :);
+    legs_start(i, :) = cube(leg_indices(i, 1), :);
+  end
+#+end_src
+
+** Verification of Height of the Stewart Platform
+If the Stewart platform is not contained in the cube, throw an error.
+
+#+begin_src matlab :results none
+  Hmax = cube(4, 3) - cube(2, 3);
+  if opts.H0 < cube(2, 3)
+    error(sprintf('H0 is not high enought. Minimum H0 = %.1f', cube(2, 3)));
+  else if opts.H0 + opts.H > cube(4, 3)
+    error(sprintf('H0+H is too high. Maximum H0+H = %.1f', cube(4, 3)));
+    error('H0+H is too high');
+  end
+#+end_src
+
+** Determinate the location of the joints
+We now determine the location of the joints on the fixed platform w.r.t the fixed frame $\{A\}$.
+$\{A\}$ is fixed to the bottom of the base.
+#+begin_src matlab :results none
+  Aa = zeros(6, 3);
+  for i = 1:6
+    t = (opts.H0-legs_start(i, 3))/(legs(i, 3));
+    Aa(i, :) = legs_start(i, :) + t*legs(i, :);
+  end
+#+end_src
+
+And the location of the joints on the mobile platform with respect to $\{A\}$.
+#+begin_src matlab :results none
+  Ab = zeros(6, 3);
+  for i = 1:6
+    t = (opts.H0+opts.H-legs_start(i, 3))/(legs(i, 3));
+    Ab(i, :) = legs_start(i, :) + t*legs(i, :);
+  end
+#+end_src
+
+And the location of the joints on the mobile platform with respect to $\{B\}$.
+#+begin_src matlab :results none
+  Bb = zeros(6, 3);
+  Bb = Ab - (opts.H0 + opts.H_tot/2 + opts.H/2)*[0, 0, 1];
+#+end_src
+
+#+begin_src matlab :results none
+  h = opts.H0 + opts.H/2 - opts.H_tot/2;
+  Aa = Aa - h*[0, 0, 1];
+  Ab = Ab - h*[0, 0, 1];
+#+end_src
+
+** Returns Stewart Structure
+#+begin_src matlab :results none
+  stewart = struct();
+  stewart.Aa = Aa;
+  stewart.Ab = Ab;
+  stewart.Bb = Bb;
+  stewart.H_tot = opts.H_tot;
+end
+#+end_src
+
+* Tests
+** First attempt to parametrisation
+#+name: fig:stewart_bottom_plate
+#+caption: Schematic of the bottom plates with all the parameters
+[[file:./figs/stewart_bottom_plate.png]]
+
+The goal is to choose $\alpha$, $\beta$, $R_\text{leg, t}$ and $R_\text{leg, b}$ in such a way that the configuration is cubic.
+
+
+The configuration is cubic if:
+\[ \overrightarrow{a_i b_i} \cdot \overrightarrow{a_j b_j} = 0, \ \forall i, j = [1, \hdots, 6], i \ne j \]
+
+Lets express $a_i$, $b_i$ and $a_j$:
+\begin{equation*}
+  a_1 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 - \alpha) \\  R_{\text{leg,b}} \cos(120 - \alpha) \\ 0\end{bmatrix} ; \quad
+  a_2 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 + \alpha) \\  R_{\text{leg,b}} \cos(120 + \alpha) \\ 0\end{bmatrix} ; \quad
+\end{equation*}
+
+\begin{equation*}
+  b_1 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 - \beta) \\  R_{\text{leg,t}} \cos(120 - \beta\\ H\end{bmatrix} ; \quad
+  b_2 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 + \beta) \\  R_{\text{leg,t}} \cos(120 + \beta\\ H\end{bmatrix} ; \quad
+\end{equation*}
+
+\[ \overrightarrow{a_1 b_1} = b_1 - a_1 = \begin{bmatrix}R_{\text{leg}} \cos(120 - \alpha) \\  R_{\text{leg}} \cos(120 - \alpha) \\ 0\end{bmatrix}\]
+
+** Second attempt
+We start with the point of a cube in space:
+\begin{align*}
+  [0, 0, 0] ; \ [0, 0, 1]; \ ...
+\end{align*}
+
+We also want the cube to point upward:
+\[ [1, 1, 1] \Rightarrow [0, 0, 1] \]
+
+Then we have the direction of all the vectors expressed in the frame of the hexapod.
+
+#+begin_src matlab :results none
+  points = [0, 0, 0; ...
+            0, 0, 1; ...
+            0, 1, 0; ...
+            0, 1, 1; ...
+            1, 0, 0; ...
+            1, 0, 1; ...
+            1, 1, 0; ...
+            1, 1, 1];
+#+end_src
+
+#+begin_src matlab :results none
+  figure;
+  plot3(points(:,1), points(:,2), points(:,3), 'ko')
+#+end_src
+
+#+begin_src matlab :results none
+  sx = cross([1, 1, 1], [1 0 0]);
+  sx = sx/norm(sx);
+
+  sy = -cross(sx, [1, 1, 1]);
+  sy = sy/norm(sy);
+
+  sz = [1, 1, 1];
+  sz = sz/norm(sz);
+
+  R = [sx', sy', sz']';
+#+end_src
+
+#+begin_src matlab :results none
+  cube = zeros(size(points));
+  for i = 1:size(points, 1)
+    cube(i, :) = R * points(i, :)';
+  end
+#+end_src
+
+#+begin_src matlab :results none
+  figure;
+  hold on;
+  plot3(points(:,1), points(:,2), points(:,3), 'ko');
+  plot3(cube(:,1), cube(:,2), cube(:,3), 'ro');
+  hold off;
+#+end_src
+
+Now we plot the legs of the hexapod.
+#+begin_src matlab :results none
+  leg_indices = [3, 4; ...
+                 2, 4; ...
+                 2, 6; ...
+                 5, 6; ...
+                 5, 7; ...
+                 3, 7]
+
+  figure;
+  hold on;
+  for i = 1:6
+    plot3(cube(leg_indices(i, :),1), cube(leg_indices(i, :),2), cube(leg_indices(i, :),3), '-');
+  end
+  hold off;
+#+end_src
+
+Vectors are:
+#+begin_src matlab :results none
+  legs = zeros(6, 3);
+  legs_start = zeros(6, 3);
+
+  for i = 1:6
+    legs(i, :) = cube(leg_indices(i, 2), :) - cube(leg_indices(i, 1), :);
+    legs_start(i, :) = cube(leg_indices(i, 1), :)
+  end
+#+end_src
+
+We now have the orientation of each leg.
+
+We here want to see if the position of the "slice" changes something.
+
+Let's first estimate the maximum height of the Stewart platform.
+#+begin_src matlab :results none
+  Hmax = cube(4, 3) - cube(2, 3);
+#+end_src
+
+Let's then estimate the middle position of the platform
+#+begin_src matlab :results none
+  Hmid = cube(8, 3)/2;
+#+end_src
+
+** Generate the Stewart platform for a Cubic configuration
+
+First we defined the height of the Hexapod.
+#+begin_src matlab :results none
+  H = Hmax/2;
+#+end_src
+
+#+begin_src matlab :results none
+  Zs = 1.2*cube(2, 3); % Height of the fixed platform
+  Ze = Zs + H; % Height of the mobile platform
+#+end_src
+
+We now determine the location of the joints on the fixed platform.
+#+begin_src matlab :results none
+  Aa = zeros(6, 3);
+  for i = 1:6
+    t = (Zs-legs_start(i, 3))/(legs(i, 3));
+    Aa(i, :) = legs_start(i, :) + t*legs(i, :);
+  end
+#+end_src
+
+And the location of the joints on the mobile platform
+#+begin_src matlab :results none
+  Ab = zeros(6, 3);
+  for i = 1:6
+    t = (Ze-legs_start(i, 3))/(legs(i, 3));
+    Ab(i, :) = legs_start(i, :) + t*legs(i, :);
+  end
+#+end_src
+
+And we plot the legs.
+#+begin_src matlab :results none
+  figure;
+  hold on;
+  for i = 1:6
+    plot3([Ab(i, 1),Aa(i, 1)], [Ab(i, 2),Aa(i, 2)], [Ab(i, 3),Aa(i, 3)], 'k-');
+  end
+  hold off;
+  xlim([-1, 1]);
+  ylim([-1, 1]);
+  zlim([0, 2]);
+#+end_src
+
+* Bibliography                                                      :ignore:
+bibliographystyle:unsrt
+bibliography:references.bib
--- a/figs/3d-cubic-stewart-aligned.png
+++ b/figs/3d-cubic-stewart-aligned.png
--- a/figs/3d-cubic-stewart-misaligned.png
+++ b/figs/3d-cubic-stewart-misaligned.png
--- a/figs/stiffness_cube_size.pdf
+++ b/figs/stiffness_cube_size.pdf
--- a/figs/stiffness_cube_size.png
+++ b/figs/stiffness_cube_size.png
--- a/figs/stiffness_cube_size.svg
+++ b/figs/stiffness_cube_size.svg
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+</g>
+</svg>
--- a/figs/stiffness_cube_size.tex
+++ b/figs/stiffness_cube_size.tex
@ -0,0 +1,54 @@
+% This file was created by matlab2tikz.
+%
+\definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
+\definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
+%
+\begin{tikzpicture}
+
+\begin{axis}[%
+width=3.23in,
+height=1.99in,
+at={(0.528in,0.42in)},
+scale only axis,
+separate axis lines,
+every outer x axis line/.append style={black},
+every x tick label/.append style={font=\color{black}},
+every x tick/.append style={black},
+xmin=250,
+xmax=350,
+xlabel={Cube Size [mm]},
+every outer y axis line/.append style={black},
+every y tick label/.append style={font=\color{black}},
+every y tick/.append style={black},
+ymin=0,
+ymax=0.0816666666666492,
+ylabel={Rotational stiffnes [normalized]},
+axis background/.style={fill=white},
+xmajorgrids,
+ymajorgrids,
+legend style={at={(0.6,2.222)}, anchor=south west, legend cell align=left, align=left, draw=black}
+]
+\addplot [color=mycolor1, line width=1.5pt]
+  table[row sep=crcr]{%
+250	0.0106166666666923\\
+270	0.0123500000000263\\
+290	0.0142166666666412\\
+310	0.0162166666666508\\
+330	0.0183499999999981\\
+350	0.0206166666666832\\
+};
+\addlegendentry{$k_{\theta_x}$}
+
+\addplot [color=mycolor2, line width=1.5pt]
+  table[row sep=crcr]{%
+250	0.0416666666666856\\
+270	0.0486000000000217\\
+290	0.0560666666666521\\
+310	0.0640666666666903\\
+330	0.0726000000000226\\
+350	0.0816666666666492\\
+};
+\addlegendentry{$k_{\theta_z}$}
+
+\end{axis}
+\end{tikzpicture}%
--- a/identification.org
+++ b/identification.org
@ -25,7 +25,7 @@
 :END:

 * Identification
-#+begin_src matlab :results none :exports none
+#+begin_src matlab :results none :exports none :noweb yes
  <<matlab-init>>
  addpath('src');
  addpath('library');
@ -37,7 +37,11 @@

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

@ -45,92 +49,111 @@ The hexapod structure and Sample structure are initialized.
  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;
-  bode(G.G_cart(1, 1));
-  bode(G.G_cart(3, 3));
+  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;
-  bode(G.G_forc(1, 1));
-  bode(G.G_forc(2, 2));
-  bode(G.G_forc(3, 3));
-  bode(G.G_forc(4, 4));
-  bode(G.G_forc(5, 5));
-  bode(G.G_forc(6, 6));
+  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;
-  bode(G.G_forc(1, 1));
-  bode(G.G_forc(1, 2));
-  bode(G.G_forc(1, 3));
-  bode(G.G_forc(1, 4));
-  bode(G.G_forc(1, 5));
-  bode(G.G_forc(1, 6));
+  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;
-  bode(G.G_legs(1, 1));
-  bode(G.G_legs(2, 2));
-  bode(G.G_legs(3, 3));
-  bode(G.G_legs(4, 4));
-  bode(G.G_legs(5, 5));
-  bode(G.G_legs(6, 6));
+  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;
-  bode(G.G_legs(1, 1));
-  bode(G.G_legs(1, 2));
-  bode(G.G_legs(1, 3));
-  bode(G.G_legs(1, 4));
-  bode(G.G_legs(1, 5));
-  bode(G.G_legs(1, 6));
+  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;
-  bode(G.G_tran(1, 1));
-  bode(G.G_tran(2, 2));
-  bode(G.G_tran(3, 3));
+  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;
-  bode(G.G_tran(4, 4));
-  bode(G.G_tran(5, 5));
-  bode(G.G_tran(6, 6));
+  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;
-  bode(G.G_tran(1, 1));
-  bode(G.G_tran(2, 1));
-  bode(G.G_tran(3, 1));
+  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
@ -139,10 +162,12 @@ From a force applied in the Cartesian frame to a relative displacement of the mo
 #+begin_src matlab :results none
  figure;
  hold on;
-  bode(G.G_comp(1, 1));
-  bode(G.G_comp(2, 2));
-  bode(G.G_comp(3, 3));
+  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
@ -151,10 +176,12 @@ From a force applied on the Cartesian frame to the absolute displacement of the
 #+begin_src matlab :results none
  figure;
  hold on;
-  bode(G.G_iner(1, 1));
-  bode(G.G_iner(2, 2));
-  bode(G.G_iner(3, 3));
+  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
@ -228,11 +255,11 @@ We defined all the Input/Output names of the identified transfer function.

 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_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_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
--- a/index.html
+++ b/index.html
@ -3,7 +3,7 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2019-03-22 ven. 12:03 -->
+<!-- 2019-03-25 lun. 18:11 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <meta name="viewport" content="width=device-width, initial-scale=1" />
 <title>Stewart Platform Studies</title>
@ -253,36 +253,37 @@ for the JavaScript code in this tag.
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#org2b3b6a5">1. Simscape Model</a></li>
-<li><a href="#org5dc817d">2. Architecture Study</a></li>
-<li><a href="#orgccde31a">3. Motion Control</a></li>
+<li><a href="#org431e1f9">1. Simscape Model</a></li>
+<li><a href="#org27d326c">2. Architecture Study</a></li>
+<li><a href="#orgd097f1e">3. Motion Control</a></li>
 </ul>
 </div>
 </div>

-<div id="outline-container-org2b3b6a5" class="outline-2">
-<h2 id="org2b3b6a5"><span class="section-number-2">1</span> Simscape Model</h2>
+<div id="outline-container-org431e1f9" class="outline-2">
+<h2 id="org431e1f9"><span class="section-number-2">1</span> Simscape Model</h2>
 <div class="outline-text-2" id="text-1">
 <ul class="org-ul">
 <li><a href="simscape-model.html">Model of the Stewart Platform</a></li>
-<li><a href="identification.html">Identification</a></li>
+<li><a href="identification.html">Identification of the Simscape Model</a></li>
 </ul>
 </div>
 </div>

-<div id="outline-container-org5dc817d" class="outline-2">
-<h2 id="org5dc817d"><span class="section-number-2">2</span> Architecture Study</h2>
+<div id="outline-container-org27d326c" class="outline-2">
+<h2 id="org27d326c"><span class="section-number-2">2</span> Architecture Study</h2>
 <div class="outline-text-2" id="text-2">
 <ul class="org-ul">
 <li><a href="kinematic-study.html">Kinematic Study</a></li>
 <li><a href="stiffness-study.html">Stiffness Matrix Study</a></li>
 <li>Jacobian Study</li>
+<li><a href="cubic-configuration.html">Cubic Architecture</a></li>
 </ul>
 </div>
 </div>

-<div id="outline-container-orgccde31a" class="outline-2">
-<h2 id="orgccde31a"><span class="section-number-2">3</span> Motion Control</h2>
+<div id="outline-container-orgd097f1e" class="outline-2">
+<h2 id="orgd097f1e"><span class="section-number-2">3</span> Motion Control</h2>
 <div class="outline-text-2" id="text-3">
 <ul class="org-ul">
 <li>Active Damping</li>
@ -294,7 +295,7 @@ for the JavaScript code in this tag.
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Thomas Dehaeze</p>
-<p class="date">Created: 2019-03-22 ven. 12:03</p>
+<p class="date">Created: 2019-03-25 lun. 18:11</p>
 <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
 </div>
 </body>
--- a/index.org
+++ b/index.org
@ -26,12 +26,13 @@

 * Simscape Model
 - [[file:simscape-model.org][Model of the Stewart Platform]]
- [[file:identification.org][Identification]]
+- [[file:identification.org][Identification of the Simscape Model]]

 * Architecture Study
 - [[file:kinematic-study.org][Kinematic Study]]
 - [[file:stiffness-study.org][Stiffness Matrix Study]]
 - Jacobian Study
+- [[file:cubic-configuration.org][Cubic Architecture]]

 * Motion Control
 - Active Damping
--- a/kinematic-study.org
+++ b/kinematic-study.org
@ -1,4 +1,28 @@
 #+TITLE: Kinematic Study of the Stewart Platform
+:DRAWER:
+#+STARTUP: overview
+
+#+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 type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
+
+#+LATEX_CLASS: cleanreport
+#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
+#+LaTeX_HEADER: \usepackage{svg}
+#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
+#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
+#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
+
+#+PROPERTY: header-args:matlab  :session *MATLAB*
+#+PROPERTY: header-args:matlab+ :comments org
+#+PROPERTY: header-args:matlab+ :exports both
+#+PROPERTY: header-args:matlab+ :eval no-export
+#+PROPERTY: header-args:matlab+ :output-dir figs
+#+PROPERTY: header-args:matlab+ :mkdirp yes
+:END:

 * Functions
  :PROPERTIES:
--- a/mat/sample.mat
+++ b/mat/sample.mat
--- a/mat/stewart.mat
+++ b/mat/stewart.mat
--- a/references.bib
+++ b/references.bib
@ -0,0 +1,37 @@
+@inproceedings{abbas14_vibrat_stewar_platf,
+  author =       {Hussain Abbas and Huang Hai},
+  title =        {Vibration isolation concepts for non-cubic Stewart Platform
+                  using modal control},
+  booktitle =    {Proceedings of 2014 11th International Bhurban Conference on
+                  Applied Sciences \& Technology (IBCAST) Islamabad, Pakistan,
+                  14th - 18th January, 2014},
+  year =         2014,
+  pages =        {nil},
+  doi =          {10.1109/ibcast.2014.6778139},
+  url =          {https://doi.org/10.1109/ibcast.2014.6778139},
+  month =        1,
+}
+
+@book{taghirad13_paral,
+  author =       {Taghirad, Hamid},
+  title =        {Parallel robots : mechanics and control},
+  year =         2013,
+  publisher =    {CRC Press},
+  address =      {Boca Raton, FL},
+  isbn =         9781466555778,
+  keywords =     {favorite},
+}
+
+@article{geng94_six_degree_of_freed_activ,
+  author =       {Z.J. Geng and L.S. Haynes},
+  title =        {Six Degree-Of-Freedom Active Vibration Control Using the
+                  Stewart Platforms},
+  journal =      {IEEE Transactions on Control Systems Technology},
+  volume =       2,
+  number =       1,
+  pages =        {45-53},
+  year =         1994,
+  doi =          {10.1109/87.273110},
+  url =          {https://doi.org/10.1109/87.273110},
+  keywords =     {},
+}
--- a/simscape-model.html
+++ b/simscape-model.html
@ -3,7 +3,7 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2019-03-22 ven. 12:03 -->
+<!-- 2019-03-25 lun. 11:18 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <meta name="viewport" content="width=device-width, initial-scale=1" />
 <title>Stewart Platform - Simscape Model</title>
@ -275,42 +275,136 @@ for the JavaScript code in this tag.
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#org9a10766">1. Function description and arguments</a></li>
-<li><a href="#orgb6911a1">2. Initialization of the stewart structure</a></li>
-<li><a href="#org030aed6">3. Bottom Plate</a></li>
-<li><a href="#orged8012a">4. Top Plate</a></li>
-<li><a href="#orgc74617a">5. Legs</a></li>
-<li><a href="#org7cd2aa5">6. Ball Joints</a></li>
-<li><a href="#org1d76ed9">7. More parameters are initialized</a></li>
-<li><a href="#orge9faa26">8. Save the Stewart Structure</a></li>
-<li><a href="#orga207d03">9. initializeParameters Function</a></li>
-<li><a href="#org724c1a1">10. initializeSample</a></li>
+<li><a href="#org527cc13">1. initializeGeneralConfiguration</a>
+<ul>
+<li><a href="#orgea5f8f5">1.1. Function description</a></li>
+<li><a href="#org2db42cb">1.2. Optional Parameters</a></li>
+<li><a href="#org2f9279a">1.3. Geometry Description</a></li>
+<li><a href="#org1409cf0">1.4. Compute Aa and Ab</a></li>
+<li><a href="#orgb91c416">1.5. Returns Stewart Structure</a></li>
+</ul>
+</li>
+<li><a href="#orgc3aa910">2. computeGeometricalProperties</a>
+<ul>
+<li><a href="#org180196f">2.1. Function description</a></li>
+<li><a href="#org12cee4f">2.2. Optional Parameters</a></li>
+<li><a href="#org0010af5">2.3. Rotation matrices</a></li>
+<li><a href="#org98f4bad">2.4. Jacobian matrices</a></li>
+</ul>
+</li>
+<li><a href="#orgb3e53d1">3. initializeMechanicalElements</a>
+<ul>
+<li><a href="#orge7f185e">3.1. Function description</a></li>
+<li><a href="#org6bd219d">3.2. Optional Parameters</a></li>
+<li><a href="#org8d0d9c0">3.3. Bottom Plate</a></li>
+<li><a href="#org23fd88c">3.4. Top Plate</a></li>
+<li><a href="#org96d7dab">3.5. Legs</a></li>
+<li><a href="#org66df86f">3.6. Ball Joints</a></li>
+</ul>
+</li>
+<li><a href="#orgf3c4474">4. initializeSample</a>
+<ul>
+<li><a href="#org1ec4152">4.1. Function description</a></li>
+<li><a href="#orgcd3268d">4.2. Optional Parameters</a></li>
+<li><a href="#org29ee9ed">4.3. Save the Sample structure</a></li>
+</ul>
+</li>
 </ul>
 </div>
 </div>

-<div id="outline-container-org9a10766" class="outline-2">
-<h2 id="org9a10766"><span class="section-number-2">1</span> Function description and arguments</h2>
-<div class="outline-text-2" id="text-1">
 <p>
-The <code>initializeHexapod</code> function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
+Stewart platforms are generated in multiple steps.
 </p>
-<div class="org-src-container">
-<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeHexapod</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
-</pre>
+
+<p>
+First, geometrical parameters are defined:
+</p>
+<ul class="org-ul">
+<li>\({}^Aa_i\) - Position of the joints fixed to the fixed base w.r.t \(\{A\}\)</li>
+<li>\({}^Ab_i\) - Position of the joints fixed to the mobile platform w.r.t \(\{A\}\)</li>
+<li>\({}^Bb_i\) - Position of the joints fixed to the mobile platform w.r.t \(\{B\}\)</li>
+<li>\(H\) - Total height of the mobile platform</li>
+</ul>
+
+<p>
+These parameter are enough to determine all the kinematic properties of the platform like the Jacobian, stroke, stiffness, &#x2026;
+These geometrical parameters can be generated using different functions: <code>initializeCubicConfiguration</code> for cubic configuration or <code>initializeGeneralConfiguration</code> for more general configuration.
+</p>
+
+<p>
+A function <code>computeGeometricalProperties</code> is then used to compute:
+</p>
+<ul class="org-ul">
+<li>\(J_f\) - Jacobian matrix for the force location</li>
+<li>\(J_d\) - Jacobian matrix for displacement estimation</li>
+<li>\(R_m\) - Rotation matrices to position the leg vectors</li>
+</ul>
+
+<p>
+Then, geometrical parameters are computed for all the mechanical elements with the function <code>initializeMechanicalElements</code>:
+</p>
+<ul class="org-ul">
+<li>Shape of the platforms
+<ul class="org-ul">
+<li>External Radius</li>
+<li>Internal Radius</li>
+<li>Density</li>
+<li>Thickness</li>
+</ul></li>
+<li>Shape of the Legs
+<ul class="org-ul">
+<li>Radius</li>
+<li>Size of ball joint</li>
+<li>Density</li>
+</ul></li>
+</ul>
+
+<p>
+Other Parameters are defined for the Simscape simulation:
+</p>
+<ul class="org-ul">
+<li>Sample mass, volume and position (<code>initializeSample</code> function)</li>
+<li>Location of the inertial sensor</li>
+<li>Location of the point for the differential measurements</li>
+<li>Location of the Jacobian point for velocity/displacement computation</li>
+</ul>
+
+<div id="outline-container-org527cc13" class="outline-2">
+<h2 id="org527cc13"><span class="section-number-2">1</span> initializeGeneralConfiguration</h2>
+<div class="outline-text-2" id="text-1">
 </div>

+<div id="outline-container-orgea5f8f5" class="outline-3">
+<h3 id="orgea5f8f5"><span class="section-number-3">1.1</span> Function description</h3>
+<div class="outline-text-3" id="text-1-1">
+<p>
+The <code>initializeGeneralConfiguration</code> function takes one structure that contains configurations for the hexapod and returns one structure representing the Hexapod.
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeGeneralConfiguration</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org2db42cb" class="outline-3">
+<h3 id="org2db42cb"><span class="section-number-3">1.2</span> Optional Parameters</h3>
+<div class="outline-text-3" id="text-1-2">
 <p>
 Default values for opts.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
-    <span style="color: #CC9393;">'height'</span>,  <span style="color: #BFEBBF;">90</span>,    <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height of the platform [mm]</span>
-    <span style="color: #CC9393;">'density'</span>, <span style="color: #BFEBBF;">8000</span>,  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Density of the material used for the hexapod [kg/m3]</span>
-    <span style="color: #CC9393;">'k_ax'</span>,    <span style="color: #BFEBBF;">1e8</span>,   <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Stiffness of each actuator [N/m]</span>
-    <span style="color: #CC9393;">'c_ax'</span>,    <span style="color: #BFEBBF;">1000</span>,   <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Damping of each actuator [N/(m/s)]</span>
-    <span style="color: #CC9393;">'stroke'</span>,  <span style="color: #BFEBBF;">50e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Maximum stroke of each actuator [m]</span>
-    <span style="color: #CC9393;">'name',    'stewart'</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Name of the file</span>
+    <span style="color: #CC9393;">'H_tot'</span>,   <span style="color: #BFEBBF;">90</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height of the platform [mm]</span>
+    <span style="color: #CC9393;">'H_joint'</span>, <span style="color: #BFEBBF;">15</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height of the joints [mm]</span>
+    <span style="color: #CC9393;">'H_plate'</span>, <span style="color: #BFEBBF;">10</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Thickness of the fixed and mobile platforms [mm]</span>
+    <span style="color: #CC9393;">'R_bot'</span>,  <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Radius where the legs articulations are positionned [mm]</span>
+    <span style="color: #CC9393;">'R_top'</span>,  <span style="color: #BFEBBF;">80</span>,  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Radius where the legs articulations are positionned [mm]</span>
+    <span style="color: #CC9393;">'a_bot'</span>,  <span style="color: #BFEBBF;">10</span>,  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Angle Offset [deg]</span>
+    <span style="color: #CC9393;">'a_top'</span>,  <span style="color: #BFEBBF;">40</span>,  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Angle Offset [deg]</span>
+    <span style="color: #CC9393;">'da_top'</span>, <span style="color: #BFEBBF;">0</span>    <span style="text-decoration: underline;">...</span> % Angle Offset from <span style="color: #BFEBBF;">0</span> position [deg]
    <span style="color: #DCDCCC;">)</span>;
 </pre>
 </div>
@ -329,37 +423,263 @@ Populate opts with input parameters
 </div>
 </div>

-<div id="outline-container-orgb6911a1" class="outline-2">
-<h2 id="orgb6911a1"><span class="section-number-2">2</span> Initialization of the stewart structure</h2>
-<div class="outline-text-2" id="text-2">
-<p>
-We initialize the Stewart structure
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">stewart = struct<span style="color: #DCDCCC;">()</span>;
-</pre>
-</div>
+<div id="outline-container-org2f9279a" class="outline-3">
+<h3 id="org2f9279a"><span class="section-number-3">1.3</span> Geometry Description</h3>
+<div class="outline-text-3" id="text-1-3">

-<p>
-And we defined its total height.
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">stewart.H = opts.height; <span style="color: #7F9F7F;">% [mm]</span>
-</pre>
-</div>
-</div>
-</div>
-
-<div id="outline-container-org030aed6" class="outline-2">
-<h2 id="org030aed6"><span class="section-number-2">3</span> Bottom Plate</h2>
-<div class="outline-text-2" id="text-3">
-
-<div id="org3d7fe71" class="figure">
+<div id="orgc30ce24" class="figure">
 <p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
 </p>
 <p><span class="figure-number">Figure 1: </span>Schematic of the bottom plates with all the parameters</p>
 </div>
+</div>
+</div>

+<div id="outline-container-org1409cf0" class="outline-3">
+<h3 id="org1409cf0"><span class="section-number-3">1.4</span> Compute Aa and Ab</h3>
+<div class="outline-text-3" id="text-1-4">
+<p>
+We compute \([a_1, a_2, a_3, a_4, a_5, a_6]^T\) and \([b_1, b_2, b_3, b_4, b_5, b_6]^T\).
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Aa = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
+Ab = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
+Bb = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">3</span>
+    Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>opts.R_bot<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">-</span> opts.a_bot<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.R_bot<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">-</span> opts.a_bot<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.H_plate<span style="color: #7CB8BB;">+</span>opts.H_joint<span style="color: #DCDCCC;">]</span>;
+    Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>   = <span style="color: #DCDCCC;">[</span>opts.R_bot<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> opts.a_bot<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.R_bot<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> opts.a_bot<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.H_plate<span style="color: #7CB8BB;">+</span>opts.H_joint<span style="color: #DCDCCC;">]</span>;
+
+    Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>opts.R_top<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> opts.da_top <span style="color: #7CB8BB;">-</span> opts.a_top<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.R_top<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> opts.da_top <span style="color: #7CB8BB;">-</span> opts.a_top<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.H_tot <span style="color: #7CB8BB;">-</span> opts.H_plate <span style="color: #7CB8BB;">-</span> opts.H_joint<span style="color: #DCDCCC;">]</span>;
+    Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>   = <span style="color: #DCDCCC;">[</span>opts.R_top<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> opts.da_top <span style="color: #7CB8BB;">+</span> opts.a_top<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.R_top<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> opts.da_top <span style="color: #7CB8BB;">+</span> opts.a_top<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
+                   opts.H_tot <span style="color: #7CB8BB;">-</span> opts.H_plate <span style="color: #7CB8BB;">-</span> opts.H_joint<span style="color: #DCDCCC;">]</span>;
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+
+Bb = Ab <span style="color: #7CB8BB;">-</span> opts.H_tot<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">]</span>;
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgb91c416" class="outline-3">
+<h3 id="orgb91c416"><span class="section-number-3">1.5</span> Returns Stewart Structure</h3>
+<div class="outline-text-3" id="text-1-5">
+<div class="org-src-container">
+<pre class="src src-matlab">  stewart = struct<span style="color: #DCDCCC;">()</span>;
+  stewart.Aa = Aa;
+  stewart.Ab = Ab;
+  stewart.Bb = Bb;
+  stewart.H_tot = opts.H_tot;
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+</pre>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgc3aa910" class="outline-2">
+<h2 id="orgc3aa910"><span class="section-number-2">2</span> computeGeometricalProperties</h2>
+<div class="outline-text-2" id="text-2">
+</div>
+
+<div id="outline-container-org180196f" class="outline-3">
+<h3 id="org180196f"><span class="section-number-3">2.1</span> Function description</h3>
+<div class="outline-text-3" id="text-2-1">
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">computeGeometricalProperties</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span>, <span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org12cee4f" class="outline-3">
+<h3 id="org12cee4f"><span class="section-number-3">2.2</span> Optional Parameters</h3>
+<div class="outline-text-3" id="text-2-2">
+<p>
+Default values for opts.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
+    <span style="color: #CC9393;">'Jd_pos'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">30</span><span style="color: #BFEBBF;">]</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
+    <span style="color: #CC9393;">'Jf_pos'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">30</span><span style="color: #BFEBBF;">]</span>  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
+    <span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+
+<p>
+Populate opts with input parameters
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">if</span> exist<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'opts_param','var'</span><span style="color: #DCDCCC;">)</span>
+    <span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">opt</span> = <span style="color: #BFEBBF;">fieldnames</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">opts_param</span><span style="color: #DCDCCC;">)</span><span style="color: #BFEBBF;">'</span>
+        opts.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span> = opts_param.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
+    <span style="color: #F0DFAF; font-weight: bold;">end</span>
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org0010af5" class="outline-3">
+<h3 id="org0010af5"><span class="section-number-3">2.3</span> Rotation matrices</h3>
+<div class="outline-text-3" id="text-2-3">
+<p>
+We initialize \(l_i\) and \(\hat{s}_i\)
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">leg_length = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
+leg_vectors = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+
+<p>
+We compute \(b_i - a_i\), and then:
+</p>
+\begin{align*}
+  l_i       &= \left|b_i - a_i\right| \\
+  \hat{s}_i &= \frac{b_i - a_i}{l_i}
+\end{align*}
+
+<div class="org-src-container">
+<pre class="src src-matlab">legs = stewart.Ab <span style="color: #7CB8BB;">-</span> stewart.Aa;
+
+<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
+    leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span> = norm<span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
+    leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = legs<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">/</span> leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span>;
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+</pre>
+</div>
+
+<p>
+We compute rotation matrices to have the orientation of the legs.
+The rotation matrix transforms the \(z\) axis to the axis of the leg. The other axis are not important here.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">stewart.Rm = struct<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'R'</span>, eye<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
+
+<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
+  sx = cross<span style="color: #DCDCCC;">(</span>leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
+  sx = sx<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sx<span style="color: #DCDCCC;">)</span>;
+
+  sy = <span style="color: #7CB8BB;">-</span>cross<span style="color: #DCDCCC;">(</span>sx, leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
+  sy = sy<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sy<span style="color: #DCDCCC;">)</span>;
+
+  sz = leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
+  sz = sz<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sz<span style="color: #DCDCCC;">)</span>;
+
+  stewart.Rm<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span>.R = <span style="color: #DCDCCC;">[</span>sx', sy', sz'<span style="color: #DCDCCC;">]</span>;
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org98f4bad" class="outline-3">
+<h3 id="org98f4bad"><span class="section-number-3">2.4</span> Jacobian matrices</h3>
+<div class="outline-text-3" id="text-2-4">
+<p>
+Compute Jacobian Matrix
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">Jd = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span>;
+
+<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
+  Jd<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
+  Jd<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">4</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span> = cross<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">001</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>stewart.Bb<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span> <span style="color: #7CB8BB;">-</span> opts.Jd_pos<span style="color: #BFEBBF;">)</span>, leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+
+stewart.Jd = Jd;
+stewart.Jd_inv = inv<span style="color: #DCDCCC;">(</span>Jd<span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Jf = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span>;
+
+<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
+  Jf<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
+  Jf<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">4</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span> = cross<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">001</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>stewart.Bb<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span> <span style="color: #7CB8BB;">-</span> opts.Jf_pos<span style="color: #BFEBBF;">)</span>, leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+
+stewart.Jf = Jf;
+stewart.Jf_inv = inv<span style="color: #DCDCCC;">(</span>Jf<span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">end</span>
+</pre>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgb3e53d1" class="outline-2">
+<h2 id="orgb3e53d1"><span class="section-number-2">3</span> initializeMechanicalElements</h2>
+<div class="outline-text-2" id="text-3">
+</div>
+
+<div id="outline-container-orge7f185e" class="outline-3">
+<h3 id="orge7f185e"><span class="section-number-3">3.1</span> Function description</h3>
+<div class="outline-text-3" id="text-3-1">
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeMechanicalElements</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span>, <span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org6bd219d" class="outline-3">
+<h3 id="org6bd219d"><span class="section-number-3">3.2</span> Optional Parameters</h3>
+<div class="outline-text-3" id="text-3-2">
+<p>
+Default values for opts.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
+    <span style="color: #CC9393;">'thickness'</span>, <span style="color: #BFEBBF;">10</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Thickness of the base and platform [mm]</span>
+    <span style="color: #CC9393;">'density'</span>,   <span style="color: #BFEBBF;">1000</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Density of the material used for the hexapod [kg/m3]</span>
+    <span style="color: #CC9393;">'k_ax'</span>,      <span style="color: #BFEBBF;">1e8</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Stiffness of each actuator [N/m]</span>
+    <span style="color: #CC9393;">'c_ax'</span>,      <span style="color: #BFEBBF;">1000</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Damping of each actuator [N/(m/s)]</span>
+    <span style="color: #CC9393;">'stroke'</span>,    <span style="color: #BFEBBF;">50e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Maximum stroke of each actuator [m]</span>
+    <span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+
+<p>
+Populate opts with input parameters
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">if</span> exist<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'opts_param','var'</span><span style="color: #DCDCCC;">)</span>
+    <span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">opt</span> = <span style="color: #BFEBBF;">fieldnames</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">opts_param</span><span style="color: #DCDCCC;">)</span><span style="color: #BFEBBF;">'</span>
+        opts.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span> = opts_param.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
+    <span style="color: #F0DFAF; font-weight: bold;">end</span>
+<span style="color: #F0DFAF; font-weight: bold;">end</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org8d0d9c0" class="outline-3">
+<h3 id="org8d0d9c0"><span class="section-number-3">3.3</span> Bottom Plate</h3>
+<div class="outline-text-3" id="text-3-3">
+
+<div id="org38598b1" class="figure">
+<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
+</p>
+<p><span class="figure-number">Figure 2: </span>Schematic of the bottom plates with all the parameters</p>
+</div>

 <p>
 The bottom plate structure is initialized.
@ -382,16 +702,7 @@ BP.Rext = <span style="color: #BFEBBF;">150</span>; <span style="color: #7F9F7F;
 We define its thickness.
 </p>
 <div class="org-src-container">
-<pre class="src src-matlab">BP.H = <span style="color: #BFEBBF;">10</span>; <span style="color: #7F9F7F;">% Thickness of the Bottom Plate [mm]</span>
-</pre>
-</div>
-
-<p>
-At which radius legs will be fixed and with that angle offset.
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">BP.Rleg  = <span style="color: #BFEBBF;">100</span>; <span style="color: #7F9F7F;">% Radius where the legs articulations are positionned [mm]</span>
-BP.alpha = <span style="color: #BFEBBF;">10</span>;  <span style="color: #7F9F7F;">% Angle Offset [deg]</span>
+<pre class="src src-matlab">BP.H = opts.thickness; <span style="color: #7F9F7F;">% Thickness of the Bottom Plate [mm]</span>
 </pre>
 </div>

@ -429,9 +740,9 @@ The structure is added to the stewart structure
 </div>
 </div>

-<div id="outline-container-orged8012a" class="outline-2">
-<h2 id="orged8012a"><span class="section-number-2">4</span> Top Plate</h2>
-<div class="outline-text-2" id="text-4">
+<div id="outline-container-org23fd88c" class="outline-3">
+<h3 id="org23fd88c"><span class="section-number-3">3.4</span> Top Plate</h3>
+<div class="outline-text-3" id="text-3-4">
 <p>
 The top plate structure is initialized.
 </p>
@ -457,16 +768,6 @@ The thickness of the top plate.
 </pre>
 </div>

-<p>
-At which radius and angle are fixed the legs.
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">TP.Rleg   = <span style="color: #BFEBBF;">100</span>; <span style="color: #7F9F7F;">% Radius where the legs articulations are positionned [mm]</span>
-TP.alpha  = <span style="color: #BFEBBF;">20</span>; <span style="color: #7F9F7F;">% Angle [deg]</span>
-TP.dalpha = <span style="color: #BFEBBF;">0</span>; % Angle Offset from <span style="color: #BFEBBF;">0</span> position [deg]
-</pre>
-</div>
-
 <p>
 The density of its material.
 </p>
@ -501,17 +802,16 @@ The structure is added to the stewart structure
 </div>
 </div>

-<div id="outline-container-orgc74617a" class="outline-2">
-<h2 id="orgc74617a"><span class="section-number-2">5</span> Legs</h2>
-<div class="outline-text-2" id="text-5">
+<div id="outline-container-org96d7dab" class="outline-3">
+<h3 id="org96d7dab"><span class="section-number-3">3.5</span> Legs</h3>
+<div class="outline-text-3" id="text-3-5">

-<div id="orgc225133" class="figure">
+<div id="orga9ade83" class="figure">
 <p><img src="./figs/stewart_legs.png" alt="stewart_legs.png" />
 </p>
-<p><span class="figure-number">Figure 2: </span>Schematic for the legs of the Stewart platform</p>
+<p><span class="figure-number">Figure 3: </span>Schematic for the legs of the Stewart platform</p>
 </div>

-
 <p>
 The leg structure is initialized.
 </p>
@ -570,6 +870,29 @@ The radius of spheres representing the ball joints are defined.
 </pre>
 </div>

+<p>
+We estimate the length of the legs.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">legs = stewart.Ab <span style="color: #7CB8BB;">-</span> stewart.Aa;
+Leg.lenght = norm<span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">5</span>;
+</pre>
+</div>
+
+<p>
+Then the shape of the bottom leg is estimated
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">Leg.shape.bot = <span style="text-decoration: underline;">...</span>
+    <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>        <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
+     Leg.Rbot <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
+     Leg.Rbot Leg.lenght; <span style="text-decoration: underline;">...</span>
+     Leg.Rtop Leg.lenght; <span style="text-decoration: underline;">...</span>
+     Leg.Rtop <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>Leg.lenght; <span style="text-decoration: underline;">...</span>
+     <span style="color: #BFEBBF;">0</span>        <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>Leg.lenght<span style="color: #DCDCCC;">]</span>;
+</pre>
+</div>
+
 <p>
 The structure is added to the stewart structure
 </p>
@ -580,14 +903,14 @@ The structure is added to the stewart structure
 </div>
 </div>

-<div id="outline-container-org7cd2aa5" class="outline-2">
-<h2 id="org7cd2aa5"><span class="section-number-2">6</span> Ball Joints</h2>
-<div class="outline-text-2" id="text-6">
+<div id="outline-container-org66df86f" class="outline-3">
+<h3 id="org66df86f"><span class="section-number-3">3.6</span> Ball Joints</h3>
+<div class="outline-text-3" id="text-3-6">

-<div id="org7b92b11" class="figure">
+<div id="org250b20b" class="figure">
 <p><img src="./figs/stewart_ball_joints.png" alt="stewart_ball_joints.png" />
 </p>
-<p><span class="figure-number">Figure 3: </span>Schematic of the support for the ball joints</p>
+<p><span class="figure-number">Figure 4: </span>Schematic of the support for the ball joints</p>
 </div>

 <p>
@ -615,7 +938,7 @@ SP.c = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% [
 Its height is defined
 </p>
 <div class="org-src-container">
-<pre class="src src-matlab">SP.H = <span style="color: #BFEBBF;">15</span>; <span style="color: #7F9F7F;">% [mm]</span>
+<pre class="src src-matlab">SP.H = stewart.Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">-</span> BP.H; <span style="color: #7F9F7F;">% [mm]</span>
 </pre>
 </div>

@ -660,195 +983,74 @@ The structure is added to the Hexapod structure
 </div>
 </div>
 </div>
-
-<div id="outline-container-org1d76ed9" class="outline-2">
-<h2 id="org1d76ed9"><span class="section-number-2">7</span> More parameters are initialized</h2>
-<div class="outline-text-2" id="text-7">
-<div class="org-src-container">
-<pre class="src src-matlab">stewart = initializeParameters<span style="color: #DCDCCC;">(</span>stewart<span style="color: #DCDCCC;">)</span>;
-</pre>
-</div>
-</div>
 </div>

-<div id="outline-container-orge9faa26" class="outline-2">
-<h2 id="orge9faa26"><span class="section-number-2">8</span> Save the Stewart Structure</h2>
-<div class="outline-text-2" id="text-8">
-<div class="org-src-container">
-<pre class="src src-matlab">save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/stewart.mat', 'stewart'</span><span style="color: #DCDCCC;">)</span>
-</pre>
-</div>
-</div>
+<div id="outline-container-orgf3c4474" class="outline-2">
+<h2 id="orgf3c4474"><span class="section-number-2">4</span> initializeSample</h2>
+<div class="outline-text-2" id="text-4">
 </div>

-<div id="outline-container-orga207d03" class="outline-2">
-<h2 id="orga207d03"><span class="section-number-2">9</span> initializeParameters Function</h2>
-<div class="outline-text-2" id="text-9">
-<div class="org-src-container">
-<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeParameters</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span>
-</pre>
-</div>
-
-<p>
-We first compute \([a_1, a_2, a_3, a_4, a_5, a_6]^T\) and \([b_1, b_2, b_3, b_4, b_5, b_6]^T\).
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">stewart.Aa = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
-stewart.Ab = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
-stewart.Bb = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
-</pre>
-</div>
-
-<div class="org-src-container">
-<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">3</span>
-    stewart.Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">-</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">-</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.BP.H<span style="color: #7CB8BB;">+</span>stewart.SP.H<span style="color: #DCDCCC;">]</span>;
-    stewart.Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>   = <span style="color: #DCDCCC;">[</span>stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.BP.H<span style="color: #7CB8BB;">+</span>stewart.SP.H<span style="color: #DCDCCC;">]</span>;
-
-    stewart.Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">-</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">-</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.H <span style="color: #7CB8BB;">-</span> stewart.TP.H <span style="color: #7CB8BB;">-</span> stewart.SP.H<span style="color: #DCDCCC;">]</span>;
-    stewart.Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>   = <span style="color: #DCDCCC;">[</span>stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">+</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">+</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
-                           stewart.H <span style="color: #7CB8BB;">-</span> stewart.TP.H <span style="color: #7CB8BB;">-</span> stewart.SP.H<span style="color: #DCDCCC;">]</span>;
-<span style="color: #F0DFAF; font-weight: bold;">end</span>
-stewart.Bb = stewart.Ab <span style="color: #7CB8BB;">-</span> stewart.H<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">]</span>;
-</pre>
-</div>
-
-<p>
-Now, we compute the leg vectors \(\hat{s}_i\) and leg position \(l_i\):
-\[ b_i - a_i = l_i \hat{s}_i \]
-</p>
-
-<p>
-We initialize \(l_i\) and \(\hat{s}_i\)
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">leg_length = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
-leg_vectors = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
-</pre>
-</div>
-
-<p>
-We compute \(b_i - a_i\), and then:
-</p>
-\begin{align*}
-  l_i       &= \left|b_i - a_i\right| \\
-  \hat{s}_i &= \frac{b_i - a_i}{l_i}
-\end{align*}
-
-<div class="org-src-container">
-<pre class="src src-matlab">legs = stewart.Ab <span style="color: #7CB8BB;">-</span> stewart.Aa;
-
-<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
-    leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span> = norm<span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
-    leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = legs<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">/</span> leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span>;
-<span style="color: #F0DFAF; font-weight: bold;">end</span>
-</pre>
-</div>
-
-<p>
-Then the shape of the bottom leg is estimated
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">stewart.Leg.lenght = leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">5</span>;
-stewart.Leg.shape.bot = <span style="text-decoration: underline;">...</span>
-    <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>                <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
-     stewart.Leg.Rbot <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
-     stewart.Leg.Rbot stewart.Leg.lenght; <span style="text-decoration: underline;">...</span>
-     stewart.Leg.Rtop stewart.Leg.lenght; <span style="text-decoration: underline;">...</span>
-     stewart.Leg.Rtop <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>stewart.Leg.lenght; <span style="text-decoration: underline;">...</span>
-     <span style="color: #BFEBBF;">0</span>                <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>stewart.Leg.lenght<span style="color: #DCDCCC;">]</span>;
-</pre>
-</div>
-
-<p>
-We compute rotation matrices to have the orientation of the legs.
-The rotation matrix transforms the \(z\) axis to the axis of the leg. The other axis are not important here.
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">stewart.Rm = struct<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'R'</span>, eye<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
-
-<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
-  sx = cross<span style="color: #DCDCCC;">(</span>leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
-  sx = sx<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sx<span style="color: #DCDCCC;">)</span>;
-
-  sy = <span style="color: #7CB8BB;">-</span>cross<span style="color: #DCDCCC;">(</span>sx, leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
-  sy = sy<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sy<span style="color: #DCDCCC;">)</span>;
-
-  sz = leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
-  sz = sz<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sz<span style="color: #DCDCCC;">)</span>;
-
-  stewart.Rm<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span>.R = <span style="color: #DCDCCC;">[</span>sx', sy', sz'<span style="color: #DCDCCC;">]</span>;
-<span style="color: #F0DFAF; font-weight: bold;">end</span>
-</pre>
-</div>
-
-<p>
-Compute Jacobian Matrix
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab">J = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span>;
-
-<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
-  J<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
-  J<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">4</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span> = cross<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">001</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>stewart.Ab<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">-</span> stewart.H<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">]</span><span style="color: #BFEBBF;">)</span>, leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
-<span style="color: #F0DFAF; font-weight: bold;">end</span>
-
-stewart.J = J;
-stewart.Jinv = inv<span style="color: #DCDCCC;">(</span>J<span style="color: #DCDCCC;">)</span>;
-</pre>
-</div>
-
-<div class="org-src-container">
-<pre class="src src-matlab">stewart.K = stewart.Leg.k_ax<span style="color: #7CB8BB;">*</span>stewart.J'<span style="color: #7CB8BB;">*</span>stewart.J;
-</pre>
-</div>
-
-<div class="org-src-container">
-<pre class="src src-matlab">  <span style="color: #F0DFAF; font-weight: bold;">end</span>
-<span style="color: #F0DFAF; font-weight: bold;">end</span>
-</pre>
-</div>
-</div>
-</div>
-<div id="outline-container-org724c1a1" class="outline-2">
-<h2 id="org724c1a1"><span class="section-number-2">10</span> initializeSample</h2>
-<div class="outline-text-2" id="text-10">
+<div id="outline-container-org1ec4152" class="outline-3">
+<h3 id="org1ec4152"><span class="section-number-3">4.1</span> Function description</h3>
+<div class="outline-text-3" id="text-4-1">
 <div class="org-src-container">
 <pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[]</span> = <span style="color: #93E0E3;">initializeSample</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
-<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Default values for opts</span>
-    sample = struct<span style="color: #DCDCCC;">(</span> <span style="text-decoration: underline;">...</span>
-        <span style="color: #CC9393;">'radius'</span>,     <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% radius of the cylinder [mm]</span>
-        <span style="color: #CC9393;">'height'</span>,     <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% height of the cylinder [mm]</span>
-        <span style="color: #CC9393;">'mass'</span>,       <span style="color: #BFEBBF;">10</span>,  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% mass of the cylinder [kg]</span>
-        <span style="color: #CC9393;">'measheight'</span>, <span style="color: #BFEBBF;">50</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% measurement point z-offset [mm]</span>
-        <span style="color: #CC9393;">'offset'</span>,     <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span>,   <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% offset position of the sample [mm]</span>
-        <span style="color: #CC9393;">'color'</span>,      <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">9</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">]</span> <span style="text-decoration: underline;">...</span>
-        <span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+</div>
+</div>

-    <span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Populate opts with input parameters</span>
-    <span style="color: #F0DFAF; font-weight: bold;">if</span> exist<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'opts_param','var'</span><span style="color: #DCDCCC;">)</span>
-        <span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">opt</span> = <span style="color: #BFEBBF;">fieldnames</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">opts_param</span><span style="color: #DCDCCC;">)</span><span style="color: #BFEBBF;">'</span>
-            sample.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span> = opts_param.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
-        <span style="color: #F0DFAF; font-weight: bold;">end</span>
+<div id="outline-container-orgcd3268d" class="outline-3">
+<h3 id="orgcd3268d"><span class="section-number-3">4.2</span> Optional Parameters</h3>
+<div class="outline-text-3" id="text-4-2">
+<p>
+Default values for opts.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">sample = struct<span style="color: #DCDCCC;">(</span> <span style="text-decoration: underline;">...</span>
+    <span style="color: #CC9393;">'radius'</span>,     <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% radius of the cylinder [mm]</span>
+    <span style="color: #CC9393;">'height'</span>,     <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% height of the cylinder [mm]</span>
+    <span style="color: #CC9393;">'mass'</span>,       <span style="color: #BFEBBF;">10</span>,  <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% mass of the cylinder [kg]</span>
+    <span style="color: #CC9393;">'measheight'</span>, <span style="color: #BFEBBF;">50</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% measurement point z-offset [mm]</span>
+    <span style="color: #CC9393;">'offset'</span>,     <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span>,   <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% offset position of the sample [mm]</span>
+    <span style="color: #CC9393;">'color'</span>,      <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">9</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">]</span> <span style="text-decoration: underline;">...</span>
+    <span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+
+<p>
+Populate opts with input parameters
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">if</span> exist<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'opts_param','var'</span><span style="color: #DCDCCC;">)</span>
+    <span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">opt</span> = <span style="color: #BFEBBF;">fieldnames</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">opts_param</span><span style="color: #DCDCCC;">)</span><span style="color: #BFEBBF;">'</span>
+        sample.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span> = opts_param.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
    <span style="color: #F0DFAF; font-weight: bold;">end</span>
-
-    <span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Save</span>
-    save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/sample.mat', 'sample'</span><span style="color: #DCDCCC;">)</span>;
 <span style="color: #F0DFAF; font-weight: bold;">end</span>
 </pre>
 </div>
 </div>
 </div>
+
+<div id="outline-container-org29ee9ed" class="outline-3">
+<h3 id="org29ee9ed"><span class="section-number-3">4.3</span> Save the Sample structure</h3>
+<div class="outline-text-3" id="text-4-3">
+<div class="org-src-container">
+<pre class="src src-matlab">save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/sample.mat', 'sample'</span><span style="color: #DCDCCC;">)</span>;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">end</span>
+</pre>
+</div>
+</div>
+</div>
+</div>
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Thomas Dehaeze</p>
-<p class="date">Created: 2019-03-22 ven. 12:03</p>
+<p class="date">Created: 2019-03-25 lun. 11:18</p>
 <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
 </div>
 </body>
--- a/simscape-model.org
+++ b/simscape-model.org
@ -17,29 +17,73 @@
 #+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}

 #+PROPERTY: header-args:matlab  :session *MATLAB*
-#+PROPERTY: header-args:matlab+ :comments no
-#+PROPERTY: header-args:matlab+ :exports bode
-#+PROPERTY: header-args:matlab+ :eval no
+#+PROPERTY: header-args:matlab+ :comments org
+#+PROPERTY: header-args:matlab+ :exports both
+#+PROPERTY: header-args:matlab+ :eval no-export
 #+PROPERTY: header-args:matlab+ :output-dir figs
 #+PROPERTY: header-args:matlab+ :mkdirp yes
-#+PROPERTY: header-args:matlab+ :tangle src/initializeHexapod.m
 :END:

-* Function description and arguments
-The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
+Stewart platforms are generated in multiple steps.
+
+First, geometrical parameters are defined:
+- ${}^Aa_i$ - Position of the joints fixed to the fixed base w.r.t $\{A\}$
+- ${}^Ab_i$ - Position of the joints fixed to the mobile platform w.r.t $\{A\}$
+- ${}^Bb_i$ - Position of the joints fixed to the mobile platform w.r.t $\{B\}$
+- $H$ - Total height of the mobile platform
+
+These parameter are enough to determine all the kinematic properties of the platform like the Jacobian, stroke, stiffness, ...
+These geometrical parameters can be generated using different functions: =initializeCubicConfiguration= for cubic configuration or =initializeGeneralConfiguration= for more general configuration.
+
+A function =computeGeometricalProperties= is then used to compute:
+- $J_f$ - Jacobian matrix for the force location
+- $J_d$ - Jacobian matrix for displacement estimation
+- $R_m$ - Rotation matrices to position the leg vectors
+
+Then, geometrical parameters are computed for all the mechanical elements with the function =initializeMechanicalElements=:
+- Shape of the platforms
+  - External Radius
+  - Internal Radius
+  - Density
+  - Thickness
+- Shape of the Legs
+  - Radius
+  - Size of ball joint
+  - Density
+
+Other Parameters are defined for the Simscape simulation:
+- Sample mass, volume and position (=initializeSample= function)
+- Location of the inertial sensor
+- Location of the point for the differential measurements
+- Location of the Jacobian point for velocity/displacement computation
+
+* initializeGeneralConfiguration
+  :PROPERTIES:
+  :HEADER-ARGS:matlab+: :exports code
+  :HEADER-ARGS:matlab+: :comments no
+  :HEADER-ARGS:matlab+: :eval no
+  :HEADER-ARGS:matlab+: :tangle src/initializeGeneralConfiguration.m
+  :END:
+
+** Function description
+The =initializeGeneralConfiguration= function takes one structure that contains configurations for the hexapod and returns one structure representing the Hexapod.
+
 #+begin_src matlab
-  function [stewart] = initializeHexapod(opts_param)
+  function [stewart] = initializeGeneralConfiguration(opts_param)
 #+end_src

+** Optional Parameters
 Default values for opts.
 #+begin_src matlab
  opts = struct(...
-      'height',  90,    ... % Height of the platform [mm]
-      'density', 8000,  ... % Density of the material used for the hexapod [kg/m3]
-      'k_ax',    1e8,   ... % Stiffness of each actuator [N/m]
-      'c_ax',    1000,   ... % Damping of each actuator [N/(m/s)]
-      'stroke',  50e-6, ... % Maximum stroke of each actuator [m]
-      'name',    'stewart' ... % Name of the file
+      'H_tot',   90, ... % Height of the platform [mm]
+      'H_joint', 15, ... % Height of the joints [mm]
+      'H_plate', 10, ... % Thickness of the fixed and mobile platforms [mm]
+      'R_bot',  100, ... % Radius where the legs articulations are positionned [mm]
+      'R_top',  80,  ... % Radius where the legs articulations are positionned [mm]
+      'a_bot',  10,  ... % Angle Offset [deg]
+      'a_top',  40,  ... % Angle Offset [deg]
+      'da_top', 0    ... % Angle Offset from 0 position [deg]
      );
 #+end_src

@ -52,22 +96,190 @@ Populate opts with input parameters
  end
 #+end_src

-* Initialization of the stewart structure
-We initialize the Stewart structure
-#+begin_src matlab
-  stewart = struct();
-#+end_src
-
-And we defined its total height.
-#+begin_src matlab
-  stewart.H = opts.height; % [mm]
-#+end_src
-
-* Bottom Plate
+** Geometry Description
 #+name: fig:stewart_bottom_plate
 #+caption: Schematic of the bottom plates with all the parameters
 [[file:./figs/stewart_bottom_plate.png]]

+** Compute Aa and Ab
+We compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
+
+#+begin_src matlab
+  Aa = zeros(6, 3); % [mm]
+  Ab = zeros(6, 3); % [mm]
+  Bb = zeros(6, 3); % [mm]
+#+end_src
+
+#+begin_src matlab
+  for i = 1:3
+      Aa(2*i-1,:) = [opts.R_bot*cos( pi/180*(120*(i-1) - opts.a_bot) ), ...
+                     opts.R_bot*sin( pi/180*(120*(i-1) - opts.a_bot) ), ...
+                     opts.H_plate+opts.H_joint];
+      Aa(2*i,:)   = [opts.R_bot*cos( pi/180*(120*(i-1) + opts.a_bot) ), ...
+                     opts.R_bot*sin( pi/180*(120*(i-1) + opts.a_bot) ), ...
+                     opts.H_plate+opts.H_joint];
+
+      Ab(2*i-1,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
+                     opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
+                     opts.H_tot - opts.H_plate - opts.H_joint];
+      Ab(2*i,:)   = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
+                     opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
+                     opts.H_tot - opts.H_plate - opts.H_joint];
+  end
+
+  Bb = Ab - opts.H_tot*[0,0,1];
+#+end_src
+
+** Returns Stewart Structure
+#+begin_src matlab :results none
+  stewart = struct();
+  stewart.Aa = Aa;
+  stewart.Ab = Ab;
+  stewart.Bb = Bb;
+  stewart.H_tot = opts.H_tot;
+end
+#+end_src
+
+* computeGeometricalProperties
+  :PROPERTIES:
+  :HEADER-ARGS:matlab+: :exports code
+  :HEADER-ARGS:matlab+: :comments no
+  :HEADER-ARGS:matlab+: :eval no
+  :HEADER-ARGS:matlab+: :tangle src/computeGeometricalProperties.m
+  :END:
+
+** Function description
+#+begin_src matlab
+  function [stewart] = computeGeometricalProperties(stewart, opts_param)
+#+end_src
+
+** Optional Parameters
+Default values for opts.
+#+begin_src matlab
+  opts = struct(...
+      'Jd_pos', [0, 0, 30], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
+      'Jf_pos', [0, 0, 30]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
+      );
+#+end_src
+
+Populate opts with input parameters
+#+begin_src matlab
+  if exist('opts_param','var')
+      for opt = fieldnames(opts_param)'
+          opts.(opt{1}) = opts_param.(opt{1});
+      end
+  end
+#+end_src
+
+** Rotation matrices
+We initialize $l_i$ and $\hat{s}_i$
+#+begin_src matlab
+  leg_length = zeros(6, 1); % [mm]
+  leg_vectors = zeros(6, 3);
+#+end_src
+
+We compute $b_i - a_i$, and then:
+\begin{align*}
+  l_i       &= \left|b_i - a_i\right| \\
+  \hat{s}_i &= \frac{b_i - a_i}{l_i}
+\end{align*}
+
+#+begin_src matlab
+  legs = stewart.Ab - stewart.Aa;
+
+  for i = 1:6
+      leg_length(i) = norm(legs(i,:));
+      leg_vectors(i,:) = legs(i,:) / leg_length(i);
+  end
+#+end_src
+
+We compute rotation matrices to have the orientation of the legs.
+The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
+#+begin_src matlab
+  stewart.Rm = struct('R', eye(3));
+
+  for i = 1:6
+    sx = cross(leg_vectors(i,:), [1 0 0]);
+    sx = sx/norm(sx);
+
+    sy = -cross(sx, leg_vectors(i,:));
+    sy = sy/norm(sy);
+
+    sz = leg_vectors(i,:);
+    sz = sz/norm(sz);
+
+    stewart.Rm(i).R = [sx', sy', sz'];
+  end
+#+end_src
+
+** Jacobian matrices
+Compute Jacobian Matrix
+#+begin_src matlab
+  Jd = zeros(6);
+
+  for i = 1:6
+    Jd(i, 1:3) = leg_vectors(i, :);
+    Jd(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jd_pos), leg_vectors(i, :));
+  end
+
+  stewart.Jd = Jd;
+  stewart.Jd_inv = inv(Jd);
+#+end_src
+
+#+begin_src matlab
+  Jf = zeros(6);
+
+  for i = 1:6
+    Jf(i, 1:3) = leg_vectors(i, :);
+    Jf(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jf_pos), leg_vectors(i, :));
+  end
+
+  stewart.Jf = Jf;
+  stewart.Jf_inv = inv(Jf);
+#+end_src
+
+#+begin_src matlab
+  end
+#+end_src
+
+* initializeMechanicalElements
+  :PROPERTIES:
+  :HEADER-ARGS:matlab+: :exports code
+  :HEADER-ARGS:matlab+: :comments no
+  :HEADER-ARGS:matlab+: :eval no
+  :HEADER-ARGS:matlab+: :tangle src/initializeMechanicalElements.m
+  :END:
+
+** Function description
+#+begin_src matlab
+  function [stewart] = initializeMechanicalElements(stewart, opts_param)
+#+end_src
+
+** Optional Parameters
+Default values for opts.
+#+begin_src matlab
+  opts = struct(...
+      'thickness', 10, ... % Thickness of the base and platform [mm]
+      'density',   1000, ... % Density of the material used for the hexapod [kg/m3]
+      'k_ax',      1e8, ... % Stiffness of each actuator [N/m]
+      'c_ax',      1000, ... % Damping of each actuator [N/(m/s)]
+      'stroke',    50e-6  ... % Maximum stroke of each actuator [m]
+      );
+#+end_src
+
+Populate opts with input parameters
+#+begin_src matlab
+  if exist('opts_param','var')
+      for opt = fieldnames(opts_param)'
+          opts.(opt{1}) = opts_param.(opt{1});
+      end
+  end
+#+end_src
+
+** Bottom Plate
+#+name: fig:stewart_bottom_plate
+#+caption: Schematic of the bottom plates with all the parameters
+[[file:./figs/stewart_bottom_plate.png]]

 The bottom plate structure is initialized.
 #+begin_src matlab
@ -82,13 +294,7 @@ We defined its internal radius (if there is a hole in the bottom plate) and its

 We define its thickness.
 #+begin_src matlab
-  BP.H = 10; % Thickness of the Bottom Plate [mm]
-#+end_src
-
-At which radius legs will be fixed and with that angle offset.
-#+begin_src matlab
-  BP.Rleg  = 100; % Radius where the legs articulations are positionned [mm]
-  BP.alpha = 10;  % Angle Offset [deg]
+  BP.H = opts.thickness; % Thickness of the Bottom Plate [mm]
 #+end_src

 We defined the density of the material of the bottom plate.
@ -111,7 +317,7 @@ The structure is added to the stewart structure
  stewart.BP = BP;
 #+end_src

-* Top Plate
+** Top Plate
 The top plate structure is initialized.
 #+begin_src matlab
  TP = struct();
@ -128,13 +334,6 @@ The thickness of the top plate.
  TP.H = 10; % [mm]
 #+end_src

-At which radius and angle are fixed the legs.
-#+begin_src matlab
-  TP.Rleg   = 100; % Radius where the legs articulations are positionned [mm]
-  TP.alpha  = 20; % Angle [deg]
-  TP.dalpha = 0; % Angle Offset from 0 position [deg]
-#+end_src
-
 The density of its material.
 #+begin_src matlab
  TP.density = opts.density; % Density of the material [kg/m3]
@ -155,12 +354,11 @@ The structure is added to the stewart structure
  stewart.TP  = TP;
 #+end_src

-* Legs
+** Legs
 #+name: fig:stewart_legs
 #+caption: Schematic for the legs of the Stewart platform
 [[file:./figs/stewart_legs.png]]

-
 The leg structure is initialized.
 #+begin_src matlab
  Leg = struct();
@ -198,12 +396,29 @@ The radius of spheres representing the ball joints are defined.
  Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
 #+end_src

+We estimate the length of the legs.
+#+begin_src matlab
+  legs = stewart.Ab - stewart.Aa;
+  Leg.lenght = norm(legs(1,:))/1.5;
+#+end_src
+
+Then the shape of the bottom leg is estimated
+#+begin_src matlab
+  Leg.shape.bot = ...
+      [0        0; ...
+       Leg.Rbot 0; ...
+       Leg.Rbot Leg.lenght; ...
+       Leg.Rtop Leg.lenght; ...
+       Leg.Rtop 0.2*Leg.lenght; ...
+       0        0.2*Leg.lenght];
+#+end_src
+
 The structure is added to the stewart structure
 #+begin_src matlab
  stewart.Leg = Leg;
 #+end_src

-* Ball Joints
+** Ball Joints
 #+name: fig:stewart_ball_joints
 #+caption: Schematic of the support for the ball joints
 [[file:./figs/stewart_ball_joints.png]]
@ -223,7 +438,7 @@ We can define its rotational stiffness and damping. For now, we use perfect join

 Its height is defined
 #+begin_src matlab
-  SP.H = 15; % [mm]
+  SP.H = stewart.Aa(1, 3) - BP.H; % [mm]
 #+end_src

 Its radius is based on the radius on the sphere at the end of the legs.
@ -253,251 +468,46 @@ The structure is added to the Hexapod structure
  stewart.SP  = SP;
 #+end_src

-* More parameters are initialized
-#+begin_src matlab
-  stewart = initializeParameters(stewart);
-#+end_src
-
-* Save the Stewart Structure
-#+begin_src matlab
-  save('./mat/stewart.mat', 'stewart')
-#+end_src
-
-* initializeParameters Function                                    :noexport:
-  :PROPERTIES:
-  :HEADER-ARGS:matlab+: :tangle no
-  :END:
-#+begin_src matlab
-  function [stewart] = initializeParameters(stewart)
-#+end_src
-
-Computation of the position of the connection points on the base and moving platform
-We first initialize =pos_base= corresponding to $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and =pos_top= corresponding to $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
-#+begin_src matlab
-  stewart.pos_base = zeros(6, 3);
-  stewart.pos_top = zeros(6, 3);
-#+end_src
-
-We estimate the height between the ball joints of the bottom platform and of the top platform.
-#+begin_src matlab
-  height = stewart.H - stewart.BP.H - stewart.TP.H - 2*stewart.SP.H; % [mm]
-#+end_src
-
-#+begin_src matlab
-      for i = 1:3
-          % base points
-          angle_m_b = 120*(i-1) - stewart.BP.alpha;
-          angle_p_b = 120*(i-1) + stewart.BP.alpha;
-
-          stewart.pos_base(2*i-1,:) = [stewart.BP.Rleg*cos(angle_m_b), stewart.BP.Rleg*sin(angle_m_b), 0.0];
-          stewart.pos_base(2*i,:)   = [stewart.BP.Rleg*cos(angle_p_b), stewart.BP.Rleg*sin(angle_p_b), 0.0];
-
-          % top points
-          angle_m_t = 120*(i-1) - stewart.TP.alpha + stewart.TP.dalpha;
-          angle_p_t = 120*(i-1) + stewart.TP.alpha + stewart.TP.dalpha;
-
-          stewart.pos_top(2*i-1,:) = [stewart.TP.Rleg*cos(angle_m_t), stewart.TP.Rleg*sin(angle_m_t), height];
-          stewart.pos_top(2*i,:) = [stewart.TP.Rleg*cos(angle_p_t), stewart.TP.Rleg*sin(angle_p_t), height];
-      end
-
-      % permute pos_top points so that legs are end points of base and top points
-      stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
-      stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
-#+end_src
-
-leg vectors
-#+begin_src matlab
-      legs = stewart.pos_top - stewart.pos_base;
-      leg_length = zeros(6, 1);
-      leg_vectors = zeros(6, 3);
-      for i = 1:6
-          leg_length(i) = norm(legs(i,:));
-          leg_vectors(i,:)  = legs(i,:) / leg_length(i);
-      end
-
-      stewart.Leg.lenght = 1000*leg_length(1)/1.5;
-      stewart.Leg.shape.bot = [0 0; ...
-                          stewart.Leg.rad.bottom 0; ...
-                          stewart.Leg.rad.bottom stewart.Leg.lenght; ...
-                          stewart.Leg.rad.top stewart.Leg.lenght; ...
-                          stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
-                          0 0.2*stewart.Leg.lenght];
-#+end_src
-
-Calculate revolute and cylindrical axes
-#+begin_src matlab
-      rev1 = zeros(6, 3);
-      rev2 = zeros(6, 3);
-      cyl1 = zeros(6, 3);
-      for i = 1:6
-          rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
-          rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
-
-          rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
-          rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
-
-          cyl1(i,:) = leg_vectors(i,:);
-      end
-#+end_src
-
-Coordinate systems
-#+begin_src matlab
-      stewart.lower_leg = struct('rotation', eye(3));
-      stewart.upper_leg = struct('rotation', eye(3));
-
-      for i = 1:6
-          stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
-          stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
-      end
-#+end_src
-
-Position Matrix
-#+begin_src matlab
-      stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.h+stewart.Leg.sphere.top+stewart.SP.h.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
-#+end_src
-
-Compute Jacobian Matrix
-#+begin_src matlab
-      %         aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.h - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
-      bb = stewart.pos_top_tranform - (stewart.TP.h + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
-      bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
-      stewart.J = getJacobianMatrix(leg_vectors', bb');
-
-      stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
-  end
-#+end_src
-
-* initializeParameters Function
-#+begin_src matlab
-  function [stewart] = initializeParameters(stewart)
-#+end_src
-
-We first compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
-#+begin_src matlab
-  stewart.Aa = zeros(6, 3); % [mm]
-  stewart.Ab = zeros(6, 3); % [mm]
-  stewart.Bb = zeros(6, 3); % [mm]
-#+end_src
-
-#+begin_src matlab
-  for i = 1:3
-      stewart.Aa(2*i-1,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
-                             stewart.BP.Rleg*sin( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
-                             stewart.BP.H+stewart.SP.H];
-      stewart.Aa(2*i,:)   = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
-                             stewart.BP.Rleg*sin( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
-                             stewart.BP.H+stewart.SP.H];
-
-      stewart.Ab(2*i-1,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
-                             stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
-                             stewart.H - stewart.TP.H - stewart.SP.H];
-      stewart.Ab(2*i,:)   = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
-                             stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
-                             stewart.H - stewart.TP.H - stewart.SP.H];
-  end
-  stewart.Bb = stewart.Ab - stewart.H*[0,0,1];
-#+end_src
-
-Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
-\[ b_i - a_i = l_i \hat{s}_i \]
-
-We initialize $l_i$ and $\hat{s}_i$
-#+begin_src matlab
-  leg_length = zeros(6, 1); % [mm]
-  leg_vectors = zeros(6, 3);
-#+end_src
-
-We compute $b_i - a_i$, and then:
-\begin{align*}
-  l_i       &= \left|b_i - a_i\right| \\
-  \hat{s}_i &= \frac{b_i - a_i}{l_i}
-\end{align*}
-
-#+begin_src matlab
-  legs = stewart.Ab - stewart.Aa;
-
-  for i = 1:6
-      leg_length(i) = norm(legs(i,:));
-      leg_vectors(i,:) = legs(i,:) / leg_length(i);
-  end
-#+end_src
-
-Then the shape of the bottom leg is estimated
-#+begin_src matlab
-  stewart.Leg.lenght = leg_length(1)/1.5;
-  stewart.Leg.shape.bot = ...
-      [0                0; ...
-       stewart.Leg.Rbot 0; ...
-       stewart.Leg.Rbot stewart.Leg.lenght; ...
-       stewart.Leg.Rtop stewart.Leg.lenght; ...
-       stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
-       0                0.2*stewart.Leg.lenght];
-#+end_src
-
-We compute rotation matrices to have the orientation of the legs.
-The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
-#+begin_src matlab
-  stewart.Rm = struct('R', eye(3));
-
-  for i = 1:6
-    sx = cross(leg_vectors(i,:), [1 0 0]);
-    sx = sx/norm(sx);
-
-    sy = -cross(sx, leg_vectors(i,:));
-    sy = sy/norm(sy);
-
-    sz = leg_vectors(i,:);
-    sz = sz/norm(sz);
-
-    stewart.Rm(i).R = [sx', sy', sz'];
-  end
-#+end_src
-
-Compute Jacobian Matrix
-#+begin_src matlab
-  J = zeros(6);
-
-  for i = 1:6
-    J(i, 1:3) = leg_vectors(i, :);
-    J(i, 4:6) = cross(0.001*(stewart.Ab(i, :)- stewart.H*[0,0,1]), leg_vectors(i, :));
-  end
-
-  stewart.J = J;
-  stewart.Jinv = inv(J);
-#+end_src
-
-#+begin_src matlab
-  stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;
-#+end_src
-
-#+begin_src matlab
-    end
-  end
-#+end_src
 * initializeSample
  :PROPERTIES:
+  :HEADER-ARGS:matlab+: :exports code
+  :HEADER-ARGS:matlab+: :comments no
+  :HEADER-ARGS:matlab+: :eval no
  :HEADER-ARGS:matlab+: :tangle src/initializeSample.m
  :END:
+
+** Function description
 #+begin_src matlab
  function [] = initializeSample(opts_param)
-  %% Default values for opts
-      sample = struct( ...
-          'radius',     100, ... % radius of the cylinder [mm]
-          'height',     100, ... % height of the cylinder [mm]
-          'mass',       10,  ... % mass of the cylinder [kg]
-          'measheight', 50, ... % measurement point z-offset [mm]
-          'offset',     [0, 0, 0],   ... % offset position of the sample [mm]
-          'color',      [0.9 0.1 0.1] ...
-          );
+#+end_src

-      %% Populate opts with input parameters
-      if exist('opts_param','var')
-          for opt = fieldnames(opts_param)'
-              sample.(opt{1}) = opts_param.(opt{1});
-          end
+** Optional Parameters
+Default values for opts.
+#+begin_src matlab
+  sample = struct( ...
+      'radius',     100, ... % radius of the cylinder [mm]
+      'height',     100, ... % height of the cylinder [mm]
+      'mass',       10,  ... % mass of the cylinder [kg]
+      'measheight', 50, ... % measurement point z-offset [mm]
+      'offset',     [0, 0, 0],   ... % offset position of the sample [mm]
+      'color',      [0.9 0.1 0.1] ...
+      );
+#+end_src
+
+Populate opts with input parameters
+#+begin_src matlab
+  if exist('opts_param','var')
+      for opt = fieldnames(opts_param)'
+          sample.(opt{1}) = opts_param.(opt{1});
      end
-
-      %% Save
-      save('./mat/sample.mat', 'sample');
+  end
+#+end_src
+
+** Save the Sample structure
+#+begin_src matlab
+  save('./mat/sample.mat', 'sample');
+#+end_src
+
+#+begin_src matlab
  end
 #+end_src
--- a/src/computeGeometricalProperties.m
+++ b/src/computeGeometricalProperties.m
@ -0,0 +1,59 @@
+function [stewart] = computeGeometricalProperties(stewart, opts_param)
+
+opts = struct(...
+    'Jd_pos', [0, 0, 30], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
+    'Jf_pos', [0, 0, 30]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
+    );
+
+if exist('opts_param','var')
+    for opt = fieldnames(opts_param)'
+        opts.(opt{1}) = opts_param.(opt{1});
+    end
+end
+
+leg_length = zeros(6, 1); % [mm]
+leg_vectors = zeros(6, 3);
+
+legs = stewart.Ab - stewart.Aa;
+
+for i = 1:6
+    leg_length(i) = norm(legs(i,:));
+    leg_vectors(i,:) = legs(i,:) / leg_length(i);
+end
+
+stewart.Rm = struct('R', eye(3));
+
+for i = 1:6
+  sx = cross(leg_vectors(i,:), [1 0 0]);
+  sx = sx/norm(sx);
+
+  sy = -cross(sx, leg_vectors(i,:));
+  sy = sy/norm(sy);
+
+  sz = leg_vectors(i,:);
+  sz = sz/norm(sz);
+
+  stewart.Rm(i).R = [sx', sy', sz'];
+end
+
+Jd = zeros(6);
+
+for i = 1:6
+  Jd(i, 1:3) = leg_vectors(i, :);
+  Jd(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jd_pos), leg_vectors(i, :));
+end
+
+stewart.Jd = Jd;
+stewart.Jd_inv = inv(Jd);
+
+Jf = zeros(6);
+
+for i = 1:6
+  Jf(i, 1:3) = leg_vectors(i, :);
+  Jf(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jf_pos), leg_vectors(i, :));
+end
+
+stewart.Jf = Jf;
+stewart.Jf_inv = inv(Jf);
+
+end
--- a/src/identifyPlant.m
+++ b/src/identifyPlant.m
@ -53,7 +53,7 @@ G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
 % identifyPlant:7 ends here

 % [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:8]]
-sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}));
+sys.G_cart = 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'}));
--- a/src/initializeCubicConfiguration.m
+++ b/src/initializeCubicConfiguration.m
@ -0,0 +1,89 @@
+function [stewart] = initializeCubicConfiguration(opts_param)
+
+opts = struct(...
+    'H_tot', 90,  ... % Total height of the Hexapod [mm]
+    'L',     110, ... % Size of the Cube [mm]
+    'H',     40,  ... % Height between base joints and platform joints [mm]
+    'H0',    75   ... % Height between the corner of the cube and the plane containing the base joints [mm]
+    );
+
+if exist('opts_param','var')
+    for opt = fieldnames(opts_param)'
+        opts.(opt{1}) = opts_param.(opt{1});
+    end
+end
+
+points = [0, 0, 0; ...
+          0, 0, 1; ...
+          0, 1, 0; ...
+          0, 1, 1; ...
+          1, 0, 0; ...
+          1, 0, 1; ...
+          1, 1, 0; ...
+          1, 1, 1];
+points = opts.L*points;
+
+sx = cross([1, 1, 1], [1 0 0]);
+sx = sx/norm(sx);
+
+sy = -cross(sx, [1, 1, 1]);
+sy = sy/norm(sy);
+
+sz = [1, 1, 1];
+sz = sz/norm(sz);
+
+R = [sx', sy', sz']';
+
+cube = zeros(size(points));
+for i = 1:size(points, 1)
+  cube(i, :) = R * points(i, :)';
+end
+
+leg_indices = [3, 4; ...
+               2, 4; ...
+               2, 6; ...
+               5, 6; ...
+               5, 7; ...
+               3, 7];
+
+legs = zeros(6, 3);
+legs_start = zeros(6, 3);
+
+for i = 1:6
+  legs(i, :) = cube(leg_indices(i, 2), :) - cube(leg_indices(i, 1), :);
+  legs_start(i, :) = cube(leg_indices(i, 1), :);
+end
+
+Hmax = cube(4, 3) - cube(2, 3);
+if opts.H0 < cube(2, 3)
+  error(sprintf('H0 is not high enought. Minimum H0 = %.1f', cube(2, 3)));
+else if opts.H0 + opts.H > cube(4, 3)
+  error(sprintf('H0+H is too high. Maximum H0+H = %.1f', cube(4, 3)));
+  error('H0+H is too high');
+end
+
+Aa = zeros(6, 3);
+for i = 1:6
+  t = (opts.H0-legs_start(i, 3))/(legs(i, 3));
+  Aa(i, :) = legs_start(i, :) + t*legs(i, :);
+end
+
+Ab = zeros(6, 3);
+for i = 1:6
+  t = (opts.H0+opts.H-legs_start(i, 3))/(legs(i, 3));
+  Ab(i, :) = legs_start(i, :) + t*legs(i, :);
+end
+
+Bb = zeros(6, 3);
+Bb = Ab - (opts.H0 + opts.H_tot/2 + opts.H/2)*[0, 0, 1];
+
+h = opts.H0 + opts.H/2 - opts.H_tot/2;
+Aa = Aa - h*[0, 0, 1];
+Ab = Ab - h*[0, 0, 1];
+
+stewart = struct();
+  stewart.Aa = Aa;
+  stewart.Ab = Ab;
+  stewart.Bb = Bb;
+  stewart.H_tot = opts.H_tot;
+end
--- a/src/initializeGeneralConfiguration.m
+++ b/src/initializeGeneralConfiguration.m
@ -0,0 +1,47 @@
+function [stewart] = initializeGeneralConfiguration(opts_param)
+
+opts = struct(...
+    'H_tot',   90, ... % Height of the platform [mm]
+    'H_joint', 15, ... % Height of the joints [mm]
+    'H_plate', 10, ... % Thickness of the fixed and mobile platforms [mm]
+    'R_bot',  100, ... % Radius where the legs articulations are positionned [mm]
+    'R_top',  80,  ... % Radius where the legs articulations are positionned [mm]
+    'a_bot',  10,  ... % Angle Offset [deg]
+    'a_top',  40,  ... % Angle Offset [deg]
+    'da_top', 0    ... % Angle Offset from 0 position [deg]
+    );
+
+if exist('opts_param','var')
+    for opt = fieldnames(opts_param)'
+        opts.(opt{1}) = opts_param.(opt{1});
+    end
+end
+
+Aa = zeros(6, 3); % [mm]
+Ab = zeros(6, 3); % [mm]
+Bb = zeros(6, 3); % [mm]
+
+for i = 1:3
+    Aa(2*i-1,:) = [opts.R_bot*cos( pi/180*(120*(i-1) - opts.a_bot) ), ...
+                   opts.R_bot*sin( pi/180*(120*(i-1) - opts.a_bot) ), ...
+                   opts.H_plate+opts.H_joint];
+    Aa(2*i,:)   = [opts.R_bot*cos( pi/180*(120*(i-1) + opts.a_bot) ), ...
+                   opts.R_bot*sin( pi/180*(120*(i-1) + opts.a_bot) ), ...
+                   opts.H_plate+opts.H_joint];
+
+    Ab(2*i-1,:) = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
+                   opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top - opts.a_top) ), ...
+                   opts.H_tot - opts.H_plate - opts.H_joint];
+    Ab(2*i,:)   = [opts.R_top*cos( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
+                   opts.R_top*sin( pi/180*(120*(i-1) + opts.da_top + opts.a_top) ), ...
+                   opts.H_tot - opts.H_plate - opts.H_joint];
+end
+
+Bb = Ab - opts.H_tot*[0,0,1];
+
+stewart = struct();
+  stewart.Aa = Aa;
+  stewart.Ab = Ab;
+  stewart.Bb = Bb;
+  stewart.H_tot = opts.H_tot;
+end
--- a/src/initializeHexapod.m
+++ b/src/initializeHexapod.m
@ -1,228 +1,86 @@
-% Function description and arguments
-% The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
-
 function [stewart] = initializeHexapod(opts_param)

-
-
-% Default values for opts.
-
 opts = struct(...
    'height',  90,    ... % Height of the platform [mm]
-    'density', 8000,  ... % Density of the material used for the hexapod [kg/m3]
+    'density', 10,  ... % Density of the material used for the hexapod [kg/m3]
    'k_ax',    1e8,   ... % Stiffness of each actuator [N/m]
    'c_ax',    1000,   ... % Damping of each actuator [N/(m/s)]
    'stroke',  50e-6, ... % Maximum stroke of each actuator [m]
    'name',    'stewart' ... % Name of the file
    );

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

-% Initialization of the stewart structure
-% We initialize the Stewart structure
-
 stewart = struct();

-
-
-% And we defined its total height.
-
 stewart.H = opts.height; % [mm]

-% Bottom Plate
-% #+name: fig:stewart_bottom_plate
-% #+caption: Schematic of the bottom plates with all the parameters
-% [[file:./figs/stewart_bottom_plate.png]]
-
-
-% The bottom plate structure is initialized.
-
 BP = struct();

-
-
-% We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
-
 BP.Rint = 0;   % Internal Radius [mm]
 BP.Rext = 150; % External Radius [mm]

-
-
-% We define its thickness.
-
 BP.H = 10; % Thickness of the Bottom Plate [mm]

-
-
-% At which radius legs will be fixed and with that angle offset.
-
 BP.Rleg  = 100; % Radius where the legs articulations are positionned [mm]
-BP.alpha = 10;  % Angle Offset [deg]
-
-
-
-% We defined the density of the material of the bottom plate.
+BP.alpha = 30;  % Angle Offset [deg]

 BP.density = opts.density; % Density of the material [kg/m3]

-
-
-% And its color.
-
 BP.color = [0.7 0.7 0.7]; % Color [RGB]

-
-
-% Then the profile of the bottom plate is computed and will be used by Simscape
-
 BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]

-
-
-% The structure is added to the stewart structure
-
 stewart.BP = BP;

-% Top Plate
-% The top plate structure is initialized.
-
 TP = struct();

-
-
-% We defined the internal and external radius of the top plate.
-
 TP.Rint = 0;   % [mm]
 TP.Rext = 100; % [mm]

-
-
-% The thickness of the top plate.
-
 TP.H = 10; % [mm]

-
-
-% At which radius and angle are fixed the legs.
-
-TP.Rleg   = 100; % Radius where the legs articulations are positionned [mm]
-TP.alpha  = 20; % Angle [deg]
+TP.Rleg   = 80; % Radius where the legs articulations are positionned [mm]
+TP.alpha  = 10; % Angle [deg]
 TP.dalpha = 0; % Angle Offset from 0 position [deg]

-
-
-% The density of its material.
-
 TP.density = opts.density; % Density of the material [kg/m3]

-
-
-% Its color.
-
 TP.color = [0.7 0.7 0.7]; % Color [RGB]

-
-
-% Then the shape of the top plate is computed
-
 TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];

-
-
-% The structure is added to the stewart structure
-
 stewart.TP  = TP;

-% Legs
-% #+name: fig:stewart_legs
-% #+caption: Schematic for the legs of the Stewart platform
-% [[file:./figs/stewart_legs.png]]
-
-
-% The leg structure is initialized.
-
 Leg = struct();

-
-
-% The maximum Stroke of each leg is defined.
-
 Leg.stroke = opts.stroke; % [m]

-
-
-% The stiffness and damping of each leg are defined
-
 Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
 Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]

-
-
-% The radius of the legs are defined
-
 Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
 Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]

-
-
-% The density of its material.
-
-Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
-
-
-
-% Its color.
+Leg.density = 0.01*opts.density; % Density of the material used for the legs [kg/m3]

 Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]

-
-
-% The radius of spheres representing the ball joints are defined.
-
 Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]

-
-
-% The structure is added to the stewart structure
-
 stewart.Leg = Leg;

-% Ball Joints
-% #+name: fig:stewart_ball_joints
-% #+caption: Schematic of the support for the ball joints
-% [[file:./figs/stewart_ball_joints.png]]
-
-% =SP= is the structure representing the support for the ball joints at the extremity of each leg.
-
-% The =SP= structure is initialized.
-
 SP = struct();

-
-
-% We can define its rotational stiffness and damping. For now, we use perfect joints.
-
 SP.k = 0; % [N*m/deg]
 SP.c = 0; % [N*m/deg]

-
-
-% Its height is defined
-
 SP.H = 15; % [mm]

-
-
-% Its radius is based on the radius on the sphere at the end of the legs.
-
 SP.R = Leg.R; % [mm]

 SP.section = [0    SP.H-SP.R;
@ -230,40 +88,18 @@ SP.section = [0    SP.H-SP.R;
              SP.R 0;
              SP.R SP.H];

-
-
-% The density of its material is defined.
-
 SP.density = opts.density; % [kg/m^3]

-
-
-% Its color is defined.
-
 SP.color = [0.7 0.7 0.7]; % [RGB]

-
-
-% The structure is added to the Hexapod structure
-
 stewart.SP  = SP;

-% More parameters are initialized
-
 stewart = initializeParameters(stewart);

-% Save the Stewart Structure
-
 save('./mat/stewart.mat', 'stewart')

-% initializeParameters Function
-
 function [stewart] = initializeParameters(stewart)

-
-
-% We first compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
-
 stewart.Aa = zeros(6, 3); % [mm]
 stewart.Ab = zeros(6, 3); % [mm]
 stewart.Bb = zeros(6, 3); % [mm]
@ -285,25 +121,9 @@ for i = 1:3
 end
 stewart.Bb = stewart.Ab - stewart.H*[0,0,1];

-
-
-% Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
-% \[ b_i - a_i = l_i \hat{s}_i \]
-
-% We initialize $l_i$ and $\hat{s}_i$
-
 leg_length = zeros(6, 1); % [mm]
 leg_vectors = zeros(6, 3);

-
-
-% We compute $b_i - a_i$, and then:
-% \begin{align*}
-%   l_i       &= \left|b_i - a_i\right| \\
-%   \hat{s}_i &= \frac{b_i - a_i}{l_i}
-% \end{align*}
-
-
 legs = stewart.Ab - stewart.Aa;

 for i = 1:6
@ -311,10 +131,6 @@ for i = 1:6
    leg_vectors(i,:) = legs(i,:) / leg_length(i);
 end

-
-
-% Then the shape of the bottom leg is estimated
-
 stewart.Leg.lenght = leg_length(1)/1.5;
 stewart.Leg.shape.bot = ...
    [0                0; ...
@ -324,11 +140,6 @@ stewart.Leg.shape.bot = ...
     stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
     0                0.2*stewart.Leg.lenght];

-
-
-% We compute rotation matrices to have the orientation of the legs.
-% The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
-
 stewart.Rm = struct('R', eye(3));

 for i = 1:6
@ -344,10 +155,6 @@ for i = 1:6
  stewart.Rm(i).R = [sx', sy', sz'];
 end

-
-
-% Compute Jacobian Matrix
-
 J = zeros(6);

 for i = 1:6
--- a/src/initializeMechanicalElements.m
+++ b/src/initializeMechanicalElements.m
@ -0,0 +1,94 @@
+function [stewart] = initializeMechanicalElements(stewart, opts_param)
+
+opts = struct(...
+    'thickness', 10, ... % Thickness of the base and platform [mm]
+    'density',   1000, ... % Density of the material used for the hexapod [kg/m3]
+    'k_ax',      1e8, ... % Stiffness of each actuator [N/m]
+    'c_ax',      1000, ... % Damping of each actuator [N/(m/s)]
+    'stroke',    50e-6  ... % Maximum stroke of each actuator [m]
+    );
+
+if exist('opts_param','var')
+    for opt = fieldnames(opts_param)'
+        opts.(opt{1}) = opts_param.(opt{1});
+    end
+end
+
+BP = struct();
+
+BP.Rint = 0;   % Internal Radius [mm]
+BP.Rext = 150; % External Radius [mm]
+
+BP.H = opts.thickness; % Thickness of the Bottom Plate [mm]
+
+BP.density = opts.density; % Density of the material [kg/m3]
+
+BP.color = [0.7 0.7 0.7]; % Color [RGB]
+
+BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
+
+stewart.BP = BP;
+
+TP = struct();
+
+TP.Rint = 0;   % [mm]
+TP.Rext = 100; % [mm]
+
+TP.H = 10; % [mm]
+
+TP.density = opts.density; % Density of the material [kg/m3]
+
+TP.color = [0.7 0.7 0.7]; % Color [RGB]
+
+TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
+
+stewart.TP  = TP;
+
+Leg = struct();
+
+Leg.stroke = opts.stroke; % [m]
+
+Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
+Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
+
+Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
+Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
+
+Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
+
+Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
+
+Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
+
+legs = stewart.Ab - stewart.Aa;
+Leg.lenght = norm(legs(1,:))/1.5;
+
+Leg.shape.bot = ...
+    [0        0; ...
+     Leg.Rbot 0; ...
+     Leg.Rbot Leg.lenght; ...
+     Leg.Rtop Leg.lenght; ...
+     Leg.Rtop 0.2*Leg.lenght; ...
+     0        0.2*Leg.lenght];
+
+stewart.Leg = Leg;
+
+SP = struct();
+
+SP.k = 0; % [N*m/deg]
+SP.c = 0; % [N*m/deg]
+
+SP.H = stewart.Aa(1, 3) - BP.H; % [mm]
+
+SP.R = Leg.R; % [mm]
+
+SP.section = [0    SP.H-SP.R;
+              0    0;
+              SP.R 0;
+              SP.R SP.H];
+
+SP.density = opts.density; % [kg/m^3]
+
+SP.color = [0.7 0.7 0.7]; % [RGB]
+
+stewart.SP  = SP;
--- a/src/initializeSample.m
+++ b/src/initializeSample.m
@ -1,21 +1,20 @@
 function [] = initializeSample(opts_param)
-%% Default values for opts
-    sample = struct( ...
-        'radius',     100, ... % radius of the cylinder [mm]
-        'height',     100, ... % height of the cylinder [mm]
-        'mass',       10,  ... % mass of the cylinder [kg]
-        'measheight', 50, ... % measurement point z-offset [mm]
-        'offset',     [0, 0, 0],   ... % offset position of the sample [mm]
-        'color',      [0.9 0.1 0.1] ...
-        );

-    %% Populate opts with input parameters
-    if exist('opts_param','var')
-        for opt = fieldnames(opts_param)'
-            sample.(opt{1}) = opts_param.(opt{1});
-        end
+sample = struct( ...
+    'radius',     100, ... % radius of the cylinder [mm]
+    'height',     100, ... % height of the cylinder [mm]
+    'mass',       10,  ... % mass of the cylinder [kg]
+    'measheight', 50, ... % measurement point z-offset [mm]
+    'offset',     [0, 0, 0],   ... % offset position of the sample [mm]
+    'color',      [0.9 0.1 0.1] ...
+    );
+
+if exist('opts_param','var')
+    for opt = fieldnames(opts_param)'
+        sample.(opt{1}) = opts_param.(opt{1});
    end
-
-    %% Save
-    save('./mat/sample.mat', 'sample');
+end
+
+save('./mat/sample.mat', 'sample');
+
 end
--- a/src/initializeSimscapeData.m
+++ b/src/initializeSimscapeData.m
@ -0,0 +1,59 @@
+function [stewart] = initializeSimscapeData(stewart, opts_param)
+
+opts = struct(...
+    'Jd_pos', [0, 0, 30], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
+    'Jf_pos', [0, 0, 30]  ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
+    );
+
+if exist('opts_param','var')
+    for opt = fieldnames(opts_param)'
+        opts.(opt{1}) = opts_param.(opt{1});
+    end
+end
+
+leg_length = zeros(6, 1); % [mm]
+leg_vectors = zeros(6, 3);
+
+legs = stewart.Ab - stewart.Aa;
+
+for i = 1:6
+    leg_length(i) = norm(legs(i,:));
+    leg_vectors(i,:) = legs(i,:) / leg_length(i);
+end
+
+stewart.Rm = struct('R', eye(3));
+
+for i = 1:6
+  sx = cross(leg_vectors(i,:), [1 0 0]);
+  sx = sx/norm(sx);
+
+  sy = -cross(sx, leg_vectors(i,:));
+  sy = sy/norm(sy);
+
+  sz = leg_vectors(i,:);
+  sz = sz/norm(sz);
+
+  stewart.Rm(i).R = [sx', sy', sz'];
+end
+
+Jd = zeros(6);
+
+for i = 1:6
+  Jd(i, 1:3) = leg_vectors(i, :);
+  Jd(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jd_pos), leg_vectors(i, :));
+end
+
+stewart.Jd = Jd;
+stewart.Jd_inv = inv(Jd);
+
+Jf = zeros(6);
+
+for i = 1:6
+  Jf(i, 1:3) = leg_vectors(i, :);
+  Jf(i, 4:6) = cross(0.001*(stewart.Bb(i, :) - opts.Jf_pos), leg_vectors(i, :));
+end
+
+stewart.Jf = Jf;
+stewart.Jf_inv = inv(Jf);
+
+end
--- a/src/initializeStewartPlatform.m
+++ b/src/initializeStewartPlatform.m
@ -0,0 +1,94 @@
+function [stewart] = initializeStewartPlatform(stewart, opts_param)
+
+opts = struct(...
+    'thickness', 10, ... % Thickness of the base and platform [mm]
+    'density',   1000, ... % Density of the material used for the hexapod [kg/m3]
+    'k_ax',      1e8, ... % Stiffness of each actuator [N/m]
+    'c_ax',      1000, ... % Damping of each actuator [N/(m/s)]
+    'stroke',    50e-6  ... % Maximum stroke of each actuator [m]
+    );
+
+if exist('opts_param','var')
+    for opt = fieldnames(opts_param)'
+        opts.(opt{1}) = opts_param.(opt{1});
+    end
+end
+
+BP = struct();
+
+BP.Rint = 0;   % Internal Radius [mm]
+BP.Rext = 150; % External Radius [mm]
+
+BP.H = opts.thickness; % Thickness of the Bottom Plate [mm]
+
+BP.density = opts.density; % Density of the material [kg/m3]
+
+BP.color = [0.7 0.7 0.7]; % Color [RGB]
+
+BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
+
+stewart.BP = BP;
+
+TP = struct();
+
+TP.Rint = 0;   % [mm]
+TP.Rext = 100; % [mm]
+
+TP.H = 10; % [mm]
+
+TP.density = opts.density; % Density of the material [kg/m3]
+
+TP.color = [0.7 0.7 0.7]; % Color [RGB]
+
+TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
+
+stewart.TP  = TP;
+
+Leg = struct();
+
+Leg.stroke = opts.stroke; % [m]
+
+Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
+Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
+
+Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
+Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
+
+Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
+
+Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
+
+Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
+
+legs = stewart.Ab - stewart.Aa;
+Leg.lenght = norm(legs(1,:))/1.5;
+
+Leg.shape.bot = ...
+    [0        0; ...
+     Leg.Rbot 0; ...
+     Leg.Rbot Leg.lenght; ...
+     Leg.Rtop Leg.lenght; ...
+     Leg.Rtop 0.2*Leg.lenght; ...
+     0        0.2*Leg.lenght];
+
+stewart.Leg = Leg;
+
+SP = struct();
+
+SP.k = 0; % [N*m/deg]
+SP.c = 0; % [N*m/deg]
+
+SP.H = stewart.Aa(1, 3) - BP.H; % [mm]
+
+SP.R = Leg.R; % [mm]
+
+SP.section = [0    SP.H-SP.R;
+              0    0;
+              SP.R 0;
+              SP.R SP.H];
+
+SP.density = opts.density; % [kg/m^3]
+
+SP.color = [0.7 0.7 0.7]; % [RGB]
+
+stewart.SP  = SP;
--- a/stewart.slx
+++ b/stewart.slx
--- a/stiffness-study.org
+++ b/stiffness-study.org
@ -1,4 +1,28 @@
 #+TITLE: Stiffness of the Stewart Platform
+:DRAWER:
+#+STARTUP: overview
+
+#+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 type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
+
+#+LATEX_CLASS: cleanreport
+#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
+#+LaTeX_HEADER: \usepackage{svg}
+#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
+#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
+#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
+
+#+PROPERTY: header-args:matlab  :session *MATLAB*
+#+PROPERTY: header-args:matlab+ :comments org
+#+PROPERTY: header-args:matlab+ :exports both
+#+PROPERTY: header-args:matlab+ :eval no-export
+#+PROPERTY: header-args:matlab+ :output-dir figs
+#+PROPERTY: header-args:matlab+ :mkdirp yes
+:END:

 * Functions
  :PROPERTIES: