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Update study: cubic configuration, renew the function for generation

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Thomas Dehaeze 2019-03-25 18:12:43 +01:00
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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

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After

Width:  |  Height:  |  Size: 141 KiB

54
figs/stiffness_cube_size.tex Normal file
View File

@ -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}%

121
identification.org
View File

@ -25,7 +25,7 @@
:END: :END:
* Identification * Identification
#+begin_src matlab :results none :exports none #+begin_src matlab :results none :exports none :noweb yes
<<matlab-init>> <<matlab-init>>
addpath('src'); addpath('src');
addpath('library'); addpath('library');
@ -37,7 +37,11 @@
The hexapod structure and Sample structure are initialized. The hexapod structure and Sample structure are initialized.
#+begin_src matlab :results none #+begin_src matlab :results none
initializeHexapod(); stewart = initializeGeneralConfiguration();
stewart = computeGeometricalProperties(stewart);
stewart = initializeMechanicalElements(stewart);
save('./mat/stewart.mat', 'stewart');
initializeSample(); initializeSample();
#+end_src #+end_src
@ -45,92 +49,111 @@ The hexapod structure and Sample structure are initialized.
G = identifyPlant(); G = identifyPlant();
#+end_src #+end_src
#+begin_src matlab :results none
freqs = logspace(2, 4, 1000);
#+end_src
* Cartesian Plot * Cartesian Plot
From a force applied in the Cartesian frame to a displacement in the Cartesian frame. From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_cart(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_cart(1, 1), freqs, 'Hz'))));
bode(G.G_cart(3, 3)); 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; 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 #+end_src
* From a force to force sensor * From a force to force sensor
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_forc(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
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));
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
legend('location', 'southeast');
#+end_src #+end_src
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_forc(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$F_{m_i}/F_{i}$');
bode(G.G_forc(1, 2)); plot(freqs, abs(squeeze(freqresp(G.G_forc(2, 1), freqs, 'Hz'))), 'k--', 'DisplayName', '$F_{m_j}/F_{i}$');
bode(G.G_forc(1, 3)); plot(freqs, abs(squeeze(freqresp(G.G_forc(3, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
bode(G.G_forc(1, 4)); plot(freqs, abs(squeeze(freqresp(G.G_forc(4, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
bode(G.G_forc(1, 5)); plot(freqs, abs(squeeze(freqresp(G.G_forc(5, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
bode(G.G_forc(1, 6)); plot(freqs, abs(squeeze(freqresp(G.G_forc(6, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [N/N]');
legend('location', 'southeast');
#+end_src #+end_src
* From a force applied in the leg to the displacement of the leg * From a force applied in the leg to the displacement of the leg
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_legs(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$D_{i}/F_{i}$');
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));
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src #+end_src
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_legs(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', '$D_{i}/F_{i}$');
bode(G.G_legs(1, 2)); plot(freqs, abs(squeeze(freqresp(G.G_legs(2, 1), freqs, 'Hz'))), 'k--', 'DisplayName', '$D_{j}/F_{i}$');
bode(G.G_legs(1, 3)); plot(freqs, abs(squeeze(freqresp(G.G_legs(3, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
bode(G.G_legs(1, 4)); plot(freqs, abs(squeeze(freqresp(G.G_legs(4, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
bode(G.G_legs(1, 5)); plot(freqs, abs(squeeze(freqresp(G.G_legs(5, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
bode(G.G_legs(1, 6)); plot(freqs, abs(squeeze(freqresp(G.G_legs(6, 1), freqs, 'Hz'))), 'k--', 'HandleVisibility', 'off');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
legend('location', 'northeast');
#+end_src #+end_src
* Transmissibility * Transmissibility
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_tran(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, 'Hz'))));
bode(G.G_tran(2, 2)); plot(freqs, abs(squeeze(freqresp(G.G_tran(2, 2), freqs, 'Hz'))));
bode(G.G_tran(3, 3)); plot(freqs, abs(squeeze(freqresp(G.G_tran(3, 3), freqs, 'Hz'))));
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/m]');
#+end_src #+end_src
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_tran(4, 4)); plot(freqs, abs(squeeze(freqresp(G.G_tran(4, 4), freqs, 'Hz'))));
bode(G.G_tran(5, 5)); plot(freqs, abs(squeeze(freqresp(G.G_tran(5, 5), freqs, 'Hz'))));
bode(G.G_tran(6, 6)); plot(freqs, abs(squeeze(freqresp(G.G_tran(6, 6), freqs, 'Hz'))));
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [$\frac{rad/s}{rad/s}$]');
#+end_src #+end_src
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_tran(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, 'Hz'))));
bode(G.G_tran(2, 1)); plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 2), freqs, 'Hz'))));
bode(G.G_tran(3, 1)); plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 3), freqs, 'Hz'))));
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/m]');
#+end_src #+end_src
* Compliance * 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 #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_comp(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_comp(1, 1), freqs, 'Hz'))));
bode(G.G_comp(2, 2)); plot(freqs, abs(squeeze(freqresp(G.G_comp(2, 2), freqs, 'Hz'))));
bode(G.G_comp(3, 3)); plot(freqs, abs(squeeze(freqresp(G.G_comp(3, 3), freqs, 'Hz'))));
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src #+end_src
* Inertial * Inertial
@ -151,10 +176,12 @@ From a force applied on the Cartesian frame to the absolute displacement of the
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
bode(G.G_iner(1, 1)); plot(freqs, abs(squeeze(freqresp(G.G_iner(1, 1), freqs, 'Hz'))));
bode(G.G_iner(2, 2)); plot(freqs, abs(squeeze(freqresp(G.G_iner(2, 2), freqs, 'Hz'))));
bode(G.G_iner(3, 3)); plot(freqs, abs(squeeze(freqresp(G.G_iner(3, 3), freqs, 'Hz'))));
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src #+end_src
* identifyPlant * 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. We split the transfer function into sub transfer functions and we compute their minimum realization.
#+begin_src matlab #+begin_src matlab
sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'})); 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_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_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_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_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
% sys.G_all = minreal(G); % sys.G_all = minreal(G);
#+end_src #+end_src

25
index.html
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <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 http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Stewart Platform Studies</title> <title>Stewart Platform Studies</title>
@ -253,36 +253,37 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org2b3b6a5">1. Simscape Model</a></li> <li><a href="#org431e1f9">1. Simscape Model</a></li>
<li><a href="#org5dc817d">2. Architecture Study</a></li> <li><a href="#org27d326c">2. Architecture Study</a></li>
<li><a href="#orgccde31a">3. Motion Control</a></li> <li><a href="#orgd097f1e">3. Motion Control</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org2b3b6a5" class="outline-2"> <div id="outline-container-org431e1f9" class="outline-2">
<h2 id="org2b3b6a5"><span class="section-number-2">1</span> Simscape Model</h2> <h2 id="org431e1f9"><span class="section-number-2">1</span> Simscape Model</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<ul class="org-ul"> <ul class="org-ul">
<li><a href="simscape-model.html">Model of the Stewart Platform</a></li> <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> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org5dc817d" class="outline-2"> <div id="outline-container-org27d326c" class="outline-2">
<h2 id="org5dc817d"><span class="section-number-2">2</span> Architecture Study</h2> <h2 id="org27d326c"><span class="section-number-2">2</span> Architecture Study</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<ul class="org-ul"> <ul class="org-ul">
<li><a href="kinematic-study.html">Kinematic Study</a></li> <li><a href="kinematic-study.html">Kinematic Study</a></li>
<li><a href="stiffness-study.html">Stiffness Matrix Study</a></li> <li><a href="stiffness-study.html">Stiffness Matrix Study</a></li>
<li>Jacobian Study</li> <li>Jacobian Study</li>
<li><a href="cubic-configuration.html">Cubic Architecture</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orgccde31a" class="outline-2"> <div id="outline-container-orgd097f1e" class="outline-2">
<h2 id="orgccde31a"><span class="section-number-2">3</span> Motion Control</h2> <h2 id="orgd097f1e"><span class="section-number-2">3</span> Motion Control</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<ul class="org-ul"> <ul class="org-ul">
<li>Active Damping</li> <li>Active Damping</li>
@ -294,7 +295,7 @@ for the JavaScript code in this tag.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p> <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> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

3
index.org
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@ -26,12 +26,13 @@
* Simscape Model * Simscape Model
- [[file:simscape-model.org][Model of the Stewart Platform]] - [[file:simscape-model.org][Model of the Stewart Platform]]
- [[file:identification.org][Identification]] - [[file:identification.org][Identification of the Simscape Model]]
* Architecture Study * Architecture Study
- [[file:kinematic-study.org][Kinematic Study]] - [[file:kinematic-study.org][Kinematic Study]]
- [[file:stiffness-study.org][Stiffness Matrix Study]] - [[file:stiffness-study.org][Stiffness Matrix Study]]
- Jacobian Study - Jacobian Study
- [[file:cubic-configuration.org][Cubic Architecture]]
* Motion Control * Motion Control
- Active Damping - Active Damping

24
kinematic-study.org
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@ -1,4 +1,28 @@
#+TITLE: Kinematic Study of the Stewart Platform #+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 * Functions
:PROPERTIES: :PROPERTIES:

BIN
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37
references.bib Normal file
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@ -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 = {},
}

716
simscape-model.html
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <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 http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Stewart Platform - Simscape Model</title> <title>Stewart Platform - Simscape Model</title>
@ -275,42 +275,136 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org9a10766">1. Function description and arguments</a></li> <li><a href="#org527cc13">1. initializeGeneralConfiguration</a>
<li><a href="#orgb6911a1">2. Initialization of the stewart structure</a></li> <ul>
<li><a href="#org030aed6">3. Bottom Plate</a></li> <li><a href="#orgea5f8f5">1.1. Function description</a></li>
<li><a href="#orged8012a">4. Top Plate</a></li> <li><a href="#org2db42cb">1.2. Optional Parameters</a></li>
<li><a href="#orgc74617a">5. Legs</a></li> <li><a href="#org2f9279a">1.3. Geometry Description</a></li>
<li><a href="#org7cd2aa5">6. Ball Joints</a></li> <li><a href="#org1409cf0">1.4. Compute Aa and Ab</a></li>
<li><a href="#org1d76ed9">7. More parameters are initialized</a></li> <li><a href="#orgb91c416">1.5. Returns Stewart Structure</a></li>
<li><a href="#orge9faa26">8. Save the Stewart Structure</a></li> </ul>
<li><a href="#orga207d03">9. initializeParameters Function</a></li> </li>
<li><a href="#org724c1a1">10. initializeSample</a></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> </ul>
</div> </div>
</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> <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> </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> <p>
</pre> 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>
<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> <p>
Default values for opts. Default values for opts.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span> <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;">'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;">'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;">'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;">'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;">'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;">'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;">'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;">'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;">'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;">'name', 'stewart'</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Name of the file</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>; <span style="color: #DCDCCC;">)</span>;
</pre> </pre>
</div> </div>
@ -329,37 +423,263 @@ Populate opts with input parameters
</div> </div>
</div> </div>
<div id="outline-container-orgb6911a1" class="outline-2"> <div id="outline-container-org2f9279a" class="outline-3">
<h2 id="orgb6911a1"><span class="section-number-2">2</span> Initialization of the stewart structure</h2> <h3 id="org2f9279a"><span class="section-number-3">1.3</span> Geometry Description</h3>
<div class="outline-text-2" id="text-2"> <div class="outline-text-3" id="text-1-3">
<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>
<p> <div id="orgc30ce24" class="figure">
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">
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" /> <p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Schematic of the bottom plates with all the parameters</p> <p><span class="figure-number">Figure 1: </span>Schematic of the bottom plates with all the parameters</p>
</div> </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> <p>
The bottom plate structure is initialized. 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. We define its thickness.
</p> </p>
<div class="org-src-container"> <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 class="src src-matlab">BP.H = opts.thickness; <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> </pre>
</div> </div>
@ -429,9 +740,9 @@ The structure is added to the stewart structure
</div> </div>
</div> </div>
<div id="outline-container-orged8012a" class="outline-2"> <div id="outline-container-org23fd88c" class="outline-3">
<h2 id="orged8012a"><span class="section-number-2">4</span> Top Plate</h2> <h3 id="org23fd88c"><span class="section-number-3">3.4</span> Top Plate</h3>
<div class="outline-text-2" id="text-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
The top plate structure is initialized. The top plate structure is initialized.
</p> </p>
@ -457,16 +768,6 @@ The thickness of the top plate.
</pre> </pre>
</div> </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> <p>
The density of its material. The density of its material.
</p> </p>
@ -501,17 +802,16 @@ The structure is added to the stewart structure
</div> </div>
</div> </div>
<div id="outline-container-orgc74617a" class="outline-2"> <div id="outline-container-org96d7dab" class="outline-3">
<h2 id="orgc74617a"><span class="section-number-2">5</span> Legs</h2> <h3 id="org96d7dab"><span class="section-number-3">3.5</span> Legs</h3>
<div class="outline-text-2" id="text-5"> <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><img src="./figs/stewart_legs.png" alt="stewart_legs.png" />
</p> </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> </div>
<p> <p>
The leg structure is initialized. The leg structure is initialized.
</p> </p>
@ -570,6 +870,29 @@ The radius of spheres representing the ball joints are defined.
</pre> </pre>
</div> </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> <p>
The structure is added to the stewart structure The structure is added to the stewart structure
</p> </p>
@ -580,14 +903,14 @@ The structure is added to the stewart structure
</div> </div>
</div> </div>
<div id="outline-container-org7cd2aa5" class="outline-2"> <div id="outline-container-org66df86f" class="outline-3">
<h2 id="org7cd2aa5"><span class="section-number-2">6</span> Ball Joints</h2> <h3 id="org66df86f"><span class="section-number-3">3.6</span> Ball Joints</h3>
<div class="outline-text-2" id="text-6"> <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><img src="./figs/stewart_ball_joints.png" alt="stewart_ball_joints.png" />
</p> </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> </div>
<p> <p>
@ -615,7 +938,7 @@ SP.c = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% [
Its height is defined Its height is defined
</p> </p>
<div class="org-src-container"> <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> </pre>
</div> </div>
@ -660,195 +983,74 @@ The structure is added to the Hexapod structure
</div> </div>
</div> </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>
<div id="outline-container-orge9faa26" class="outline-2"> <div id="outline-container-orgf3c4474" class="outline-2">
<h2 id="orge9faa26"><span class="section-number-2">8</span> Save the Stewart Structure</h2> <h2 id="orgf3c4474"><span class="section-number-2">4</span> initializeSample</h2>
<div class="outline-text-2" id="text-8"> <div class="outline-text-2" id="text-4">
<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> </div>
<div id="outline-container-orga207d03" class="outline-2"> <div id="outline-container-org1ec4152" class="outline-3">
<h2 id="orga207d03"><span class="section-number-2">9</span> initializeParameters Function</h2> <h3 id="org1ec4152"><span class="section-number-3">4.1</span> Function description</h3>
<div class="outline-text-2" id="text-9"> <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: #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 class="org-src-container"> <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> <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> </pre>
sample = struct<span style="color: #DCDCCC;">(</span> <span style="text-decoration: underline;">...</span> </div>
<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> </div>
<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> </div>
<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>;
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Populate opts with input parameters</span> <div id="outline-container-orgcd3268d" class="outline-3">
<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> <h3 id="orgcd3268d"><span class="section-number-3">4.2</span> Optional Parameters</h3>
<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> <div class="outline-text-3" id="text-4-2">
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>; <p>
<span style="color: #F0DFAF; font-weight: bold;">end</span> 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: #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> <span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre> </pre>
</div> </div>
</div> </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>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p> <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> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

574
simscape-model.org
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@ -17,29 +17,73 @@
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com} #+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
#+PROPERTY: header-args:matlab :session *MATLAB* #+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments no #+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports bode #+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no #+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs #+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :mkdirp yes #+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :tangle src/initializeHexapod.m
:END: :END:
* Function description and arguments Stewart platforms are generated in multiple steps.
The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
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 #+begin_src matlab
function [stewart] = initializeHexapod(opts_param) function [stewart] = initializeGeneralConfiguration(opts_param)
#+end_src #+end_src
** Optional Parameters
Default values for opts. Default values for opts.
#+begin_src matlab #+begin_src matlab
opts = struct(... opts = struct(...
'height', 90, ... % Height of the platform [mm] 'H_tot', 90, ... % Height of the platform [mm]
'density', 8000, ... % Density of the material used for the hexapod [kg/m3] 'H_joint', 15, ... % Height of the joints [mm]
'k_ax', 1e8, ... % Stiffness of each actuator [N/m] 'H_plate', 10, ... % Thickness of the fixed and mobile platforms [mm]
'c_ax', 1000, ... % Damping of each actuator [N/(m/s)] 'R_bot', 100, ... % Radius where the legs articulations are positionned [mm]
'stroke', 50e-6, ... % Maximum stroke of each actuator [m] 'R_top', 80, ... % Radius where the legs articulations are positionned [mm]
'name', 'stewart' ... % Name of the file 'a_bot', 10, ... % Angle Offset [deg]
'a_top', 40, ... % Angle Offset [deg]
'da_top', 0 ... % Angle Offset from 0 position [deg]
); );
#+end_src #+end_src
@ -52,22 +96,190 @@ Populate opts with input parameters
end end
#+end_src #+end_src
* Initialization of the stewart structure ** Geometry Description
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
#+name: fig:stewart_bottom_plate #+name: fig:stewart_bottom_plate
#+caption: Schematic of the bottom plates with all the parameters #+caption: Schematic of the bottom plates with all the parameters
[[file:./figs/stewart_bottom_plate.png]] [[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. The bottom plate structure is initialized.
#+begin_src matlab #+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. We define its thickness.
#+begin_src matlab #+begin_src matlab
BP.H = 10; % Thickness of the Bottom Plate [mm] BP.H = opts.thickness; % 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]
#+end_src #+end_src
We defined the density of the material of the bottom plate. 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; stewart.BP = BP;
#+end_src #+end_src
* Top Plate ** Top Plate
The top plate structure is initialized. The top plate structure is initialized.
#+begin_src matlab #+begin_src matlab
TP = struct(); TP = struct();
@ -128,13 +334,6 @@ The thickness of the top plate.
TP.H = 10; % [mm] TP.H = 10; % [mm]
#+end_src #+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. The density of its material.
#+begin_src matlab #+begin_src matlab
TP.density = opts.density; % Density of the material [kg/m3] 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; stewart.TP = TP;
#+end_src #+end_src
* Legs ** Legs
#+name: fig:stewart_legs #+name: fig:stewart_legs
#+caption: Schematic for the legs of the Stewart platform #+caption: Schematic for the legs of the Stewart platform
[[file:./figs/stewart_legs.png]] [[file:./figs/stewart_legs.png]]
The leg structure is initialized. The leg structure is initialized.
#+begin_src matlab #+begin_src matlab
Leg = struct(); 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] Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
#+end_src #+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 The structure is added to the stewart structure
#+begin_src matlab #+begin_src matlab
stewart.Leg = Leg; stewart.Leg = Leg;
#+end_src #+end_src
* Ball Joints ** Ball Joints
#+name: fig:stewart_ball_joints #+name: fig:stewart_ball_joints
#+caption: Schematic of the support for the ball joints #+caption: Schematic of the support for the ball joints
[[file:./figs/stewart_ball_joints.png]] [[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 Its height is defined
#+begin_src matlab #+begin_src matlab
SP.H = 15; % [mm] SP.H = stewart.Aa(1, 3) - BP.H; % [mm]
#+end_src #+end_src
Its radius is based on the radius on the sphere at the end of the legs. 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; stewart.SP = SP;
#+end_src #+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 * initializeSample
:PROPERTIES: :PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :eval no
:HEADER-ARGS:matlab+: :tangle src/initializeSample.m :HEADER-ARGS:matlab+: :tangle src/initializeSample.m
:END: :END:
** Function description
#+begin_src matlab #+begin_src matlab
function [] = initializeSample(opts_param) function [] = initializeSample(opts_param)
%% Default values for opts #+end_src
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 ** Optional Parameters
if exist('opts_param','var') Default values for opts.
for opt = fieldnames(opts_param)' #+begin_src matlab
sample.(opt{1}) = opts_param.(opt{1}); sample = struct( ...
end '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 end
end
%% Save #+end_src
save('./mat/sample.mat', 'sample');
** Save the Sample structure
#+begin_src matlab
save('./mat/sample.mat', 'sample');
#+end_src
#+begin_src matlab
end end
#+end_src #+end_src

59
src/computeGeometricalProperties.m Normal file
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@ -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

2
src/identifyPlant.m
View File

@ -53,7 +53,7 @@ G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
% identifyPlant:7 ends here % identifyPlant:7 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:8]] % [[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_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_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_tran = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));

89
src/initializeCubicConfiguration.m Normal file
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@ -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

47
src/initializeGeneralConfiguration.m Normal file
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@ -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

203
src/initializeHexapod.m
View File

@ -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) function [stewart] = initializeHexapod(opts_param)
% Default values for opts.
opts = struct(... opts = struct(...
'height', 90, ... % Height of the platform [mm] '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] 'k_ax', 1e8, ... % Stiffness of each actuator [N/m]
'c_ax', 1000, ... % Damping of each actuator [N/(m/s)] 'c_ax', 1000, ... % Damping of each actuator [N/(m/s)]
'stroke', 50e-6, ... % Maximum stroke of each actuator [m] 'stroke', 50e-6, ... % Maximum stroke of each actuator [m]
'name', 'stewart' ... % Name of the file 'name', 'stewart' ... % Name of the file
); );
% Populate opts with input parameters
if exist('opts_param','var') if exist('opts_param','var')
for opt = fieldnames(opts_param)' for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1}); opts.(opt{1}) = opts_param.(opt{1});
end end
end end
% Initialization of the stewart structure
% We initialize the Stewart structure
stewart = struct(); stewart = struct();
% And we defined its total height.
stewart.H = opts.height; % [mm] 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(); 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.Rint = 0; % Internal Radius [mm]
BP.Rext = 150; % External Radius [mm] BP.Rext = 150; % External Radius [mm]
% We define its thickness.
BP.H = 10; % Thickness of the Bottom Plate [mm] 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.Rleg = 100; % Radius where the legs articulations are positionned [mm]
BP.alpha = 10; % Angle Offset [deg] BP.alpha = 30; % Angle Offset [deg]
% We defined the density of the material of the bottom plate.
BP.density = opts.density; % Density of the material [kg/m3] BP.density = opts.density; % Density of the material [kg/m3]
% And its color.
BP.color = [0.7 0.7 0.7]; % Color [RGB] 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] 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; stewart.BP = BP;
% Top Plate
% The top plate structure is initialized.
TP = struct(); TP = struct();
% We defined the internal and external radius of the top plate.
TP.Rint = 0; % [mm] TP.Rint = 0; % [mm]
TP.Rext = 100; % [mm] TP.Rext = 100; % [mm]
% The thickness of the top plate.
TP.H = 10; % [mm] TP.H = 10; % [mm]
TP.Rleg = 80; % Radius where the legs articulations are positionned [mm]
TP.alpha = 10; % Angle [deg]
% 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.dalpha = 0; % Angle Offset from 0 position [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] TP.density = opts.density; % Density of the material [kg/m3]
% Its color.
TP.color = [0.7 0.7 0.7]; % Color [RGB] 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]; 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; 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(); Leg = struct();
% The maximum Stroke of each leg is defined.
Leg.stroke = opts.stroke; % [m] 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.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.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.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.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
Leg.density = 0.01*opts.density; % Density of the material used for the legs [kg/m3]
% The density of its material.
Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
% Its color.
Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB] 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] 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; 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(); SP = struct();
% We can define its rotational stiffness and damping. For now, we use perfect joints.
SP.k = 0; % [N*m/deg] SP.k = 0; % [N*m/deg]
SP.c = 0; % [N*m/deg] SP.c = 0; % [N*m/deg]
% Its height is defined
SP.H = 15; % [mm] 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.R = Leg.R; % [mm]
SP.section = [0 SP.H-SP.R; SP.section = [0 SP.H-SP.R;
@ -230,40 +88,18 @@ SP.section = [0 SP.H-SP.R;
SP.R 0; SP.R 0;
SP.R SP.H]; SP.R SP.H];
% The density of its material is defined.
SP.density = opts.density; % [kg/m^3] SP.density = opts.density; % [kg/m^3]
% Its color is defined.
SP.color = [0.7 0.7 0.7]; % [RGB] SP.color = [0.7 0.7 0.7]; % [RGB]
% The structure is added to the Hexapod structure
stewart.SP = SP; stewart.SP = SP;
% More parameters are initialized
stewart = initializeParameters(stewart); stewart = initializeParameters(stewart);
% Save the Stewart Structure
save('./mat/stewart.mat', 'stewart') save('./mat/stewart.mat', 'stewart')
% initializeParameters Function
function [stewart] = initializeParameters(stewart) 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.Aa = zeros(6, 3); % [mm]
stewart.Ab = zeros(6, 3); % [mm] stewart.Ab = zeros(6, 3); % [mm]
stewart.Bb = zeros(6, 3); % [mm] stewart.Bb = zeros(6, 3); % [mm]
@ -285,25 +121,9 @@ for i = 1:3
end end
stewart.Bb = stewart.Ab - stewart.H*[0,0,1]; 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_length = zeros(6, 1); % [mm]
leg_vectors = zeros(6, 3); 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; legs = stewart.Ab - stewart.Aa;
for i = 1:6 for i = 1:6
@ -311,10 +131,6 @@ for i = 1:6
leg_vectors(i,:) = legs(i,:) / leg_length(i); leg_vectors(i,:) = legs(i,:) / leg_length(i);
end end
% Then the shape of the bottom leg is estimated
stewart.Leg.lenght = leg_length(1)/1.5; stewart.Leg.lenght = leg_length(1)/1.5;
stewart.Leg.shape.bot = ... stewart.Leg.shape.bot = ...
[0 0; ... [0 0; ...
@ -324,11 +140,6 @@ stewart.Leg.shape.bot = ...
stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ... stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
0 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)); stewart.Rm = struct('R', eye(3));
for i = 1:6 for i = 1:6
@ -344,10 +155,6 @@ for i = 1:6
stewart.Rm(i).R = [sx', sy', sz']; stewart.Rm(i).R = [sx', sy', sz'];
end end
% Compute Jacobian Matrix
J = zeros(6); J = zeros(6);
for i = 1:6 for i = 1:6

94
src/initializeMechanicalElements.m Normal file
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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;

33
src/initializeSample.m
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function [] = initializeSample(opts_param) 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 sample = struct( ...
if exist('opts_param','var') 'radius', 100, ... % radius of the cylinder [mm]
for opt = fieldnames(opts_param)' 'height', 100, ... % height of the cylinder [mm]
sample.(opt{1}) = opts_param.(opt{1}); 'mass', 10, ... % mass of the cylinder [kg]
end '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 end
end
%% Save
save('./mat/sample.mat', 'sample'); save('./mat/sample.mat', 'sample');
end end

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src/initializeSimscapeData.m Normal file
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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

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src/initializeStewartPlatform.m Normal file
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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;

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stewart.slx
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24
stiffness-study.org
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#+TITLE: Stiffness of the Stewart Platform #+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 * Functions
:PROPERTIES: :PROPERTIES: