tdehaeze/stewart-simscape
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Add analysis of the cube's size

Browse Source
This commit is contained in:
Thomas Dehaeze 2020-02-12 10:37:20 +01:00
parent 747fe5549c
commit f4abfbe85c
4 changed files with 129 additions and 49 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

109
docs/cubic-configuration.html
View File

@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-02-12 mer. 10:22 -->
<!-- 2020-02-12 mer. 10:37 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Cubic configuration for the Stewart Platform</title>
@ -278,9 +278,10 @@ for the JavaScript code in this tag.
<li><a href="#org8afa645">1.6. Having Cube&rsquo;s center above the top platform</a></li>
</ul>
</li>
<li><a href="#org3044455">2. Functions</a>
<li><a href="#orgcc4ecce">2. Cubic size analysis</a></li>
<li><a href="#org3044455">3. Functions</a>
<ul>
<li><a href="#org56504f1">2.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
<li><a href="#org56504f1">3.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
<ul>
<li><a href="#orga5a9ba8">Function description</a></li>
<li><a href="#org3253792">Documentation</a></li>
@ -324,7 +325,7 @@ This topology provides a uniform control capability and a uniform stiffness in a
</blockquote>
<p>
To generate and study the Cubic configuration, <code>generateCubicConfiguration</code> is used (description in section <a href="#orga8311d3">2.1</a>).
To generate and study the Cubic configuration, <code>generateCubicConfiguration</code> is used (description in section <a href="#orga8311d3">3.1</a>).
The goal is to study the benefits of using a cubic configuration:
</p>
<ul class="org-ul">
@ -1132,17 +1133,105 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
</div>
</div>
<div id="outline-container-org3044455" class="outline-2">
<h2 id="org3044455"><span class="section-number-2">2</span> Functions</h2>
<div id="outline-container-orgcc4ecce" class="outline-2">
<h2 id="orgcc4ecce"><span class="section-number-2">2</span> Cubic size analysis</h2>
<div class="outline-text-2" id="text-2">
<p>
We here study the effect of the size of the cube used for the Stewart Cubic configuration.
</p>
<p>
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform and the frames \(\{A\}\) and \(\{B\}\) are also taken at the center of the cube.
</p>
<p>
We only vary the size of the cube.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Hcs = 1e<span class="org-type">-</span>3<span class="org-type">*</span>[250<span class="org-type">:</span>20<span class="org-type">:</span>350]; <span class="org-comment">% Heights for the Cube [m]</span>
Ks = zeros(6, 6, length(Hcs));
</pre>
</div>
<p>
The height of the Stewart platform is fixed:
</p>
<div class="org-src-container">
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
</pre>
</div>
<p>
The frames \(\{A\}\) and \(\{B\}\) are positioned at the Stewart platform center as well as the cube&rsquo;s center:
</p>
<div class="org-src-container">
<pre class="src src-matlab">MO_B = <span class="org-type">-</span>50e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Hcs)</span>
Hc = Hcs(<span class="org-constant">i</span>);
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
stewart = computeJacobian(stewart);
Ks(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = stewart.kinematics.K;
<span class="org-keyword">end</span>
</pre>
</div>
<p>
We find that for all the cube&rsquo;s size, \(k_x = k_y = k_z = k\) where \(k\) is the strut stiffness.
We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varying with the cube&rsquo;s size (figure <a href="#orgf5b4a80">9</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(Hcs, squeeze(Ks(4, 4, <span class="org-type">:</span>)), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$k_{\theta_x} = k_{\theta_y}$'</span>);
plot(Hcs, squeeze(Ks(6, 6, <span class="org-type">:</span>)), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$k_{\theta_z}$'</span>);
hold off;
legend(<span class="org-string">'location'</span>, <span class="org-string">'northwest'</span>);
xlabel(<span class="org-string">'Cube Size [m]'</span>); ylabel(<span class="org-string">'Rotational stiffnes [normalized]'</span>);
</pre>
</div>
<div id="orgf5b4a80" class="figure">
<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
</p>
<p><span class="figure-number">Figure 9: </span>\(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube</p>
</div>
<p>
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
</p>
<div class="important">
<p>
In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
</p>
</div>
</div>
</div>
<div id="outline-container-org3044455" class="outline-2">
<h2 id="org3044455"><span class="section-number-2">3</span> Functions</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org28ba607"></a>
</p>
</div>
<div id="outline-container-org56504f1" class="outline-3">
<h3 id="org56504f1"><span class="section-number-3">2.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
<div class="outline-text-3" id="text-2-1">
<h3 id="org56504f1"><span class="section-number-3">3.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="orga8311d3"></a>
</p>
@ -1186,7 +1275,7 @@ This Matlab function is accessible <a href="../src/generateCubicConfiguration.m"
<div id="org8a7f3d8" class="figure">
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Cubic Configuration</p>
<p><span class="figure-number">Figure 10: </span>Cubic Configuration</p>
</div>
</div>
</div>
@ -1291,7 +1380,7 @@ stewart.platform_M.Mb = Mb;
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-12 mer. 10:22</p>
<p class="date">Created: 2020-02-12 mer. 10:37</p>
</div>
</body>
</html>

BIN
docs/figs/stiffness_cube_size.pdf
View File

Binary file not shown.

BIN
docs/figs/stiffness_cube_size.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 23 KiB

After

Width:  |  Height:  |  Size: 25 KiB

69
org/cubic-configuration.org
View File

@ -431,60 +431,53 @@ However, the rotational stiffnesses are increasing with the cube's size but the
| -8e-17 | 0 | -3e-17 | -6.1e-19 | 0.094 | 0 |
| -6.2e-18 | 7.2e-17 | 5.6e-17 | 2.3e-17 | 0 | 0.37 |
* TODO Cubic size analysis :noexport:
We here study the effect of the size of the cube used for the Stewart configuration.
* Cubic size analysis
We here study the effect of the size of the cube used for the Stewart Cubic configuration.
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform.
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform and the frames $\{A\}$ and $\{B\}$ are also taken at the center of the cube.
We only vary the size of the cube.
#+begin_src matlab :results silent
H_cubes = 250:20:350;
stewarts = {zeros(length(H_cubes), 1)};
Hcs = 1e-3*[250:20:350]; % Heights for the Cube [m]
Ks = zeros(6, 6, length(Hcs));
#+end_src
The height of the Stewart platform is fixed:
#+begin_src matlab
H = 100e-3; % height of the Stewart platform [m]
#+end_src
The frames $\{A\}$ and $\{B\}$ are positioned at the Stewart platform center as well as the cube's center:
#+begin_src matlab
MO_B = -50e-3; % Position {B} with respect to {M} [m]
FOc = H + MO_B; % Center of the cube with respect to {F}
#+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};
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
for i = 1:length(Hcs)
Hc = Hcs(i);
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
stewart = computeJacobian(stewart);
Ks(:,:,i) = stewart.kinematics.K;
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
We find that for all the cube's size, $k_x = k_y = k_z = k$ where $k$ is the strut stiffness.
We also find that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ are varying with the cube's size (figure [[fig:stiffness_cube_size]]).
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', '$k_{\theta_z}$');
plot(Hcs, squeeze(Ks(4, 4, :)), 'DisplayName', '$k_{\theta_x} = k_{\theta_y}$');
plot(Hcs, squeeze(Ks(6, 6, :)), 'DisplayName', '$k_{\theta_z}$');
hold off;
legend('location', 'northwest');
xlabel('Cube Size [mm]'); ylabel('Rotational stiffnes [normalized]');
xlabel('Cube Size [m]'); ylabel('Rotational stiffnes [normalized]');
#+end_src
#+NAME: fig:stiffness_cube_size
@ -498,12 +491,10 @@ Finally, we plot $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$
#+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
* Functions