1350 lines
83 KiB
HTML
1350 lines
83 KiB
HTML
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<title>Cubic configuration for the Stewart Platform</title>
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<h1 class="title">Cubic configuration for the Stewart Platform</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgec4f5e2">1. Questions we wish to answer with this analysis</a></li>
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<li><a href="#orgef11581">2. Configuration Analysis - Stiffness Matrix</a>
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<ul>
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<li><a href="#org4203cad">2.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org3344772">2.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
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<li><a href="#org52de20d">2.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#orgd7e1449">2.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
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<li><a href="#orgf16b788">2.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orga2cd408">3. Cubic size analysis</a></li>
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<li><a href="#org9220275">4. initializeCubicConfiguration</a>
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<ul>
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<li><a href="#orgdee5436">4.1. Function description</a></li>
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<li><a href="#org68794ca">4.2. Optional Parameters</a></li>
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<li><a href="#org93d8028">4.3. Cube Creation</a></li>
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<li><a href="#org00e16e1">4.4. Vectors of each leg</a></li>
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<li><a href="#orgd7df0cc">4.5. Verification of Height of the Stewart Platform</a></li>
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<li><a href="#orgc4b765a">4.6. Determinate the location of the joints</a></li>
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<li><a href="#org0a84b4d">4.7. Returns Stewart Structure</a></li>
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</ul>
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</li>
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<li><a href="#orgcb21f88">5. Tests</a>
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<ul>
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<li><a href="#orgc47f87d">5.1. First attempt to parametrisation</a></li>
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<li><a href="#orgff4f69c">5.2. Second attempt</a></li>
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<li><a href="#org011ab94">5.3. Generate the Stewart platform for a Cubic configuration</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<p>
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The discovery of the Cubic configuration is done in. 1
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The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
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</p>
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<p>
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To generate and study the Cubic configuration, <code>initializeCubicConfiguration</code> is used (description in section <a href="#org8876664">4</a>).
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</p>
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<div id="outline-container-orgec4f5e2" class="outline-2">
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<h2 id="orgec4f5e2"><span class="section-number-2">1</span> Questions we wish to answer with this analysis</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The goal is to study the benefits of using a cubic configuration:
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</p>
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<ul class="org-ul">
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<li>Equal stiffness in all the degrees of freedom?</li>
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<li>No coupling between the actuators?</li>
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<li>Is the center of the cube an important point?</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgef11581" class="outline-2">
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<h2 id="orgef11581"><span class="section-number-2">2</span> Configuration Analysis - Stiffness Matrix</h2>
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<div class="outline-text-2" id="text-2">
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</div>
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<div id="outline-container-org4203cad" class="outline-3">
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<h3 id="org4203cad"><span class="section-number-3">2.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We create a cubic Stewart platform (figure <a href="#org620d9b9">1</a>) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
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The Jacobian matrix is estimated at the location of the center of the cube.
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</p>
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<div id="org620d9b9" class="figure">
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<p><img src="./figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
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<span style="color: #CC9393;">'H_tot'</span>, <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
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<span style="color: #CC9393;">'L'</span>, <span style="color: #BFEBBF;">200</span><span style="color: #7CB8BB;">/</span>sqrt<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
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<span style="color: #CC9393;">'H'</span>, <span style="color: #BFEBBF;">60</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
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<span style="color: #CC9393;">'H0'</span>, <span style="color: #BFEBBF;">200</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">60</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
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<span style="color: #DCDCCC;">)</span>;
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stewart = initializeCubicConfiguration<span style="color: #DCDCCC;">(</span>opts<span style="color: #DCDCCC;">)</span>;
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opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
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<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: #7CB8BB;">-</span><span style="color: #BFEBBF;">50</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>
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<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: #7CB8BB;">-</span><span style="color: #BFEBBF;">50</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>
|
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<span style="color: #DCDCCC;">)</span>;
|
|
stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
|
|
|
|
save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/stewart.mat', 'stewart'</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jd'<span style="color: #7CB8BB;">*</span>stewart.Jd;
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">1.9e-18</td>
|
|
<td class="org-right">-2.3e-17</td>
|
|
<td class="org-right">1.8e-18</td>
|
|
<td class="org-right">5.5e-17</td>
|
|
<td class="org-right">-1.5e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.9e-18</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">6.8e-18</td>
|
|
<td class="org-right">-6.1e-17</td>
|
|
<td class="org-right">-1.6e-18</td>
|
|
<td class="org-right">4.8e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-2.3e-17</td>
|
|
<td class="org-right">6.8e-18</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-6.7e-18</td>
|
|
<td class="org-right">4.9e-18</td>
|
|
<td class="org-right">5.3e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.8e-18</td>
|
|
<td class="org-right">-6.1e-17</td>
|
|
<td class="org-right">-6.7e-18</td>
|
|
<td class="org-right">0.0067</td>
|
|
<td class="org-right">-2.3e-20</td>
|
|
<td class="org-right">-6.1e-20</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5.5e-17</td>
|
|
<td class="org-right">-1.6e-18</td>
|
|
<td class="org-right">4.9e-18</td>
|
|
<td class="org-right">-2.3e-20</td>
|
|
<td class="org-right">0.0067</td>
|
|
<td class="org-right">1e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.5e-17</td>
|
|
<td class="org-right">4.8e-18</td>
|
|
<td class="org-right">5.3e-19</td>
|
|
<td class="org-right">-6.1e-20</td>
|
|
<td class="org-right">1e-18</td>
|
|
<td class="org-right">0.027</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3344772" class="outline-3">
|
|
<h3 id="org3344772"><span class="section-number-3">2.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org620d9b9">1</a>).
|
|
The Jacobian matrix is not estimated at the location of the center of the cube.
|
|
</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;">'H_tot'</span>, <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
|
|
<span style="color: #CC9393;">'L'</span>, <span style="color: #BFEBBF;">200</span><span style="color: #7CB8BB;">/</span>sqrt<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
|
|
<span style="color: #CC9393;">'H'</span>, <span style="color: #BFEBBF;">60</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
|
|
<span style="color: #CC9393;">'H0'</span>, <span style="color: #BFEBBF;">200</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">60</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
<span style="color: #DCDCCC;">)</span>;
|
|
stewart = initializeCubicConfiguration<span style="color: #DCDCCC;">(</span>opts<span style="color: #DCDCCC;">)</span>;
|
|
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;">0</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;">0</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>;
|
|
stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jd'<span style="color: #7CB8BB;">*</span>stewart.Jd;
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">1.9e-18</td>
|
|
<td class="org-right">-2.3e-17</td>
|
|
<td class="org-right">1.5e-18</td>
|
|
<td class="org-right">-0.1</td>
|
|
<td class="org-right">-1.5e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.9e-18</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">6.8e-18</td>
|
|
<td class="org-right">0.1</td>
|
|
<td class="org-right">-1.6e-18</td>
|
|
<td class="org-right">4.8e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-2.3e-17</td>
|
|
<td class="org-right">6.8e-18</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-5.1e-19</td>
|
|
<td class="org-right">-5.5e-18</td>
|
|
<td class="org-right">5.3e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.5e-18</td>
|
|
<td class="org-right">0.1</td>
|
|
<td class="org-right">-5.1e-19</td>
|
|
<td class="org-right">0.012</td>
|
|
<td class="org-right">-3e-19</td>
|
|
<td class="org-right">3.1e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-0.1</td>
|
|
<td class="org-right">-1.6e-18</td>
|
|
<td class="org-right">-5.5e-18</td>
|
|
<td class="org-right">-3e-19</td>
|
|
<td class="org-right">0.012</td>
|
|
<td class="org-right">1.9e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.5e-17</td>
|
|
<td class="org-right">4.8e-18</td>
|
|
<td class="org-right">5.3e-19</td>
|
|
<td class="org-right">3.1e-19</td>
|
|
<td class="org-right">1.9e-18</td>
|
|
<td class="org-right">0.027</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org52de20d" class="outline-3">
|
|
<h3 id="org52de20d"><span class="section-number-3">2.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
Here, the "center" of the Stewart platform is not at the cube center (figure <a href="#org283dc40">2</a>).
|
|
The Jacobian is estimated at the cube center.
|
|
</p>
|
|
|
|
|
|
<div id="org283dc40" class="figure">
|
|
<p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Not centered cubic configuration</p>
|
|
</div>
|
|
|
|
<p>
|
|
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 \(z = 110 - 175 = -65\).
|
|
</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;">'H_tot'</span>, <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
|
|
<span style="color: #CC9393;">'L'</span>, <span style="color: #BFEBBF;">220</span><span style="color: #7CB8BB;">/</span>sqrt<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
|
|
<span style="color: #CC9393;">'H'</span>, <span style="color: #BFEBBF;">60</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
|
|
<span style="color: #CC9393;">'H0'</span>, <span style="color: #BFEBBF;">75</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
<span style="color: #DCDCCC;">)</span>;
|
|
stewart = initializeCubicConfiguration<span style="color: #DCDCCC;">(</span>opts<span style="color: #DCDCCC;">)</span>;
|
|
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: #7CB8BB;">-</span><span style="color: #BFEBBF;">65</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: #7CB8BB;">-</span><span style="color: #BFEBBF;">65</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>;
|
|
stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jd'<span style="color: #7CB8BB;">*</span>stewart.Jd;
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-1.8e-17</td>
|
|
<td class="org-right">2.6e-17</td>
|
|
<td class="org-right">3.3e-18</td>
|
|
<td class="org-right">0.04</td>
|
|
<td class="org-right">1.7e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.8e-17</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">1.9e-16</td>
|
|
<td class="org-right">-0.04</td>
|
|
<td class="org-right">2.2e-19</td>
|
|
<td class="org-right">-5.3e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2.6e-17</td>
|
|
<td class="org-right">1.9e-16</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-8.9e-18</td>
|
|
<td class="org-right">6.5e-19</td>
|
|
<td class="org-right">-5.8e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.3e-18</td>
|
|
<td class="org-right">-0.04</td>
|
|
<td class="org-right">-8.9e-18</td>
|
|
<td class="org-right">0.0089</td>
|
|
<td class="org-right">-9.3e-20</td>
|
|
<td class="org-right">9.8e-20</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0.04</td>
|
|
<td class="org-right">2.2e-19</td>
|
|
<td class="org-right">6.5e-19</td>
|
|
<td class="org-right">-9.3e-20</td>
|
|
<td class="org-right">0.0089</td>
|
|
<td class="org-right">-2.4e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.7e-19</td>
|
|
<td class="org-right">-5.3e-19</td>
|
|
<td class="org-right">-5.8e-19</td>
|
|
<td class="org-right">9.8e-20</td>
|
|
<td class="org-right">-2.4e-18</td>
|
|
<td class="org-right">0.032</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiffness matrix is not diagonal.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd7e1449" class="outline-3">
|
|
<h3 id="orgd7e1449"><span class="section-number-3">2.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
Here, the "center" of the Stewart platform is not at the cube center.
|
|
The Jacobian is estimated at the center of the Stewart platform.
|
|
</p>
|
|
|
|
<p>
|
|
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 \(z = 110 - 175 = -65\).
|
|
</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;">'H_tot'</span>, <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
|
|
<span style="color: #CC9393;">'L'</span>, <span style="color: #BFEBBF;">220</span><span style="color: #7CB8BB;">/</span>sqrt<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
|
|
<span style="color: #CC9393;">'H'</span>, <span style="color: #BFEBBF;">60</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
|
|
<span style="color: #CC9393;">'H0'</span>, <span style="color: #BFEBBF;">75</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
<span style="color: #DCDCCC;">)</span>;
|
|
stewart = initializeCubicConfiguration<span style="color: #DCDCCC;">(</span>opts<span style="color: #DCDCCC;">)</span>;
|
|
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: #7CB8BB;">-</span><span style="color: #BFEBBF;">60</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: #7CB8BB;">-</span><span style="color: #BFEBBF;">60</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>;
|
|
stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jd'<span style="color: #7CB8BB;">*</span>stewart.Jd;
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-1.8e-17</td>
|
|
<td class="org-right">2.6e-17</td>
|
|
<td class="org-right">-5.7e-19</td>
|
|
<td class="org-right">0.03</td>
|
|
<td class="org-right">1.7e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.8e-17</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">1.9e-16</td>
|
|
<td class="org-right">-0.03</td>
|
|
<td class="org-right">2.2e-19</td>
|
|
<td class="org-right">-5.3e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2.6e-17</td>
|
|
<td class="org-right">1.9e-16</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-1.5e-17</td>
|
|
<td class="org-right">6.5e-19</td>
|
|
<td class="org-right">-5.8e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-5.7e-19</td>
|
|
<td class="org-right">-0.03</td>
|
|
<td class="org-right">-1.5e-17</td>
|
|
<td class="org-right">0.0085</td>
|
|
<td class="org-right">4.9e-20</td>
|
|
<td class="org-right">1.7e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0.03</td>
|
|
<td class="org-right">2.2e-19</td>
|
|
<td class="org-right">6.5e-19</td>
|
|
<td class="org-right">4.9e-20</td>
|
|
<td class="org-right">0.0085</td>
|
|
<td class="org-right">-1.1e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.7e-19</td>
|
|
<td class="org-right">-5.3e-19</td>
|
|
<td class="org-right">-5.8e-19</td>
|
|
<td class="org-right">1.7e-19</td>
|
|
<td class="org-right">-1.1e-18</td>
|
|
<td class="org-right">0.032</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiffness matrix is not diagonal.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf16b788" class="outline-3">
|
|
<h3 id="orgf16b788"><span class="section-number-3">2.5</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-5">
|
|
<div class="important">
|
|
<ul class="org-ul">
|
|
<li>The cubic configuration permits to have \(k_x = k_y = k_z\) and \(k_{\theta\x} = k_{\theta_y}\)</li>
|
|
<li>The stiffness matrix \(K\) is diagonal for the cubic configuration if the Stewart platform and the cube are centered <b>and</b> the Jacobian is estimated at the cube center</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga2cd408" class="outline-2">
|
|
<h2 id="orga2cd408"><span class="section-number-2">3</span> Cubic size analysis</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
We here study the effect of the size of the cube used for the Stewart configuration.
|
|
</p>
|
|
|
|
<p>
|
|
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform.
|
|
</p>
|
|
|
|
<p>
|
|
We only vary the size of the cube.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H_cubes = <span style="color: #BFEBBF;">250</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">20</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">350</span>;
|
|
stewarts = <span style="color: #DCDCCC;">{</span>zeros<span style="color: #BFEBBF;">(</span>length<span style="color: #D0BF8F;">(</span>H_cubes<span style="color: #D0BF8F;">)</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</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;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:length</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">H_cubes</span><span style="color: #DCDCCC;">)</span>
|
|
H_cube = H_cubes<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span>;
|
|
H_tot = <span style="color: #BFEBBF;">100</span>;
|
|
H = <span style="color: #BFEBBF;">80</span>;
|
|
|
|
opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
|
|
<span style="color: #CC9393;">'H_tot'</span>, H_tot, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
|
|
<span style="color: #CC9393;">'L'</span>, H_cube<span style="color: #7CB8BB;">/</span>sqrt<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
|
|
<span style="color: #CC9393;">'H'</span>, H, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
|
|
<span style="color: #CC9393;">'H0'</span>, H_cube<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">-</span>H<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
<span style="color: #DCDCCC;">)</span>;
|
|
stewart = initializeCubicConfiguration<span style="color: #DCDCCC;">(</span>opts<span style="color: #DCDCCC;">)</span>;
|
|
|
|
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>, H_cube<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">-</span>opts.H0<span style="color: #7CB8BB;">-</span>opts.H_tot<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>, H_cube<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">-</span>opts.H0<span style="color: #7CB8BB;">-</span>opts.H_tot<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>;
|
|
stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
|
|
stewarts<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">{</span>stewart<span style="color: #DCDCCC;">}</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
The Stiffness matrix is computed for all generated Stewart platforms.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ks = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">6</span>, length<span style="color: #BFEBBF;">(</span>H_cube<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;">:length</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">H_cubes</span><span style="color: #DCDCCC;">)</span>
|
|
Ks<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span> = stewarts<span style="color: #DCDCCC;">{</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">}</span>.Jd'<span style="color: #7CB8BB;">*</span>stewarts<span style="color: #DCDCCC;">{</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">}</span>.Jd;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The only elements of \(K\) that vary are \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\).
|
|
</p>
|
|
|
|
<p>
|
|
Finally, we plot \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\)
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
hold on;
|
|
plot<span style="color: #DCDCCC;">(</span>H_cubes, squeeze<span style="color: #BFEBBF;">(</span>Ks<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">4</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'DisplayName', '</span>$k_<span style="color: #BFEBBF;">{</span><span style="color: #7CB8BB;">\</span>theta_x<span style="color: #BFEBBF;">}</span>$'<span style="color: #DCDCCC;">)</span>;
|
|
plot<span style="color: #DCDCCC;">(</span>H_cubes, squeeze<span style="color: #BFEBBF;">(</span>Ks<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">6</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'DisplayName', '</span>$k_<span style="color: #BFEBBF;">{</span><span style="color: #7CB8BB;">\</span>theta_z<span style="color: #BFEBBF;">}</span>$'<span style="color: #DCDCCC;">)</span>;
|
|
hold off;
|
|
legend<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'location', 'northwest'</span><span style="color: #DCDCCC;">)</span>;
|
|
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Cube Size </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">mm</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Rotational stiffnes </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">normalized</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org859b371" class="figure">
|
|
<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </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.
|
|
In that case, the legs will the further separated. Size of the cube is then limited by allowed space.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9220275" class="outline-2">
|
|
<h2 id="org9220275"><span class="section-number-2">4</span> initializeCubicConfiguration</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org8876664"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgdee5436" class="outline-3">
|
|
<h3 id="orgdee5436"><span class="section-number-3">4.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeCubicConfiguration</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-org68794ca" class="outline-3">
|
|
<h3 id="org68794ca"><span class="section-number-3">4.2</span> Optional Parameters</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
Default values for opts.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
|
|
<span style="color: #CC9393;">'H_tot'</span>, <span style="color: #BFEBBF;">90</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
|
|
<span style="color: #CC9393;">'L'</span>, <span style="color: #BFEBBF;">110</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
|
|
<span style="color: #CC9393;">'H'</span>, <span style="color: #BFEBBF;">40</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
|
|
<span style="color: #CC9393;">'H0'</span>, <span style="color: #BFEBBF;">75</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [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-org93d8028" class="outline-3">
|
|
<h3 id="org93d8028"><span class="section-number-3">4.3</span> Cube Creation</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">points = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">]</span>;
|
|
points = opts.L<span style="color: #7CB8BB;">*</span>points;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We create the rotation matrix to rotate the cube
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sx = cross<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</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, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</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 = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</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>;
|
|
|
|
R = <span style="color: #DCDCCC;">[</span>sx', sy', sz'<span style="color: #DCDCCC;">]</span>';
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We use to rotation matrix to rotate the cube
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">cube = zeros<span style="color: #DCDCCC;">(</span>size<span style="color: #BFEBBF;">(</span>points<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;">:size</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">points, </span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>
|
|
cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = R <span style="color: #7CB8BB;">*</span> points<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>';
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org00e16e1" class="outline-3">
|
|
<h3 id="org00e16e1"><span class="section-number-3">4.4</span> Vectors of each leg</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">leg_indices = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">4</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">4</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">6</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">6</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">7</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">7</span><span style="color: #DCDCCC;">]</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Vectors are:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">legs = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
|
|
legs_start = 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: #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>
|
|
legs<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = cube<span style="color: #DCDCCC;">(</span>leg_indices<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">-</span> cube<span style="color: #DCDCCC;">(</span>leg_indices<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
|
|
legs_start<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = cube<span style="color: #DCDCCC;">(</span>leg_indices<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd7df0cc" class="outline-3">
|
|
<h3 id="orgd7df0cc"><span class="section-number-3">4.5</span> Verification of Height of the Stewart Platform</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
<p>
|
|
If the Stewart platform is not contained in the cube, throw an error.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hmax = cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">-</span> cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">if</span> opts.H0 <span style="color: #7CB8BB;"><</span> cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>
|
|
error<span style="color: #DCDCCC;">(</span>sprintf<span style="color: #BFEBBF;">(</span>'H0 is not high enought. Minimum H0 = %.<span style="color: #BFEBBF;">1f</span>', cube(<span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">3</span>)));
|
|
<span style="color: #F0DFAF; font-weight: bold;">else</span> <span style="color: #F0DFAF; font-weight: bold;">if</span> opts.H0 <span style="color: #7CB8BB;">+</span> opts.H <span style="color: #7CB8BB;">></span> cube<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">3</span><span style="color: #D0BF8F;">)</span>
|
|
error<span style="color: #D0BF8F;">(</span>sprintf<span style="color: #93E0E3;">(</span>'H0<span style="color: #7CB8BB;">+</span>H is too high. Maximum H0<span style="color: #7CB8BB;">+</span>H = %.<span style="color: #BFEBBF;">1f</span>', cube(<span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">3</span>)));
|
|
error<span style="color: #9FC59F;">(</span><span style="color: #CC9393;">'H0+H is too high'</span><span style="color: #9FC59F;">)</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc4b765a" class="outline-3">
|
|
<h3 id="orgc4b765a"><span class="section-number-3">4.6</span> Determinate the location of the joints</h3>
|
|
<div class="outline-text-3" id="text-4-6">
|
|
<p>
|
|
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.
|
|
</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: #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>
|
|
t = <span style="color: #DCDCCC;">(</span>opts.H0<span style="color: #7CB8BB;">-</span>legs_start<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = legs_start<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> t<span style="color: #7CB8BB;">*</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: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the location of the joints on the mobile platform with respect to \(\{A\}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">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: #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>
|
|
t = <span style="color: #DCDCCC;">(</span>opts.H0<span style="color: #7CB8BB;">+</span>opts.H<span style="color: #7CB8BB;">-</span>legs_start<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = legs_start<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> t<span style="color: #7CB8BB;">*</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: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the location of the joints on the mobile platform with respect to \(\{B\}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Bb = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
|
|
Bb = Ab <span style="color: #7CB8BB;">-</span> <span style="color: #DCDCCC;">(</span>opts.H0 <span style="color: #7CB8BB;">+</span> opts.H_tot<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> opts.H<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span><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 class="org-src-container">
|
|
<pre class="src src-matlab">h = opts.H0 <span style="color: #7CB8BB;">+</span> opts.H<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">-</span> opts.H_tot<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span>;
|
|
Aa = Aa <span style="color: #7CB8BB;">-</span> 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>;
|
|
Ab = Ab <span style="color: #7CB8BB;">-</span> 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>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0a84b4d" class="outline-3">
|
|
<h3 id="org0a84b4d"><span class="section-number-3">4.7</span> Returns Stewart Structure</h3>
|
|
<div class="outline-text-3" id="text-4-7">
|
|
<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-orgcb21f88" class="outline-2">
|
|
<h2 id="orgcb21f88"><span class="section-number-2">5</span> Tests</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
</div>
|
|
<div id="outline-container-orgc47f87d" class="outline-3">
|
|
<h3 id="orgc47f87d"><span class="section-number-3">5.1</span> First attempt to parametrisation</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
|
|
<div id="orgb15dddd" class="figure">
|
|
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Schematic of the bottom plates with all the parameters</p>
|
|
</div>
|
|
|
|
<p>
|
|
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.
|
|
</p>
|
|
|
|
|
|
<p>
|
|
The configuration is cubic if:
|
|
\[ \overrightarrow{a_i b_i} \cdot \overrightarrow{a_j b_j} = 0, \ \forall i, j = [1, \hdots, 6], i \ne j \]
|
|
</p>
|
|
|
|
<p>
|
|
Lets express \(a_i\), \(b_i\) and \(a_j\):
|
|
</p>
|
|
\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*}
|
|
|
|
<p>
|
|
\[ \overrightarrow{a_1 b_1} = b_1 - a_1 = \begin{bmatrix}R_{\text{leg}} \cos(120 - \alpha) \\ R_{\text{leg}} \cos(120 - \alpha) \\ 0\end{bmatrix}\]
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgff4f69c" class="outline-3">
|
|
<h3 id="orgff4f69c"><span class="section-number-3">5.2</span> Second attempt</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
<p>
|
|
We start with the point of a cube in space:
|
|
</p>
|
|
\begin{align*}
|
|
[0, 0, 0] ; \ [0, 0, 1]; \ ...
|
|
\end{align*}
|
|
|
|
<p>
|
|
We also want the cube to point upward:
|
|
\[ [1, 1, 1] \Rightarrow [0, 0, 1] \]
|
|
</p>
|
|
|
|
<p>
|
|
Then we have the direction of all the vectors expressed in the frame of the hexapod.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">points = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">]</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
plot3<span style="color: #DCDCCC;">(</span>points<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, points<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span>, points<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'ko'</span><span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sx = cross<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</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, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</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 = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</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>;
|
|
|
|
R = <span style="color: #DCDCCC;">[</span>sx', sy', sz'<span style="color: #DCDCCC;">]</span>';
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">cube = zeros<span style="color: #DCDCCC;">(</span>size<span style="color: #BFEBBF;">(</span>points<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;">:size</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">points, </span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>
|
|
cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = R <span style="color: #7CB8BB;">*</span> points<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>';
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
hold on;
|
|
plot3<span style="color: #DCDCCC;">(</span>points<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, points<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span>, points<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'ko'</span><span style="color: #DCDCCC;">)</span>;
|
|
plot3<span style="color: #DCDCCC;">(</span>cube<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, cube<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span>, cube<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>,<span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'ro'</span><span style="color: #DCDCCC;">)</span>;
|
|
hold off;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Now we plot the legs of the hexapod.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">leg_indices = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">4</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">4</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">6</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">6</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">7</span>; <span style="text-decoration: underline;">...</span>
|
|
<span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">7</span><span style="color: #DCDCCC;">]</span>
|
|
|
|
<span style="color: #7CB8BB;">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot3<span style="color: #DCDCCC;">(</span>cube<span style="color: #BFEBBF;">(</span>leg_indices<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, cube<span style="color: #BFEBBF;">(</span>leg_indices<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span>, cube<span style="color: #BFEBBF;">(</span>leg_indices<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'-'</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
hold off;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Vectors are:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">legs = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
|
|
legs_start = 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: #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>
|
|
legs<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = cube<span style="color: #DCDCCC;">(</span>leg_indices<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">-</span> cube<span style="color: #DCDCCC;">(</span>leg_indices<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
|
|
legs_start<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = cube<span style="color: #DCDCCC;">(</span>leg_indices<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We now have the orientation of each leg.
|
|
</p>
|
|
|
|
<p>
|
|
We here want to see if the position of the "slice" changes something.
|
|
</p>
|
|
|
|
<p>
|
|
Let's first estimate the maximum height of the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hmax = cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">-</span> cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let's then estimate the middle position of the platform
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hmid = cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">8</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org011ab94" class="outline-3">
|
|
<h3 id="org011ab94"><span class="section-number-3">5.3</span> Generate the Stewart platform for a Cubic configuration</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
<p>
|
|
First we defined the height of the Hexapod.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = Hmax<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Zs = <span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>cube<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Height of the fixed platform</span>
|
|
Ze = Zs <span style="color: #7CB8BB;">+</span> H; <span style="color: #7F9F7F;">% Height of the mobile platform</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We now determine the location of the joints on the fixed platform.
|
|
</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: #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>
|
|
t = <span style="color: #DCDCCC;">(</span>Zs<span style="color: #7CB8BB;">-</span>legs_start<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = legs_start<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> t<span style="color: #7CB8BB;">*</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: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the location of the joints on the mobile platform
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">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: #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>
|
|
t = <span style="color: #DCDCCC;">(</span>Ze<span style="color: #7CB8BB;">-</span>legs_start<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = legs_start<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> t<span style="color: #7CB8BB;">*</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: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we plot the legs.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot3<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>Ab<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span>,Aa<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">[</span>Ab<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>,Aa<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">[</span>Ab<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #D0BF8F;">)</span>,Aa<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">3</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span>, <span style="color: #CC9393;">'k-'</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
hold off;
|
|
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
|
|
ylim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
|
|
zlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
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<ol class="org-ol">
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<li>Z.J. Geng, and L.S. Haynes, , <i>Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms</i>, IEEE Transactions on Control Systems Technology, 2<b>(1)</b>, pp. 45-53 (1994). <a href="http://dx.doi.org/10.1109/87.273110">http://dx.doi.org/10.1109/87.273110</a>.</li>
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<p class="author">Author: Thomas Dehaeze</p>
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<p class="date">Created: 2019-03-25 lun. 18:11</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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