1375 lines
54 KiB
HTML
1375 lines
54 KiB
HTML
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<a accesskey="h" href="./index.html"> UP </a>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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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="#org86c83bf">1. Questions we wish to answer with this analysis</a></li>
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<li><a href="#org0b05973">2. <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</a>
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<ul>
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<li><a href="#org3f035e8">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="#org77ecb36">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="#org42ea8ad">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="#org38870ce">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="#org08c7461">2.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgc4c2abd">3. <span class="todo TODO">TODO</span> Cubic size analysis</a></li>
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<li><a href="#org36a27e6">4. <span class="todo TODO">TODO</span> initializeCubicConfiguration</a>
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<ul>
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<li><a href="#orgf299c5c">4.1. Function description</a></li>
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<li><a href="#org46c8589">4.2. Optional Parameters</a></li>
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<li><a href="#orgd8d9b14">4.3. Cube Creation</a></li>
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<li><a href="#org181d1d8">4.4. Vectors of each leg</a></li>
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<li><a href="#orgb396e98">4.5. Verification of Height of the Stewart Platform</a></li>
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<li><a href="#orgf38af83">4.6. Determinate the location of the joints</a></li>
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<li><a href="#orgdf9e3cf">4.7. Returns Stewart Structure</a></li>
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</ul>
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</li>
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<li><a href="#orgf8fb731">5. <span class="todo TODO">TODO</span> Tests</a>
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<ul>
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<li><a href="#org4434fe5">5.1. First attempt to parametrisation</a></li>
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<li><a href="#org723e6eb">5.2. Second attempt</a></li>
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<li><a href="#orgcc173ac">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 <a class='org-ref-reference' href="#geng94_six_degree_of_freed_activ">geng94_six_degree_of_freed_activ</a>.
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Further analysis is conducted in <a class='org-ref-reference' href="#jafari03_orthog_gough_stewar_platf_microm">jafari03_orthog_gough_stewar_platf_microm</a>.
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</p>
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<p>
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People using orthogonal/cubic configuration: <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>.
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</p>
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<p>
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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="#org8b1f609">4</a>).
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</p>
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<p>
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According to <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>, the cubic configuration provides a uniform stiffness in all directions and <b>minimizes the crosscoupling</b> from actuator to sensor of different legs (being orthogonal to each other).
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</p>
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<div id="outline-container-org86c83bf" class="outline-2">
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<h2 id="org86c83bf"><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-org0b05973" class="outline-2">
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<h2 id="org0b05973"><span class="section-number-2">2</span> <span class="todo TODO">TODO</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-org3f035e8" class="outline-3">
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<h3 id="org3f035e8"><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="#org1effc0f">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="org1effc0f" 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(...
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<span class="org-string">'H_tot'</span>, 100, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
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<span class="org-string">'L'</span>, 200<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
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<span class="org-string">'H'</span>, 60, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
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<span class="org-string">'H0'</span>, 200<span class="org-type">/</span>2<span class="org-type">-</span>60<span class="org-type">/</span>2 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
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);
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stewart = initializeCubicConfiguration(opts);
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opts = struct(...
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<span class="org-string">'Jd_pos'</span>, [0, 0, <span class="org-type">-</span>50], ...<span class="org-comment"> % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
|
|
<span class="org-string">'Jf_pos'</span>, [0, 0, <span class="org-type">-</span>50] ...<span class="org-comment"> % Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
|
|
);
|
|
stewart = computeGeometricalProperties(stewart, opts);
|
|
|
|
save(<span class="org-string">'./mat/stewart.mat'</span>, <span class="org-string">'stewart'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jf<span class="org-type">'*</span>stewart.Jf;
|
|
</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-org77ecb36" class="outline-3">
|
|
<h3 id="org77ecb36"><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="#org1effc0f">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 class="org-string">'H_tot'</span>, 100, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
|
<span class="org-string">'L'</span>, 200<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
|
<span class="org-string">'H'</span>, 60, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
|
<span class="org-string">'H0'</span>, 200<span class="org-type">/</span>2<span class="org-type">-</span>60<span class="org-type">/</span>2 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
);
|
|
stewart = initializeCubicConfiguration(opts);
|
|
opts = struct(...
|
|
<span class="org-string">'Jd_pos'</span>, [0, 0, 0], ...<span class="org-comment"> % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
|
|
<span class="org-string">'Jf_pos'</span>, [0, 0, 0] ...<span class="org-comment"> % Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
|
|
);
|
|
stewart = computeGeometricalProperties(stewart, opts);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jf<span class="org-type">'*</span>stewart.Jf;
|
|
</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-org42ea8ad" class="outline-3">
|
|
<h3 id="org42ea8ad"><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="#org3f10bc2">2</a>).
|
|
The Jacobian is estimated at the cube center.
|
|
</p>
|
|
|
|
|
|
<div id="org3f10bc2" 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 class="org-string">'H_tot'</span>, 100, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
|
<span class="org-string">'L'</span>, 220<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
|
<span class="org-string">'H'</span>, 60, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
|
<span class="org-string">'H0'</span>, 75 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
);
|
|
stewart = initializeCubicConfiguration(opts);
|
|
opts = struct(...
|
|
<span class="org-string">'Jd_pos'</span>, [0, 0, <span class="org-type">-</span>65], ...<span class="org-comment"> % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
|
|
<span class="org-string">'Jf_pos'</span>, [0, 0, <span class="org-type">-</span>65] ...<span class="org-comment"> % Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
|
|
);
|
|
stewart = computeGeometricalProperties(stewart, opts);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jf<span class="org-type">'*</span>stewart.Jf;
|
|
</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-org38870ce" class="outline-3">
|
|
<h3 id="org38870ce"><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 class="org-string">'H_tot'</span>, 100, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
|
<span class="org-string">'L'</span>, 220<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
|
<span class="org-string">'H'</span>, 60, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
|
<span class="org-string">'H0'</span>, 75 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
);
|
|
stewart = initializeCubicConfiguration(opts);
|
|
opts = struct(...
|
|
<span class="org-string">'Jd_pos'</span>, [0, 0, <span class="org-type">-</span>60], ...<span class="org-comment"> % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
|
|
<span class="org-string">'Jf_pos'</span>, [0, 0, <span class="org-type">-</span>60] ...<span class="org-comment"> % Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
|
|
);
|
|
stewart = computeGeometricalProperties(stewart, opts);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = stewart.Jf<span class="org-type">'*</span>stewart.Jf;
|
|
</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-org08c7461" class="outline-3">
|
|
<h3 id="org08c7461"><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-orgc4c2abd" class="outline-2">
|
|
<h2 id="orgc4c2abd"><span class="section-number-2">3</span> <span class="todo TODO">TODO</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 = 250<span class="org-type">:</span>20<span class="org-type">:</span>350;
|
|
stewarts = {zeros(length(H_cubes), 1)};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(H_cubes)</span>
|
|
H_cube = H_cubes(<span class="org-constant">i</span>);
|
|
H_tot = 100;
|
|
H = 80;
|
|
|
|
opts = struct(...
|
|
<span class="org-string">'H_tot'</span>, H_tot, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
|
<span class="org-string">'L'</span>, H_cube<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
|
<span class="org-string">'H'</span>, H, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
|
<span class="org-string">'H0'</span>, H_cube<span class="org-type">/</span>2<span class="org-type">-</span>H<span class="org-type">/</span>2 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
);
|
|
stewart = initializeCubicConfiguration(opts);
|
|
|
|
opts = struct(...
|
|
<span class="org-string">'Jd_pos'</span>, [0, 0, H_cube<span class="org-type">/</span>2<span class="org-type">-</span>opts.H0<span class="org-type">-</span>opts.H_tot], ...<span class="org-comment"> % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
|
|
<span class="org-string">'Jf_pos'</span>, [0, 0, H_cube<span class="org-type">/</span>2<span class="org-type">-</span>opts.H0<span class="org-type">-</span>opts.H_tot] ...<span class="org-comment"> % Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
|
|
);
|
|
stewart = computeGeometricalProperties(stewart, opts);
|
|
stewarts(<span class="org-constant">i</span>) = {stewart};
|
|
<span class="org-keyword">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(6, 6, length(H_cube));
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(H_cubes)</span>
|
|
Ks(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = stewarts{<span class="org-constant">i</span>}.Jd<span class="org-type">'*</span>stewarts{<span class="org-constant">i</span>}.Jd;
|
|
<span class="org-keyword">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 class="org-type">figure</span>;
|
|
hold on;
|
|
plot(H_cubes, squeeze(Ks(4, 4, <span class="org-type">:</span>)), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$k_{\theta_x}$'</span>);
|
|
plot(H_cubes, 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 [mm]'</span>); ylabel(<span class="org-string">'Rotational stiffnes [normalized]'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org659a01f" 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-org36a27e6" class="outline-2">
|
|
<h2 id="org36a27e6"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> initializeCubicConfiguration</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org8b1f609"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf299c5c" class="outline-3">
|
|
<h3 id="orgf299c5c"><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 class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">initializeCubicConfiguration</span>(<span class="org-variable-name">opts_param</span>)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org46c8589" class="outline-3">
|
|
<h3 id="org46c8589"><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 class="org-string">'H_tot'</span>, 90, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
|
<span class="org-string">'L'</span>, 110, ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
|
<span class="org-string">'H'</span>, 40, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
|
<span class="org-string">'H0'</span>, 75 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
|
);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Populate opts with input parameters
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">if</span> exist(<span class="org-string">'opts_param'</span>,<span class="org-string">'var'</span>)
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">opt</span> = <span class="org-constant">fieldnames(opts_param)'</span>
|
|
opts.(opt{1}) = opts_param.(opt{1});
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd8d9b14" class="outline-3">
|
|
<h3 id="orgd8d9b14"><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 = [0, 0, 0; ...
|
|
0, 0, 1; ...
|
|
0, 1, 0; ...
|
|
0, 1, 1; ...
|
|
1, 0, 0; ...
|
|
1, 0, 1; ...
|
|
1, 1, 0; ...
|
|
1, 1, 1];
|
|
points = opts.L<span class="org-type">*</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([1, 1, 1], [1 0 0]);
|
|
sx = sx<span class="org-type">/</span>norm(sx);
|
|
|
|
sy = <span class="org-type">-</span>cross(sx, [1, 1, 1]);
|
|
sy = sy<span class="org-type">/</span>norm(sy);
|
|
|
|
sz = [1, 1, 1];
|
|
sz = sz<span class="org-type">/</span>norm(sz);
|
|
|
|
R = [sx<span class="org-type">'</span>, sy<span class="org-type">'</span>, sz<span class="org-type">'</span>]<span class="org-type">'</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(size(points));
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:size(points, 1)</span>
|
|
cube(<span class="org-constant">i</span>, <span class="org-type">:</span>) = R <span class="org-type">*</span> points(<span class="org-constant">i</span>, <span class="org-type">:</span>)<span class="org-type">'</span>;
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org181d1d8" class="outline-3">
|
|
<h3 id="org181d1d8"><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 = [3, 4; ...
|
|
2, 4; ...
|
|
2, 6; ...
|
|
5, 6; ...
|
|
5, 7; ...
|
|
3, 7];
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Vectors are:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">legs = zeros(6, 3);
|
|
legs_start = zeros(6, 3);
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
legs(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 2), <span class="org-type">:</span>) <span class="org-type">-</span> cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>);
|
|
legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb396e98" class="outline-3">
|
|
<h3 id="orgb396e98"><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(4, 3) <span class="org-type">-</span> cube(2, 3);
|
|
<span class="org-keyword">if</span> opts.H0 <span class="org-type"><</span> cube(2, 3)
|
|
error(sprintf(<span class="org-string">'H0 is not high enought. Minimum H0 = %.1f'</span>, cube(2, 3)));
|
|
<span class="org-keyword">else</span> <span class="org-keyword">if</span> opts.H0 <span class="org-type">+</span> opts.H <span class="org-type">></span> cube(4, 3)
|
|
error(sprintf(<span class="org-string">'H0+H is too high. Maximum H0+H = %.1f'</span>, cube(4, 3)));
|
|
error(<span class="org-string">'H0+H is too high'</span>);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf38af83" class="outline-3">
|
|
<h3 id="orgf38af83"><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(6, 3);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
t = (opts.H0<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
|
Aa(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
|
<span class="org-keyword">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(6, 3);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
t = (opts.H0<span class="org-type">+</span>opts.H<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
|
Ab(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
|
<span class="org-keyword">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(6, 3);
|
|
Bb = Ab <span class="org-type">-</span> (opts.H0 <span class="org-type">+</span> opts.H_tot<span class="org-type">/</span>2 <span class="org-type">+</span> opts.H<span class="org-type">/</span>2)<span class="org-type">*</span>[0, 0, 1];
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h = opts.H0 <span class="org-type">+</span> opts.H<span class="org-type">/</span>2 <span class="org-type">-</span> opts.H_tot<span class="org-type">/</span>2;
|
|
Aa = Aa <span class="org-type">-</span> h<span class="org-type">*</span>[0, 0, 1];
|
|
Ab = Ab <span class="org-type">-</span> h<span class="org-type">*</span>[0, 0, 1];
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgdf9e3cf" class="outline-3">
|
|
<h3 id="orgdf9e3cf"><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();
|
|
stewart.Aa = Aa;
|
|
stewart.Ab = Ab;
|
|
stewart.Bb = Bb;
|
|
stewart.H_tot = opts.H_tot;
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf8fb731" class="outline-2">
|
|
<h2 id="orgf8fb731"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> Tests</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
</div>
|
|
<div id="outline-container-org4434fe5" class="outline-3">
|
|
<h3 id="org4434fe5"><span class="section-number-3">5.1</span> First attempt to parametrisation</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
|
|
<div id="org8dfcb96" 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-org723e6eb" class="outline-3">
|
|
<h3 id="org723e6eb"><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 = [0, 0, 0; ...
|
|
0, 0, 1; ...
|
|
0, 1, 0; ...
|
|
0, 1, 1; ...
|
|
1, 0, 0; ...
|
|
1, 0, 1; ...
|
|
1, 1, 0; ...
|
|
1, 1, 1];
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
plot3(points(<span class="org-type">:</span>,1), points(<span class="org-type">:</span>,2), points(<span class="org-type">:</span>,3), <span class="org-string">'ko'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sx = cross([1, 1, 1], [1 0 0]);
|
|
sx = sx<span class="org-type">/</span>norm(sx);
|
|
|
|
sy = <span class="org-type">-</span>cross(sx, [1, 1, 1]);
|
|
sy = sy<span class="org-type">/</span>norm(sy);
|
|
|
|
sz = [1, 1, 1];
|
|
sz = sz<span class="org-type">/</span>norm(sz);
|
|
|
|
R = [sx<span class="org-type">'</span>, sy<span class="org-type">'</span>, sz<span class="org-type">'</span>]<span class="org-type">'</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">cube = zeros(size(points));
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:size(points, 1)</span>
|
|
cube(<span class="org-constant">i</span>, <span class="org-type">:</span>) = R <span class="org-type">*</span> points(<span class="org-constant">i</span>, <span class="org-type">:</span>)<span class="org-type">'</span>;
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
plot3(points(<span class="org-type">:</span>,1), points(<span class="org-type">:</span>,2), points(<span class="org-type">:</span>,3), <span class="org-string">'ko'</span>);
|
|
plot3(cube(<span class="org-type">:</span>,1), cube(<span class="org-type">:</span>,2), cube(<span class="org-type">:</span>,3), <span class="org-string">'ro'</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 = [3, 4; ...
|
|
2, 4; ...
|
|
2, 6; ...
|
|
5, 6; ...
|
|
5, 7; ...
|
|
3, 7]
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
plot3(cube(leg_indices(<span class="org-constant">i</span>, <span class="org-type">:</span>),1), cube(leg_indices(<span class="org-constant">i</span>, <span class="org-type">:</span>),2), cube(leg_indices(<span class="org-constant">i</span>, <span class="org-type">:</span>),3), <span class="org-string">'-'</span>);
|
|
<span class="org-keyword">end</span>
|
|
hold off;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Vectors are:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">legs = zeros(6, 3);
|
|
legs_start = zeros(6, 3);
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
legs(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 2), <span class="org-type">:</span>) <span class="org-type">-</span> cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>);
|
|
legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>)
|
|
<span class="org-keyword">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(4, 3) <span class="org-type">-</span> cube(2, 3);
|
|
</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(8, 3)<span class="org-type">/</span>2;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcc173ac" class="outline-3">
|
|
<h3 id="orgcc173ac"><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 class="org-type">/</span>2;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Zs = 1.2<span class="org-type">*</span>cube(2, 3); <span class="org-comment">% Height of the fixed platform</span>
|
|
Ze = Zs <span class="org-type">+</span> H; <span class="org-comment">% 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(6, 3);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
t = (Zs<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
|
Aa(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
|
<span class="org-keyword">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(6, 3);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
t = (Ze<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
|
Ab(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we plot the legs.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
plot3([Ab(<span class="org-constant">i</span>, 1),Aa(<span class="org-constant">i</span>, 1)], [Ab(<span class="org-constant">i</span>, 2),Aa(<span class="org-constant">i</span>, 2)], [Ab(<span class="org-constant">i</span>, 3),Aa(<span class="org-constant">i</span>, 3)], <span class="org-string">'k-'</span>);
|
|
<span class="org-keyword">end</span>
|
|
hold off;
|
|
xlim([<span class="org-type">-</span>1, 1]);
|
|
ylim([<span class="org-type">-</span>1, 1]);
|
|
zlim([0, 2]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<p>
|
|
|
|
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
|
<ul class='org-ref-bib'><li><a id="geng94_six_degree_of_freed_activ">[geng94_six_degree_of_freed_activ]</a> <a name="geng94_six_degree_of_freed_activ"></a>Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, <i>IEEE Transactions on Control Systems Technology</i>, <b>2(1)</b>, 45-53 (1994). <a href="https://doi.org/10.1109/87.273110">link</a>. <a href="http://dx.doi.org/10.1109/87.273110">doi</a>.</li>
|
|
<li><a id="jafari03_orthog_gough_stewar_platf_microm">[jafari03_orthog_gough_stewar_platf_microm]</a> <a name="jafari03_orthog_gough_stewar_platf_microm"></a>Jafari & McInroy, Orthogonal Gough-Stewart Platforms for Micromanipulation, <i>IEEE Transactions on Robotics and Automation</i>, <b>19(4)</b>, 595-603 (2003). <a href="https://doi.org/10.1109/tra.2003.814506">link</a>. <a href="http://dx.doi.org/10.1109/tra.2003.814506">doi</a>.</li>
|
|
<li><a id="preumont07_six_axis_singl_stage_activ">[preumont07_six_axis_singl_stage_activ]</a> <a name="preumont07_six_axis_singl_stage_activ"></a>Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, <i>Journal of Sound and Vibration</i>, <b>300(3-5)</b>, 644-661 (2007). <a href="https://doi.org/10.1016/j.jsv.2006.07.050">link</a>. <a href="http://dx.doi.org/10.1016/j.jsv.2006.07.050">doi</a>.</li>
|
|
</ul>
|
|
</p>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-01-27 lun. 17:41</p>
|
|
</div>
|
|
</body>
|
|
</html>
|