1046 lines
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1046 lines
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HTML
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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="#org8c6677e">1. Configuration Analysis - Stiffness Matrix</a>
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<ul>
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<li><a href="#orgf6f7ad2">1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#orga88e79a">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
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<li><a href="#orge02ec88">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org43fd7e4">1.4. 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="#orgd35acc0">1.5. Conclusion</a></li>
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<li><a href="#org8afa645">1.6. Having Cube’s center above the top platform</a></li>
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</ul>
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</li>
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<li><a href="#org3044455">2. Functions</a>
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<ul>
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<li><a href="#org56504f1">2.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
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<ul>
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<li><a href="#orga5a9ba8">Function description</a></li>
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<li><a href="#org3253792">Documentation</a></li>
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<li><a href="#org154b5fb">Optional Parameters</a></li>
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<li><a href="#orgbb480a6">Check the <code>stewart</code> structure elements</a></li>
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<li><a href="#org771c630">Position of the Cube</a></li>
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<li><a href="#org3a2f468">Compute the pose</a></li>
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<li><a href="#org8c1af4f">Populate the <code>stewart</code> structure</a></li>
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</ul>
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</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
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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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The cubic (or orthogonal) configuration of the Stewart platform is now widely used (<a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>,<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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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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<p>
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To generate and study the Cubic configuration, <code>generateCubicConfiguration</code> is used (description in section <a href="#orga8311d3">2.1</a>).
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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 id="outline-container-org8c6677e" class="outline-2">
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<h2 id="org8c6677e"><span class="section-number-2">1</span> Configuration Analysis - Stiffness Matrix</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-orgf6f7ad2" class="outline-3">
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<h3 id="orgf6f7ad2"><span class="section-number-3">1.1</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-1-1">
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<p>
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We create a cubic Stewart platform (figure <a href="#org9454f54">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 class="org-src-container">
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<pre class="src src-matlab">stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, <span class="org-type">-</span>50e<span class="org-type">-</span>3);
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stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
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stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
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</pre>
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</div>
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<div id="org9454f54" 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 id="orgaba20c8" class="figure">
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<p><img src="figs/cubic_conf_centered_J_center.png" alt="cubic_conf_centered_J_center.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_center.pdf">pdf</a>)</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<tbody>
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<tr>
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<td class="org-right">2</td>
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<td class="org-right">0</td>
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<td class="org-right">-2.5e-16</td>
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<td class="org-right">0</td>
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<td class="org-right">2.1e-17</td>
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<td class="org-right">0</td>
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|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-7.8e-19</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-2.5e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-2.4e-18</td>
|
|
<td class="org-right">-1.4e-17</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-7.8e-19</td>
|
|
<td class="org-right">-2.4e-18</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">-4.3e-19</td>
|
|
<td class="org-right">1.7e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.8e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-1.1e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6.6e-18</td>
|
|
<td class="org-right">-3.3e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">1.7e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga88e79a" class="outline-3">
|
|
<h3 id="orga88e79a"><span class="section-number-3">1.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
<p>
|
|
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org9454f54">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">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 0);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org47f8142" class="figure">
|
|
<p><img src="figs/cubic_conf_centered_J_not_center.png" alt="cubic_conf_centered_J_not_center.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center (<a href="./figs/cubic_conf_centered_J_not_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_not_center.pdf">pdf</a>)</p>
|
|
</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">0</td>
|
|
<td class="org-right">-2.5e-16</td>
|
|
<td class="org-right">1.4e-17</td>
|
|
<td class="org-right">-0.1</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.1</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-2.5e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">3.4e-18</td>
|
|
<td class="org-right">-1.4e-17</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.4e-17</td>
|
|
<td class="org-right">0.1</td>
|
|
<td class="org-right">3.4e-18</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">1.1e-20</td>
|
|
<td class="org-right">3.4e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-0.1</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-1.4e-17</td>
|
|
<td class="org-right">1.4e-19</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">-1.7e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6.6e-18</td>
|
|
<td class="org-right">-3.3e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">3.6e-18</td>
|
|
<td class="org-right">-1.7e-18</td>
|
|
<td class="org-right">0.06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge02ec88" class="outline-3">
|
|
<h3 id="orge02ec88"><span class="section-number-3">1.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<p>
|
|
Here, the “center” of the Stewart platform is not at the cube center (figure <a href="#org97b319c">4</a>).
|
|
The Jacobian is estimated at the cube center.
|
|
</p>
|
|
|
|
|
|
<div id="org97b319c" class="figure">
|
|
<p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Not centered cubic configuration</p>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 80e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, <span class="org-type">-</span>40e<span class="org-type">-</span>3);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org0235d3a" class="figure">
|
|
<p><img src="figs/cubic_conf_not_centered_J_center.png" alt="cubic_conf_not_centered_J_center.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_not_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_center.pdf">pdf</a>)</p>
|
|
</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">0</td>
|
|
<td class="org-right">-1.7e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-0.02</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2.8e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.7e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">1.2e-19</td>
|
|
<td class="org-right">-1.4e-17</td>
|
|
<td class="org-right">1.4e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-0.02</td>
|
|
<td class="org-right">1.2e-19</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">-4.3e-19</td>
|
|
<td class="org-right">1.7e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-7.3e-18</td>
|
|
<td class="org-right">-3.3e-21</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6.6e-18</td>
|
|
<td class="org-right">2.5e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.06</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-org43fd7e4" class="outline-3">
|
|
<h3 id="org43fd7e4"><span class="section-number-3">1.4</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-1-4">
|
|
<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">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 80e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, <span class="org-type">-</span>40e<span class="org-type">-</span>3);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgbe766b3" class="figure">
|
|
<p><img src="figs/cubic_conf_not_centered_J_stewart_center.png" alt="cubic_conf_not_centered_J_stewart_center.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center (<a href="./figs/cubic_conf_not_centered_J_stewart_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_stewart_center.pdf">pdf</a>)</p>
|
|
</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">0</td>
|
|
<td class="org-right">-1.7e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-0.02</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2.8e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.7e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">1.2e-19</td>
|
|
<td class="org-right">-1.4e-17</td>
|
|
<td class="org-right">1.4e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-0.02</td>
|
|
<td class="org-right">1.2e-19</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">-4.3e-19</td>
|
|
<td class="org-right">1.7e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-7.3e-18</td>
|
|
<td class="org-right">-3.3e-21</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6.6e-18</td>
|
|
<td class="org-right">2.5e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd35acc0" class="outline-3">
|
|
<h3 id="orgd35acc0"><span class="section-number-3">1.5</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-1-5">
|
|
<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 Jacobian is estimated at the cube center.</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8afa645" class="outline-3">
|
|
<h3 id="org8afa645"><span class="section-number-3">1.6</span> Having Cube’s center above the top platform</h3>
|
|
<div class="outline-text-3" id="text-1-6">
|
|
<p>
|
|
Let’s say we want to have a decouple dynamics above the top platform.
|
|
Thus, we want the cube’s center to be located above the top center.
|
|
This is possible, to do so:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>The position of the center of the cube should be positioned at A</li>
|
|
<li>The Height of the “useful” part of the cube should be at least equal to two times the distance from F to A.
|
|
It is possible to have small cube, but then to configuration is a little bit strange.</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 50e<span class="org-type">-</span>3);
|
|
FOc = stewart.H <span class="org-type">+</span> stewart.MO_B(3);
|
|
Hc = 2<span class="org-type">*</span>(stewart.H <span class="org-type">+</span> stewart.MO_B(3));
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 10e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 10e<span class="org-type">-</span>3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
</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">0</td>
|
|
<td class="org-right">-3.2e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">3.1e-16</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-1.2e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-3.2e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">5e-18</td>
|
|
<td class="org-right">-5.6e-17</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-1.2e-16</td>
|
|
<td class="org-right">5e-18</td>
|
|
<td class="org-right">0.14</td>
|
|
<td class="org-right">3.5e-18</td>
|
|
<td class="org-right">1.4e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-5.4e-17</td>
|
|
<td class="org-right">2.1e-19</td>
|
|
<td class="org-right">0.14</td>
|
|
<td class="org-right">-6.9e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.4e-19</td>
|
|
<td class="org-right">-2.6e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">1.3e-17</td>
|
|
<td class="org-right">-6.9e-18</td>
|
|
<td class="org-right">0.54</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>
|
|
|
|
<div id="outline-container-org3044455" class="outline-2">
|
|
<h2 id="org3044455"><span class="section-number-2">2</span> Functions</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="org28ba607"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org56504f1" class="outline-3">
|
|
<h3 id="org56504f1"><span class="section-number-3">2.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
<a id="orga8311d3"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/generateCubicConfiguration.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orga5a9ba8" class="outline-4">
|
|
<h4 id="orga5a9ba8">Function description</h4>
|
|
<div class="outline-text-4" id="text-orga5a9ba8">
|
|
<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">generateCubicConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = generateCubicConfiguration(stewart, args)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - stewart - A structure with the following fields</span>
|
|
<span class="org-comment">% - geometry.H [1x1] - Total height of the platform [m]</span>
|
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
|
<span class="org-comment">% - Hc [1x1] - Height of the "useful" part of the cube [m]</span>
|
|
<span class="org-comment">% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]</span>
|
|
<span class="org-comment">% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]</span>
|
|
<span class="org-comment">% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
|
<span class="org-comment">% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
|
<span class="org-comment">% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3253792" class="outline-4">
|
|
<h4 id="org3253792">Documentation</h4>
|
|
<div class="outline-text-4" id="text-org3253792">
|
|
|
|
<div id="org8a7f3d8" class="figure">
|
|
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Cubic Configuration</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org154b5fb" class="outline-4">
|
|
<h4 id="org154b5fb">Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-org154b5fb">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e<span class="org-type">-</span>3
|
|
args.FOc (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
|
|
args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e<span class="org-type">-</span>3
|
|
args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e<span class="org-type">-</span>3
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbb480a6" class="outline-4">
|
|
<h4 id="orgbb480a6">Check the <code>stewart</code> structure elements</h4>
|
|
<div class="outline-text-4" id="text-orgbb480a6">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'H'</span>), <span class="org-string">'stewart.geometry should have attribute H'</span>)
|
|
H = stewart.geometry.H;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org771c630" class="outline-4">
|
|
<h4 id="org771c630">Position of the Cube</h4>
|
|
<div class="outline-text-4" id="text-org771c630">
|
|
<p>
|
|
We define the useful points of the cube with respect to the Cube’s center.
|
|
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
|
|
located at the center of the cube and aligned with {F} and {M}.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sx = [ 2; <span class="org-type">-</span>1; <span class="org-type">-</span>1];
|
|
sy = [ 0; 1; <span class="org-type">-</span>1];
|
|
sz = [ 1; 1; 1];
|
|
|
|
R = [sx, sy, sz]<span class="org-type">./</span>vecnorm([sx, sy, sz]);
|
|
|
|
L = args.Hc<span class="org-type">*</span>sqrt(3);
|
|
|
|
Cc = R<span class="org-type">'*</span>[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] <span class="org-type">-</span> [0;0;1.5<span class="org-type">*</span>args.Hc];
|
|
|
|
CCf = [Cc(<span class="org-type">:</span>,1), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,1)]; <span class="org-comment">% CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg</span>
|
|
CCm = [Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,6), Cc(<span class="org-type">:</span>,6)]; <span class="org-comment">% CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3a2f468" class="outline-4">
|
|
<h4 id="org3a2f468">Compute the pose</h4>
|
|
<div class="outline-text-4" id="text-org3a2f468">
|
|
<p>
|
|
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">CSi = (CCm <span class="org-type">-</span> CCf)<span class="org-type">./</span>vecnorm(CCm <span class="org-type">-</span> CCf);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Fa = CCf <span class="org-type">+</span> [0; 0; args.FOc] <span class="org-type">+</span> ((args.FHa<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
|
Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org-type">-</span>H] <span class="org-type">+</span> ((H<span class="org-type">-</span>args.MHb<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8c1af4f" class="outline-4">
|
|
<h4 id="org8c1af4f">Populate the <code>stewart</code> structure</h4>
|
|
<div class="outline-text-4" id="text-org8c1af4f">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.platform_F.Fa = Fa;
|
|
stewart.platform_M.Mb = Mb;
|
|
</pre>
|
|
</div>
|
|
</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="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>
|
|
<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>
|
|
</ul>
|
|
</p>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-02-11 mar. 15:50</p>
|
|
</div>
|
|
</body>
|
|
</html>
|