1035 lines
41 KiB
HTML
1035 lines
41 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="#org75c6951">1. <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</a>
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<ul>
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<li><a href="#orga823f72">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="#org4261310">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="#orgf297eb8">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="#orgfeaf9c1">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="#org24bdf29">1.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org2cb2ab0">2. <span class="todo TODO">TODO</span> Cubic size analysis</a></li>
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<li><a href="#orgeec7b47">3. Functions</a>
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<ul>
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<li><a href="#org92224ef">3.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
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<ul>
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<li><a href="#org715472d">Function description</a></li>
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<li><a href="#orgbab37f8">Documentation</a></li>
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<li><a href="#orgddbe42e">Optional Parameters</a></li>
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<li><a href="#org66dd074">Position of the Cube</a></li>
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<li><a href="#org388f35d">Compute the pose</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="#orgb0ae4eb">3.1</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-org75c6951" class="outline-2">
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<h2 id="org75c6951"><span class="section-number-2">1</span> <span class="todo TODO">TODO</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-orga823f72" class="outline-3">
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<h3 id="orga823f72"><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="#org66ade8d">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="org66ade8d" 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>
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<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>
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);
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stewart = computeGeometricalProperties(stewart, opts);
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save(<span class="org-string">'./mat/stewart.mat'</span>, <span class="org-string">'stewart'</span>);
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">K = stewart.Jf<span class="org-type">'*</span>stewart.Jf;
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</pre>
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<td class="org-right">2</td>
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<td class="org-right">1.9e-18</td>
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<td class="org-right">-2.3e-17</td>
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<td class="org-right">1.8e-18</td>
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<td class="org-right">5.5e-17</td>
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<td class="org-right">-1.5e-17</td>
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|
<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-org4261310" class="outline-3">
|
|
<h3 id="org4261310"><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="#org66ade8d">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-orgf297eb8" class="outline-3">
|
|
<h3 id="orgf297eb8"><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="#org4492663">2</a>).
|
|
The Jacobian is estimated at the cube center.
|
|
</p>
|
|
|
|
|
|
<div id="org4492663" 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-orgfeaf9c1" class="outline-3">
|
|
<h3 id="orgfeaf9c1"><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">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-org24bdf29" class="outline-3">
|
|
<h3 id="org24bdf29"><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 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-org2cb2ab0" class="outline-2">
|
|
<h2 id="org2cb2ab0"><span class="section-number-2">2</span> <span class="todo TODO">TODO</span> Cubic size analysis</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
We here study the effect of the size of the cube used for the Stewart 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="org009489e" 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-orgeec7b47" class="outline-2">
|
|
<h2 id="orgeec7b47"><span class="section-number-2">3</span> Functions</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgb108018"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org92224ef" class="outline-3">
|
|
<h3 id="org92224ef"><span class="section-number-3">3.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
<a id="orgb0ae4eb"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/generateCubicConfiguration.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org715472d" class="outline-4">
|
|
<h4 id="org715472d">Function description</h4>
|
|
<div class="outline-text-4" id="text-org715472d">
|
|
<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">% - 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">% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
|
<span class="org-comment">% - 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-orgbab37f8" class="outline-4">
|
|
<h4 id="orgbab37f8">Documentation</h4>
|
|
<div class="outline-text-4" id="text-orgbab37f8">
|
|
|
|
<div id="org946a873" class="figure">
|
|
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Cubic Configuration</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgddbe42e" class="outline-4">
|
|
<h4 id="orgddbe42e">Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-orgddbe42e">
|
|
<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, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org66dd074" class="outline-4">
|
|
<h4 id="org66dd074">Position of the Cube</h4>
|
|
<div class="outline-text-4" id="text-org66dd074">
|
|
<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-org388f35d" class="outline-4">
|
|
<h4 id="org388f35d">Compute the pose</h4>
|
|
<div class="outline-text-4" id="text-org388f35d">
|
|
<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">stewart.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;
|
|
stewart.Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org-type">-</span>stewart.H] <span class="org-type">+</span> ((stewart.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>
|
|
</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: 2020-02-06 jeu. 17:29</p>
|
|
</div>
|
|
</body>
|
|
</html>
|