1945 lines
85 KiB
HTML
1945 lines
85 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="#org3d18192">1. Stiffness Matrix for the Cubic configuration</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="#org510da86">1.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgd70418b">2. Configuration with the Cube’s center above the mobile platform</a>
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<ul>
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<li><a href="#org8afa645">2.1. Having Cube’s center above the top platform</a></li>
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<li><a href="#org949a403">2.2. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgcc4ecce">3. Cubic size analysis</a>
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<ul>
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<li><a href="#org0029d8c">3.1. Analysis</a></li>
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<li><a href="#orgfc7135f">3.2. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgf09da67">4. Dynamic Coupling in the Cartesian Frame</a>
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<ul>
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<li><a href="#org5fe01ec">4.1. Cube’s center at the Center of Mass of the mobile platform</a></li>
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<li><a href="#org4cb2a36">4.2. Cube’s center not coincident with the Mass of the Mobile platform</a></li>
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<li><a href="#org2e09bcb">4.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org8f26dc0">5. Dynamic Coupling between actuators and sensors of each strut</a>
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<ul>
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<li><a href="#org6e391c9">5.1. Coupling between the actuators and sensors - Cubic Architecture</a></li>
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<li><a href="#orgafd808d">5.2. Coupling between the actuators and sensors - Non-Cubic Architecture</a></li>
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<li><a href="#org8c1a310">5.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org3044455">6. Functions</a>
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<ul>
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<li><a href="#org56504f1">6.1. <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 Cubic configuration for the Stewart platform was first proposed in <a class='org-ref-reference' href="#geng94_six_degree_of_freed_activ">geng94_six_degree_of_freed_activ</a>.
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This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube.
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This configuration 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 offers the following advantages:
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</p>
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<blockquote>
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<p>
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This topology provides a uniform control capability and a uniform stiffness in all directions, and it minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other).
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</p>
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</blockquote>
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<p>
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In this document, the cubic architecture is analyzed:
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</p>
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<ul class="org-ul">
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<li>In section <a href="#orgda0ee50">1</a>, we study the link between the Stiffness matrix and the cubic architecture and we find what are the conditions to obtain a diagonal stiffness matrix</li>
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<li>In section <a href="#orgb73265d">2</a>, we study cubic configurations where the cube’s center is located above the mobile platform</li>
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<li>In section <a href="#org348ec7d">3</a>, we study the effect of the cube’s size on the Stewart platform properties</li>
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<li>In section <a href="#org00d3816">4</a>, we study the dynamic coupling of the cubic configuration in the cartesian frame</li>
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<li>In section <a href="#org5b5c8a9">5</a>, we study the dynamic coupling of the cubic configuration from actuators to sensors of each strut</li>
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<li>In section <a href="#org28ba607">6</a>, function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function <code>generateCubicConfiguration</code> is used (described <a href="#orga8311d3">here</a>).</li>
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</ul>
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<div id="outline-container-org3d18192" class="outline-2">
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<h2 id="org3d18192"><span class="section-number-2">1</span> Stiffness Matrix for the Cubic configuration</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="orgda0ee50"></a>
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</p>
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<p>
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First, we have to understand what is the physical meaning of the Stiffness matrix \(\bm{K}\).
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</p>
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<p>
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The Stiffness matrix links forces \(\bm{f}\) and torques \(\bm{n}\) applied on the mobile platform at \(\{B\}\) to the displacement \(\Delta\bm{\mathcal{X}}\) of the mobile platform represented by \(\{B\}\) with respect to \(\{A\}\):
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\[ \bm{\mathcal{F}} = \bm{K} \Delta\bm{\mathcal{X}} \]
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</p>
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<p>
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with:
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</p>
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<ul class="org-ul">
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<li>\(\bm{\mathcal{F}} = [\bm{f}\ \bm{n}]^{T}\)</li>
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<li>\(\Delta\bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_{x}, \delta \theta_{y}, \delta \theta_{z}]^{T}\)</li>
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</ul>
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<p>
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If the stiffness matrix is inversible, its inverse is the compliance matrix: \(\bm{C} = \bm{K}^{-1\) and:
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\[ \Delta \bm{\mathcal{X}} = C \bm{\mathcal{F}} \]
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</p>
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<p>
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Thus, if the stiffness matrix is diagonal, the compliance matrix is also diagonal and a force (resp. torque) \(\bm{\mathcal{F}}_i\) applied on the mobile platform at \(\{B\}\) will induce a pure translation (resp. rotation) of the mobile platform represented by \(\{B\}\) with respect to \(\{A\}\).
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</p>
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<p>
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One has to note that this is only valid in a static way.
|
|
</p>
|
|
|
|
<p>
|
|
We here study what makes the Stiffness matrix diagonal when using a cubic configuration.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgf6f7ad2" class="outline-3">
|
|
<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>
|
|
<div class="outline-text-3" id="text-1-1">
|
|
<p>
|
|
We create a cubic Stewart platform (figure <a href="#orgaba20c8">1</a>) in such a way that the center of the cube (black star) is located at the center of the Stewart platform (blue dot).
|
|
The Jacobian matrix is estimated at the location of the center of the cube.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = <span class="org-type">-</span>H<span class="org-type">/</span>2; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
Hc = H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
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="orgaba20c8" class="figure">
|
|
<p><img src="figs/cubic_conf_centered_J_center.png" alt="cubic_conf_centered_J_center.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </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>
|
|
</div>
|
|
|
|
<table id="org4baf591" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 1:</span> Stiffness Matrix</caption>
|
|
|
|
<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">0</td>
|
|
<td class="org-right">2.1e-17</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">-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="#org47f8142">2</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">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = 20e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
Hc = H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H<span class="org-type">/</span>2; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
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 2: </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 id="org5cc2020" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 2:</span> Stiffness Matrix</caption>
|
|
|
|
<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">0</td>
|
|
<td class="org-right">-0.14</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">0.14</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">-5.3e-19</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.14</td>
|
|
<td class="org-right">-5.3e-19</td>
|
|
<td class="org-right">0.025</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">8.7e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-0.14</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2.6e-18</td>
|
|
<td class="org-right">1.6e-19</td>
|
|
<td class="org-right">0.025</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">8.9e-19</td>
|
|
<td class="org-right">0</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="#org0235d3a">3</a>).
|
|
The Jacobian is estimated at the cube center.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 80e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = <span class="org-type">-</span>30e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
Hc = 100e<span class="org-type">-</span>3; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
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 3: </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 id="org6b3d8b1" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 3:</span> Stiffness Matrix</caption>
|
|
|
|
<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">4.9e-17</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">-2.2e-17</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">1.1e-18</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">-2.2e-17</td>
|
|
<td class="org-right">1.1e-18</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">3.5e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.4e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-1.4e-17</td>
|
|
<td class="org-right">-5.7e-20</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">-8.7e-19</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">3.5e-18</td>
|
|
<td class="org-right">-8.7e-19</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">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = <span class="org-type">-</span>H<span class="org-type">/</span>2; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
Hc = 1.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H<span class="org-type">/</span>2 <span class="org-type">+</span> 10e<span class="org-type">-</span>3; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 215e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 195e<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 4: </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 id="org846d51c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 4:</span> Stiffness Matrix</caption>
|
|
|
|
<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">1.5e-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">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.5e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-3e-18</td>
|
|
<td class="org-right">-2.8e-17</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-0.02</td>
|
|
<td class="org-right">-3e-18</td>
|
|
<td class="org-right">0.034</td>
|
|
<td class="org-right">-8.7e-19</td>
|
|
<td class="org-right">5.2e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-2.2e-17</td>
|
|
<td class="org-right">-4.4e-19</td>
|
|
<td class="org-right">0.034</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5.9e-18</td>
|
|
<td class="org-right">-7.5e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">3.5e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.14</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org510da86" class="outline-3">
|
|
<h3 id="org510da86"><span class="section-number-3">1.5</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-1-5">
|
|
<div class="important">
|
|
<p>
|
|
Here are the conclusion about the Stiffness matrix for the Cubic configuration:
|
|
</p>
|
|
<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>
|
|
|
|
<div id="outline-container-orgd70418b" class="outline-2">
|
|
<h2 id="orgd70418b"><span class="section-number-2">2</span> Configuration with the Cube’s center above the mobile platform</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="orgb73265d"></a>
|
|
</p>
|
|
<p>
|
|
We saw in section <a href="#orgda0ee50">1</a> that in order to have a diagonal stiffness matrix, we need the cube’s center to be located at frames \(\{A\}\) and \(\{B\}\).
|
|
Or, we usually want to have \(\{A\}\) and \(\{B\}\) located above the top platform where forces are applied and where displacements are expressed.
|
|
</p>
|
|
|
|
<p>
|
|
We here see if the cubic configuration can provide a diagonal stiffness matrix when \(\{A\}\) and \(\{B\}\) are above the mobile platform.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org8afa645" class="outline-3">
|
|
<h3 id="org8afa645"><span class="section-number-3">2.1</span> Having Cube’s center above the top platform</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
Let’s say we want to have a diagonal stiffness matrix when \(\{A\}\) and \(\{B\}\) are located above the top platform.
|
|
Thus, we want the cube’s center to be located above the top center.
|
|
</p>
|
|
|
|
<p>
|
|
Let’s fix the Height of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = 20e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We find the several Cubic configuration for the Stewart platform where the center of the cube is located at frame \(\{A\}\).
|
|
The differences between the configuration are the cube’s size:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Small Cube Size in Figure <a href="#org105635f">5</a></li>
|
|
<li>Medium Cube Size in Figure <a href="#org264ab9c">6</a></li>
|
|
<li>Large Cube Size in Figure <a href="#org52254fe">7</a></li>
|
|
</ul>
|
|
|
|
<p>
|
|
For each of the configuration, the Stiffness matrix is diagonal with \(k_x = k_y = k_y = 2k\) with \(k\) is the stiffness of each strut.
|
|
However, the rotational stiffnesses are increasing with the cube’s size but the required size of the platform is also increasing, so there is a trade-off here.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hc = 0.4<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org105635f" class="figure">
|
|
<p><img src="figs/stewart_cubic_conf_type_1.png" alt="stewart_cubic_conf_type_1.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Cubic Configuration for the Stewart Platform - Small Cube Size (<a href="./figs/stewart_cubic_conf_type_1.png">png</a>, <a href="./figs/stewart_cubic_conf_type_1.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<table id="org91f89e4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 5:</span> Stiffness Matrix</caption>
|
|
|
|
<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">-2.8e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2.4e-17</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">-2.3e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-2.8e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-2.1e-19</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-2.3e-17</td>
|
|
<td class="org-right">-2.1e-19</td>
|
|
<td class="org-right">0.0024</td>
|
|
<td class="org-right">-5.4e-20</td>
|
|
<td class="org-right">6.5e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2.4e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">4.9e-19</td>
|
|
<td class="org-right">-2.3e-20</td>
|
|
<td class="org-right">0.0024</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.2e-18</td>
|
|
<td class="org-right">1.1e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">6.2e-19</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.0096</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hc = 1.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org264ab9c" class="figure">
|
|
<p><img src="figs/stewart_cubic_conf_type_2.png" alt="stewart_cubic_conf_type_2.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Cubic Configuration for the Stewart Platform - Medium Cube Size (<a href="./figs/stewart_cubic_conf_type_2.png">png</a>, <a href="./figs/stewart_cubic_conf_type_2.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<table id="orgcf84781" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 6:</span> Stiffness Matrix</caption>
|
|
|
|
<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">-1.9e-16</td>
|
|
<td class="org-right">0</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">2</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-7.6e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.9e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">2.5e-18</td>
|
|
<td class="org-right">2.8e-17</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-7.6e-17</td>
|
|
<td class="org-right">2.5e-18</td>
|
|
<td class="org-right">0.034</td>
|
|
<td class="org-right">8.7e-19</td>
|
|
<td class="org-right">8.7e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5.7e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">3.2e-17</td>
|
|
<td class="org-right">2.9e-19</td>
|
|
<td class="org-right">0.034</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1e-18</td>
|
|
<td class="org-right">-1.3e-17</td>
|
|
<td class="org-right">5.6e-17</td>
|
|
<td class="org-right">8.4e-18</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.14</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org52254fe" class="figure">
|
|
<p><img src="figs/stewart_cubic_conf_type_3.png" alt="stewart_cubic_conf_type_3.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Cubic Configuration for the Stewart Platform - Large Cube Size (<a href="./figs/stewart_cubic_conf_type_3.png">png</a>, <a href="./figs/stewart_cubic_conf_type_3.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<table id="org02f7789" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 7:</span> Stiffness Matrix</caption>
|
|
|
|
<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">-3e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-8.3e-17</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">-2.2e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">5.6e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-3e-16</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">-9.3e-19</td>
|
|
<td class="org-right">-2.8e-17</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-2.2e-17</td>
|
|
<td class="org-right">-9.3e-19</td>
|
|
<td class="org-right">0.094</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">2.1e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-8e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">-3e-17</td>
|
|
<td class="org-right">-6.1e-19</td>
|
|
<td class="org-right">0.094</td>
|
|
<td class="org-right">0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-6.2e-18</td>
|
|
<td class="org-right">7.2e-17</td>
|
|
<td class="org-right">5.6e-17</td>
|
|
<td class="org-right">2.3e-17</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">0.37</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org949a403" class="outline-3">
|
|
<h3 id="org949a403"><span class="section-number-3">2.2</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<div class="important">
|
|
<p>
|
|
We found that we can have a diagonal stiffness matrix using the cubic architecture when \(\{A\}\) and \(\{B\}\) are located above the top platform.
|
|
Depending on the cube’s size, we obtain 3 different configurations.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcc4ecce" class="outline-2">
|
|
<h2 id="orgcc4ecce"><span class="section-number-2">3</span> Cubic size analysis</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org348ec7d"></a>
|
|
</p>
|
|
<p>
|
|
We here study the effect of the size of the cube used for the Stewart Cubic configuration.
|
|
</p>
|
|
|
|
<p>
|
|
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform and the frames \(\{A\}\) and \(\{B\}\) are also taken at the center of the cube.
|
|
</p>
|
|
|
|
<p>
|
|
We only vary the size of the cube.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org0029d8c" class="outline-3">
|
|
<h3 id="org0029d8c"><span class="section-number-3">3.1</span> Analysis</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
We initialize the wanted cube’s size.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hcs = 1e<span class="org-type">-</span>3<span class="org-type">*</span>[250<span class="org-type">:</span>20<span class="org-type">:</span>350]; <span class="org-comment">% Heights for the Cube [m]</span>
|
|
Ks = zeros(6, 6, length(Hcs));
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The height of the Stewart platform is fixed:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The frames \(\{A\}\) and \(\{B\}\) are positioned at the Stewart platform center as well as the cube’s center:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">MO_B = <span class="org-type">-</span>50e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We find that for all the cube’s size, \(k_x = k_y = k_z = k\) where \(k\) is the strut stiffness.
|
|
We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varying with the cube’s size (figure <a href="#orgf5b4a80">8</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgf5b4a80" class="figure">
|
|
<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>\(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfc7135f" class="outline-3">
|
|
<h3 id="orgfc7135f"><span class="section-number-3">3.2</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
|
|
</p>
|
|
|
|
<div class="important">
|
|
<p>
|
|
In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf09da67" class="outline-2">
|
|
<h2 id="orgf09da67"><span class="section-number-2">4</span> Dynamic Coupling in the Cartesian Frame</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org00d3816"></a>
|
|
</p>
|
|
<p>
|
|
In this section, we study the dynamics of the platform in the cartesian frame.
|
|
</p>
|
|
|
|
<p>
|
|
We here suppose that there is one relative motion sensor in each strut (\(\delta\bm{\mathcal{L}}\) is measured) and we would like to control the position of the top platform pose \(\delta \bm{\mathcal{X}}\).
|
|
</p>
|
|
|
|
<p>
|
|
Thanks to the Jacobian matrix, we can use the “architecture” shown in Figure <a href="#org76f24a0">9</a> to obtain the dynamics of the system from forces/torques applied by the actuators on the top platform to translations/rotations of the top platform.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
|
<span class="org-font-latex-sedate">\node</span>[block] (Jt) at (0, 0) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{J}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-T}$</span>};
|
|
<span class="org-font-latex-sedate">\node</span>[block, right= of Jt] (G) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{G}$</span>};
|
|
<span class="org-font-latex-sedate">\node</span>[block, right= of G] (J) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{J}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}$</span>};
|
|
|
|
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(Jt.west)+(-0.8, 0)$</span>) -- (Jt.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{F}}$</span>};
|
|
<span class="org-font-latex-sedate">\draw</span>[->] (Jt.east) -- (G.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tau</span></span><span class="org-font-latex-math">}$</span>};
|
|
<span class="org-font-latex-sedate">\draw</span>[->] (G.east) -- (J.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\delta\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{L}}$</span>};
|
|
<span class="org-font-latex-sedate">\draw</span>[->] (J.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\delta\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{X}}$</span>};
|
|
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org76f24a0" class="figure">
|
|
<p><img src="figs/local_to_cartesian_coordinates.png" alt="local_to_cartesian_coordinates.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>From Strut coordinate to Cartesian coordinate using the Jacobian matrix</p>
|
|
</div>
|
|
|
|
<p>
|
|
We here study the dynamics from \(\bm{\mathcal{F}}\) to \(\delta\bm{\mathcal{X}}\).
|
|
</p>
|
|
|
|
<p>
|
|
One has to note that when considering the static behavior:
|
|
\[ \bm{G}(s = 0) = \begin{bmatrix}
|
|
1/k_1 & & 0 \\
|
|
& \ddots & 0 \\
|
|
0 & & 1/k_6
|
|
\end{bmatrix}\]
|
|
</p>
|
|
|
|
<p>
|
|
And thus:
|
|
\[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \]
|
|
</p>
|
|
|
|
<p>
|
|
We conclude that the <b>static</b> behavior of the platform depends on the stiffness matrix.
|
|
For the cubic configuration, we have a diagonal stiffness matrix is the frames \(\{A\}\) and \(\{B\}\) are coincident with the cube’s center.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org5fe01ec" class="outline-3">
|
|
<h3 id="org5fe01ec"><span class="section-number-3">4.1</span> Cube’s center at the Center of Mass of the mobile platform</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
Let’s create a Cubic Stewart Platform where the <b>Center of Mass of the mobile platform is located at the center of the cube</b>.
|
|
</p>
|
|
|
|
<p>
|
|
We define the size of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Now, we set the cube’s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with \(\{A\}\) and \(\{B\}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
|
<span class="org-string">'Mpm'</span>, 10, ...
|
|
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
|
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we set small mass for the struts.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
No flexibility below the Stewart platform and no payload.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtain geometry is shown in figure <a href="#orgc92a65b">10</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgc92a65b" class="figure">
|
|
<p><img src="figs/stewart_cubic_conf_decouple_dynamics.png" alt="stewart_cubic_conf_decouple_dynamics.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Geometry used for the simulations - The cube’s center, the frames \(\{A\}\) and \(\{B\}\) and the Center of mass of the mobile platform are coincident (<a href="./figs/stewart_cubic_conf_decouple_dynamics.png">png</a>, <a href="./figs/stewart_cubic_conf_decouple_dynamics.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform_model.slx'</span>)
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
|
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Now, thanks to the Jacobian (Figure <a href="#org76f24a0">9</a>), we compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
|
|
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
|
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure <a href="#orgcb3ac4d">11</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgcb3ac4d" class="figure">
|
|
<p><img src="figs/stewart_cubic_decoupled_dynamics_cartesian.png" alt="stewart_cubic_decoupled_dynamics_cartesian.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (<a href="./figs/stewart_cubic_decoupled_dynamics_cartesian.png">png</a>, <a href="./figs/stewart_cubic_decoupled_dynamics_cartesian.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<div class="important">
|
|
<p>
|
|
The dynamics is well decoupled at all frequencies.
|
|
</p>
|
|
|
|
<p>
|
|
We have the same dynamics for:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(D_x/F_x\), \(D_y/F_y\) and \(D_z/F_z\)</li>
|
|
<li>\(R_x/M_x\) and \(D_y/F_y\)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The Dynamics from \(F_i\) to \(D_i\) is just a 1-dof mass-spring-damper system.
|
|
</p>
|
|
|
|
<p>
|
|
This is because the Mass, Damping and Stiffness matrices are all diagonal.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4cb2a36" class="outline-3">
|
|
<h3 id="org4cb2a36"><span class="section-number-3">4.2</span> Cube’s center not coincident with the Mass of the Mobile platform</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
Let’s create a Stewart platform with a cubic architecture where the cube’s center is at the center of the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = <span class="org-type">-</span>100e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Now, we set the cube’s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
However, the Center of Mass of the mobile platform is <b>not</b> located at the cube’s center.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
|
<span class="org-string">'Mpm'</span>, 10, ...
|
|
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
|
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we set small mass for the struts.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
No flexibility below the Stewart platform and no payload.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtain geometry is shown in figure <a href="#orgfce7805">12</a>.
|
|
</p>
|
|
|
|
<div id="orgfce7805" class="figure">
|
|
<p><img src="figs/stewart_cubic_conf_mass_above.png" alt="stewart_cubic_conf_mass_above.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Geometry used for the simulations - The cube’s center is coincident with the frames \(\{A\}\) and \(\{B\}\) but not with the Center of mass of the mobile platform (<a href="./figs/stewart_cubic_conf_mass_above.png">png</a>, <a href="./figs/stewart_cubic_conf_mass_above.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform_model.slx'</span>)
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
|
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we use the Jacobian to compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
|
|
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
|
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure <a href="#org7a04d45">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org7a04d45" class="figure">
|
|
<p><img src="figs/stewart_conf_coupling_mass_matrix.png" alt="stewart_conf_coupling_mass_matrix.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Obtained Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (<a href="./figs/stewart_conf_coupling_mass_matrix.png">png</a>, <a href="./figs/stewart_conf_coupling_mass_matrix.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<div class="important">
|
|
<p>
|
|
The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is <b>not</b> decoupled at all frequencies.
|
|
</p>
|
|
|
|
<p>
|
|
This was expected as the mass matrix is not diagonal (the Center of Mass of the mobile platform not being coincident with the frame \(\{B\}\)).
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2e09bcb" class="outline-3">
|
|
<h3 id="org2e09bcb"><span class="section-number-3">4.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="important">
|
|
<p>
|
|
Some conclusions can be drawn from the above analysis:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Static Decoupling <=> Diagonal Stiffness matrix <=> {A} and {B} at the cube’s center</li>
|
|
<li>Dynamic Decoupling <=> Static Decoupling + CoM of mobile platform coincident with {A} and {B}.</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8f26dc0" class="outline-2">
|
|
<h2 id="org8f26dc0"><span class="section-number-2">5</span> Dynamic Coupling between actuators and sensors of each strut</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org5b5c8a9"></a>
|
|
</p>
|
|
<p>
|
|
From <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>, the cubic configuration “<i>minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other)</i>”.
|
|
</p>
|
|
|
|
<p>
|
|
In this section, we wish to study such properties of the cubic architecture.
|
|
</p>
|
|
|
|
<p>
|
|
We will compare the transfer function from sensors to actuators in each strut for a cubic architecture and for a non-cubic architecture (where the struts are not orthogonal with each other).
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org6e391c9" class="outline-3">
|
|
<h3 id="org6e391c9"><span class="section-number-3">5.1</span> Coupling between the actuators and sensors - Cubic Architecture</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
Let’s generate a Cubic architecture where the cube’s center and the frames \(\{A\}\) and \(\{B\}\) are coincident.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
|
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
|
<span class="org-string">'Mpm'</span>, 10, ...
|
|
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
|
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
|
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
No flexibility below the Stewart platform and no payload.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org67d7284" class="figure">
|
|
<p><img src="figs/stewart_architecture_coupling_struts_cubic.png" alt="stewart_architecture_coupling_struts_cubic.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Geometry of the generated Stewart platform (<a href="./figs/stewart_architecture_coupling_struts_cubic.png">png</a>, <a href="./figs/stewart_architecture_coupling_struts_cubic.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure <a href="#orga20cd7d">15</a>) and to the force sensors \(\tau_{m,i}\) (Figure <a href="#org645e6c3">16</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orga20cd7d" class="figure">
|
|
<p><img src="figs/coupling_struts_relative_sensor_cubic.png" alt="coupling_struts_relative_sensor_cubic.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Dynamics from the force actuators to the relative motion sensors (<a href="./figs/coupling_struts_relative_sensor_cubic.png">png</a>, <a href="./figs/coupling_struts_relative_sensor_cubic.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org645e6c3" class="figure">
|
|
<p><img src="figs/coupling_struts_force_sensor_cubic.png" alt="coupling_struts_force_sensor_cubic.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Dynamics from the force actuators to the force sensors (<a href="./figs/coupling_struts_force_sensor_cubic.png">png</a>, <a href="./figs/coupling_struts_force_sensor_cubic.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgafd808d" class="outline-3">
|
|
<h3 id="orgafd808d"><span class="section-number-3">5.2</span> Coupling between the actuators and sensors - Non-Cubic Architecture</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
<p>
|
|
Now we generate a Stewart platform which is not cubic but with approximately the same size as the previous cubic architecture.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
|
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
|
stewart = generateGeneralConfiguration(stewart, <span class="org-string">'FR'</span>, 250e<span class="org-type">-</span>3, <span class="org-string">'MR'</span>, 150e<span class="org-type">-</span>3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
|
<span class="org-string">'Mpm'</span>, 10, ...
|
|
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
|
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
|
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
|
stewart = initializeInertialSensor(stewart);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
No flexibility below the Stewart platform and no payload.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org14d3492" class="figure">
|
|
<p><img src="figs/stewart_architecture_coupling_struts_non_cubic.png" alt="stewart_architecture_coupling_struts_non_cubic.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Geometry of the generated Stewart platform (<a href="./figs/stewart_architecture_coupling_struts_non_cubic.png">png</a>, <a href="./figs/stewart_architecture_coupling_struts_non_cubic.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure <a href="#orgff23a38">18</a>) and to the force sensors \(\tau_{m,i}\) (Figure <a href="#orgd802951">19</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgff23a38" class="figure">
|
|
<p><img src="figs/coupling_struts_relative_sensor_non_cubic.png" alt="coupling_struts_relative_sensor_non_cubic.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Dynamics from the force actuators to the relative motion sensors (<a href="./figs/coupling_struts_relative_sensor_non_cubic.png">png</a>, <a href="./figs/coupling_struts_relative_sensor_non_cubic.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgd802951" class="figure">
|
|
<p><img src="figs/coupling_struts_force_sensor_non_cubic.png" alt="coupling_struts_force_sensor_non_cubic.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Dynamics from the force actuators to the force sensors (<a href="./figs/coupling_struts_force_sensor_non_cubic.png">png</a>, <a href="./figs/coupling_struts_force_sensor_non_cubic.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8c1a310" class="outline-3">
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<h3 id="org8c1a310"><span class="section-number-3">5.3</span> Conclusion</h3>
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<div class="outline-text-3" id="text-5-3">
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<div class="important">
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<p>
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The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut and thus provides no advantages for decentralized control.
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</p>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org3044455" class="outline-2">
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<h2 id="org3044455"><span class="section-number-2">6</span> Functions</h2>
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<div class="outline-text-2" id="text-6">
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<p>
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<a id="org28ba607"></a>
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</p>
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</div>
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<div id="outline-container-org56504f1" class="outline-3">
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<h3 id="org56504f1"><span class="section-number-3">6.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
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<div class="outline-text-3" id="text-6-1">
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<p>
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<a id="orga8311d3"></a>
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</p>
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<p>
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This Matlab function is accessible <a href="../src/generateCubicConfiguration.m">here</a>.
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</p>
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</div>
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<div id="outline-container-orga5a9ba8" class="outline-4">
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<h4 id="orga5a9ba8">Function description</h4>
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<div class="outline-text-4" id="text-orga5a9ba8">
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<div class="org-src-container">
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<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>)
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<span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Syntax: [stewart] = generateCubicConfiguration(stewart, args)</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Inputs:</span>
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<span class="org-comment">% - stewart - A structure with the following fields</span>
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<span class="org-comment">% - geometry.H [1x1] - Total height of the platform [m]</span>
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<span class="org-comment">% - args - Can have the following fields:</span>
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<span class="org-comment">% - Hc [1x1] - Height of the "useful" part of the cube [m]</span>
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<span class="org-comment">% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]</span>
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<span class="org-comment">% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]</span>
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<span class="org-comment">% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Outputs:</span>
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<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
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<span class="org-comment">% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
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<span class="org-comment">% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org3253792" class="outline-4">
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<h4 id="org3253792">Documentation</h4>
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<div class="outline-text-4" id="text-org3253792">
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|
|
|
<div id="org8a7f3d8" class="figure">
|
|
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
|
|
</p>
|
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<p><span class="figure-number">Figure 20: </span>Cubic Configuration</p>
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|
</div>
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|
</div>
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|
</div>
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|
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<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: 2020-02-13 jeu. 15:01</p>
|
|
</div>
|
|
</body>
|
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</html>
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