1907 lines
66 KiB
HTML
1907 lines
66 KiB
HTML
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<!-- 2020-08-05 mer. 13:27 -->
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<title>Cubic configuration for the Stewart Platform</title>
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<body>
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<div id="org-div-home-and-up">
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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>
|
||
<li><a href="#orgf6f7ad2">1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
|
||
<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="#org3e0c3da">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>
|
||
<li><a href="#org8afa645">2.1. Having Cube’s center above the top platform</a></li>
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||
<li><a href="#org25d045b">2.2. Size of the platforms</a></li>
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||
<li><a href="#org2972f78">2.3. 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="#orga34a8ab">3.2. Conclusion</a></li>
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||
</ul>
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||
</li>
|
||
<li><a href="#orgf09da67">4. Dynamic Coupling in the Cartesian Frame</a>
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||
<ul>
|
||
<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="#orgacfeac7">4.3. Conclusion</a></li>
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||
</ul>
|
||
</li>
|
||
<li><a href="#org8f26dc0">5. Dynamic Coupling between actuators and sensors of each strut</a>
|
||
<ul>
|
||
<li><a href="#org6e391c9">5.1. Coupling between the actuators and sensors - Cubic Architecture</a></li>
|
||
<li><a href="#orgafd808d">5.2. Coupling between the actuators and sensors - Non-Cubic Architecture</a></li>
|
||
<li><a href="#org4413be4">5.3. Conclusion</a></li>
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||
</ul>
|
||
</li>
|
||
<li><a href="#org3044455">6. Functions</a>
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||
<ul>
|
||
<li><a href="#org56504f1">6.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
|
||
<ul>
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||
<li><a href="#orga5a9ba8">Function description</a></li>
|
||
<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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||
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||
<p>
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||
The Cubic configuration for the Stewart platform was first proposed in (<a href="#citeproc_bib_item_2">Geng and Haynes 1994</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 href="#citeproc_bib_item_5">Preumont et al. 2007</a>; <a href="#citeproc_bib_item_3">Jafari and McInroy 2003</a>)).
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||
</p>
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||
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||
<p>
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||
According to (<a href="#citeproc_bib_item_5">Preumont et al. 2007</a>), the cubic configuration offers the following advantages:
|
||
</p>
|
||
<blockquote>
|
||
<p>
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||
This topology provides a <b>uniform control capability</b> and a <b>uniform stiffness</b> in all directions, and it <b>minimizes the cross-coupling amongst actuators and sensors of different legs</b> (being orthogonal to each other).
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||
</p>
|
||
</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 <b>uniform stiffness</b> of such configuration and we find the conditions to obtain a diagonal stiffness matrix</li>
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||
<li>In section <a href="#orgb73265d">2</a>, we find 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 dynamics 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 <b>cross-coupling</b> 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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||
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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>
|
||
<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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||
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<div class="note">
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||
<p>
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||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_stiffnessl.m">here</a>.
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||
</p>
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<p>
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||
To run the script, open the Simulink Project, and type <code>run cubic_conf_stiffness.m</code>.
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||
</p>
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||
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||
</div>
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||
<p>
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||
First, we have to understand what is the physical meaning of the Stiffness matrix \(\bm{K}\).
|
||
</p>
|
||
|
||
<p>
|
||
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 \(\{A\}\):
|
||
\[ \bm{\mathcal{F}} = \bm{K} \Delta\bm{\mathcal{X}} \]
|
||
</p>
|
||
|
||
<p>
|
||
with:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<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>
|
||
</ul>
|
||
|
||
<p>
|
||
If the stiffness matrix is inversible, its inverse is the compliance matrix: \(\bm{C} = \bm{K}^{-1\) and:
|
||
\[ \Delta \bm{\mathcal{X}} = C \bm{\mathcal{F}} \]
|
||
</p>
|
||
|
||
<p>
|
||
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 \(\{A\}\).
|
||
</p>
|
||
|
||
<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-3; % height of the Stewart platform [m]
|
||
MO_B = -H/2; % Position {B} with respect to {M} [m]
|
||
Hc = H; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
|
||
stewart = computeJacobian(stewart);
|
||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-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-3; % height of the Stewart platform [m]
|
||
MO_B = 20e-3; % Position {B} with respect to {M} [m]
|
||
Hc = H; % Size of the useful part of the cube [m]
|
||
FOc = H/2; % Center of the cube with respect to {F}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
|
||
stewart = computeJacobian(stewart);
|
||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-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-3; % height of the Stewart platform [m]
|
||
MO_B = -30e-3; % Position {B} with respect to {M} [m]
|
||
Hc = 100e-3; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
|
||
stewart = computeJacobian(stewart);
|
||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-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-3; % height of the Stewart platform [m]
|
||
MO_B = -H/2; % Position {B} with respect to {M} [m]
|
||
Hc = 1.5*H; % Size of the useful part of the cube [m]
|
||
FOc = H/2 + 10e-3; % Center of the cube with respect to {F}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
|
||
stewart = computeJacobian(stewart);
|
||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 215e-3, 'Mpr', 195e-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-org3e0c3da" class="outline-3">
|
||
<h3 id="org3e0c3da"><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>
|
||
|
||
<div class="note">
|
||
<p>
|
||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_above_platforml.m">here</a>.
|
||
</p>
|
||
|
||
<p>
|
||
To run the script, open the Simulink Project, and type <code>run cubic_conf_above_platform.m</code>.
|
||
</p>
|
||
|
||
</div>
|
||
<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-3; % height of the Stewart platform [m]
|
||
MO_B = 20e-3; % Position {B} with respect to {M} [m]
|
||
</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*H; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</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*H; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</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*H; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</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-org25d045b" class="outline-3">
|
||
<h3 id="org25d045b"><span class="section-number-3">2.2</span> Size of the platforms</h3>
|
||
<div class="outline-text-3" id="text-2-2">
|
||
<p>
|
||
The minimum size of the platforms depends on the cube’s size and the height between the platform and the cube’s center.
|
||
</p>
|
||
|
||
<p>
|
||
Let’s denote:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>\(H\) the height between the cube’s center and the considered platform</li>
|
||
<li>\(D\) the size of the cube’s edges</li>
|
||
</ul>
|
||
|
||
<p>
|
||
Let’s denote by \(a\) and \(b\) the points of both ends of one of the cube’s edge.
|
||
</p>
|
||
|
||
<p>
|
||
Initially, we have:
|
||
</p>
|
||
\begin{align}
|
||
a &= \frac{D}{2} \begin{bmatrix}-1 \\ -1 \\ 1\end{bmatrix} \\
|
||
b &= \frac{D}{2} \begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix}
|
||
\end{align}
|
||
|
||
<p>
|
||
We rotate the cube around its center (origin of the rotated frame) such that one of its diagonal is vertical.
|
||
\[ R = \begin{bmatrix}
|
||
\frac{2}{\sqrt{6}} & 0 & \frac{1}{\sqrt{3}} \\
|
||
\frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\
|
||
\frac{-1}{\sqrt{6}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{3}}
|
||
\end{bmatrix} \]
|
||
</p>
|
||
|
||
<p>
|
||
After rotation, the points \(a\) and \(b\) become:
|
||
</p>
|
||
\begin{align}
|
||
a &= \frac{D}{2} \begin{bmatrix}-\frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ -\frac{1}{\sqrt{3}}\end{bmatrix} \\
|
||
b &= \frac{D}{2} \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ \frac{1}{\sqrt{3}}\end{bmatrix}
|
||
\end{align}
|
||
|
||
<p>
|
||
Points \(a\) and \(b\) define a vector \(u = b - a\) that gives the orientation of one of the Stewart platform strut:
|
||
\[ u = \frac{D}{\sqrt{3}} \begin{bmatrix} -\sqrt{2} \\ 0 \\ -1\end{bmatrix} \]
|
||
</p>
|
||
|
||
<p>
|
||
Then we want to find the intersection between the line that defines the strut with the plane defined by the height \(H\) from the cube’s center.
|
||
To do so, we first find \(g\) such that:
|
||
\[ a_z + g u_z = -H \]
|
||
We obtain:
|
||
</p>
|
||
\begin{align}
|
||
g &= - \frac{H + a_z}{u_z} \\
|
||
&= \sqrt{3} \frac{H}{D} - \frac{1}{2}
|
||
\end{align}
|
||
|
||
<p>
|
||
Then, the intersection point \(P\) is given by:
|
||
</p>
|
||
\begin{align}
|
||
P &= a + g u \\
|
||
&= \begin{bmatrix}
|
||
H \sqrt{2} \\
|
||
D \frac{1}{\sqrt{2}} \\
|
||
H
|
||
\end{bmatrix}
|
||
\end{align}
|
||
|
||
<p>
|
||
Finally, the circle can contains the intersection point has a radius \(r\):
|
||
</p>
|
||
\begin{align}
|
||
r &= \sqrt{P_x^2 + P_y^2} \\
|
||
&= \sqrt{2 H^2 + \frac{1}{2}D^2}
|
||
\end{align}
|
||
|
||
<p>
|
||
By symmetry, we can show that all the other intersection points will also be on the circle with a radius \(r\).
|
||
</p>
|
||
|
||
<p>
|
||
For a small cube:
|
||
\[ r \approx \sqrt{2} H \]
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org2972f78" class="outline-3">
|
||
<h3 id="org2972f78"><span class="section-number-3">2.3</span> Conclusion</h3>
|
||
<div class="outline-text-3" id="text-2-3">
|
||
<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>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-left" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left">Cube’s Size</th>
|
||
<th scope="col" class="org-left">Paper with the corresponding cubic architecture</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Small</td>
|
||
<td class="org-left">(<a href="#citeproc_bib_item_1">Furutani, Suzuki, and Kudoh 2004</a>)</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Medium</td>
|
||
<td class="org-left">(<a href="#citeproc_bib_item_6">Yang et al. 2019</a>)</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Large</td>
|
||
<td class="org-left"> </td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
</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>
|
||
|
||
<div class="note">
|
||
<p>
|
||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_size_analysisl.m">here</a>.
|
||
</p>
|
||
|
||
<p>
|
||
To run the script, open the Simulink Project, and type <code>run cubic_conf_size_analysis.m</code>.
|
||
</p>
|
||
|
||
</div>
|
||
<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-3*[250:20:350]; % Heights for the Cube [m]
|
||
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-3; % height of the Stewart platform [m]
|
||
</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 = -50e-3; % Position {B} with respect to {M} [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</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-orga34a8ab" class="outline-3">
|
||
<h3 id="orga34a8ab"><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>
|
||
|
||
<div class="note">
|
||
<p>
|
||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_coupling_cartesianl.m">here</a>.
|
||
</p>
|
||
|
||
<p>
|
||
To run the script, open the Simulink Project, and type <code>run cubic_conf_coupling_cartesian.m</code>.
|
||
</p>
|
||
|
||
</div>
|
||
<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 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-3; % height of the Stewart platform [m]
|
||
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
||
</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*H; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
||
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, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
||
'Mpm', 10, ...
|
||
'Mph', 20e-3, ...
|
||
'Mpr', 1.2*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, 'Fsm', 1e-3, 'Msm', 1e-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('type', 'none');
|
||
payload = initializePayload('type', 'none');
|
||
controller = initializeController('type', 'open-loop');
|
||
</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('stewart_platform_model.slx')
|
||
|
||
%% Options for Linearized
|
||
options = linearizeOptions;
|
||
options.SampleTime = 0;
|
||
|
||
%% Name of the Simulink File
|
||
mdl = 'stewart_platform_model';
|
||
|
||
%% Input/Output definition
|
||
clear io; io_i = 1;
|
||
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
||
|
||
%% Run the linearization
|
||
G = linearize(mdl, io, options);
|
||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||
</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)*G*inv(stewart.kinematics.J');
|
||
Gc = inv(stewart.kinematics.J)*G*stewart.kinematics.J;
|
||
Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||
Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
||
</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>
|
||
|
||
<p>
|
||
It is interesting to note here that the system shown in Figure <a href="#org9e58bc5">12</a> also yield a decoupled system (explained in section 1.3.3 in (<a href="#citeproc_bib_item_4">Li 2001</a>)).
|
||
</p>
|
||
|
||
|
||
<div id="org9e58bc5" class="figure">
|
||
<p><img src="figs/local_to_cartesian_coordinates_bis.png" alt="local_to_cartesian_coordinates_bis.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 12: </span>Alternative way to decouple the system</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-3; % height of the Stewart platform [m]
|
||
MO_B = -100e-3; % Position {B} with respect to {M} [m]
|
||
</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*H; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
||
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, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
||
'Mpm', 10, ...
|
||
'Mph', 20e-3, ...
|
||
'Mpr', 1.2*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, 'Fsm', 1e-3, 'Msm', 1e-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('type', 'none');
|
||
payload = initializePayload('type', 'none');
|
||
controller = initializeController('type', 'open-loop');
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The obtain geometry is shown in figure <a href="#orgfce7805">13</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 13: </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('stewart_platform_model.slx')
|
||
|
||
%% Options for Linearized
|
||
options = linearizeOptions;
|
||
options.SampleTime = 0;
|
||
|
||
%% Name of the Simulink File
|
||
mdl = 'stewart_platform_model';
|
||
|
||
%% Input/Output definition
|
||
clear io; io_i = 1;
|
||
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
||
|
||
%% Run the linearization
|
||
G = linearize(mdl, io, options);
|
||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||
</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)*G*inv(stewart.kinematics.J');
|
||
Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||
Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
||
</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">14</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 14: </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-orgacfeac7" class="outline-3">
|
||
<h3 id="orgacfeac7"><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>
|
||
|
||
<div class="note">
|
||
<p>
|
||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_coupling_strutsl.m">here</a>.
|
||
</p>
|
||
|
||
<p>
|
||
To run the script, open the Simulink Project, and type <code>run cubic_conf_coupling_struts.m</code>.
|
||
</p>
|
||
|
||
</div>
|
||
<p>
|
||
From (<a href="#citeproc_bib_item_5">Preumont et al. 2007</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-3; % height of the Stewart platform [m]
|
||
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
||
Hc = 2.5*H; % Size of the useful part of the cube [m]
|
||
FOc = H + MO_B; % Center of the cube with respect to {F}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
||
stewart = computeJacobian(stewart);
|
||
stewart = initializeStewartPose(stewart);
|
||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
||
'Mpm', 10, ...
|
||
'Mph', 20e-3, ...
|
||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
||
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-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('type', 'none');
|
||
payload = initializePayload('type', 'none');
|
||
controller = initializeController('type', 'open-loop');
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">disturbances = initializeDisturbances();
|
||
references = initializeReferences(stewart);
|
||
</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 15: </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">16</a>) and to the force sensors \(\tau_{m,i}\) (Figure <a href="#org645e6c3">17</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 16: </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 17: </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-3; % height of the Stewart platform [m]
|
||
MO_B = -10e-3; % Position {B} with respect to {M} [m]
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
|
||
stewart = generateGeneralConfiguration(stewart, 'FR', 250e-3, 'MR', 150e-3);
|
||
stewart = computeJointsPose(stewart);
|
||
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
|
||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
|
||
stewart = computeJacobian(stewart);
|
||
stewart = initializeStewartPose(stewart);
|
||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
|
||
'Mpm', 10, ...
|
||
'Mph', 20e-3, ...
|
||
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
|
||
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-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('type', 'none');
|
||
payload = initializePayload('type', 'none');
|
||
controller = initializeController('type', 'open-loop');
|
||
</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 18: </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">19</a>) and to the force sensors \(\tau_{m,i}\) (Figure <a href="#orgd802951">20</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 19: </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 20: </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-org4413be4" class="outline-3">
|
||
<h3 id="org4413be4"><span class="section-number-3">5.3</span> Conclusion</h3>
|
||
<div class="outline-text-3" id="text-5-3">
|
||
<div class="important">
|
||
<p>
|
||
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.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3044455" class="outline-2">
|
||
<h2 id="org3044455"><span class="section-number-2">6</span> Functions</h2>
|
||
<div class="outline-text-2" id="text-6">
|
||
<p>
|
||
<a id="org28ba607"></a>
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org56504f1" class="outline-3">
|
||
<h3 id="org56504f1"><span class="section-number-3">6.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
|
||
<div class="outline-text-3" id="text-6-1">
|
||
<p>
|
||
<a id="orga8311d3"></a>
|
||
</p>
|
||
|
||
<p>
|
||
This Matlab function is accessible <a href="../src/generateCubicConfiguration.m">here</a>.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-orga5a9ba8" class="outline-4">
|
||
<h4 id="orga5a9ba8">Function description</h4>
|
||
<div class="outline-text-4" id="text-orga5a9ba8">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">function [stewart] = generateCubicConfiguration(stewart, args)
|
||
% generateCubicConfiguration - Generate a Cubic Configuration
|
||
%
|
||
% Syntax: [stewart] = generateCubicConfiguration(stewart, args)
|
||
%
|
||
% Inputs:
|
||
% - stewart - A structure with the following fields
|
||
% - geometry.H [1x1] - Total height of the platform [m]
|
||
% - args - Can have the following fields:
|
||
% - Hc [1x1] - Height of the "useful" part of the cube [m]
|
||
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
|
||
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
|
||
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
|
||
%
|
||
% Outputs:
|
||
% - stewart - updated Stewart structure with the added fields:
|
||
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
|
||
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3253792" class="outline-4">
|
||
<h4 id="org3253792">Documentation</h4>
|
||
<div class="outline-text-4" id="text-org3253792">
|
||
|
||
<div id="org8a7f3d8" class="figure">
|
||
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 21: </span>Cubic Configuration</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org154b5fb" class="outline-4">
|
||
<h4 id="org154b5fb">Optional Parameters</h4>
|
||
<div class="outline-text-4" id="text-org154b5fb">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">arguments
|
||
stewart
|
||
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
|
||
args.FOc (1,1) double {mustBeNumeric} = 50e-3
|
||
args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
|
||
args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
|
||
end
|
||
</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, 'H'), 'stewart.geometry should have attribute H')
|
||
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; -1; -1];
|
||
sy = [ 0; 1; -1];
|
||
sz = [ 1; 1; 1];
|
||
|
||
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
|
||
|
||
L = args.Hc*sqrt(3);
|
||
|
||
Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];
|
||
|
||
CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
|
||
CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
|
||
</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 - CCf)./vecnorm(CCm - 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 + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
|
||
Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*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>
|
||
|
||
</p>
|
||
|
||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
|
||
<div class="csl-bib-body">
|
||
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” <i>Measurement Science and Technology</i> 15 (2):467–74. <a href="https://doi.org/10.1088/0957-0233/15/2/022">https://doi.org/10.1088/0957-0233/15/2/022</a>.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_2"></a>Geng, Z.J., and L.S. Haynes. 1994. “Six Degree-of-Freedom Active Vibration Control Using the Stewart Platforms.” <i>IEEE Transactions on Control Systems Technology</i> 2 (1):45–53. <a href="https://doi.org/10.1109/87.273110">https://doi.org/10.1109/87.273110</a>.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_3"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” <i>IEEE Transactions on Robotics and Automation</i> 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <a href="https://doi.org/10.1109/tra.2003.814506">https://doi.org/10.1109/tra.2003.814506</a>.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_4"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_5"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” <i>Journal of Sound and Vibration</i> 300 (3-5):644–61. <a href="https://doi.org/10.1016/j.jsv.2006.07.050">https://doi.org/10.1016/j.jsv.2006.07.050</a>.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_6"></a>Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” <i>Journal of Sound and Vibration</i> 439 (January). Elsevier BV:398–412. <a href="https://doi.org/10.1016/j.jsv.2018.10.007">https://doi.org/10.1016/j.jsv.2018.10.007</a>.</div>
|
||
</div>
|
||
</div>
|
||
<div id="postamble" class="status">
|
||
<p class="author">Author: Dehaeze Thomas</p>
|
||
<p class="date">Created: 2020-08-05 mer. 13:27</p>
|
||
</div>
|
||
</body>
|
||
</html>
|