4270 lines
128 KiB
HTML
4270 lines
128 KiB
HTML
<?xml version="1.0" encoding="utf-8"?>
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<!-- 2020-11-12 jeu. 10:34 -->
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<title>Finite Element Model with Simscape</title>
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<meta name="author" content="Dehaeze Thomas" />
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<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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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">Finite Element Model with Simscape</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="#org5a554a0">1. Amplified Piezoelectric Actuator - 3D elements</a>
|
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<ul>
|
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<li><a href="#org29056ab">1.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
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<li><a href="#orgccb0a56">1.2. Output parameters</a></li>
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||
<li><a href="#orgdd7c7ba">1.3. Piezoelectric parameters</a></li>
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||
<li><a href="#orgb2959d5">1.4. Identification of the Dynamics</a></li>
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||
<li><a href="#org1e29135">1.5. Comparison with Ansys</a></li>
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<li><a href="#org3f1fc2e">1.6. Force Sensor</a></li>
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<li><a href="#org4f1753f">1.7. Distributed Actuator</a></li>
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<li><a href="#org21246e5">1.8. Distributed Actuator and Force Sensor</a></li>
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||
<li><a href="#orgd0b2aaa">1.9. Dynamics from input voltage to displacement</a></li>
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||
<li><a href="#org5452b7e">1.10. Dynamics from input voltage to output voltage</a></li>
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||
<li><a href="#orgaff4afc">1.11. Identification for a simpler model</a></li>
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</ul>
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||
</li>
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||
<li><a href="#org3dc5d8d">2. APA300ML</a>
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||
<ul>
|
||
<li><a href="#org3eaf978">2.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
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||
<li><a href="#org160ca26">2.2. Output parameters</a></li>
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||
<li><a href="#org7932f3c">2.3. Piezoelectric parameters</a></li>
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||
<li><a href="#org18316cd">2.4. Identification of the APA Characteristics</a>
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<ul>
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||
<li><a href="#orgc0281f1">2.4.1. Stiffness</a></li>
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||
<li><a href="#orgcebe0f9">2.4.2. Resonance Frequency</a></li>
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||
<li><a href="#orgda4f233">2.4.3. Amplification factor</a></li>
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||
<li><a href="#org59829b6">2.4.4. Stroke</a></li>
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||
</ul>
|
||
</li>
|
||
<li><a href="#org1cbc8a6">2.5. Identification of the Dynamics</a></li>
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||
<li><a href="#org44a32d5">2.6. IFF</a></li>
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||
<li><a href="#org3e558c6">2.7. DVF</a></li>
|
||
<li><a href="#orgad3fdd9">2.8. Identification for a simpler model</a></li>
|
||
<li><a href="#orge0b9f5a">2.9. Identification of the stiffness properties</a>
|
||
<ul>
|
||
<li><a href="#org52ddecb">2.9.1. APA Alone</a></li>
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||
<li><a href="#org02b6855">2.9.2. See how the global stiffness is changing with the flexible joints</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org34de703">2.10. Effect of APA300ML in the flexibility of the leg</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#orgb6c0ee0">3. Flexible Joint</a>
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||
<ul>
|
||
<li><a href="#orgd7b1d5f">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
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||
<li><a href="#org9778a32">3.2. Output parameters</a></li>
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||
<li><a href="#orgcb9bad1">3.3. Flexible Joint Characteristics</a></li>
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||
<li><a href="#org4dadc02">3.4. Identification of the parameters using Simscape</a></li>
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||
<li><a href="#org30336a6">3.5. Simpler Model</a></li>
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||
</ul>
|
||
</li>
|
||
<li><a href="#orgd9d5aff">4. Optimal Flexible Joint</a>
|
||
<ul>
|
||
<li><a href="#org83c1679">4.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
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||
<li><a href="#orgbee4a84">4.2. Output parameters</a></li>
|
||
<li><a href="#org7609951">4.3. Flexible Joint Characteristics</a></li>
|
||
<li><a href="#org8bf4f56">4.4. Identification of the parameters using Simscape</a></li>
|
||
<li><a href="#orgd8cb8ff">4.5. Simpler Model</a></li>
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||
</ul>
|
||
</li>
|
||
<li><a href="#org7f2d76d">5. Integral Force Feedback with Amplified Piezo</a>
|
||
<ul>
|
||
<li><a href="#orgd9dc7be">5.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
|
||
<li><a href="#org3671fca">5.2. IFF Plant</a></li>
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||
<li><a href="#org1cbfb66">5.3. IFF controller</a></li>
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||
<li><a href="#org7b29313">5.4. Closed Loop System</a></li>
|
||
</ul>
|
||
</li>
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||
<li><a href="#org1272d3f">6. Complete Strut with Encoder</a>
|
||
<ul>
|
||
<li><a href="#orgddf8d43">6.1. Introduction</a></li>
|
||
<li><a href="#org4742c38">6.2. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
|
||
<li><a href="#org332b172">6.3. Output parameters</a></li>
|
||
<li><a href="#orgadca4a4">6.4. Piezoelectric parameters</a></li>
|
||
<li><a href="#org226d3f3">6.5. Identification of the Dynamics</a></li>
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||
</ul>
|
||
</li>
|
||
</ul>
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||
</div>
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||
</div>
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|
||
<div id="outline-container-org5a554a0" class="outline-2">
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||
<h2 id="org5a554a0"><span class="section-number-2">1</span> Amplified Piezoelectric Actuator - 3D elements</h2>
|
||
<div class="outline-text-2" id="text-1">
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||
<p>
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||
The idea here is to:
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||
</p>
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<ul class="org-ul">
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||
<li>export a FEM of an amplified piezoelectric actuator from Ansys to Matlab</li>
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||
<li>import it into a Simscape model</li>
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||
<li>compare the obtained dynamics</li>
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||
<li>add 10kg mass on top of the actuator and identify the dynamics</li>
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||
<li>compare with results from Ansys where 10kg are directly added to the FEM</li>
|
||
</ul>
|
||
</div>
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||
|
||
<div id="outline-container-org29056ab" class="outline-3">
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||
<h3 id="org29056ab"><span class="section-number-3">1.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
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||
<div class="outline-text-3" id="text-1-1">
|
||
<p>
|
||
We first extract the stiffness and mass matrices.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = extractMatrix(<span class="org-string">'piezo_amplified_3d_K.txt'</span>);
|
||
M = extractMatrix(<span class="org-string">'piezo_amplified_3d_M.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we extract the coordinates of the interface nodes.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'piezo_amplified_3d.txt'</span>);
|
||
</pre>
|
||
</div>
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||
|
||
<div class="org-src-container">
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||
<pre class="src src-matlab">save(<span class="org-string">'./mat/piezo_amplified_3d.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgccb0a56" class="outline-3">
|
||
<h3 id="orgccb0a56"><span class="section-number-3">1.2</span> Output parameters</h3>
|
||
<div class="outline-text-3" id="text-1-2">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">load(<span class="org-string">'./mat/piezo_amplified_3d.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
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||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Total number of Nodes</td>
|
||
<td class="org-right">168959</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of interface Nodes</td>
|
||
<td class="org-right">13</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of Modes</td>
|
||
<td class="org-right">30</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Size of M and K matrices</td>
|
||
<td class="org-right">108</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<div id="orgf427fec" class="figure">
|
||
<p><img src="figs/amplified_piezo_interface_nodes.png" alt="amplified_piezo_interface_nodes.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 1: </span>Interface Nodes for the Amplified Piezo Actuator</p>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 1:</span> Coordinates of the interface nodes</caption>
|
||
|
||
<colgroup>
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-right">Node i</th>
|
||
<th scope="col" class="org-right">Node Number</th>
|
||
<th scope="col" class="org-right">x [m]</th>
|
||
<th scope="col" class="org-right">y [m]</th>
|
||
<th scope="col" class="org-right">z [m]</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">1.0</td>
|
||
<td class="org-right">168947.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.03</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2.0</td>
|
||
<td class="org-right">168949.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">-0.03</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">3.0</td>
|
||
<td class="org-right">168950.0</td>
|
||
<td class="org-right">-0.035</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">4.0</td>
|
||
<td class="org-right">168951.0</td>
|
||
<td class="org-right">-0.028</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">5.0</td>
|
||
<td class="org-right">168952.0</td>
|
||
<td class="org-right">-0.021</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">6.0</td>
|
||
<td class="org-right">168953.0</td>
|
||
<td class="org-right">-0.014</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">7.0</td>
|
||
<td class="org-right">168954.0</td>
|
||
<td class="org-right">-0.007</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">8.0</td>
|
||
<td class="org-right">168955.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">9.0</td>
|
||
<td class="org-right">168956.0</td>
|
||
<td class="org-right">0.007</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">10.0</td>
|
||
<td class="org-right">168957.0</td>
|
||
<td class="org-right">0.014</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">11.0</td>
|
||
<td class="org-right">168958.0</td>
|
||
<td class="org-right">0.021</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">12.0</td>
|
||
<td class="org-right">168959.0</td>
|
||
<td class="org-right">0.035</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">13.0</td>
|
||
<td class="org-right">168960.0</td>
|
||
<td class="org-right">0.028</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 2:</span> First 10x10 elements of the 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">300000000.0</td>
|
||
<td class="org-right">-30000.0</td>
|
||
<td class="org-right">8000.0</td>
|
||
<td class="org-right">-200.0</td>
|
||
<td class="org-right">-30.0</td>
|
||
<td class="org-right">-60000.0</td>
|
||
<td class="org-right">20000000.0</td>
|
||
<td class="org-right">-4000.0</td>
|
||
<td class="org-right">500.0</td>
|
||
<td class="org-right">8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-30000.0</td>
|
||
<td class="org-right">100000000.0</td>
|
||
<td class="org-right">400.0</td>
|
||
<td class="org-right">30.0</td>
|
||
<td class="org-right">200.0</td>
|
||
<td class="org-right">-1</td>
|
||
<td class="org-right">4000.0</td>
|
||
<td class="org-right">-8000000.0</td>
|
||
<td class="org-right">800.0</td>
|
||
<td class="org-right">7</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">8000.0</td>
|
||
<td class="org-right">400.0</td>
|
||
<td class="org-right">50000000.0</td>
|
||
<td class="org-right">-800000.0</td>
|
||
<td class="org-right">-300.0</td>
|
||
<td class="org-right">-40.0</td>
|
||
<td class="org-right">300.0</td>
|
||
<td class="org-right">100.0</td>
|
||
<td class="org-right">5000000.0</td>
|
||
<td class="org-right">40000.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-200.0</td>
|
||
<td class="org-right">30.0</td>
|
||
<td class="org-right">-800000.0</td>
|
||
<td class="org-right">20000.0</td>
|
||
<td class="org-right">5</td>
|
||
<td class="org-right">1</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">-2</td>
|
||
<td class="org-right">-40000.0</td>
|
||
<td class="org-right">-300.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-30.0</td>
|
||
<td class="org-right">200.0</td>
|
||
<td class="org-right">-300.0</td>
|
||
<td class="org-right">5</td>
|
||
<td class="org-right">40000.0</td>
|
||
<td class="org-right">0.3</td>
|
||
<td class="org-right">-4</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">40.0</td>
|
||
<td class="org-right">0.4</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-60000.0</td>
|
||
<td class="org-right">-1</td>
|
||
<td class="org-right">-40.0</td>
|
||
<td class="org-right">1</td>
|
||
<td class="org-right">0.3</td>
|
||
<td class="org-right">3000.0</td>
|
||
<td class="org-right">7000.0</td>
|
||
<td class="org-right">0.8</td>
|
||
<td class="org-right">-1</td>
|
||
<td class="org-right">0.0003</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">20000000.0</td>
|
||
<td class="org-right">4000.0</td>
|
||
<td class="org-right">300.0</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">-4</td>
|
||
<td class="org-right">7000.0</td>
|
||
<td class="org-right">300000000.0</td>
|
||
<td class="org-right">20000.0</td>
|
||
<td class="org-right">3000.0</td>
|
||
<td class="org-right">80.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-4000.0</td>
|
||
<td class="org-right">-8000000.0</td>
|
||
<td class="org-right">100.0</td>
|
||
<td class="org-right">-2</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">0.8</td>
|
||
<td class="org-right">20000.0</td>
|
||
<td class="org-right">100000000.0</td>
|
||
<td class="org-right">-4000.0</td>
|
||
<td class="org-right">-100.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">500.0</td>
|
||
<td class="org-right">800.0</td>
|
||
<td class="org-right">5000000.0</td>
|
||
<td class="org-right">-40000.0</td>
|
||
<td class="org-right">40.0</td>
|
||
<td class="org-right">-1</td>
|
||
<td class="org-right">3000.0</td>
|
||
<td class="org-right">-4000.0</td>
|
||
<td class="org-right">50000000.0</td>
|
||
<td class="org-right">800000.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">8</td>
|
||
<td class="org-right">7</td>
|
||
<td class="org-right">40000.0</td>
|
||
<td class="org-right">-300.0</td>
|
||
<td class="org-right">0.4</td>
|
||
<td class="org-right">0.0003</td>
|
||
<td class="org-right">80.0</td>
|
||
<td class="org-right">-100.0</td>
|
||
<td class="org-right">800000.0</td>
|
||
<td class="org-right">20000.0</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 3:</span> First 10x10 elements of the Mass 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">0.03</td>
|
||
<td class="org-right">2e-06</td>
|
||
<td class="org-right">-2e-07</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">2e-08</td>
|
||
<td class="org-right">0.0002</td>
|
||
<td class="org-right">-0.001</td>
|
||
<td class="org-right">2e-07</td>
|
||
<td class="org-right">-8e-08</td>
|
||
<td class="org-right">-9e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2e-06</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-5e-07</td>
|
||
<td class="org-right">7e-09</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">2e-08</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">0.0003</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">1e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-2e-07</td>
|
||
<td class="org-right">-5e-07</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-9e-05</td>
|
||
<td class="org-right">4e-09</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">2e-07</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">-0.0006</td>
|
||
<td class="org-right">-5e-06</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">7e-09</td>
|
||
<td class="org-right">-9e-05</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">6e-11</td>
|
||
<td class="org-right">4e-10</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">3e-11</td>
|
||
<td class="org-right">5e-06</td>
|
||
<td class="org-right">3e-08</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2e-08</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">4e-09</td>
|
||
<td class="org-right">6e-11</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">-7e-09</td>
|
||
<td class="org-right">-4e-11</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.0002</td>
|
||
<td class="org-right">2e-08</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">4e-10</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">2e-06</td>
|
||
<td class="org-right">-2e-06</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-7e-10</td>
|
||
<td class="org-right">-9e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-0.001</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">2e-07</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">-2e-06</td>
|
||
<td class="org-right">0.03</td>
|
||
<td class="org-right">-2e-06</td>
|
||
<td class="org-right">-1e-07</td>
|
||
<td class="org-right">-5e-09</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2e-07</td>
|
||
<td class="org-right">0.0003</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">3e-11</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-2e-06</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-8e-07</td>
|
||
<td class="org-right">-1e-08</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-8e-08</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">-0.0006</td>
|
||
<td class="org-right">5e-06</td>
|
||
<td class="org-right">-7e-09</td>
|
||
<td class="org-right">-7e-10</td>
|
||
<td class="org-right">-1e-07</td>
|
||
<td class="org-right">-8e-07</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">9e-05</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-9e-10</td>
|
||
<td class="org-right">1e-10</td>
|
||
<td class="org-right">-5e-06</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">-4e-11</td>
|
||
<td class="org-right">-9e-12</td>
|
||
<td class="org-right">-5e-09</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">9e-05</td>
|
||
<td class="org-right">1e-06</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-orgdd7c7ba" class="outline-3">
|
||
<h3 id="orgdd7c7ba"><span class="section-number-3">1.3</span> Piezoelectric parameters</h3>
|
||
<div class="outline-text-3" id="text-1-3">
|
||
<p>
|
||
Parameters for the APA95ML:
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">d33 = 3e<span class="org-type">-</span>10; <span class="org-comment">% Strain constant [m/V]</span>
|
||
n = 80; <span class="org-comment">% Number of layers per stack</span>
|
||
eT = 1.6e<span class="org-type">-</span>7; <span class="org-comment">% Permittivity under constant stress [F/m]</span>
|
||
sD = 2e<span class="org-type">-</span>11; <span class="org-comment">% Elastic compliance under constant electric displacement [m2/N]</span>
|
||
ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
|
||
C = 5e<span class="org-type">-</span>6; <span class="org-comment">% Stack capactiance [F]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">na = 2; <span class="org-comment">% Number of stacks used as actuator</span>
|
||
ns = 1; <span class="org-comment">% Number of stacks used as force sensor</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V].
|
||
We denote this constant by \(g_a\) and:
|
||
\[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">d33<span class="org-type">*</span>(na<span class="org-type">*</span>n)<span class="org-type">*</span>(ka<span class="org-type">/</span>(na <span class="org-type">+</span> ns)) <span class="org-comment">% [N/V]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
3.76
|
||
</pre>
|
||
|
||
|
||
<p>
|
||
From (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>) (page 123), the relation between relative displacement and generated voltage is:
|
||
\[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \]
|
||
where:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>\(V_s\): measured voltage [V]</li>
|
||
<li>\(d_{33}\): strain constant [m/V]</li>
|
||
<li>\(\epsilon^T\): permittivity under constant stress [F/m]</li>
|
||
<li>\(s^D\): elastic compliance under constant electric displacement [m^2/N]</li>
|
||
<li>\(n\): number of layers</li>
|
||
<li>\(\Delta h\): relative displacement [m]</li>
|
||
</ul>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">1e<span class="org-type">-</span>6<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>ns<span class="org-type">*</span>n) <span class="org-comment">% [V/um]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
1.1719
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgb2959d5" class="outline-3">
|
||
<h3 id="orgb2959d5"><span class="section-number-3">1.4</span> Identification of the Dynamics</h3>
|
||
<div class="outline-text-3" id="text-1-4">
|
||
<p>
|
||
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> simscape block.
|
||
</p>
|
||
|
||
<p>
|
||
To model the actuator, an <code>Internal Force</code> block is added between the nodes 3 and 12.
|
||
A <code>Relative Motion Sensor</code> block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
|
||
</p>
|
||
|
||
<p>
|
||
One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
|
||
</p>
|
||
|
||
<p>
|
||
We first set the mass to be zero.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 0.01;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The dynamics is identified from the applied force to the measured relative displacement.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d'</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">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gh = <span class="org-type">-</span>linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we add 10Kg of mass:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 5;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
And the dynamics is identified.
|
||
</p>
|
||
|
||
<p>
|
||
The two identified dynamics are compared in Figure <a href="#orgd90f204">2</a>.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d'</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">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Ghm = <span class="org-type">-</span>linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orgd90f204" class="figure">
|
||
<p><img src="figs/dynamics_act_disp_comp_mass.png" alt="dynamics_act_disp_comp_mass.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 2: </span>Dynamics from \(F\) to \(d\) without a payload and with a 10kg payload</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1e29135" class="outline-3">
|
||
<h3 id="org1e29135"><span class="section-number-3">1.5</span> Comparison with Ansys</h3>
|
||
<div class="outline-text-3" id="text-1-5">
|
||
<p>
|
||
Let’s import the results from an Harmonic response analysis in Ansys.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Gresp0 = readtable(<span class="org-string">'FEA_HarmResponse_00kg.txt'</span>);
|
||
Gresp10 = readtable(<span class="org-string">'FEA_HarmResponse_10kg.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The obtained dynamics from the Simscape model and from the Ansys analysis are compare in Figure <a href="#org32de39c">3</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org32de39c" class="figure">
|
||
<p><img src="figs/dynamics_force_disp_comp_anasys.png" alt="dynamics_force_disp_comp_anasys.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 3: </span>Comparison of the obtained dynamics using Simscape with the harmonic response analysis using Ansys</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3f1fc2e" class="outline-3">
|
||
<h3 id="org3f1fc2e"><span class="section-number-3">1.6</span> Force Sensor</h3>
|
||
<div class="outline-text-3" id="text-1-6">
|
||
<p>
|
||
The dynamics is identified from internal forces applied between nodes 3 and 11 to the relative displacement of nodes 11 and 13.
|
||
</p>
|
||
|
||
<p>
|
||
The obtained dynamics is shown in Figure <a href="#org5ab3306">4</a>.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 0;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d'</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">'/Fa'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gf = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 10;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d'</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">'/Fa'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gfm = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org5ab3306" class="figure">
|
||
<p><img src="figs/dynamics_force_force_sensor_comp_mass.png" alt="dynamics_force_force_sensor_comp_mass.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 4: </span>Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org4f1753f" class="outline-3">
|
||
<h3 id="org4f1753f"><span class="section-number-3">1.7</span> Distributed Actuator</h3>
|
||
<div class="outline-text-3" id="text-1-7">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 0;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The dynamics is identified from the applied force to the measured relative displacement.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d_distri'</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">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gd = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we add 10Kg of mass:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 10;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
And the dynamics is identified.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d_distri'</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">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gdm = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org21246e5" class="outline-3">
|
||
<h3 id="org21246e5"><span class="section-number-3">1.8</span> Distributed Actuator and Force Sensor</h3>
|
||
<div class="outline-text-3" id="text-1-8">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 0;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d_distri_act_sens'</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">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/Fm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gfd = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 10;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d_distri_act_sens'</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">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/Fm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gfdm = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd0b2aaa" class="outline-3">
|
||
<h3 id="orgd0b2aaa"><span class="section-number-3">1.9</span> Dynamics from input voltage to displacement</h3>
|
||
<div class="outline-text-3" id="text-1-9">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 5;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
And the dynamics is identified.
|
||
</p>
|
||
|
||
<p>
|
||
The two identified dynamics are compared in Figure <a href="#orgd90f204">2</a>.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d'</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">'/V'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
G = <span class="org-type">-</span>linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">save(<span class="org-string">'../test-bench-apa/mat/fem_model_5kg.mat'</span>, <span class="org-string">'G'</span>)
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org5452b7e" class="outline-3">
|
||
<h3 id="org5452b7e"><span class="section-number-3">1.10</span> Dynamics from input voltage to output voltage</h3>
|
||
<div class="outline-text-3" id="text-1-10">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 5;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_3d'</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">'/Fa'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/dL'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
G = <span class="org-type">-</span>linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgaff4afc" class="outline-3">
|
||
<h3 id="orgaff4afc"><span class="section-number-3">1.11</span> Identification for a simpler model</h3>
|
||
<div class="outline-text-3" id="text-1-11">
|
||
<p>
|
||
The goal in this section is to identify the parameters of a simple APA model from the FEM.
|
||
This can be useful is a lower order model is to be used for simulations.
|
||
</p>
|
||
|
||
<p>
|
||
The presented model is based on (<a href="#citeproc_bib_item_2">Souleille et al. 2018</a>).
|
||
</p>
|
||
|
||
<p>
|
||
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#org73ab5e6">5</a>).
|
||
The parameters are shown in the table below.
|
||
</p>
|
||
|
||
|
||
<div id="org73ab5e6" class="figure">
|
||
<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 5: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 4:</span> Parameters used for the model of the APA 100M</caption>
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-left" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"> </th>
|
||
<th scope="col" class="org-left">Meaning</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">\(k_e\)</td>
|
||
<td class="org-left">Stiffness used to adjust the pole of the isolator</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">\(k_1\)</td>
|
||
<td class="org-left">Stiffness of the metallic suspension when the stack is removed</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">\(k_a\)</td>
|
||
<td class="org-left">Stiffness of the actuator</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">\(c_1\)</td>
|
||
<td class="org-left">Added viscous damping</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
The goal is to determine \(k_e\), \(k_a\) and \(k_1\) so that the simplified model fits the FEM model.
|
||
</p>
|
||
|
||
<p>
|
||
\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
|
||
\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
|
||
</p>
|
||
|
||
<p>
|
||
If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with:
|
||
\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
|
||
\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
|
||
</p>
|
||
|
||
<p>
|
||
From the identified dynamics, compute \(\alpha\) and \(\beta\)
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">alpha = abs(dcgain(G(<span class="org-string">'y'</span>, <span class="org-string">'Fa'</span>)));
|
||
beta = abs(dcgain(G(<span class="org-string">'y'</span>, <span class="org-string">'Fd'</span>)));
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
\(k_a\) is estimated using the following formula:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">ka = 0.9<span class="org-type">/</span>abs(dcgain(G(<span class="org-string">'y'</span>, <span class="org-string">'Fa'</span>)));
|
||
</pre>
|
||
</div>
|
||
<p>
|
||
The factor can be adjusted to better match the curves.
|
||
</p>
|
||
|
||
<p>
|
||
Then \(k_e\) and \(k_1\) are computed.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">ke = ka<span class="org-type">/</span>(beta<span class="org-type">/</span>alpha <span class="org-type">-</span> 1);
|
||
k1 = 1<span class="org-type">/</span>beta <span class="org-type">-</span> ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka);
|
||
</pre>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"> </th>
|
||
<th scope="col" class="org-right">Value [N/um]</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">ka</td>
|
||
<td class="org-right">54.9</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">ke</td>
|
||
<td class="org-right">25.1</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">k1</td>
|
||
<td class="org-right">4.3</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
The damping in the system is adjusted to match the FEM model if necessary.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">c1 = 1e2;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Analytical model of the simpler system:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ga = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> k1 <span class="org-type">+</span> c1<span class="org-type">*</span>s <span class="org-type">+</span> ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka)) <span class="org-type">*</span> ...
|
||
[ 1 , k1 <span class="org-type">+</span> c1<span class="org-type">*</span>s <span class="org-type">+</span> ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka) , ke<span class="org-type">/</span>(ke <span class="org-type">+</span> ka) ;
|
||
<span class="org-type">-</span>ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka), ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka)<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2 , <span class="org-type">-</span>ke<span class="org-type">/</span>(ke <span class="org-type">+</span> ka)<span class="org-type">*</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c1<span class="org-type">*</span>s <span class="org-type">+</span> k1)];
|
||
|
||
Ga.InputName = {<span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>, <span class="org-string">'Fa'</span>};
|
||
Ga.OutputName = {<span class="org-string">'y'</span>, <span class="org-string">'Fs'</span>};
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Adjust the DC gain for the force sensor:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">F_gain = dcgain(G(<span class="org-string">'Fs'</span>, <span class="org-string">'Fd'</span>))<span class="org-type">/</span>dcgain(Ga(<span class="org-string">'Fs'</span>, <span class="org-string">'Fd'</span>));
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org629ff2d" class="figure">
|
||
<p><img src="figs/apa95ml_comp_simpler_model.png" alt="apa95ml_comp_simpler_model.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 6: </span>Comparison of the Dynamics between the FEM model and the simplified one</p>
|
||
</div>
|
||
|
||
<p>
|
||
We save the parameters of the simplified model for the APA95ML:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">save(<span class="org-string">'./mat/APA95ML_simplified_model.mat'</span>, <span class="org-string">'ka'</span>, <span class="org-string">'ke'</span>, <span class="org-string">'k1'</span>, <span class="org-string">'c1'</span>, <span class="org-string">'F_gain'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3dc5d8d" class="outline-2">
|
||
<h2 id="org3dc5d8d"><span class="section-number-2">2</span> APA300ML</h2>
|
||
<div class="outline-text-2" id="text-2">
|
||
|
||
<div id="org60aa4c9" class="figure">
|
||
<p><img src="figs/apa300ml_ansys.jpg" alt="apa300ml_ansys.jpg" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 7: </span>Ansys FEM of the APA300ML</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3eaf978" class="outline-3">
|
||
<h3 id="org3eaf978"><span class="section-number-3">2.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
|
||
<div class="outline-text-3" id="text-2-1">
|
||
<p>
|
||
We first extract the stiffness and mass matrices.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = readmatrix(<span class="org-string">'mat_K.CSV'</span>);
|
||
M = readmatrix(<span class="org-string">'mat_M.CSV'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we extract the coordinates of the interface nodes.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'out_nodes_3D.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">save(<span class="org-string">'./mat/APA300ML.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org160ca26" class="outline-3">
|
||
<h3 id="org160ca26"><span class="section-number-3">2.2</span> Output parameters</h3>
|
||
<div class="outline-text-3" id="text-2-2">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">load(<span class="org-string">'./mat/APA300ML.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Total number of Nodes</td>
|
||
<td class="org-right">7</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of interface Nodes</td>
|
||
<td class="org-right">7</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of Modes</td>
|
||
<td class="org-right">120</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Size of M and K matrices</td>
|
||
<td class="org-right">162</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 5:</span> Coordinates of the interface nodes</caption>
|
||
|
||
<colgroup>
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-right">Node i</th>
|
||
<th scope="col" class="org-right">Node Number</th>
|
||
<th scope="col" class="org-right">x [m]</th>
|
||
<th scope="col" class="org-right">y [m]</th>
|
||
<th scope="col" class="org-right">z [m]</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">1.0</td>
|
||
<td class="org-right">697783.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">-0.015</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2.0</td>
|
||
<td class="org-right">697784.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.015</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">3.0</td>
|
||
<td class="org-right">697785.0</td>
|
||
<td class="org-right">-0.0325</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">4.0</td>
|
||
<td class="org-right">697786.0</td>
|
||
<td class="org-right">-0.0125</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">5.0</td>
|
||
<td class="org-right">697787.0</td>
|
||
<td class="org-right">-0.0075</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">6.0</td>
|
||
<td class="org-right">697788.0</td>
|
||
<td class="org-right">0.0125</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">7.0</td>
|
||
<td class="org-right">697789.0</td>
|
||
<td class="org-right">0.0325</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 6:</span> First 10x10 elements of the 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">200000000.0</td>
|
||
<td class="org-right">30000.0</td>
|
||
<td class="org-right">-20000.0</td>
|
||
<td class="org-right">-70.0</td>
|
||
<td class="org-right">300000.0</td>
|
||
<td class="org-right">40.0</td>
|
||
<td class="org-right">10000000.0</td>
|
||
<td class="org-right">10000.0</td>
|
||
<td class="org-right">-6000.0</td>
|
||
<td class="org-right">30.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">30000.0</td>
|
||
<td class="org-right">30000000.0</td>
|
||
<td class="org-right">2000.0</td>
|
||
<td class="org-right">-200000.0</td>
|
||
<td class="org-right">60.0</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">4000.0</td>
|
||
<td class="org-right">2000000.0</td>
|
||
<td class="org-right">-500.0</td>
|
||
<td class="org-right">9000.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-20000.0</td>
|
||
<td class="org-right">2000.0</td>
|
||
<td class="org-right">7000000.0</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">-30.0</td>
|
||
<td class="org-right">10.0</td>
|
||
<td class="org-right">6000.0</td>
|
||
<td class="org-right">900.0</td>
|
||
<td class="org-right">-500000.0</td>
|
||
<td class="org-right">3</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-70.0</td>
|
||
<td class="org-right">-200000.0</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">1000.0</td>
|
||
<td class="org-right">-0.1</td>
|
||
<td class="org-right">0.08</td>
|
||
<td class="org-right">-20.0</td>
|
||
<td class="org-right">-9000.0</td>
|
||
<td class="org-right">3</td>
|
||
<td class="org-right">-30.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">300000.0</td>
|
||
<td class="org-right">60.0</td>
|
||
<td class="org-right">-30.0</td>
|
||
<td class="org-right">-0.1</td>
|
||
<td class="org-right">900.0</td>
|
||
<td class="org-right">0.1</td>
|
||
<td class="org-right">30000.0</td>
|
||
<td class="org-right">20.0</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">0.06</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">40.0</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">10.0</td>
|
||
<td class="org-right">0.08</td>
|
||
<td class="org-right">0.1</td>
|
||
<td class="org-right">10000.0</td>
|
||
<td class="org-right">20.0</td>
|
||
<td class="org-right">9</td>
|
||
<td class="org-right">-5</td>
|
||
<td class="org-right">0.03</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">10000000.0</td>
|
||
<td class="org-right">4000.0</td>
|
||
<td class="org-right">6000.0</td>
|
||
<td class="org-right">-20.0</td>
|
||
<td class="org-right">30000.0</td>
|
||
<td class="org-right">20.0</td>
|
||
<td class="org-right">200000000.0</td>
|
||
<td class="org-right">10000.0</td>
|
||
<td class="org-right">9000.0</td>
|
||
<td class="org-right">50.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">10000.0</td>
|
||
<td class="org-right">2000000.0</td>
|
||
<td class="org-right">900.0</td>
|
||
<td class="org-right">-9000.0</td>
|
||
<td class="org-right">20.0</td>
|
||
<td class="org-right">9</td>
|
||
<td class="org-right">10000.0</td>
|
||
<td class="org-right">30000000.0</td>
|
||
<td class="org-right">-500.0</td>
|
||
<td class="org-right">200000.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-6000.0</td>
|
||
<td class="org-right">-500.0</td>
|
||
<td class="org-right">-500000.0</td>
|
||
<td class="org-right">3</td>
|
||
<td class="org-right">-10.0</td>
|
||
<td class="org-right">-5</td>
|
||
<td class="org-right">9000.0</td>
|
||
<td class="org-right">-500.0</td>
|
||
<td class="org-right">7000000.0</td>
|
||
<td class="org-right">-2</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">30.0</td>
|
||
<td class="org-right">9000.0</td>
|
||
<td class="org-right">3</td>
|
||
<td class="org-right">-30.0</td>
|
||
<td class="org-right">0.06</td>
|
||
<td class="org-right">0.03</td>
|
||
<td class="org-right">50.0</td>
|
||
<td class="org-right">200000.0</td>
|
||
<td class="org-right">-2</td>
|
||
<td class="org-right">1000.0</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 7:</span> First 10x10 elements of the Mass 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">0.01</td>
|
||
<td class="org-right">-2e-06</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">6e-09</td>
|
||
<td class="org-right">5e-05</td>
|
||
<td class="org-right">-5e-09</td>
|
||
<td class="org-right">-0.0005</td>
|
||
<td class="org-right">-7e-07</td>
|
||
<td class="org-right">6e-07</td>
|
||
<td class="org-right">-3e-09</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-2e-06</td>
|
||
<td class="org-right">0.01</td>
|
||
<td class="org-right">8e-07</td>
|
||
<td class="org-right">-2e-05</td>
|
||
<td class="org-right">-8e-09</td>
|
||
<td class="org-right">2e-09</td>
|
||
<td class="org-right">-9e-07</td>
|
||
<td class="org-right">-0.0002</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">-9e-07</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">8e-07</td>
|
||
<td class="org-right">0.009</td>
|
||
<td class="org-right">5e-10</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-5e-07</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">6e-05</td>
|
||
<td class="org-right">1e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">6e-09</td>
|
||
<td class="org-right">-2e-05</td>
|
||
<td class="org-right">5e-10</td>
|
||
<td class="org-right">3e-07</td>
|
||
<td class="org-right">2e-11</td>
|
||
<td class="org-right">-3e-12</td>
|
||
<td class="org-right">3e-09</td>
|
||
<td class="org-right">9e-07</td>
|
||
<td class="org-right">-4e-10</td>
|
||
<td class="org-right">3e-09</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">5e-05</td>
|
||
<td class="org-right">-8e-09</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">2e-11</td>
|
||
<td class="org-right">6e-07</td>
|
||
<td class="org-right">-4e-11</td>
|
||
<td class="org-right">-1e-06</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">-8e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-5e-09</td>
|
||
<td class="org-right">2e-09</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-3e-12</td>
|
||
<td class="org-right">-4e-11</td>
|
||
<td class="org-right">1e-07</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-4e-10</td>
|
||
<td class="org-right">-5e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-0.0005</td>
|
||
<td class="org-right">-9e-07</td>
|
||
<td class="org-right">-5e-07</td>
|
||
<td class="org-right">3e-09</td>
|
||
<td class="org-right">-1e-06</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">0.01</td>
|
||
<td class="org-right">1e-07</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">-2e-08</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-7e-07</td>
|
||
<td class="org-right">-0.0002</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">9e-07</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">1e-07</td>
|
||
<td class="org-right">0.01</td>
|
||
<td class="org-right">-4e-07</td>
|
||
<td class="org-right">2e-05</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">6e-07</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">6e-05</td>
|
||
<td class="org-right">-4e-10</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">-4e-10</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">-4e-07</td>
|
||
<td class="org-right">0.009</td>
|
||
<td class="org-right">-2e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-3e-09</td>
|
||
<td class="org-right">-9e-07</td>
|
||
<td class="org-right">1e-10</td>
|
||
<td class="org-right">3e-09</td>
|
||
<td class="org-right">-8e-12</td>
|
||
<td class="org-right">-5e-12</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">2e-05</td>
|
||
<td class="org-right">-2e-10</td>
|
||
<td class="org-right">3e-07</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org7932f3c" class="outline-3">
|
||
<h3 id="org7932f3c"><span class="section-number-3">2.3</span> Piezoelectric parameters</h3>
|
||
<div class="outline-text-3" id="text-2-3">
|
||
<p>
|
||
Parameters for the APA300ML:
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">d33 = 3e<span class="org-type">-</span>10; <span class="org-comment">% Strain constant [m/V]</span>
|
||
n = 80; <span class="org-comment">% Number of layers per stack</span>
|
||
eT = 1.6e<span class="org-type">-</span>8; <span class="org-comment">% Permittivity under constant stress [F/m]</span>
|
||
sD = 2e<span class="org-type">-</span>11; <span class="org-comment">% Elastic compliance under constant electric displacement [m2/N]</span>
|
||
ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
|
||
C = 5e<span class="org-type">-</span>6; <span class="org-comment">% Stack capactiance [F]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">na = 2; <span class="org-comment">% Number of stacks used as actuator</span>
|
||
ns = 1; <span class="org-comment">% Number of stacks used as force sensor</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V].
|
||
We denote this constant by \(g_a\) and:
|
||
\[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">d33<span class="org-type">*</span>(na<span class="org-type">*</span>n)<span class="org-type">*</span>(ka<span class="org-type">/</span>(na <span class="org-type">+</span> ns)) <span class="org-comment">% [N/V]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
3.76
|
||
</pre>
|
||
|
||
|
||
<p>
|
||
From (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>) (page 123), the relation between relative displacement and generated voltage is:
|
||
\[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \]
|
||
where:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>\(V_s\): measured voltage [V]</li>
|
||
<li>\(d_{33}\): strain constant [m/V]</li>
|
||
<li>\(\epsilon^T\): permittivity under constant stress [F/m]</li>
|
||
<li>\(s^D\): elastic compliance under constant electric displacement [m^2/N]</li>
|
||
<li>\(n\): number of layers</li>
|
||
<li>\(\Delta h\): relative displacement [m]</li>
|
||
</ul>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">1e<span class="org-type">-</span>6<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>ns<span class="org-type">*</span>n) <span class="org-comment">% [V/um]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
11.719
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org18316cd" class="outline-3">
|
||
<h3 id="org18316cd"><span class="section-number-3">2.4</span> Identification of the APA Characteristics</h3>
|
||
<div class="outline-text-3" id="text-2-4">
|
||
</div>
|
||
<div id="outline-container-orgc0281f1" class="outline-4">
|
||
<h4 id="orgc0281f1"><span class="section-number-4">2.4.1</span> Stiffness</h4>
|
||
<div class="outline-text-4" id="text-2-4-1">
|
||
<p>
|
||
The transfer function from vertical external force to the relative vertical displacement is identified.
|
||
</p>
|
||
|
||
<p>
|
||
The inverse of its DC gain is the axial stiffness of the APA:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">1e<span class="org-type">-</span>6<span class="org-type">/</span>dcgain(G) <span class="org-comment">% [N/um]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
1.753
|
||
</pre>
|
||
|
||
|
||
<p>
|
||
The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgcebe0f9" class="outline-4">
|
||
<h4 id="orgcebe0f9"><span class="section-number-4">2.4.2</span> Resonance Frequency</h4>
|
||
<div class="outline-text-4" id="text-2-4-2">
|
||
<p>
|
||
The resonance frequency is specified to be between 650Hz and 840Hz.
|
||
This is also the case for the FEM model (Figure <a href="#orgbb4a26e">8</a>).
|
||
</p>
|
||
|
||
|
||
<div id="orgbb4a26e" class="figure">
|
||
<p><img src="figs/apa300ml_resonance.png" alt="apa300ml_resonance.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 8: </span>First resonance is around 800Hz</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgda4f233" class="outline-4">
|
||
<h4 id="orgda4f233"><span class="section-number-4">2.4.3</span> Amplification factor</h4>
|
||
<div class="outline-text-4" id="text-2-4-3">
|
||
<p>
|
||
The amplification factor is the ratio of the axial displacement to the stack displacement.
|
||
</p>
|
||
|
||
<p>
|
||
The ratio of the two displacement is computed from the FEM model.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">abs(dcgain(G(1,1))<span class="org-type">./</span>dcgain(G(2,1)))
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
5.0749
|
||
</pre>
|
||
|
||
|
||
<p>
|
||
If we take the ratio of the piezo height and length (approximation of the amplification factor):
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">75<span class="org-type">/</span>15
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
5
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org59829b6" class="outline-4">
|
||
<h4 id="org59829b6"><span class="section-number-4">2.4.4</span> Stroke</h4>
|
||
<div class="outline-text-4" id="text-2-4-4">
|
||
<p>
|
||
Estimation of the actuator stroke:
|
||
\[ \Delta H = A n \Delta L \]
|
||
with:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>\(\Delta H\) Axial Stroke of the APA</li>
|
||
<li>\(A\) Amplification factor (5 for the APA300ML)</li>
|
||
<li>\(n\) Number of stack used</li>
|
||
<li>\(\Delta L\) Stroke of the stack (0.1% of its length)</li>
|
||
</ul>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">1e6 <span class="org-type">*</span> 5 <span class="org-type">*</span> 3 <span class="org-type">*</span> 20e<span class="org-type">-</span>3 <span class="org-type">*</span> 0.1e<span class="org-type">-</span>2
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
300
|
||
</pre>
|
||
|
||
|
||
<p>
|
||
This is exactly the specified stroke in the data-sheet.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1cbc8a6" class="outline-3">
|
||
<h3 id="org1cbc8a6"><span class="section-number-3">2.5</span> Identification of the Dynamics</h3>
|
||
<div class="outline-text-3" id="text-2-5">
|
||
<p>
|
||
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> simscape block.
|
||
</p>
|
||
|
||
<p>
|
||
To model the actuator, an <code>Internal Force</code> block is added between the nodes 3 and 12.
|
||
A <code>Relative Motion Sensor</code> block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
|
||
</p>
|
||
|
||
<p>
|
||
One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
|
||
</p>
|
||
|
||
<p>
|
||
We first set the mass to be zero.
|
||
The dynamics is identified from the applied force to the measured relative displacement.
|
||
The same dynamics is identified for a payload mass of 10Kg.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 10;
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orgb07566c" class="figure">
|
||
<p><img src="figs/apa300ml_plant_dynamics.png" alt="apa300ml_plant_dynamics.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 9: </span>Transfer function from forces applied by the stack to the axial displacement of the APA</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org44a32d5" class="outline-3">
|
||
<h3 id="org44a32d5"><span class="section-number-3">2.6</span> IFF</h3>
|
||
<div class="outline-text-3" id="text-2-6">
|
||
<p>
|
||
Let’s use 2 stacks as actuators and 1 stack as force sensor.
|
||
</p>
|
||
|
||
<p>
|
||
The transfer function from actuator to sensors is identified and shown in Figure <a href="#org0cb4e3c">10</a>.
|
||
</p>
|
||
|
||
<div id="org0cb4e3c" class="figure">
|
||
<p><img src="figs/apa300ml_iff_plant.png" alt="apa300ml_iff_plant.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 10: </span>Transfer function from actuator to force sensor</p>
|
||
</div>
|
||
|
||
<p>
|
||
For root locus corresponding to IFF is shown in Figure <a href="#org57c3b0d">11</a>.
|
||
</p>
|
||
|
||
<div id="org57c3b0d" class="figure">
|
||
<p><img src="figs/apa300ml_iff_root_locus.png" alt="apa300ml_iff_root_locus.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 11: </span>Root Locus for IFF</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3e558c6" class="outline-3">
|
||
<h3 id="org3e558c6"><span class="section-number-3">2.7</span> DVF</h3>
|
||
<div class="outline-text-3" id="text-2-7">
|
||
<p>
|
||
Now the dynamics from the stack actuator to the relative motion sensor is identified and shown in Figure <a href="#org828d315">12</a>.
|
||
</p>
|
||
|
||
<div id="org828d315" class="figure">
|
||
<p><img src="figs/apa300ml_dvf_plant.png" alt="apa300ml_dvf_plant.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 12: </span>Transfer function from stack actuator to relative motion sensor</p>
|
||
</div>
|
||
|
||
<p>
|
||
The root locus for DVF is shown in Figure <a href="#orgf2f0551">13</a>.
|
||
</p>
|
||
|
||
|
||
<div id="orgf2f0551" class="figure">
|
||
<p><img src="figs/apa300ml_dvf_root_locus.png" alt="apa300ml_dvf_root_locus.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 13: </span>Root Locus for Direct Velocity Feedback</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgad3fdd9" class="outline-3">
|
||
<h3 id="orgad3fdd9"><span class="section-number-3">2.8</span> Identification for a simpler model</h3>
|
||
<div class="outline-text-3" id="text-2-8">
|
||
<p>
|
||
The goal in this section is to identify the parameters of a simple APA model from the FEM.
|
||
This can be useful is a lower order model is to be used for simulations.
|
||
</p>
|
||
|
||
<p>
|
||
The presented model is based on (<a href="#citeproc_bib_item_2">Souleille et al. 2018</a>).
|
||
</p>
|
||
|
||
<p>
|
||
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#org73ab5e6">5</a>).
|
||
The parameters are shown in the table below.
|
||
</p>
|
||
|
||
|
||
<div id="org2d53ab0" class="figure">
|
||
<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 14: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 8:</span> Parameters used for the model of the APA 100M</caption>
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-left" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"> </th>
|
||
<th scope="col" class="org-left">Meaning</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">\(k_e\)</td>
|
||
<td class="org-left">Stiffness used to adjust the pole of the isolator</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">\(k_1\)</td>
|
||
<td class="org-left">Stiffness of the metallic suspension when the stack is removed</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">\(k_a\)</td>
|
||
<td class="org-left">Stiffness of the actuator</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">\(c_1\)</td>
|
||
<td class="org-left">Added viscous damping</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
The goal is to determine \(k_e\), \(k_a\) and \(k_1\) so that the simplified model fits the FEM model.
|
||
</p>
|
||
|
||
<p>
|
||
\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
|
||
\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
|
||
</p>
|
||
|
||
<p>
|
||
If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with:
|
||
\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
|
||
\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
|
||
</p>
|
||
|
||
<p>
|
||
From the identified dynamics, compute \(\alpha\) and \(\beta\)
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">alpha = abs(dcgain(G(<span class="org-string">'y'</span>, <span class="org-string">'Fa'</span>)));
|
||
beta = abs(dcgain(G(<span class="org-string">'y'</span>, <span class="org-string">'Fd'</span>)));
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
\(k_a\) is estimated using the following formula:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">ka = 0.8<span class="org-type">/</span>abs(dcgain(G(<span class="org-string">'y'</span>, <span class="org-string">'Fa'</span>)));
|
||
</pre>
|
||
</div>
|
||
<p>
|
||
The factor can be adjusted to better match the curves.
|
||
</p>
|
||
|
||
<p>
|
||
Then \(k_e\) and \(k_1\) are computed.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">ke = ka<span class="org-type">/</span>(beta<span class="org-type">/</span>alpha <span class="org-type">-</span> 1);
|
||
k1 = 1<span class="org-type">/</span>beta <span class="org-type">-</span> ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka);
|
||
</pre>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"> </th>
|
||
<th scope="col" class="org-right">Value [N/um]</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">ka</td>
|
||
<td class="org-right">40.5</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">ke</td>
|
||
<td class="org-right">1.5</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">k1</td>
|
||
<td class="org-right">0.4</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
The damping in the system is adjusted to match the FEM model if necessary.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">c1 = 1e2;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Analytical model of the simpler system:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ga = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> k1 <span class="org-type">+</span> c1<span class="org-type">*</span>s <span class="org-type">+</span> ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka)) <span class="org-type">*</span> ...
|
||
[ 1 , k1 <span class="org-type">+</span> c1<span class="org-type">*</span>s <span class="org-type">+</span> ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka) , ke<span class="org-type">/</span>(ke <span class="org-type">+</span> ka) ;
|
||
<span class="org-type">-</span>ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka), ke<span class="org-type">*</span>ka<span class="org-type">/</span>(ke <span class="org-type">+</span> ka)<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2 , <span class="org-type">-</span>ke<span class="org-type">/</span>(ke <span class="org-type">+</span> ka)<span class="org-type">*</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c1<span class="org-type">*</span>s <span class="org-type">+</span> k1)];
|
||
|
||
Ga.InputName = {<span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>, <span class="org-string">'Fa'</span>};
|
||
Ga.OutputName = {<span class="org-string">'y'</span>, <span class="org-string">'Fs'</span>};
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Adjust the DC gain for the force sensor:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">F_gain = dcgain(G(<span class="org-string">'Fs'</span>, <span class="org-string">'Fd'</span>))<span class="org-type">/</span>dcgain(Ga(<span class="org-string">'Fs'</span>, <span class="org-string">'Fd'</span>));
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org6c57210" class="figure">
|
||
<p><img src="figs/apa300ml_comp_simpler_model.png" alt="apa300ml_comp_simpler_model.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 15: </span>Comparison of the Dynamics between the FEM model and the simplified one</p>
|
||
</div>
|
||
|
||
<p>
|
||
We now compare the FEM model with the simplified simscape model.
|
||
</p>
|
||
|
||
<div id="orgdec5f8c" class="figure">
|
||
<p><img src="figs/apa300ml_comp_simpler_simscape.png" alt="apa300ml_comp_simpler_simscape.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 16: </span>Comparison of the Dynamics between the FEM model and the simplified simscape model</p>
|
||
</div>
|
||
|
||
<p>
|
||
We save the parameters of the simplified model for the APA300ML:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">save(<span class="org-string">'./mat/APA300ML_simplified_model.mat'</span>, <span class="org-string">'ka'</span>, <span class="org-string">'ke'</span>, <span class="org-string">'k1'</span>, <span class="org-string">'c1'</span>, <span class="org-string">'F_gain'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orge0b9f5a" class="outline-3">
|
||
<h3 id="orge0b9f5a"><span class="section-number-3">2.9</span> Identification of the stiffness properties</h3>
|
||
<div class="outline-text-3" id="text-2-9">
|
||
</div>
|
||
<div id="outline-container-org52ddecb" class="outline-4">
|
||
<h4 id="org52ddecb"><span class="section-number-4">2.9.1</span> APA Alone</h4>
|
||
<div class="outline-text-4" id="text-2-9-1">
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"><b>Caracteristics</b></th>
|
||
<th scope="col" class="org-right"><b>Value</b></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Kx [N/um]</td>
|
||
<td class="org-right">0.8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Ky [N/um]</td>
|
||
<td class="org-right">1.6</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Kz [N/um]</td>
|
||
<td class="org-right">1.8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Rx [Nm/rad]</td>
|
||
<td class="org-right">71.4</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Ry [Nm/rad]</td>
|
||
<td class="org-right">148.2</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Rz [Nm/rad]</td>
|
||
<td class="org-right">4241.8</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org02b6855" class="outline-4">
|
||
<h4 id="org02b6855"><span class="section-number-4">2.9.2</span> See how the global stiffness is changing with the flexible joints</h4>
|
||
<div class="outline-text-4" id="text-2-9-2">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">flex = load(<span class="org-string">'./mat/flexor_ID16.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"><b>Caracteristic</b></th>
|
||
<th scope="col" class="org-right"><b>Value</b></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Kx [N/um]</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Ky [N/um]</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Kz [N/um]</td>
|
||
<td class="org-right">1.8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Rx [Nm/rad]</td>
|
||
<td class="org-right">722.9</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Ry [Nm/rad]</td>
|
||
<td class="org-right">129.6</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Rz [Nm/rad]</td>
|
||
<td class="org-right">115.3</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">freqs = logspace(<span class="org-type">-</span>2, 5, 1000);
|
||
|
||
<span class="org-type">figure</span>;
|
||
hold on;
|
||
plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'APA'</span>);
|
||
plot(freqs, abs(squeeze(freqresp(Gf(2,2), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'Flex'</span>);
|
||
hold off;
|
||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/N]'</span>);
|
||
hold off;
|
||
legend(<span class="org-string">'location'</span>, <span class="org-string">'northeast'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">freqs = logspace(<span class="org-type">-</span>2, 5, 1000);
|
||
|
||
<span class="org-type">figure</span>;
|
||
hold on;
|
||
plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'APA'</span>);
|
||
plot(freqs, abs(squeeze(freqresp(Gf(3,3), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'Flex'</span>);
|
||
hold off;
|
||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/N]'</span>);
|
||
hold off;
|
||
legend(<span class="org-string">'location'</span>, <span class="org-string">'northeast'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org34de703" class="outline-3">
|
||
<h3 id="org34de703"><span class="section-number-3">2.10</span> Effect of APA300ML in the flexibility of the leg</h3>
|
||
<div class="outline-text-3" id="text-2-10">
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"><b>Caracteristic</b></th>
|
||
<th scope="col" class="org-right"><b>Rigid APA</b></th>
|
||
<th scope="col" class="org-right"><b>Flexible APA</b></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Kx [N/um]</td>
|
||
<td class="org-right">0.018</td>
|
||
<td class="org-right">0.019</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Ky [N/um]</td>
|
||
<td class="org-right">0.018</td>
|
||
<td class="org-right">0.018</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Kz [N/um]</td>
|
||
<td class="org-right">60.0</td>
|
||
<td class="org-right">2.647</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Rx [Nm/rad]</td>
|
||
<td class="org-right">16.705</td>
|
||
<td class="org-right">557.682</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Ry [Nm/rad]</td>
|
||
<td class="org-right">16.535</td>
|
||
<td class="org-right">185.939</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Rz [Nm/rad]</td>
|
||
<td class="org-right">118.0</td>
|
||
<td class="org-right">114.803</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgb6c0ee0" class="outline-2">
|
||
<h2 id="orgb6c0ee0"><span class="section-number-2">3</span> Flexible Joint</h2>
|
||
<div class="outline-text-2" id="text-3">
|
||
<p>
|
||
The studied flexor is shown in Figure <a href="#orgdff9d67">17</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The stiffness and mass matrices representing the dynamics of the flexor are exported from a FEM.
|
||
It is then imported into Simscape.
|
||
</p>
|
||
|
||
<p>
|
||
A simplified model of the flexor is then developped.
|
||
</p>
|
||
|
||
|
||
<div id="orgdff9d67" class="figure">
|
||
<p><img src="figs/flexor_id16_screenshot.png" alt="flexor_id16_screenshot.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 17: </span>Flexor studied</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd7b1d5f" class="outline-3">
|
||
<h3 id="orgd7b1d5f"><span class="section-number-3">3.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
|
||
<div class="outline-text-3" id="text-3-1">
|
||
<p>
|
||
We first extract the stiffness and mass matrices.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = extractMatrix(<span class="org-string">'mat_K_6modes_2MDoF.matrix'</span>);
|
||
M = extractMatrix(<span class="org-string">'mat_M_6modes_2MDoF.matrix'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we extract the coordinates of the interface nodes.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'out_nodes_3D.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">save(<span class="org-string">'./mat/flexor_ID16.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org9778a32" class="outline-3">
|
||
<h3 id="org9778a32"><span class="section-number-3">3.2</span> Output parameters</h3>
|
||
<div class="outline-text-3" id="text-3-2">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">load(<span class="org-string">'./mat/flexor_ID16.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Total number of Nodes</td>
|
||
<td class="org-right">2</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of interface Nodes</td>
|
||
<td class="org-right">2</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of Modes</td>
|
||
<td class="org-right">6</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Size of M and K matrices</td>
|
||
<td class="org-right">18</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 9:</span> Coordinates of the interface nodes</caption>
|
||
|
||
<colgroup>
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-right">Node i</th>
|
||
<th scope="col" class="org-right">Node Number</th>
|
||
<th scope="col" class="org-right">x [m]</th>
|
||
<th scope="col" class="org-right">y [m]</th>
|
||
<th scope="col" class="org-right">z [m]</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">1.0</td>
|
||
<td class="org-right">181278.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2.0</td>
|
||
<td class="org-right">181279.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">-0.0</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 10:</span> First 10x10 elements of the 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">11200000.0</td>
|
||
<td class="org-right">195.0</td>
|
||
<td class="org-right">2220.0</td>
|
||
<td class="org-right">-0.719</td>
|
||
<td class="org-right">-265.0</td>
|
||
<td class="org-right">1.59</td>
|
||
<td class="org-right">-11200000.0</td>
|
||
<td class="org-right">-213.0</td>
|
||
<td class="org-right">-2220.0</td>
|
||
<td class="org-right">0.147</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">195.0</td>
|
||
<td class="org-right">11400000.0</td>
|
||
<td class="org-right">1290.0</td>
|
||
<td class="org-right">-148.0</td>
|
||
<td class="org-right">-0.188</td>
|
||
<td class="org-right">2.41</td>
|
||
<td class="org-right">-212.0</td>
|
||
<td class="org-right">-11400000.0</td>
|
||
<td class="org-right">-1290.0</td>
|
||
<td class="org-right">148.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2220.0</td>
|
||
<td class="org-right">1290.0</td>
|
||
<td class="org-right">119000000.0</td>
|
||
<td class="org-right">1.31</td>
|
||
<td class="org-right">1.49</td>
|
||
<td class="org-right">1.79</td>
|
||
<td class="org-right">-2220.0</td>
|
||
<td class="org-right">-1290.0</td>
|
||
<td class="org-right">-119000000.0</td>
|
||
<td class="org-right">-1.31</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-0.719</td>
|
||
<td class="org-right">-148.0</td>
|
||
<td class="org-right">1.31</td>
|
||
<td class="org-right">33.0</td>
|
||
<td class="org-right">0.000488</td>
|
||
<td class="org-right">-0.000977</td>
|
||
<td class="org-right">0.141</td>
|
||
<td class="org-right">148.0</td>
|
||
<td class="org-right">-1.31</td>
|
||
<td class="org-right">-33.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-265.0</td>
|
||
<td class="org-right">-0.188</td>
|
||
<td class="org-right">1.49</td>
|
||
<td class="org-right">0.000488</td>
|
||
<td class="org-right">33.0</td>
|
||
<td class="org-right">0.00293</td>
|
||
<td class="org-right">266.0</td>
|
||
<td class="org-right">0.154</td>
|
||
<td class="org-right">-1.49</td>
|
||
<td class="org-right">0.00026</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">1.59</td>
|
||
<td class="org-right">2.41</td>
|
||
<td class="org-right">1.79</td>
|
||
<td class="org-right">-0.000977</td>
|
||
<td class="org-right">0.00293</td>
|
||
<td class="org-right">236.0</td>
|
||
<td class="org-right">-1.32</td>
|
||
<td class="org-right">-2.55</td>
|
||
<td class="org-right">-1.79</td>
|
||
<td class="org-right">0.000379</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-11200000.0</td>
|
||
<td class="org-right">-212.0</td>
|
||
<td class="org-right">-2220.0</td>
|
||
<td class="org-right">0.141</td>
|
||
<td class="org-right">266.0</td>
|
||
<td class="org-right">-1.32</td>
|
||
<td class="org-right">11400000.0</td>
|
||
<td class="org-right">24600.0</td>
|
||
<td class="org-right">1640.0</td>
|
||
<td class="org-right">120.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-213.0</td>
|
||
<td class="org-right">-11400000.0</td>
|
||
<td class="org-right">-1290.0</td>
|
||
<td class="org-right">148.0</td>
|
||
<td class="org-right">0.154</td>
|
||
<td class="org-right">-2.55</td>
|
||
<td class="org-right">24600.0</td>
|
||
<td class="org-right">11400000.0</td>
|
||
<td class="org-right">1290.0</td>
|
||
<td class="org-right">-72.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-2220.0</td>
|
||
<td class="org-right">-1290.0</td>
|
||
<td class="org-right">-119000000.0</td>
|
||
<td class="org-right">-1.31</td>
|
||
<td class="org-right">-1.49</td>
|
||
<td class="org-right">-1.79</td>
|
||
<td class="org-right">1640.0</td>
|
||
<td class="org-right">1290.0</td>
|
||
<td class="org-right">119000000.0</td>
|
||
<td class="org-right">1.32</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.147</td>
|
||
<td class="org-right">148.0</td>
|
||
<td class="org-right">-1.31</td>
|
||
<td class="org-right">-33.0</td>
|
||
<td class="org-right">0.00026</td>
|
||
<td class="org-right">0.000379</td>
|
||
<td class="org-right">120.0</td>
|
||
<td class="org-right">-72.0</td>
|
||
<td class="org-right">1.32</td>
|
||
<td class="org-right">34.7</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 11:</span> First 10x10 elements of the Mass 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">-4e-08</td>
|
||
<td class="org-right">-1e-10</td>
|
||
<td class="org-right">0.0002</td>
|
||
<td class="org-right">-3e-11</td>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">5e-08</td>
|
||
<td class="org-right">7e-08</td>
|
||
<td class="org-right">1e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">-0.0002</td>
|
||
<td class="org-right">-1e-10</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">2e-08</td>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">3e-07</td>
|
||
<td class="org-right">1e-05</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-4e-08</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">7e-10</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">3e-07</td>
|
||
<td class="org-right">7e-08</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">1e-09</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-1e-10</td>
|
||
<td class="org-right">-0.0002</td>
|
||
<td class="org-right">7e-10</td>
|
||
<td class="org-right">4e-06</td>
|
||
<td class="org-right">-1e-12</td>
|
||
<td class="org-right">-6e-13</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">-7e-06</td>
|
||
<td class="org-right">-8e-10</td>
|
||
<td class="org-right">-1e-09</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.0002</td>
|
||
<td class="org-right">-1e-10</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">-1e-12</td>
|
||
<td class="org-right">3e-06</td>
|
||
<td class="org-right">2e-13</td>
|
||
<td class="org-right">9e-06</td>
|
||
<td class="org-right">4e-11</td>
|
||
<td class="org-right">2e-09</td>
|
||
<td class="org-right">-3e-13</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-3e-11</td>
|
||
<td class="org-right">-2e-09</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">-6e-13</td>
|
||
<td class="org-right">2e-13</td>
|
||
<td class="org-right">4e-07</td>
|
||
<td class="org-right">8e-11</td>
|
||
<td class="org-right">9e-10</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">2e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">2e-08</td>
|
||
<td class="org-right">3e-07</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">9e-06</td>
|
||
<td class="org-right">8e-11</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-7e-08</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">-2e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">5e-08</td>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">7e-08</td>
|
||
<td class="org-right">-7e-06</td>
|
||
<td class="org-right">4e-11</td>
|
||
<td class="org-right">9e-10</td>
|
||
<td class="org-right">-7e-08</td>
|
||
<td class="org-right">0.01</td>
|
||
<td class="org-right">-4e-08</td>
|
||
<td class="org-right">0.0002</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">7e-08</td>
|
||
<td class="org-right">3e-07</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">-8e-10</td>
|
||
<td class="org-right">2e-09</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">-4e-08</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-1e-09</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">1e-10</td>
|
||
<td class="org-right">1e-05</td>
|
||
<td class="org-right">1e-09</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">-3e-13</td>
|
||
<td class="org-right">2e-12</td>
|
||
<td class="org-right">-2e-10</td>
|
||
<td class="org-right">0.0002</td>
|
||
<td class="org-right">-1e-09</td>
|
||
<td class="org-right">2e-06</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgcb9bad1" class="outline-3">
|
||
<h3 id="orgcb9bad1"><span class="section-number-3">3.3</span> Flexible Joint Characteristics</h3>
|
||
<div class="outline-text-3" id="text-3-3">
|
||
<p>
|
||
The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
|
||
</p>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"><b>Caracteristic</b></th>
|
||
<th scope="col" class="org-right"><b>Value</b></th>
|
||
<th scope="col" class="org-right"><b>Estimation by Francois</b></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Axial Stiffness [N/um]</td>
|
||
<td class="org-right">119</td>
|
||
<td class="org-right">60</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Shear Stiffness [N/um]</td>
|
||
<td class="org-right">11</td>
|
||
<td class="org-right">0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness [Nm/rad]</td>
|
||
<td class="org-right">33</td>
|
||
<td class="org-right">15</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness [Nm/rad]</td>
|
||
<td class="org-right">33</td>
|
||
<td class="org-right">15</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Torsion Stiffness [Nm/rad]</td>
|
||
<td class="org-right">236</td>
|
||
<td class="org-right">20</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org4dadc02" class="outline-3">
|
||
<h3 id="org4dadc02"><span class="section-number-3">3.4</span> Identification of the parameters using Simscape</h3>
|
||
<div class="outline-text-3" id="text-3-4">
|
||
<p>
|
||
The flexor is now imported into Simscape and its parameters are estimated using an identification.
|
||
</p>
|
||
|
||
<p>
|
||
The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
|
||
And we find the same parameters as the one estimated from the Stiffness matrix.
|
||
</p>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"><b>Caracteristic</b></th>
|
||
<th scope="col" class="org-right"><b>Value</b></th>
|
||
<th scope="col" class="org-right"><b>Identification</b></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Axial Stiffness Dz [N/um]</td>
|
||
<td class="org-right">119</td>
|
||
<td class="org-right">119</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness Rx [Nm/rad]</td>
|
||
<td class="org-right">33</td>
|
||
<td class="org-right">34</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness Ry [Nm/rad]</td>
|
||
<td class="org-right">33</td>
|
||
<td class="org-right">126</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Torsion Stiffness Rz [Nm/rad]</td>
|
||
<td class="org-right">236</td>
|
||
<td class="org-right">238</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org30336a6" class="outline-3">
|
||
<h3 id="org30336a6"><span class="section-number-3">3.5</span> Simpler Model</h3>
|
||
<div class="outline-text-3" id="text-3-5">
|
||
<p>
|
||
Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure <a href="#org9c5b090">18</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org9c5b090" class="figure">
|
||
<p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 18: </span>Bushing Joint used to model the flexible joint</p>
|
||
</div>
|
||
|
||
<p>
|
||
The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kx = K(1,1); <span class="org-comment">% [N/m]</span>
|
||
Ky = K(2,2); <span class="org-comment">% [N/m]</span>
|
||
Kz = K(3,3); <span class="org-comment">% [N/m]</span>
|
||
Krx = K(4,4); <span class="org-comment">% [Nm/rad]</span>
|
||
Kry = K(5,5); <span class="org-comment">% [Nm/rad]</span>
|
||
Krz = K(6,6); <span class="org-comment">% [Nm/rad]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
|
||
The two obtained dynamics are compared in Figure
|
||
</p>
|
||
|
||
|
||
<div id="org6baee4c" class="figure">
|
||
<p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 19: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd9d5aff" class="outline-2">
|
||
<h2 id="orgd9d5aff"><span class="section-number-2">4</span> Optimal Flexible Joint</h2>
|
||
<div class="outline-text-2" id="text-4">
|
||
|
||
<div id="org47739fa" class="figure">
|
||
<p><img src="data/flexor_circ_025/CS.jpg" alt="CS.jpg" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 20: </span>Flexor studied</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org83c1679" class="outline-3">
|
||
<h3 id="org83c1679"><span class="section-number-3">4.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
|
||
<div class="outline-text-3" id="text-4-1">
|
||
<p>
|
||
We first extract the stiffness and mass matrices.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = readmatrix(<span class="org-string">'mat_K.CSV'</span>);
|
||
M = readmatrix(<span class="org-string">'mat_M.CSV'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we extract the coordinates of the interface nodes.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'out_nodes_3D.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">save(<span class="org-string">'./mat/flexor_025.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgbee4a84" class="outline-3">
|
||
<h3 id="orgbee4a84"><span class="section-number-3">4.2</span> Output parameters</h3>
|
||
<div class="outline-text-3" id="text-4-2">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">load(<span class="org-string">'./mat/flexor_025.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Total number of Nodes</td>
|
||
<td class="org-right">2</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of interface Nodes</td>
|
||
<td class="org-right">2</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of Modes</td>
|
||
<td class="org-right">6</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Size of M and K matrices</td>
|
||
<td class="org-right">18</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 12:</span> Coordinates of the interface nodes</caption>
|
||
|
||
<colgroup>
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-right">Node i</th>
|
||
<th scope="col" class="org-right">Node Number</th>
|
||
<th scope="col" class="org-right">x [m]</th>
|
||
<th scope="col" class="org-right">y [m]</th>
|
||
<th scope="col" class="org-right">z [m]</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">1.0</td>
|
||
<td class="org-right">528875.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2.0</td>
|
||
<td class="org-right">528876.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">-0.0</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 13:</span> First 10x10 elements of the 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">12700000.0</td>
|
||
<td class="org-right">-18.5</td>
|
||
<td class="org-right">-26.8</td>
|
||
<td class="org-right">0.00162</td>
|
||
<td class="org-right">-4.63</td>
|
||
<td class="org-right">64.0</td>
|
||
<td class="org-right">-12700000.0</td>
|
||
<td class="org-right">18.3</td>
|
||
<td class="org-right">26.7</td>
|
||
<td class="org-right">0.00234</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-18.5</td>
|
||
<td class="org-right">12700000.0</td>
|
||
<td class="org-right">-499.0</td>
|
||
<td class="org-right">-132.0</td>
|
||
<td class="org-right">0.00414</td>
|
||
<td class="org-right">-0.495</td>
|
||
<td class="org-right">18.4</td>
|
||
<td class="org-right">-12700000.0</td>
|
||
<td class="org-right">499.0</td>
|
||
<td class="org-right">132.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-26.8</td>
|
||
<td class="org-right">-499.0</td>
|
||
<td class="org-right">94000000.0</td>
|
||
<td class="org-right">-470.0</td>
|
||
<td class="org-right">0.00771</td>
|
||
<td class="org-right">-0.855</td>
|
||
<td class="org-right">26.8</td>
|
||
<td class="org-right">498.0</td>
|
||
<td class="org-right">-94000000.0</td>
|
||
<td class="org-right">470.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.00162</td>
|
||
<td class="org-right">-132.0</td>
|
||
<td class="org-right">-470.0</td>
|
||
<td class="org-right">4.83</td>
|
||
<td class="org-right">2.61e-07</td>
|
||
<td class="org-right">0.000123</td>
|
||
<td class="org-right">-0.00163</td>
|
||
<td class="org-right">132.0</td>
|
||
<td class="org-right">470.0</td>
|
||
<td class="org-right">-4.83</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-4.63</td>
|
||
<td class="org-right">0.00414</td>
|
||
<td class="org-right">0.00771</td>
|
||
<td class="org-right">2.61e-07</td>
|
||
<td class="org-right">4.83</td>
|
||
<td class="org-right">4.43e-05</td>
|
||
<td class="org-right">4.63</td>
|
||
<td class="org-right">-0.00413</td>
|
||
<td class="org-right">-0.00772</td>
|
||
<td class="org-right">-4.3e-07</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">64.0</td>
|
||
<td class="org-right">-0.495</td>
|
||
<td class="org-right">-0.855</td>
|
||
<td class="org-right">0.000123</td>
|
||
<td class="org-right">4.43e-05</td>
|
||
<td class="org-right">260.0</td>
|
||
<td class="org-right">-64.0</td>
|
||
<td class="org-right">0.495</td>
|
||
<td class="org-right">0.855</td>
|
||
<td class="org-right">-0.000124</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-12700000.0</td>
|
||
<td class="org-right">18.4</td>
|
||
<td class="org-right">26.8</td>
|
||
<td class="org-right">-0.00163</td>
|
||
<td class="org-right">4.63</td>
|
||
<td class="org-right">-64.0</td>
|
||
<td class="org-right">12700000.0</td>
|
||
<td class="org-right">-18.2</td>
|
||
<td class="org-right">-26.7</td>
|
||
<td class="org-right">-0.00234</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">18.3</td>
|
||
<td class="org-right">-12700000.0</td>
|
||
<td class="org-right">498.0</td>
|
||
<td class="org-right">132.0</td>
|
||
<td class="org-right">-0.00413</td>
|
||
<td class="org-right">0.495</td>
|
||
<td class="org-right">-18.2</td>
|
||
<td class="org-right">12700000.0</td>
|
||
<td class="org-right">-498.0</td>
|
||
<td class="org-right">-132.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">26.7</td>
|
||
<td class="org-right">499.0</td>
|
||
<td class="org-right">-94000000.0</td>
|
||
<td class="org-right">470.0</td>
|
||
<td class="org-right">-0.00772</td>
|
||
<td class="org-right">0.855</td>
|
||
<td class="org-right">-26.7</td>
|
||
<td class="org-right">-498.0</td>
|
||
<td class="org-right">94000000.0</td>
|
||
<td class="org-right">-470.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.00234</td>
|
||
<td class="org-right">132.0</td>
|
||
<td class="org-right">470.0</td>
|
||
<td class="org-right">-4.83</td>
|
||
<td class="org-right">-4.3e-07</td>
|
||
<td class="org-right">-0.000124</td>
|
||
<td class="org-right">-0.00234</td>
|
||
<td class="org-right">-132.0</td>
|
||
<td class="org-right">-470.0</td>
|
||
<td class="org-right">4.83</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 14:</span> First 10x10 elements of the Mass 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">0.006</td>
|
||
<td class="org-right">8e-09</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">-1e-10</td>
|
||
<td class="org-right">3e-05</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">-3e-09</td>
|
||
<td class="org-right">9e-09</td>
|
||
<td class="org-right">2e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">8e-09</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">1e-07</td>
|
||
<td class="org-right">-3e-05</td>
|
||
<td class="org-right">1e-11</td>
|
||
<td class="org-right">6e-10</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">-5e-08</td>
|
||
<td class="org-right">3e-09</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">1e-07</td>
|
||
<td class="org-right">0.01</td>
|
||
<td class="org-right">-6e-08</td>
|
||
<td class="org-right">-6e-11</td>
|
||
<td class="org-right">-8e-12</td>
|
||
<td class="org-right">-1e-07</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">-1e-08</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-1e-10</td>
|
||
<td class="org-right">-3e-05</td>
|
||
<td class="org-right">-6e-08</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">7e-14</td>
|
||
<td class="org-right">6e-13</td>
|
||
<td class="org-right">1e-10</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">3e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">3e-05</td>
|
||
<td class="org-right">1e-11</td>
|
||
<td class="org-right">-6e-11</td>
|
||
<td class="org-right">7e-14</td>
|
||
<td class="org-right">2e-07</td>
|
||
<td class="org-right">1e-10</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">-7e-12</td>
|
||
<td class="org-right">6e-11</td>
|
||
<td class="org-right">-6e-16</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">6e-10</td>
|
||
<td class="org-right">-8e-12</td>
|
||
<td class="org-right">6e-13</td>
|
||
<td class="org-right">1e-10</td>
|
||
<td class="org-right">5e-07</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">-5e-10</td>
|
||
<td class="org-right">-1e-11</td>
|
||
<td class="org-right">1e-13</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">-1e-07</td>
|
||
<td class="org-right">1e-10</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">1e-07</td>
|
||
<td class="org-right">-4e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-3e-09</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">1e-08</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">-7e-12</td>
|
||
<td class="org-right">-5e-10</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">0.006</td>
|
||
<td class="org-right">-8e-08</td>
|
||
<td class="org-right">3e-05</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">9e-09</td>
|
||
<td class="org-right">-5e-08</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">6e-11</td>
|
||
<td class="org-right">-1e-11</td>
|
||
<td class="org-right">1e-07</td>
|
||
<td class="org-right">-8e-08</td>
|
||
<td class="org-right">0.01</td>
|
||
<td class="org-right">-6e-08</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2e-12</td>
|
||
<td class="org-right">3e-09</td>
|
||
<td class="org-right">-1e-08</td>
|
||
<td class="org-right">3e-10</td>
|
||
<td class="org-right">-6e-16</td>
|
||
<td class="org-right">1e-13</td>
|
||
<td class="org-right">-4e-12</td>
|
||
<td class="org-right">3e-05</td>
|
||
<td class="org-right">-6e-08</td>
|
||
<td class="org-right">2e-07</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org7609951" class="outline-3">
|
||
<h3 id="org7609951"><span class="section-number-3">4.3</span> Flexible Joint Characteristics</h3>
|
||
<div class="outline-text-3" id="text-4-3">
|
||
<p>
|
||
The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
|
||
</p>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"><b>Caracteristic</b></th>
|
||
<th scope="col" class="org-right"><b>Value</b></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Axial Stiffness [N/um]</td>
|
||
<td class="org-right">94.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Shear Stiffness [N/um]</td>
|
||
<td class="org-right">12.7</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness [Nm/rad]</td>
|
||
<td class="org-right">4.8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness [Nm/rad]</td>
|
||
<td class="org-right">4.8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Torsion Stiffness [Nm/rad]</td>
|
||
<td class="org-right">260.2</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org8bf4f56" class="outline-3">
|
||
<h3 id="org8bf4f56"><span class="section-number-3">4.4</span> Identification of the parameters using Simscape</h3>
|
||
<div class="outline-text-3" id="text-4-4">
|
||
<p>
|
||
The flexor is now imported into Simscape and its parameters are estimated using an identification.
|
||
</p>
|
||
|
||
<p>
|
||
The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
|
||
And we find the same parameters as the one estimated from the Stiffness matrix.
|
||
</p>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-left"><b>Caracteristic</b></th>
|
||
<th scope="col" class="org-right"><b>Value</b></th>
|
||
<th scope="col" class="org-right"><b>Identification</b></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Axial Stiffness Dz [N/um]</td>
|
||
<td class="org-right">94.0</td>
|
||
<td class="org-right">93.9</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness Rx [Nm/rad]</td>
|
||
<td class="org-right">4.8</td>
|
||
<td class="org-right">4.8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Bending Stiffness Ry [Nm/rad]</td>
|
||
<td class="org-right">4.8</td>
|
||
<td class="org-right">4.8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Torsion Stiffness Rz [Nm/rad]</td>
|
||
<td class="org-right">260.2</td>
|
||
<td class="org-right">260.2</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd8cb8ff" class="outline-3">
|
||
<h3 id="orgd8cb8ff"><span class="section-number-3">4.5</span> Simpler Model</h3>
|
||
<div class="outline-text-3" id="text-4-5">
|
||
<p>
|
||
Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure <a href="#org9c5b090">18</a>.
|
||
</p>
|
||
|
||
|
||
<div id="orgc5e5982" class="figure">
|
||
<p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 21: </span>Bushing Joint used to model the flexible joint</p>
|
||
</div>
|
||
|
||
<p>
|
||
The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kx = K(1,1); <span class="org-comment">% [N/m]</span>
|
||
Ky = K(2,2); <span class="org-comment">% [N/m]</span>
|
||
Kz = K(3,3); <span class="org-comment">% [N/m]</span>
|
||
Krx = K(4,4); <span class="org-comment">% [Nm/rad]</span>
|
||
Kry = K(5,5); <span class="org-comment">% [Nm/rad]</span>
|
||
Krz = K(6,6); <span class="org-comment">% [Nm/rad]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
|
||
The two obtained dynamics are compared in Figure
|
||
</p>
|
||
|
||
|
||
<div id="org2936555" class="figure">
|
||
<p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 22: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org7f2d76d" class="outline-2">
|
||
<h2 id="org7f2d76d"><span class="section-number-2">5</span> Integral Force Feedback with Amplified Piezo</h2>
|
||
<div class="outline-text-2" id="text-5">
|
||
<p>
|
||
In this section, we try to replicate the results obtained in (<a href="#citeproc_bib_item_2">Souleille et al. 2018</a>).
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd9dc7be" class="outline-3">
|
||
<h3 id="orgd9dc7be"><span class="section-number-3">5.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
|
||
<div class="outline-text-3" id="text-5-1">
|
||
<p>
|
||
We first extract the stiffness and mass matrices.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = extractMatrix(<span class="org-string">'piezo_amplified_IFF_K.txt'</span>);
|
||
M = extractMatrix(<span class="org-string">'piezo_amplified_IFF_M.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we extract the coordinates of the interface nodes.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'piezo_amplified_IFF.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3671fca" class="outline-3">
|
||
<h3 id="org3671fca"><span class="section-number-3">5.2</span> IFF Plant</h3>
|
||
<div class="outline-text-3" id="text-5-2">
|
||
<p>
|
||
The transfer function from the force actuator to the force sensor is identified and shown in Figure <a href="#org294cdfb">23</a>.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kiff = tf(0);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 0;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_IFF'</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">'/Kiff'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/G'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Gf = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 10;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Gfm = linearize(mdl, io);
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org294cdfb" class="figure">
|
||
<p><img src="figs/piezo_amplified_iff_plant.png" alt="piezo_amplified_iff_plant.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 23: </span>IFF Plant</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1cbfb66" class="outline-3">
|
||
<h3 id="org1cbfb66"><span class="section-number-3">5.3</span> IFF controller</h3>
|
||
<div class="outline-text-3" id="text-5-3">
|
||
<p>
|
||
The controller is defined and the loop gain is shown in Figure <a href="#orgbcc66e9">24</a>.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kiff = <span class="org-type">-</span>1e12<span class="org-type">/</span>s;
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orgbcc66e9" class="figure">
|
||
<p><img src="figs/piezo_amplified_iff_loop_gain.png" alt="piezo_amplified_iff_loop_gain.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 24: </span>IFF Loop Gain</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org7b29313" class="outline-3">
|
||
<h3 id="org7b29313"><span class="section-number-3">5.4</span> Closed Loop System</h3>
|
||
<div class="outline-text-3" id="text-5-4">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 10;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kiff = <span class="org-type">-</span>1e12<span class="org-type">/</span>s;
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||
mdl = <span class="org-string">'piezo_amplified_IFF'</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">'/Dw'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/Fd'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/d'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
io(io_i) = linio([mdl, <span class="org-string">'/G'</span>], 1, <span class="org-string">'output'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||
|
||
Giff = linearize(mdl, io);
|
||
Giff.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'f'</span>, <span class="org-string">'F'</span>};
|
||
Giff.OutputName = {<span class="org-string">'x1'</span>, <span class="org-string">'Fs'</span>};
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kiff = tf(0);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">G = linearize(mdl, io);
|
||
G.InputName = {<span class="org-string">'w'</span>, <span class="org-string">'f'</span>, <span class="org-string">'F'</span>};
|
||
G.OutputName = {<span class="org-string">'x1'</span>, <span class="org-string">'Fs'</span>};
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org9317d23" class="figure">
|
||
<p><img src="figs/piezo_amplified_iff_comp.png" alt="piezo_amplified_iff_comp.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 25: </span>OL and CL transfer functions</p>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="org7f216a9" class="figure">
|
||
<p><img src="figs/souleille18_results.png" alt="souleille18_results.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 26: </span>Results obtained in <a class='org-ref-reference' href="#souleille18_concep_activ_mount_space_applic">souleille18_concep_activ_mount_space_applic</a></p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1272d3f" class="outline-2">
|
||
<h2 id="org1272d3f"><span class="section-number-2">6</span> Complete Strut with Encoder</h2>
|
||
<div class="outline-text-2" id="text-6">
|
||
</div>
|
||
<div id="outline-container-orgddf8d43" class="outline-3">
|
||
<h3 id="orgddf8d43"><span class="section-number-3">6.1</span> Introduction</h3>
|
||
<div class="outline-text-3" id="text-6-1">
|
||
|
||
<div id="org1af2e05" class="figure">
|
||
<p><img src="data/strut_encoder/points3.jpg" alt="points3.jpg" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 27: </span>Interface points</p>
|
||
</div>
|
||
|
||
<p>
|
||
Flexible joints have 0.25mm width.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org4742c38" class="outline-3">
|
||
<h3 id="org4742c38"><span class="section-number-3">6.2</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
|
||
<div class="outline-text-3" id="text-6-2">
|
||
<p>
|
||
We first extract the stiffness and mass matrices.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = readmatrix(<span class="org-string">'mat_K.CSV'</span>);
|
||
M = readmatrix(<span class="org-string">'mat_M.CSV'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Then, we extract the coordinates of the interface nodes.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'out_nodes_3D.txt'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">save(<span class="org-string">'./mat/strut_encoder.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org332b172" class="outline-3">
|
||
<h3 id="org332b172"><span class="section-number-3">6.3</span> Output parameters</h3>
|
||
<div class="outline-text-3" id="text-6-3">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">load(<span class="org-string">'./mat/strut_encoder.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
|
||
</pre>
|
||
</div>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
||
|
||
<colgroup>
|
||
<col class="org-left" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-left">Total number of Nodes</td>
|
||
<td class="org-right">8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of interface Nodes</td>
|
||
<td class="org-right">8</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Number of Modes</td>
|
||
<td class="org-right">6</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-left">Size of M and K matrices</td>
|
||
<td class="org-right">54</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 15:</span> Coordinates of the interface nodes</caption>
|
||
|
||
<colgroup>
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr>
|
||
<th scope="col" class="org-right">Node i</th>
|
||
<th scope="col" class="org-right">Node Number</th>
|
||
<th scope="col" class="org-right">x [m]</th>
|
||
<th scope="col" class="org-right">y [m]</th>
|
||
<th scope="col" class="org-right">z [m]</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">1.0</td>
|
||
<td class="org-right">504411.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0405</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2.0</td>
|
||
<td class="org-right">504412.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">-0.0405</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">3.0</td>
|
||
<td class="org-right">504413.0</td>
|
||
<td class="org-right">-0.0325</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">4.0</td>
|
||
<td class="org-right">504414.0</td>
|
||
<td class="org-right">-0.0125</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">5.0</td>
|
||
<td class="org-right">504415.0</td>
|
||
<td class="org-right">-0.0075</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">6.0</td>
|
||
<td class="org-right">504416.0</td>
|
||
<td class="org-right">0.0325</td>
|
||
<td class="org-right">0.0</td>
|
||
<td class="org-right">0.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">7.0</td>
|
||
<td class="org-right">504417.0</td>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">0.0145</td>
|
||
<td class="org-right">-0.00175</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">8.0</td>
|
||
<td class="org-right">504418.0</td>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">0.0166</td>
|
||
<td class="org-right">-0.00175</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 16:</span> First 10x10 elements of the 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">2000000.0</td>
|
||
<td class="org-right">1000000.0</td>
|
||
<td class="org-right">-3000000.0</td>
|
||
<td class="org-right">-400.0</td>
|
||
<td class="org-right">300.0</td>
|
||
<td class="org-right">200.0</td>
|
||
<td class="org-right">-30.0</td>
|
||
<td class="org-right">2000.0</td>
|
||
<td class="org-right">-10000.0</td>
|
||
<td class="org-right">0.3</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">1000000.0</td>
|
||
<td class="org-right">4000000.0</td>
|
||
<td class="org-right">-8000000.0</td>
|
||
<td class="org-right">-900.0</td>
|
||
<td class="org-right">400.0</td>
|
||
<td class="org-right">-50.0</td>
|
||
<td class="org-right">-6000.0</td>
|
||
<td class="org-right">10000.0</td>
|
||
<td class="org-right">-20000.0</td>
|
||
<td class="org-right">3</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-3000000.0</td>
|
||
<td class="org-right">-8000000.0</td>
|
||
<td class="org-right">20000000.0</td>
|
||
<td class="org-right">2000.0</td>
|
||
<td class="org-right">-900.0</td>
|
||
<td class="org-right">200.0</td>
|
||
<td class="org-right">-10000.0</td>
|
||
<td class="org-right">20000.0</td>
|
||
<td class="org-right">-300000.0</td>
|
||
<td class="org-right">7</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-400.0</td>
|
||
<td class="org-right">-900.0</td>
|
||
<td class="org-right">2000.0</td>
|
||
<td class="org-right">5</td>
|
||
<td class="org-right">-0.1</td>
|
||
<td class="org-right">0.05</td>
|
||
<td class="org-right">1</td>
|
||
<td class="org-right">-3</td>
|
||
<td class="org-right">6</td>
|
||
<td class="org-right">-0.0007</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">300.0</td>
|
||
<td class="org-right">400.0</td>
|
||
<td class="org-right">-900.0</td>
|
||
<td class="org-right">-0.1</td>
|
||
<td class="org-right">5</td>
|
||
<td class="org-right">0.04</td>
|
||
<td class="org-right">-0.1</td>
|
||
<td class="org-right">0.5</td>
|
||
<td class="org-right">-3</td>
|
||
<td class="org-right">0.0001</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">200.0</td>
|
||
<td class="org-right">-50.0</td>
|
||
<td class="org-right">200.0</td>
|
||
<td class="org-right">0.05</td>
|
||
<td class="org-right">0.04</td>
|
||
<td class="org-right">300.0</td>
|
||
<td class="org-right">4</td>
|
||
<td class="org-right">-0.01</td>
|
||
<td class="org-right">-1</td>
|
||
<td class="org-right">3e-05</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-30.0</td>
|
||
<td class="org-right">-6000.0</td>
|
||
<td class="org-right">-10000.0</td>
|
||
<td class="org-right">1</td>
|
||
<td class="org-right">-0.1</td>
|
||
<td class="org-right">4</td>
|
||
<td class="org-right">3000000.0</td>
|
||
<td class="org-right">-1000000.0</td>
|
||
<td class="org-right">-2000000.0</td>
|
||
<td class="org-right">-300.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2000.0</td>
|
||
<td class="org-right">10000.0</td>
|
||
<td class="org-right">20000.0</td>
|
||
<td class="org-right">-3</td>
|
||
<td class="org-right">0.5</td>
|
||
<td class="org-right">-0.01</td>
|
||
<td class="org-right">-1000000.0</td>
|
||
<td class="org-right">6000000.0</td>
|
||
<td class="org-right">7000000.0</td>
|
||
<td class="org-right">1000.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-10000.0</td>
|
||
<td class="org-right">-20000.0</td>
|
||
<td class="org-right">-300000.0</td>
|
||
<td class="org-right">6</td>
|
||
<td class="org-right">-3</td>
|
||
<td class="org-right">-1</td>
|
||
<td class="org-right">-2000000.0</td>
|
||
<td class="org-right">7000000.0</td>
|
||
<td class="org-right">20000000.0</td>
|
||
<td class="org-right">2000.0</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.3</td>
|
||
<td class="org-right">3</td>
|
||
<td class="org-right">7</td>
|
||
<td class="org-right">-0.0007</td>
|
||
<td class="org-right">0.0001</td>
|
||
<td class="org-right">3e-05</td>
|
||
<td class="org-right">-300.0</td>
|
||
<td class="org-right">1000.0</td>
|
||
<td class="org-right">2000.0</td>
|
||
<td class="org-right">5</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
<caption class="t-above"><span class="table-number">Table 17:</span> First 10x10 elements of the Mass 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" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
|
||
<col class="org-right" />
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td class="org-right">0.04</td>
|
||
<td class="org-right">-0.005</td>
|
||
<td class="org-right">0.007</td>
|
||
<td class="org-right">2e-06</td>
|
||
<td class="org-right">0.0001</td>
|
||
<td class="org-right">-5e-07</td>
|
||
<td class="org-right">-1e-05</td>
|
||
<td class="org-right">-9e-07</td>
|
||
<td class="org-right">8e-05</td>
|
||
<td class="org-right">-5e-10</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-0.005</td>
|
||
<td class="org-right">0.03</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-0.0001</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">3e-05</td>
|
||
<td class="org-right">-0.0001</td>
|
||
<td class="org-right">8e-05</td>
|
||
<td class="org-right">-3e-08</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.007</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">0.08</td>
|
||
<td class="org-right">-6e-06</td>
|
||
<td class="org-right">-5e-06</td>
|
||
<td class="org-right">-7e-07</td>
|
||
<td class="org-right">4e-05</td>
|
||
<td class="org-right">-0.0001</td>
|
||
<td class="org-right">0.0005</td>
|
||
<td class="org-right">-3e-08</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">2e-06</td>
|
||
<td class="org-right">-0.0001</td>
|
||
<td class="org-right">-6e-06</td>
|
||
<td class="org-right">2e-06</td>
|
||
<td class="org-right">-4e-10</td>
|
||
<td class="org-right">2e-11</td>
|
||
<td class="org-right">-8e-09</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">6e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">0.0001</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">-5e-06</td>
|
||
<td class="org-right">-4e-10</td>
|
||
<td class="org-right">3e-06</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">-3e-09</td>
|
||
<td class="org-right">3e-09</td>
|
||
<td class="org-right">-7e-09</td>
|
||
<td class="org-right">6e-13</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-5e-07</td>
|
||
<td class="org-right">-3e-07</td>
|
||
<td class="org-right">-7e-07</td>
|
||
<td class="org-right">2e-11</td>
|
||
<td class="org-right">2e-10</td>
|
||
<td class="org-right">5e-07</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">5e-09</td>
|
||
<td class="org-right">-5e-09</td>
|
||
<td class="org-right">1e-12</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-1e-05</td>
|
||
<td class="org-right">3e-05</td>
|
||
<td class="org-right">4e-05</td>
|
||
<td class="org-right">-8e-09</td>
|
||
<td class="org-right">-3e-09</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">0.04</td>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">1e-06</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-9e-07</td>
|
||
<td class="org-right">-0.0001</td>
|
||
<td class="org-right">-0.0001</td>
|
||
<td class="org-right">3e-08</td>
|
||
<td class="org-right">3e-09</td>
|
||
<td class="org-right">5e-09</td>
|
||
<td class="org-right">0.004</td>
|
||
<td class="org-right">0.02</td>
|
||
<td class="org-right">-0.02</td>
|
||
<td class="org-right">0.0001</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">8e-05</td>
|
||
<td class="org-right">8e-05</td>
|
||
<td class="org-right">0.0005</td>
|
||
<td class="org-right">-2e-08</td>
|
||
<td class="org-right">-7e-09</td>
|
||
<td class="org-right">-5e-09</td>
|
||
<td class="org-right">0.003</td>
|
||
<td class="org-right">-0.02</td>
|
||
<td class="org-right">0.08</td>
|
||
<td class="org-right">-5e-06</td>
|
||
</tr>
|
||
|
||
<tr>
|
||
<td class="org-right">-5e-10</td>
|
||
<td class="org-right">-3e-08</td>
|
||
<td class="org-right">-3e-08</td>
|
||
<td class="org-right">6e-12</td>
|
||
<td class="org-right">6e-13</td>
|
||
<td class="org-right">1e-12</td>
|
||
<td class="org-right">1e-06</td>
|
||
<td class="org-right">0.0001</td>
|
||
<td class="org-right">-5e-06</td>
|
||
<td class="org-right">2e-06</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
<p>
|
||
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
|
||
<div id="outline-container-orgadca4a4" class="outline-3">
|
||
<h3 id="orgadca4a4"><span class="section-number-3">6.4</span> Piezoelectric parameters</h3>
|
||
<div class="outline-text-3" id="text-6-4">
|
||
<p>
|
||
Parameters for the APA300ML:
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">d33 = 3e<span class="org-type">-</span>10; <span class="org-comment">% Strain constant [m/V]</span>
|
||
n = 80; <span class="org-comment">% Number of layers per stack</span>
|
||
eT = 1.6e<span class="org-type">-</span>8; <span class="org-comment">% Permittivity under constant stress [F/m]</span>
|
||
sD = 2e<span class="org-type">-</span>11; <span class="org-comment">% Elastic compliance under constant electric displacement [m2/N]</span>
|
||
ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
|
||
C = 5e<span class="org-type">-</span>6; <span class="org-comment">% Stack capactiance [F]</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">na = 2; <span class="org-comment">% Number of stacks used as actuator</span>
|
||
ns = 1; <span class="org-comment">% Number of stacks used as force sensor</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org226d3f3" class="outline-3">
|
||
<h3 id="org226d3f3"><span class="section-number-3">6.5</span> Identification of the Dynamics</h3>
|
||
<div class="outline-text-3" id="text-6-5">
|
||
<p>
|
||
The dynamics is identified from the applied force to the measured relative displacement.
|
||
The same dynamics is identified for a payload mass of 10Kg.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 10;
|
||
</pre>
|
||
</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>Fleming, Andrew J., and Kam K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. <a href="https://doi.org/10.1007/978-3-319-06617-2">https://doi.org/10.1007/978-3-319-06617-2</a>.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_2"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>CEAS Space Journal</i> 10 (2). Springer:157–65.</div>
|
||
</div>
|
||
</div>
|
||
<div id="postamble" class="status">
|
||
<p class="author">Author: Dehaeze Thomas</p>
|
||
<p class="date">Created: 2020-11-12 jeu. 10:34</p>
|
||
</div>
|
||
</body>
|
||
</html>
|