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774 lines
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<!-- 2021-06-10 jeu. 17:52 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Nano-Hexapod - Test Bench</title>
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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">Nano-Hexapod - Test Bench</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="#org8cdd071">1. Encoders fixed to the Struts</a>
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<ul>
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<li><a href="#org3d72b93">1.1. Introduction</a></li>
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<li><a href="#orgcb16db3">1.2. Identification of the dynamics</a>
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<ul>
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<li><a href="#org7810966">1.2.1. Load Data</a></li>
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<li><a href="#orge39bc92">1.2.2. Spectral Analysis - Setup</a></li>
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<li><a href="#orge971c61">1.2.3. DVF Plant</a></li>
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<li><a href="#orgfe174fa">1.2.4. IFF Plant</a></li>
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</ul>
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</li>
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<li><a href="#orgaa25438">1.3. Comparison with the Simscape Model</a>
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<ul>
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<li><a href="#org8c6e670">1.3.1. Dynamics from Actuator to Force Sensors</a></li>
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<li><a href="#orgb40c194">1.3.2. Dynamics from Actuator to Encoder</a></li>
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</ul>
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</li>
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<li><a href="#org07b83f4">1.4. Integral Force Feedback</a>
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<ul>
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<li><a href="#org32df075">1.4.1. Root Locus and Decentralized Loop gain</a></li>
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<li><a href="#orge5e9cea">1.4.2. Multiple Gains - Simulation</a></li>
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<li><a href="#orga581915">1.4.3. Experimental Results - Gains</a>
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<ul>
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<li><a href="#org7089ea1">1.4.3.1. Load Data</a></li>
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<li><a href="#org54efadb">1.4.3.2. Spectral Analysis - Setup</a></li>
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<li><a href="#org5262511">1.4.3.3. DVF Plant</a></li>
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<li><a href="#org47db15b">1.4.3.4. Experimental Results - Comparison of the un-damped and fully damped system</a></li>
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</ul>
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</li>
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<li><a href="#orgd48530e">1.4.4. Experimental Results - Damped Plant with Optimal gain</a>
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<ul>
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<li><a href="#orgf0af7da">1.4.4.1. Load Data</a></li>
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<li><a href="#org1ce1b15">1.4.4.2. Spectral Analysis - Setup</a></li>
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<li><a href="#orgac97ace">1.4.4.3. DVF Plant</a></li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orgba38b08">2. Encoders fixed to the plates</a></li>
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</ul>
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</div>
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</div>
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<hr>
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<p>This report is also available as a <a href="./test-bench-nano-hexapod.pdf">pdf</a>.</p>
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<hr>
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<p>
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In this document, the dynamics of the nano-hexapod shown in Figure <a href="#org67868f6">1</a> is identified.
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</p>
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<div class="note" id="org61a7813">
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<p>
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Here are the documentation of the equipment used for this test bench:
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</p>
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<ul class="org-ul">
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<li>Voltage Amplifier: PiezoDrive <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
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<li>Amplified Piezoelectric Actuator: Cedrat <a href="doc/APA300ML.pdf">APA300ML</a></li>
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<li>DAC/ADC: Speedgoat <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
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<li>Encoder: Renishaw <a href="doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf">Vionic</a> and used <a href="doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf">Ruler</a></li>
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<li>Interferometers: Attocube</li>
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</ul>
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</div>
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<div id="org67868f6" class="figure">
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<p><img src="figs/IMG_20210608_152917.jpg" alt="IMG_20210608_152917.jpg" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Nano-Hexapod</p>
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</div>
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<div id="org492c735" class="figure">
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<p><img src="figs/IMG_20210608_154722.jpg" alt="IMG_20210608_154722.jpg" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Nano-Hexapod and the control electronics</p>
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</div>
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<div id="orgab57ea9" class="figure">
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<p><img src="figs/nano_hexapod_signals.png" alt="nano_hexapod_signals.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Block diagram of the system with named signals</p>
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</div>
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<table id="org3c0425e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> List of signals</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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<col class="org-left" />
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<col class="org-left" />
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<col class="org-left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left"> </th>
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<th scope="col" class="org-left"><b>Unit</b></th>
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<th scope="col" class="org-left"><b>Matlab</b></th>
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<th scope="col" class="org-left"><b>Vector</b></th>
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<th scope="col" class="org-left"><b>Elements</b></th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">Control Input (wanted DAC voltage)</td>
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<td class="org-left"><code>[V]</code></td>
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<td class="org-left"><code>u</code></td>
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<td class="org-left">\(\bm{u}\)</td>
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<td class="org-left">\(u_i\)</td>
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</tr>
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<tr>
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<td class="org-left">DAC Output Voltage</td>
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<td class="org-left"><code>[V]</code></td>
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<td class="org-left"><code>u</code></td>
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<td class="org-left">\(\tilde{\bm{u}}\)</td>
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<td class="org-left">\(\tilde{u}_i\)</td>
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</tr>
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<tr>
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<td class="org-left">PD200 Output Voltage</td>
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<td class="org-left"><code>[V]</code></td>
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<td class="org-left"><code>ua</code></td>
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<td class="org-left">\(\bm{u}_a\)</td>
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<td class="org-left">\(u_{a,i}\)</td>
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</tr>
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<tr>
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<td class="org-left">Actuator applied force</td>
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<td class="org-left"><code>[N]</code></td>
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<td class="org-left"><code>tau</code></td>
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<td class="org-left">\(\bm{\tau}\)</td>
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<td class="org-left">\(\tau_i\)</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left">Strut motion</td>
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<td class="org-left"><code>[m]</code></td>
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<td class="org-left"><code>dL</code></td>
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<td class="org-left">\(d\bm{\mathcal{L}}\)</td>
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<td class="org-left">\(d\mathcal{L}_i\)</td>
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</tr>
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<tr>
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<td class="org-left">Encoder measured displacement</td>
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<td class="org-left"><code>[m]</code></td>
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<td class="org-left"><code>dLm</code></td>
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<td class="org-left">\(d\bm{\mathcal{L}}_m\)</td>
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<td class="org-left">\(d\mathcal{L}_{m,i}\)</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left">Force Sensor strain</td>
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<td class="org-left"><code>[m]</code></td>
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<td class="org-left"><code>epsilon</code></td>
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<td class="org-left">\(\bm{\epsilon}\)</td>
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<td class="org-left">\(\epsilon_i\)</td>
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</tr>
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<tr>
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<td class="org-left">Force Sensor Generated Voltage</td>
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<td class="org-left"><code>[V]</code></td>
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<td class="org-left"><code>taum</code></td>
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<td class="org-left">\(\tilde{\bm{\tau}}_m\)</td>
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<td class="org-left">\(\tilde{\tau}_{m,i}\)</td>
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</tr>
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<tr>
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<td class="org-left">Measured Generated Voltage</td>
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<td class="org-left"><code>[V]</code></td>
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<td class="org-left"><code>taum</code></td>
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<td class="org-left">\(\bm{\tau}_m\)</td>
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<td class="org-left">\(\tau_{m,i}\)</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left">Motion of the top platform</td>
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<td class="org-left"><code>[m,rad]</code></td>
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<td class="org-left"><code>dX</code></td>
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<td class="org-left">\(d\bm{\mathcal{X}}\)</td>
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<td class="org-left">\(d\mathcal{X}_i\)</td>
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</tr>
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<tr>
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<td class="org-left">Metrology measured displacement</td>
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<td class="org-left"><code>[m,rad]</code></td>
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<td class="org-left"><code>dXm</code></td>
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<td class="org-left">\(d\bm{\mathcal{X}}_m\)</td>
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<td class="org-left">\(d\mathcal{X}_{m,i}\)</td>
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</tr>
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</tbody>
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</table>
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<div id="outline-container-org8cdd071" class="outline-2">
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<h2 id="org8cdd071"><span class="section-number-2">1</span> Encoders fixed to the Struts</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org3d72b93" class="outline-3">
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<h3 id="org3d72b93"><span class="section-number-3">1.1</span> Introduction</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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In this section, the encoders are fixed to the struts.
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</p>
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</div>
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</div>
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<div id="outline-container-orgcb16db3" class="outline-3">
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<h3 id="orgcb16db3"><span class="section-number-3">1.2</span> Identification of the dynamics</h3>
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<div class="outline-text-3" id="text-1-2">
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</div>
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<div id="outline-container-org7810966" class="outline-4">
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<h4 id="org7810966"><span class="section-number-4">1.2.1</span> Load Data</h4>
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<div class="outline-text-4" id="text-1-2-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load Identification Data</span></span>
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meas_data_lf = {};
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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meas_data_lf(<span class="org-constant">i</span>) = {load(sprintf(<span class="org-string">'mat/frf_data_exc_strut_%i_noise_lf.mat'</span>, <span class="org-constant">i</span>), <span class="org-string">'t'</span>, <span class="org-string">'Va'</span>, <span class="org-string">'Vs'</span>, <span class="org-string">'de'</span>)};
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meas_data_hf(<span class="org-constant">i</span>) = {load(sprintf(<span class="org-string">'mat/frf_data_exc_strut_%i_noise_hf.mat'</span>, <span class="org-constant">i</span>), <span class="org-string">'t'</span>, <span class="org-string">'Va'</span>, <span class="org-string">'Vs'</span>, <span class="org-string">'de'</span>)};
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<span class="org-keyword">end</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orge39bc92" class="outline-4">
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<h4 id="orge39bc92"><span class="section-number-4">1.2.2</span> Spectral Analysis - Setup</h4>
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<div class="outline-text-4" id="text-1-2-2">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Setup useful variables</span></span>
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<span class="org-comment">% Sampling Time [s]</span>
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Ts = (meas_data_lf{1}.t(end) <span class="org-type">-</span> (meas_data_lf{1}.t(1)))<span class="org-type">/</span>(length(meas_data_lf{1}.t)<span class="org-type">-</span>1);
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<span class="org-comment">% Sampling Frequency [Hz]</span>
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Fs = 1<span class="org-type">/</span>Ts;
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<span class="org-comment">% Hannning Windows</span>
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win = hanning(ceil(1<span class="org-type">*</span>Fs));
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<span class="org-comment">% And we get the frequency vector</span>
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[<span class="org-type">~</span>, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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i_lf = f <span class="org-type"><</span> 250; <span class="org-comment">% Points for low frequency excitation</span>
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i_hf = f <span class="org-type">></span> 250; <span class="org-comment">% Points for high frequency excitation</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orge971c61" class="outline-4">
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<h4 id="orge971c61"><span class="section-number-4">1.2.3</span> DVF Plant</h4>
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<div class="outline-text-4" id="text-1-2-3">
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<p>
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First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#orgc8a5209">4</a>).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Coherence</span></span>
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coh_dvf_lf = zeros(length(f), 6, 6);
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coh_dvf_hf = zeros(length(f), 6, 6);
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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coh_dvf_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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coh_dvf_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="orgc8a5209" class="figure">
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<p><img src="figs/enc_struts_dvf_coh.png" alt="enc_struts_dvf_coh.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Obtained coherence for the DVF plant</p>
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</div>
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<p>
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Then the 6x6 transfer function matrix is estimated (Figure <a href="#orgb9f3fd5">5</a>).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% DVF Plant (transfer function from u to dLm)</span></span>
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G_dvf_lf = zeros(length(f), 6, 6);
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G_dvf_hf = zeros(length(f), 6, 6);
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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G_dvf_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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G_dvf_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="orgb9f3fd5" class="figure">
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<p><img src="figs/enc_struts_dvf_frf.png" alt="enc_struts_dvf_frf.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Measured FRF for the DVF plant</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgfe174fa" class="outline-4">
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<h4 id="orgfe174fa"><span class="section-number-4">1.2.4</span> IFF Plant</h4>
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<div class="outline-text-4" id="text-1-2-4">
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<p>
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First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#orgb8bd5d5">6</a>).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Coherence for the IFF plant</span></span>
|
|
coh_iff_lf = zeros(length(f), 6, 6);
|
|
coh_iff_hf = zeros(length(f), 6, 6);
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
coh_iff_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
coh_iff_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
<span class="org-keyword">end</span>
|
|
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgb8bd5d5" class="figure">
|
|
<p><img src="figs/enc_struts_iff_coh.png" alt="enc_struts_iff_coh.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Obtained coherence for the IFF plant</p>
|
|
</div>
|
|
|
|
<p>
|
|
Then the 6x6 transfer function matrix is estimated (Figure <a href="#org7d9a20b">7</a>).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF Plant</span></span>
|
|
G_iff_lf = zeros(length(f), 6, 6);
|
|
G_iff_hf = zeros(length(f), 6, 6);
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
G_iff_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
G_iff_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org7d9a20b" class="figure">
|
|
<p><img src="figs/enc_struts_iff_frf.png" alt="enc_struts_iff_frf.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Measured FRF for the IFF plant</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaa25438" class="outline-3">
|
|
<h3 id="orgaa25438"><span class="section-number-3">1.3</span> Comparison with the Simscape Model</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<p>
|
|
In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org8c6e670" class="outline-4">
|
|
<h4 id="org8c6e670"><span class="section-number-4">1.3.1</span> Dynamics from Actuator to Force Sensors</h4>
|
|
<div class="outline-text-4" id="text-1-3-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Initialize Nano-Hexapod</span></span>
|
|
n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ...
|
|
<span class="org-string">'flex_top_type'</span>, <span class="org-string">'4dof'</span>, ...
|
|
<span class="org-string">'motion_sensor_type'</span>, <span class="org-string">'struts'</span>, ...
|
|
<span class="org-string">'actuator_type'</span>, <span class="org-string">'2dof'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Identify the IFF Plant (transfer function from u to taum)</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; <span class="org-comment">% Actuator Inputs</span>
|
|
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; <span class="org-comment">% Force Sensors</span>
|
|
|
|
Giff = exp(<span class="org-type">-</span>s<span class="org-type">*</span>Ts)<span class="org-type">*</span>linearize(mdl, io, 0.0, options);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgf514a0e" class="figure">
|
|
<p><img src="figs/enc_struts_iff_comp_simscape.png" alt="enc_struts_iff_comp_simscape.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Diagonal elements of the IFF Plant</p>
|
|
</div>
|
|
|
|
|
|
<div id="org5cb4798" class="figure">
|
|
<p><img src="figs/enc_struts_iff_comp_offdiag_simscape.png" alt="enc_struts_iff_comp_offdiag_simscape.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Off diagonal elements of the IFF Plant</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb40c194" class="outline-4">
|
|
<h4 id="orgb40c194"><span class="section-number-4">1.3.2</span> Dynamics from Actuator to Encoder</h4>
|
|
<div class="outline-text-4" id="text-1-3-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Initialization of the Nano-Hexapod</span></span>
|
|
n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ...
|
|
<span class="org-string">'flex_top_type'</span>, <span class="org-string">'4dof'</span>, ...
|
|
<span class="org-string">'motion_sensor_type'</span>, <span class="org-string">'struts'</span>, ...
|
|
<span class="org-string">'actuator_type'</span>, <span class="org-string">'2dof'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Identify the DVF Plant (transfer function from u to dLm)</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; <span class="org-comment">% Actuator Inputs</span>
|
|
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; <span class="org-comment">% Encoders</span>
|
|
|
|
Gdvf = exp(<span class="org-type">-</span>s<span class="org-type">*</span>Ts)<span class="org-type">*</span>linearize(mdl, io, 0.0, options);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org0adec37" class="figure">
|
|
<p><img src="figs/enc_struts_dvf_comp_simscape.png" alt="enc_struts_dvf_comp_simscape.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Diagonal elements of the DVF Plant</p>
|
|
</div>
|
|
|
|
|
|
<div id="org97225e9" class="figure">
|
|
<p><img src="figs/enc_struts_dvf_comp_offdiag_simscape.png" alt="enc_struts_dvf_comp_offdiag_simscape.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Off diagonal elements of the DVF Plant</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org07b83f4" class="outline-3">
|
|
<h3 id="org07b83f4"><span class="section-number-3">1.4</span> Integral Force Feedback</h3>
|
|
<div class="outline-text-3" id="text-1-4">
|
|
</div>
|
|
<div id="outline-container-org32df075" class="outline-4">
|
|
<h4 id="org32df075"><span class="section-number-4">1.4.1</span> Root Locus and Decentralized Loop gain</h4>
|
|
<div class="outline-text-4" id="text-1-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF Controller</span></span>
|
|
Kiff_g1 = (1<span class="org-type">/</span>(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>40))<span class="org-type">*</span>...<span class="org-comment"> % Low pass filter (provides integral action above 40Hz)</span>
|
|
(s<span class="org-type">/</span>(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30))<span class="org-type">*</span>...<span class="org-comment"> % High pass filter to limit low frequency gain</span>
|
|
(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>500))<span class="org-type">*</span>...<span class="org-comment"> % Low pass filter to be more robust to high frequency resonances</span>
|
|
eye(6); <span class="org-comment">% Diagonal 6x6 controller</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org605753e" class="figure">
|
|
<p><img src="figs/enc_struts_iff_root_locus.png" alt="enc_struts_iff_root_locus.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Root Locus for the IFF control strategy</p>
|
|
</div>
|
|
|
|
<p>
|
|
Then the “optimal” IFF controller is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF controller with Optimal gain</span></span>
|
|
Kiff = g<span class="org-type">*</span>Kiff_g1;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgc1b7b46" class="figure">
|
|
<p><img src="figs/enc_struts_iff_opt_loop_gain.png" alt="enc_struts_iff_opt_loop_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Bode plot of the “decentralized loop gain” \(G_\text{iff}(i,i) \times K_\text{iff}(i,i)\)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge5e9cea" class="outline-4">
|
|
<h4 id="orge5e9cea"><span class="section-number-4">1.4.2</span> Multiple Gains - Simulation</h4>
|
|
<div class="outline-text-4" id="text-1-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Tested IFF gains</span></span>
|
|
iff_gains = [4, 10, 20, 40, 100, 200, 400];
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Initialize the Simscape model in closed loop</span></span>
|
|
n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ...
|
|
<span class="org-string">'flex_top_type'</span>, <span class="org-string">'4dof'</span>, ...
|
|
<span class="org-string">'motion_sensor_type'</span>, <span class="org-string">'struts'</span>, ...
|
|
<span class="org-string">'actuator_type'</span>, <span class="org-string">'2dof'</span>, ...
|
|
<span class="org-string">'controller_type'</span>, <span class="org-string">'iff'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain</span></span>
|
|
Gd_iff = {zeros(1, length(iff_gains))};
|
|
|
|
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; <span class="org-comment">% Actuator Inputs</span>
|
|
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; <span class="org-comment">% Strut Displacement (encoder)</span>
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(iff_gains)</span>
|
|
Kiff = iff_gains(<span class="org-constant">i</span>)<span class="org-type">*</span>Kiff_g1<span class="org-type">*</span>eye(6); <span class="org-comment">% IFF Controller</span>
|
|
Gd_iff(<span class="org-constant">i</span>) = {exp(<span class="org-type">-</span>s<span class="org-type">*</span>Ts)<span class="org-type">*</span>linearize(mdl, io, 0.0, options)};
|
|
|
|
isstable(Gd_iff{<span class="org-constant">i</span>})
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgb361d62" class="figure">
|
|
<p><img src="figs/enc_struts_iff_gains_effect_dvf_plant.png" alt="enc_struts_iff_gains_effect_dvf_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Effect of the IFF gain \(g\) on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga581915" class="outline-4">
|
|
<h4 id="orga581915"><span class="section-number-4">1.4.3</span> Experimental Results - Gains</h4>
|
|
<div class="outline-text-4" id="text-1-4-3">
|
|
<p>
|
|
Let’s look at the damping introduced by IFF as a function of the IFF gain and compare that with the results obtained using the Simscape model.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org7089ea1" class="outline-5">
|
|
<h5 id="org7089ea1"><span class="section-number-5">1.4.3.1</span> Load Data</h5>
|
|
<div class="outline-text-5" id="text-1-4-3-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load Identification Data</span></span>
|
|
meas_iff_gains = {};
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(iff_gains)</span>
|
|
meas_iff_gains(<span class="org-constant">i</span>) = {load(sprintf(<span class="org-string">'mat/iff_strut_1_noise_g_%i.mat'</span>, iff_gains(<span class="org-constant">i</span>)), <span class="org-string">'t'</span>, <span class="org-string">'Vexc'</span>, <span class="org-string">'Vs'</span>, <span class="org-string">'de'</span>, <span class="org-string">'u'</span>)};
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org54efadb" class="outline-5">
|
|
<h5 id="org54efadb"><span class="section-number-5">1.4.3.2</span> Spectral Analysis - Setup</h5>
|
|
<div class="outline-text-5" id="text-1-4-3-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Setup useful variables</span></span>
|
|
<span class="org-comment">% Sampling Time [s]</span>
|
|
Ts = (meas_iff_gains{1}.t(end) <span class="org-type">-</span> (meas_iff_gains{1}.t(1)))<span class="org-type">/</span>(length(meas_iff_gains{1}.t)<span class="org-type">-</span>1);
|
|
|
|
<span class="org-comment">% Sampling Frequency [Hz]</span>
|
|
Fs = 1<span class="org-type">/</span>Ts;
|
|
|
|
<span class="org-comment">% Hannning Windows</span>
|
|
win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
|
|
|
<span class="org-comment">% And we get the frequency vector</span>
|
|
[<span class="org-type">~</span>, f] = tfestimate(meas_iff_gains{1}.Vexc, meas_iff_gains{1}.de, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5262511" class="outline-5">
|
|
<h5 id="org5262511"><span class="section-number-5">1.4.3.3</span> DVF Plant</h5>
|
|
<div class="outline-text-5" id="text-1-4-3-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% DVF Plant (transfer function from u to dLm)</span></span>
|
|
G_iff_gains = {};
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(iff_gains)</span>
|
|
G_iff_gains{<span class="org-constant">i</span>} = tfestimate(meas_iff_gains{<span class="org-constant">i</span>}.Vexc, meas_iff_gains{<span class="org-constant">i</span>}.de(<span class="org-type">:</span>,1), win, [], [], 1<span class="org-type">/</span>Ts);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgecf391c" class="figure">
|
|
<p><img src="figs/comp_iff_gains_dvf_plant.png" alt="comp_iff_gains_dvf_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgf79da15" class="figure">
|
|
<p><img src="figs/comp_iff_gains_dvf_plant_zoom.png" alt="comp_iff_gains_dvf_plant_zoom.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain (Zoom)</p>
|
|
</div>
|
|
|
|
<div class="important" id="orgbb44640">
|
|
<p>
|
|
The IFF control strategy is very effective for the damping of the suspension modes.
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It however does not damp the modes at 200Hz, 300Hz and 400Hz (flexible modes of the APA).
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This is very logical.
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</p>
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<p>
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Also, the experimental results and the models obtained from the Simscape model are in agreement.
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</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org47db15b" class="outline-5">
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<h5 id="org47db15b"><span class="section-number-5">1.4.3.4</span> Experimental Results - Comparison of the un-damped and fully damped system</h5>
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<div class="outline-text-5" id="text-1-4-3-4">
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<div id="org4f580a8" class="figure">
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<p><img src="figs/comp_undamped_opt_iff_gain_diagonal.png" alt="comp_undamped_opt_iff_gain_diagonal.png" />
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</p>
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<p><span class="figure-number">Figure 17: </span>Comparison of the diagonal elements of the tranfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) without active damping and with optimal IFF gain</p>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-orgd48530e" class="outline-4">
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<h4 id="orgd48530e"><span class="section-number-4">1.4.4</span> Experimental Results - Damped Plant with Optimal gain</h4>
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|
<div class="outline-text-4" id="text-1-4-4">
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<p>
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Let’s now look at the \(6 \times 6\) damped plant with the optimal gain \(g = 400\).
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</p>
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</div>
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<div id="outline-container-orgf0af7da" class="outline-5">
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<h5 id="orgf0af7da"><span class="section-number-5">1.4.4.1</span> Load Data</h5>
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|
<div class="outline-text-5" id="text-1-4-4-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load Identification Data</span></span>
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meas_iff_struts = {};
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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|
meas_iff_struts(<span class="org-constant">i</span>) = {load(sprintf(<span class="org-string">'mat/iff_strut_%i_noise_g_400.mat'</span>, <span class="org-constant">i</span>), <span class="org-string">'t'</span>, <span class="org-string">'Vexc'</span>, <span class="org-string">'Vs'</span>, <span class="org-string">'de'</span>, <span class="org-string">'u'</span>)};
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|
<span class="org-keyword">end</span>
|
|
</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org1ce1b15" class="outline-5">
|
|
<h5 id="org1ce1b15"><span class="section-number-5">1.4.4.2</span> Spectral Analysis - Setup</h5>
|
|
<div class="outline-text-5" id="text-1-4-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Setup useful variables</span></span>
|
|
<span class="org-comment">% Sampling Time [s]</span>
|
|
Ts = (meas_iff_struts{1}.t(end) <span class="org-type">-</span> (meas_iff_struts{1}.t(1)))<span class="org-type">/</span>(length(meas_iff_struts{1}.t)<span class="org-type">-</span>1);
|
|
|
|
<span class="org-comment">% Sampling Frequency [Hz]</span>
|
|
Fs = 1<span class="org-type">/</span>Ts;
|
|
|
|
<span class="org-comment">% Hannning Windows</span>
|
|
win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
|
|
|
<span class="org-comment">% And we get the frequency vector</span>
|
|
[<span class="org-type">~</span>, f] = tfestimate(meas_iff_struts{1}.Vexc, meas_iff_struts{1}.de, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgac97ace" class="outline-5">
|
|
<h5 id="orgac97ace"><span class="section-number-5">1.4.4.3</span> DVF Plant</h5>
|
|
<div class="outline-text-5" id="text-1-4-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% DVF Plant (transfer function from u to dLm)</span></span>
|
|
G_iff_opt = {};
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
G_iff_opt{<span class="org-constant">i</span>} = tfestimate(meas_iff_struts{<span class="org-constant">i</span>}.Vexc, meas_iff_struts{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org50d06b6" class="figure">
|
|
<p><img src="figs/damped_iff_plant_comp_diagonal.png" alt="damped_iff_plant_comp_diagonal.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Comparison of the diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgc9f6f9f" class="figure">
|
|
<p><img src="figs/damped_iff_plant_comp_off_diagonal.png" alt="damped_iff_plant_comp_off_diagonal.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Comparison of the off-diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)</p>
|
|
</div>
|
|
|
|
<div class="important" id="org067c4cc">
|
|
<p>
|
|
With the IFF control strategy applied and the optimal gain used, the suspension modes are very well dapmed.
|
|
Remains the undamped flexible modes of the APA, and the modes of the plates.
|
|
</p>
|
|
|
|
<p>
|
|
The Simscape model and the experimental results are in very good agreement.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgba38b08" class="outline-2">
|
|
<h2 id="orgba38b08"><span class="section-number-2">2</span> Encoders fixed to the plates</h2>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2021-06-10 jeu. 17:52</p>
|
|
</div>
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</body>
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</html>
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