252 lines
9.9 KiB
HTML
252 lines
9.9 KiB
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<!-- 2020-04-17 ven. 14:10 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Decentralize control to add virtual mass</title>
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<meta name="author" content="Dehaeze Thomas" />
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macros: {bm: ["\\boldsymbol{#1}",1],}
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="./index.html"> UP </a>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Decentralize control to add virtual mass</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="#org982b263">1. Initialization</a></li>
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<li><a href="#org35a3822">2. Identification</a>
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<ul>
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<li><a href="#org33f35d2">2.1. Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\)</a></li>
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<li><a href="#org6663ed2">2.2. Identification of the Primary plant without virtual add of mass</a></li>
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</ul>
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</li>
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<li><a href="#orgd6fc719">3. Adding Virtual Mass in the Leg’s Space</a>
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<ul>
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<li><a href="#orgc37faa7">3.1. Plant</a></li>
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<li><a href="#org4ae3263">3.2. Controller Design</a></li>
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<li><a href="#orgb270293">3.3. Identification of the Primary Plant</a></li>
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</ul>
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</li>
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<li><a href="#orgc9131d0">4. Adding Virtual Mass in the Task Space</a>
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<ul>
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<li><a href="#org9ed2d4c">4.1. Plant</a></li>
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<li><a href="#org4f03a34">4.2. Controller Design</a></li>
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<li><a href="#org2fe0ce0">4.3. Identification of the Primary Plant</a></li>
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</ul>
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</li>
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</ul>
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</div>
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<div id="outline-container-org982b263" class="outline-2">
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<h2 id="org982b263"><span class="section-number-2">1</span> Initialization</h2>
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<div class="outline-text-2" id="text-1">
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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initializeSimscapeConfiguration();
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initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
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initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
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initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
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</pre>
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</div>
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<p>
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We set the stiffness of the payload fixation:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Kp = 1e8; <span class="org-comment">% [N/m]</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org35a3822" class="outline-2">
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<h2 id="org35a3822"><span class="section-number-2">2</span> Identification</h2>
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<div class="outline-text-2" id="text-2">
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</div>
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<div id="outline-container-org33f35d2" class="outline-3">
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<h3 id="org33f35d2"><span class="section-number-3">2.1</span> Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\)</h3>
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<div class="outline-text-3" id="text-2-1">
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<div class="org-src-container">
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<pre class="src src-matlab">K = tf(zeros(6));
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Kdvf = tf(zeros(6));
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</pre>
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</div>
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<p>
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We identify the system for the following payload masses:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Ms = [1, 10, 50];
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</pre>
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</div>
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<p>
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The nano-hexapod has the following leg’s stiffness and damping.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'k'</span>, 1e5, <span class="org-string">'c'</span>, 2e2);
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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-org6663ed2" class="outline-3">
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<h3 id="org6663ed2"><span class="section-number-3">2.2</span> Identification of the Primary plant without virtual add of mass</h3>
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</div>
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</div>
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<div id="outline-container-orgd6fc719" class="outline-2">
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<h2 id="orgd6fc719"><span class="section-number-2">3</span> Adding Virtual Mass in the Leg’s Space</h2>
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<div class="outline-text-2" id="text-3">
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</div>
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<div id="outline-container-orgc37faa7" class="outline-3">
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<h3 id="orgc37faa7"><span class="section-number-3">3.1</span> Plant</h3>
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<div class="outline-text-3" id="text-3-1">
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<div id="org98e7ba8" class="figure">
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<p><img src="figs/virtual_mass_plant_L.png" alt="virtual_mass_plant_L.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Transfer function from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org4ae3263" class="outline-3">
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<h3 id="org4ae3263"><span class="section-number-3">3.2</span> Controller Design</h3>
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<div class="outline-text-3" id="text-3-2">
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<div class="org-src-container">
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<pre class="src src-matlab">Kdvf = 10<span class="org-type">*</span>s<span class="org-type">^</span>2<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>2<span class="org-type">*</span>eye(6);
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</pre>
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</div>
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<div id="orgccb3b9e" class="figure">
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<p><img src="figs/virtual_mass_loop_gain_L.png" alt="virtual_mass_loop_gain_L.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Loop Gain for the addition of virtual mass in the leg’s space</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgb270293" class="outline-3">
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<h3 id="orgb270293"><span class="section-number-3">3.3</span> Identification of the Primary Plant</h3>
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<div class="outline-text-3" id="text-3-3">
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<div id="orgd49505e" class="figure">
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<p><img src="figs/virtual_mass_L_primary_plant_X.png" alt="virtual_mass_L_primary_plant_X.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the leg’s space</p>
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</div>
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<div id="org2281744" class="figure">
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<p><img src="figs/virtual_mass_L_primary_plant_L.png" alt="virtual_mass_L_primary_plant_L.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the leg’s space</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-orgc9131d0" class="outline-2">
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<h2 id="orgc9131d0"><span class="section-number-2">4</span> Adding Virtual Mass in the Task Space</h2>
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<div class="outline-text-2" id="text-4">
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</div>
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<div id="outline-container-org9ed2d4c" class="outline-3">
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<h3 id="org9ed2d4c"><span class="section-number-3">4.1</span> Plant</h3>
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<div class="outline-text-3" id="text-4-1">
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<p>
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Let’s look at the transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\):
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\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \]
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</p>
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<div id="org6488b4c" class="figure">
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<p><img src="figs/virtual_mass_plant_X.png" alt="virtual_mass_plant_X.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Dynamics from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) used for virtual mass addition in the task space</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org4f03a34" class="outline-3">
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<h3 id="org4f03a34"><span class="section-number-3">4.2</span> Controller Design</h3>
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<div class="outline-text-3" id="text-4-2">
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<div class="org-src-container">
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<pre class="src src-matlab">KmX = (s<span class="org-type">^</span>2<span class="org-type">*</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>2<span class="org-type">*</span>diag([1 1 50 0 0 0]));
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</pre>
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</div>
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<div id="orgf411330" class="figure">
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<p><img src="figs/virtual_mass_loop_gain_X.png" alt="virtual_mass_loop_gain_X.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Loop gain for virtual mass addition in the task space</p>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">Kdvf = inv(nano_hexapod.J<span class="org-type">'</span>)<span class="org-type">*</span>KmX<span class="org-type">*</span>inv(nano_hexapod.J);
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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-org2fe0ce0" class="outline-3">
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<h3 id="org2fe0ce0"><span class="section-number-3">4.3</span> Identification of the Primary Plant</h3>
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<div class="outline-text-3" id="text-4-3">
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<div id="orge1df87b" class="figure">
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<p><img src="figs/virtual_mass_X_primary_plant_X.png" alt="virtual_mass_X_primary_plant_X.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the task space</p>
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</div>
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<div id="org647b748" class="figure">
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<p><img src="figs/virtual_mass_X_primary_plant_L.png" alt="virtual_mass_X_primary_plant_L.png" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the task space</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>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-04-17 ven. 14:10</p>
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</div>
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</html>
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