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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-11-06 ven. 12:22 --> <!-- 2020-11-06 ven. 15:06 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>SVD Control</title> <title>SVD Control</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -35,56 +35,56 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org40c86ca">1. Gravimeter - Simscape Model</a> <li><a href="#org588d944">1. Gravimeter - Simscape Model</a>
<ul> <ul>
<li><a href="#orgac27a65">1.1. Introduction</a></li> <li><a href="#org91ed3f1">1.1. Introduction</a></li>
<li><a href="#org991b9ad">1.2. Simscape Model - Parameters</a></li> <li><a href="#org2a3289b">1.2. Simscape Model - Parameters</a></li>
<li><a href="#org7417c14">1.3. System Identification - Without Gravity</a></li> <li><a href="#orge1533ee">1.3. System Identification - Without Gravity</a></li>
<li><a href="#org3ac74c3">1.4. System Identification - With Gravity</a></li> <li><a href="#orgbcef719">1.4. System Identification - With Gravity</a></li>
<li><a href="#org13de6f7">1.5. Analytical Model</a> <li><a href="#org24c3a91">1.5. Analytical Model</a>
<ul> <ul>
<li><a href="#orgef157da">1.5.1. Parameters</a></li> <li><a href="#orgfdc2987">1.5.1. Parameters</a></li>
<li><a href="#orgb72d17d">1.5.2. Generation of the State Space Model</a></li> <li><a href="#org620e32a">1.5.2. Generation of the State Space Model</a></li>
<li><a href="#org3b77585">1.5.3. Comparison with the Simscape Model</a></li> <li><a href="#orgfe0c577">1.5.3. Comparison with the Simscape Model</a></li>
<li><a href="#org2f7cb8f">1.5.4. Analysis</a></li> <li><a href="#orga854866">1.5.4. Analysis</a></li>
<li><a href="#org218243e">1.5.5. Control Section</a></li> <li><a href="#org95a6eba">1.5.5. Control Section</a></li>
<li><a href="#orgad11a63">1.5.6. Greshgorin radius</a></li> <li><a href="#org9b1baf2">1.5.6. Greshgorin radius</a></li>
<li><a href="#orga23d907">1.5.7. Injecting ground motion in the system to have the output</a></li> <li><a href="#org80e1355">1.5.7. Injecting ground motion in the system to have the output</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org23fa18d">2. Gravimeter - Functions</a> <li><a href="#org4c3e754">2. Gravimeter - Functions</a>
<ul> <ul>
<li><a href="#org81c3333">2.1. <code>align</code></a></li> <li><a href="#org790312c">2.1. <code>align</code></a></li>
<li><a href="#org8b6878d">2.2. <code>pzmap_testCL</code></a></li> <li><a href="#orge6969fe">2.2. <code>pzmap_testCL</code></a></li>
</ul> </ul>
</li> </li>
<li><a href="#org50746f8">3. Stewart Platform - Simscape Model</a> <li><a href="#org9d512a7">3. Stewart Platform - Simscape Model</a>
<ul> <ul>
<li><a href="#orga12724f">3.1. Simscape Model - Parameters</a></li> <li><a href="#org1235f4d">3.1. Simscape Model - Parameters</a></li>
<li><a href="#org820527f">3.2. Identification of the plant</a></li> <li><a href="#org8c80aff">3.2. Identification of the plant</a></li>
<li><a href="#orga58761b">3.3. Obtained Dynamics</a></li> <li><a href="#orgffd8770">3.3. Obtained Dynamics</a></li>
<li><a href="#orgb3d55c6">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li> <li><a href="#org639dffa">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#org2f2890a">3.5. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li> <li><a href="#org0cb963a">3.5. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#org70b5fa2">3.6. Decoupled Plant</a></li> <li><a href="#org1e039d4">3.6. Decoupled Plant</a></li>
<li><a href="#orgc23974f">3.7. Diagonal Controller</a></li> <li><a href="#orga66d3f9">3.7. Diagonal Controller</a></li>
<li><a href="#org6e4ced6">3.8. Closed-Loop system Performances</a></li> <li><a href="#orgdeb9b20">3.8. Closed-Loop system Performances</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org40c86ca" class="outline-2"> <div id="outline-container-org588d944" class="outline-2">
<h2 id="org40c86ca"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2> <h2 id="org588d944"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-orgac27a65" class="outline-3"> <div id="outline-container-org91ed3f1" class="outline-3">
<h3 id="orgac27a65"><span class="section-number-3">1.1</span> Introduction</h3> <h3 id="org91ed3f1"><span class="section-number-3">1.1</span> Introduction</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<div id="orgfaa8196" class="figure"> <div id="orgb33269b" class="figure">
<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" /> <p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p> <p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
@ -92,8 +92,8 @@
</div> </div>
</div> </div>
<div id="outline-container-org991b9ad" class="outline-3"> <div id="outline-container-org2a3289b" class="outline-3">
<h3 id="org991b9ad"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3> <h3 id="org2a3289b"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>) <pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>)
@ -124,8 +124,8 @@ g = 0; <span class="org-comment">% Gravity [m/s2]</span>
</div> </div>
</div> </div>
<div id="outline-container-org7417c14" class="outline-3"> <div id="outline-container-orge1533ee" class="outline-3">
<h3 id="org7417c14"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3> <h3 id="orge1533ee"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<div class="org-src-container"> <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> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
@ -147,7 +147,7 @@ G.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">
</pre> </pre>
</div> </div>
<pre class="example" id="orgefbf7cd"> <pre class="example" id="org554e6db">
pole(G) pole(G)
ans = ans =
-0.000473481142385795 + 21.7596190728632i -0.000473481142385795 + 21.7596190728632i
@ -172,7 +172,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
<div id="orgfe2be7d" class="figure"> <div id="org238cc1e" class="figure">
<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" /> <p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p> <p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
@ -180,8 +180,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org3ac74c3" class="outline-3"> <div id="outline-container-orgbcef719" class="outline-3">
<h3 id="org3ac74c3"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3> <h3 id="orgbcef719"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span> <pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span>
@ -198,7 +198,7 @@ Gg.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string"
<p> <p>
We can now see that the system is unstable due to gravity. We can now see that the system is unstable due to gravity.
</p> </p>
<pre class="example" id="org9de3a30"> <pre class="example" id="orgc834be0">
pole(Gg) pole(Gg)
ans = ans =
-10.9848275341252 + 0i -10.9848275341252 + 0i
@ -210,7 +210,7 @@ ans =
</pre> </pre>
<div id="org295f713" class="figure"> <div id="orge2ad788" class="figure">
<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" /> <p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p> <p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
@ -218,12 +218,12 @@ ans =
</div> </div>
</div> </div>
<div id="outline-container-org13de6f7" class="outline-3"> <div id="outline-container-org24c3a91" class="outline-3">
<h3 id="org13de6f7"><span class="section-number-3">1.5</span> Analytical Model</h3> <h3 id="org24c3a91"><span class="section-number-3">1.5</span> Analytical Model</h3>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-3" id="text-1-5">
</div> </div>
<div id="outline-container-orgef157da" class="outline-4"> <div id="outline-container-orgfdc2987" class="outline-4">
<h4 id="orgef157da"><span class="section-number-4">1.5.1</span> Parameters</h4> <h4 id="orgfdc2987"><span class="section-number-4">1.5.1</span> Parameters</h4>
<div class="outline-text-4" id="text-1-5-1"> <div class="outline-text-4" id="text-1-5-1">
<p> <p>
Bode options. Bode options.
@ -255,8 +255,8 @@ Frequency vector.
</div> </div>
</div> </div>
<div id="outline-container-orgb72d17d" class="outline-4"> <div id="outline-container-org620e32a" class="outline-4">
<h4 id="orgb72d17d"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4> <h4 id="org620e32a"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
<div class="outline-text-4" id="text-1-5-2"> <div class="outline-text-4" id="text-1-5-2">
<p> <p>
Mass matrix Mass matrix
@ -357,11 +357,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org3b77585" class="outline-4"> <div id="outline-container-orgfe0c577" class="outline-4">
<h4 id="org3b77585"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4> <h4 id="orgfe0c577"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
<div class="outline-text-4" id="text-1-5-3"> <div class="outline-text-4" id="text-1-5-3">
<div id="org8f52253" class="figure"> <div id="orgc91e57a" class="figure">
<p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" /> <p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p> <p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p>
@ -369,8 +369,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org2f7cb8f" class="outline-4"> <div id="outline-container-orga854866" class="outline-4">
<h4 id="org2f7cb8f"><span class="section-number-4">1.5.4</span> Analysis</h4> <h4 id="orga854866"><span class="section-number-4">1.5.4</span> Analysis</h4>
<div class="outline-text-4" id="text-1-5-4"> <div class="outline-text-4" id="text-1-5-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% figure</span> <pre class="src src-matlab"><span class="org-comment">% figure</span>
@ -438,8 +438,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org218243e" class="outline-4"> <div id="outline-container-org95a6eba" class="outline-4">
<h4 id="org218243e"><span class="section-number-4">1.5.5</span> Control Section</h4> <h4 id="org95a6eba"><span class="section-number-4">1.5.5</span> Control Section</h4>
<div class="outline-text-4" id="text-1-5-5"> <div class="outline-text-4" id="text-1-5-5">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10); <pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10);
@ -579,8 +579,8 @@ legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'D
</div> </div>
</div> </div>
<div id="outline-container-orgad11a63" class="outline-4"> <div id="outline-container-org9b1baf2" class="outline-4">
<h4 id="orgad11a63"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4> <h4 id="org9b1baf2"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
<div class="outline-text-4" id="text-1-5-6"> <div class="outline-text-4" id="text-1-5-6">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w); <pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w);
@ -626,8 +626,8 @@ ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
</div> </div>
</div> </div>
<div id="outline-container-orga23d907" class="outline-4"> <div id="outline-container-org80e1355" class="outline-4">
<h4 id="orga23d907"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4> <h4 id="org80e1355"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
<div class="outline-text-4" id="text-1-5-7"> <div class="outline-text-4" id="text-1-5-7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3); <pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3);
@ -683,15 +683,15 @@ rot = PHI(<span class="org-type">:</span>,11,11);
</div> </div>
</div> </div>
<div id="outline-container-org23fa18d" class="outline-2"> <div id="outline-container-org4c3e754" class="outline-2">
<h2 id="org23fa18d"><span class="section-number-2">2</span> Gravimeter - Functions</h2> <h2 id="org4c3e754"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
</div> </div>
<div id="outline-container-org81c3333" class="outline-3"> <div id="outline-container-org790312c" class="outline-3">
<h3 id="org81c3333"><span class="section-number-3">2.1</span> <code>align</code></h3> <h3 id="org790312c"><span class="section-number-3">2.1</span> <code>align</code></h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
<a id="org303d818"></a> <a id="org0505783"></a>
</p> </p>
<p> <p>
@ -720,11 +720,11 @@ This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
</div> </div>
<div id="outline-container-org8b6878d" class="outline-3"> <div id="outline-container-orge6969fe" class="outline-3">
<h3 id="org8b6878d"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3> <h3 id="orge6969fe"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
<a id="org7c6ecb8"></a> <a id="orga422981"></a>
</p> </p>
<p> <p>
@ -773,15 +773,15 @@ This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org50746f8" class="outline-2"> <div id="outline-container-org9d512a7" class="outline-2">
<h2 id="org50746f8"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2> <h2 id="org9d512a7"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org9c6bf2d">5</a>. In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#orge1e9c00">5</a>.
</p> </p>
<div id="org9c6bf2d" class="figure"> <div id="orge1e9c00" class="figure">
<p><img src="figs/SP_assembly.png" alt="SP_assembly.png" /> <p><img src="figs/SP_assembly.png" alt="SP_assembly.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Stewart Platform CAD View</p> <p><span class="figure-number">Figure 5: </span>Stewart Platform CAD View</p>
@ -791,21 +791,21 @@ In this analysis, we wish to applied SVD control to the Stewart Platform shown i
The analysis of the SVD control applied to the Stewart platform is performed in the following sections: The analysis of the SVD control applied to the Stewart platform is performed in the following sections:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org5932d29">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li> <li>Section <a href="#org1f1154c">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
<li>Section <a href="#org7980ba7">3.2</a>: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian</li> <li>Section <a href="#org76fc591">3.2</a>: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian</li>
<li>Section <a href="#orgb9c44bf">3.3</a>: The identified Dynamics is shown</li> <li>Section <a href="#org4d48d60">3.3</a>: The identified Dynamics is shown</li>
<li>Section <a href="#orgea1a70b">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li> <li>Section <a href="#orgf063500">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li>
<li>Section <a href="#org0cd9585">3.5</a>: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii</li> <li>Section <a href="#org6d984d9">3.5</a>: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii</li>
<li>Section <a href="#org6e20bec">3.6</a>: The dynamics of the decoupled plant is shown</li> <li>Section <a href="#org083c541">3.6</a>: The dynamics of the decoupled plant is shown</li>
<li>Section <a href="#org7c9ebe2">3.7</a>: A diagonal controller is defined to control the decoupled plant</li> <li>Section <a href="#org7fb568e">3.7</a>: A diagonal controller is defined to control the decoupled plant</li>
<li>Section <a href="#orgfaeace7">3.8</a>: Finally, the closed loop system properties are studied</li> <li>Section <a href="#org3072cea">3.8</a>: Finally, the closed loop system properties are studied</li>
</ul> </ul>
</div> </div>
<div id="outline-container-orga12724f" class="outline-3"> <div id="outline-container-org1235f4d" class="outline-3">
<h3 id="orga12724f"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3> <h3 id="org1235f4d"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
<a id="org5932d29"></a> <a id="org1f1154c"></a>
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>); <pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>);
@ -841,14 +841,24 @@ We load the Jacobian (previously computed from the geometry).
<pre class="src src-matlab">load(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">'Aa'</span>, <span class="org-string">'Ab'</span>, <span class="org-string">'As'</span>, <span class="org-string">'l'</span>, <span class="org-string">'J'</span>); <pre class="src src-matlab">load(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">'Aa'</span>, <span class="org-string">'Ab'</span>, <span class="org-string">'As'</span>, <span class="org-string">'l'</span>, <span class="org-string">'J'</span>);
</pre> </pre>
</div> </div>
<p>
We initialize other parameters:
</p>
<div class="org-src-container">
<pre class="src src-matlab">U = eye(6);
V = eye(6);
Kc = tf(zeros(6));
</pre>
</div>
</div> </div>
</div> </div>
<div id="outline-container-org820527f" class="outline-3"> <div id="outline-container-org8c80aff" class="outline-3">
<h3 id="org820527f"><span class="section-number-3">3.2</span> Identification of the plant</h3> <h3 id="org8c80aff"><span class="section-number-3">3.2</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
<a id="org7980ba7"></a> <a id="org76fc591"></a>
</p> </p>
<p> <p>
@ -861,7 +871,7 @@ mdl = <span class="org-string">'drone_platform'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> <span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1; 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">'/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">'/u'</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">'/V-T'</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">'/Inertial Sensor'</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">'/Inertial Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
G = linearize(mdl, io); G = linearize(mdl, io);
@ -885,11 +895,11 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
<p> <p>
The &ldquo;centralized&rdquo; plant \(\bm{G}_x\) is now computed (Figure <a href="#org249f9cd">6</a>). The &ldquo;centralized&rdquo; plant \(\bm{G}_x\) is now computed (Figure <a href="#org5fb072e">6</a>).
</p> </p>
<div id="org249f9cd" class="figure"> <div id="org5fb072e" class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" /> <p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Centralized control architecture</p> <p><span class="figure-number">Figure 6: </span>Centralized control architecture</p>
@ -907,22 +917,22 @@ Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string"
</div> </div>
</div> </div>
<div id="outline-container-orga58761b" class="outline-3"> <div id="outline-container-orgffd8770" class="outline-3">
<h3 id="orga58761b"><span class="section-number-3">3.3</span> Obtained Dynamics</h3> <h3 id="orgffd8770"><span class="section-number-3">3.3</span> Obtained Dynamics</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
<a id="orgb9c44bf"></a> <a id="org4d48d60"></a>
</p> </p>
<div id="org6d21a96" class="figure"> <div id="orgdb3fa27" class="figure">
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" /> <p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p> <p><span class="figure-number">Figure 7: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
</div> </div>
<div id="orge724936" class="figure"> <div id="org1b6e945" class="figure">
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" /> <p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p> <p><span class="figure-number">Figure 8: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
@ -930,11 +940,11 @@ Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string"
</div> </div>
</div> </div>
<div id="outline-container-orgb3d55c6" class="outline-3"> <div id="outline-container-org639dffa" class="outline-3">
<h3 id="orgb3d55c6"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3> <h3 id="org639dffa"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
<a id="orgea1a70b"></a> <a id="orgf063500"></a>
</p> </p>
<p> <p>
@ -1114,11 +1124,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown
</div> </div>
</div> </div>
<div id="outline-container-org2f2890a" class="outline-3"> <div id="outline-container-org0cb963a" class="outline-3">
<h3 id="org2f2890a"><span class="section-number-3">3.5</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3> <h3 id="org0cb963a"><span class="section-number-3">3.5</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-3-5">
<p> <p>
<a id="org0cd9585"></a> <a id="org6d984d9"></a>
</p> </p>
<p> <p>
@ -1187,7 +1197,7 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
</div> </div>
<div id="org4e85b3b" class="figure"> <div id="org3cf0ede" class="figure">
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" /> <p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Gershgorin Radii of the Coupled and Decoupled plants</p> <p><span class="figure-number">Figure 9: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
@ -1195,11 +1205,11 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
</div> </div>
</div> </div>
<div id="outline-container-org70b5fa2" class="outline-3"> <div id="outline-container-org1e039d4" class="outline-3">
<h3 id="org70b5fa2"><span class="section-number-3">3.6</span> Decoupled Plant</h3> <h3 id="org1e039d4"><span class="section-number-3">3.6</span> Decoupled Plant</h3>
<div class="outline-text-3" id="text-3-6"> <div class="outline-text-3" id="text-3-6">
<p> <p>
<a id="org6e20bec"></a> <a id="org083c541"></a>
</p> </p>
<p> <p>
@ -1208,14 +1218,14 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</p> </p>
<div id="org82227b9" class="figure"> <div id="orgcc74e6b" class="figure">
<p><img src="figs/simscape_model_decoupled_plant_svd.png" alt="simscape_model_decoupled_plant_svd.png" /> <p><img src="figs/simscape_model_decoupled_plant_svd.png" alt="simscape_model_decoupled_plant_svd.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Decoupled Plant using SVD</p> <p><span class="figure-number">Figure 10: </span>Decoupled Plant using SVD</p>
</div> </div>
<div id="org2cf3f8e" class="figure"> <div id="orgaf3df78" class="figure">
<p><img src="figs/simscape_model_decoupled_plant_jacobian.png" alt="simscape_model_decoupled_plant_jacobian.png" /> <p><img src="figs/simscape_model_decoupled_plant_jacobian.png" alt="simscape_model_decoupled_plant_jacobian.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Decoupled Plant using the Jacobian</p> <p><span class="figure-number">Figure 11: </span>Decoupled Plant using the Jacobian</p>
@ -1223,11 +1233,11 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</div> </div>
</div> </div>
<div id="outline-container-orgc23974f" class="outline-3"> <div id="outline-container-orga66d3f9" class="outline-3">
<h3 id="orgc23974f"><span class="section-number-3">3.7</span> Diagonal Controller</h3> <h3 id="orga66d3f9"><span class="section-number-3">3.7</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-3-7"> <div class="outline-text-3" id="text-3-7">
<p> <p>
<a id="org7c9ebe2"></a> <a id="org7fb568e"></a>
</p> </p>
<p> <p>
@ -1238,12 +1248,12 @@ The controller \(K\) is a diagonal controller consisting a low pass filters with
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; <span class="org-comment">% Crossover Frequency [rad/s]</span> <pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; <span class="org-comment">% Crossover Frequency [rad/s]</span>
C_g = 50; <span class="org-comment">% DC Gain</span> C_g = 50; <span class="org-comment">% DC Gain</span>
K = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<span class="org-type">+</span>wc); Kc = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<span class="org-type">+</span>wc);
</pre> </pre>
</div> </div>
<p> <p>
The control diagram for the centralized control is shown in Figure <a href="#org249f9cd">6</a>. The control diagram for the centralized control is shown in Figure <a href="#org5fb072e">6</a>.
</p> </p>
<p> <p>
@ -1252,7 +1262,7 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
</p> </p>
<div id="org6e49f6b" class="figure"> <div id="orge11b6b2" class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" /> <p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Control Diagram for the Centralized control</p> <p><span class="figure-number">Figure 12: </span>Control Diagram for the Centralized control</p>
@ -1262,16 +1272,16 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
The feedback system is computed as shown below. The feedback system is computed as shown below.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">G_cen = feedback(G, inv(J<span class="org-type">'</span>)<span class="org-type">*</span>K, [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]); <pre class="src src-matlab">G_cen = feedback(G, inv(J<span class="org-type">'</span>)<span class="org-type">*</span>Kc, [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
</pre> </pre>
</div> </div>
<p> <p>
The SVD control architecture is shown in Figure <a href="#org98507fe">13</a>. The SVD control architecture is shown in Figure <a href="#orgef128af">13</a>.
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\). The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
</p> </p>
<div id="org98507fe" class="figure"> <div id="orgef128af" class="figure">
<p><img src="figs/svd_control.png" alt="svd_control.png" /> <p><img src="figs/svd_control.png" alt="svd_control.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Control Diagram for the SVD control</p> <p><span class="figure-number">Figure 13: </span>Control Diagram for the SVD control</p>
@ -1281,17 +1291,17 @@ The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
The feedback system is computed as shown below. The feedback system is computed as shown below.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">G_svd = feedback(G, pinv(V<span class="org-type">'</span>)<span class="org-type">*</span>K<span class="org-type">*</span>pinv(U), [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]); <pre class="src src-matlab">G_svd = feedback(G, pinv(V<span class="org-type">'</span>)<span class="org-type">*</span>Kc<span class="org-type">*</span>pinv(U), [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
</pre> </pre>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org6e4ced6" class="outline-3"> <div id="outline-container-orgdeb9b20" class="outline-3">
<h3 id="org6e4ced6"><span class="section-number-3">3.8</span> Closed-Loop system Performances</h3> <h3 id="orgdeb9b20"><span class="section-number-3">3.8</span> Closed-Loop system Performances</h3>
<div class="outline-text-3" id="text-3-8"> <div class="outline-text-3" id="text-3-8">
<p> <p>
<a id="orgfaeace7"></a> <a id="org3072cea"></a>
</p> </p>
<p> <p>
@ -1322,11 +1332,11 @@ ans =
<p> <p>
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org500fc7e">14</a>. The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org9b356fe">14</a>.
</p> </p>
<div id="org500fc7e" class="figure"> <div id="org9b356fe" class="figure">
<p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" /> <p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Obtained Transmissibility</p> <p><span class="figure-number">Figure 14: </span>Obtained Transmissibility</p>
@ -1337,7 +1347,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-06 ven. 12:22</p> <p class="date">Created: 2020-11-06 ven. 15:06</p>
</div> </div>
</body> </body>
</html> </html>

8
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@ -727,15 +727,15 @@ The analysis of the SVD control applied to the Stewart platform is performed in
** Jacobian :noexport: ** Jacobian :noexport:
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed. First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
#+begin_src matlab #+begin_src matlab :tangle no
open('drone_platform_jacobian.slx'); open('drone_platform_jacobian.slx');
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab :tangle no
sim('drone_platform_jacobian'); sim('drone_platform_jacobian');
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab :tangle no
Aa = [a1.Data(1,:); Aa = [a1.Data(1,:);
a2.Data(1,:); a2.Data(1,:);
a3.Data(1,:); a3.Data(1,:);
@ -804,7 +804,7 @@ The dynamics is identified from forces applied by each legs to the measured acce
%% Input/Output definition %% Input/Output definition
clear io; io_i = 1; clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io); G = linearize(mdl, io);