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Update Stewart Platform figures

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Thomas Dehaeze 2020-11-06 12:22:37 +01:00
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-11-06 ven. 12:04 -->
<!-- 2020-11-06 ven. 12:22 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>SVD Control</title>
<meta name="generator" content="Org mode" />
@ -35,56 +35,56 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org29eb71f">1. Gravimeter - Simscape Model</a>
<li><a href="#org40c86ca">1. Gravimeter - Simscape Model</a>
<ul>
<li><a href="#org3d08142">1.1. Introduction</a></li>
<li><a href="#org0e81328">1.2. Simscape Model - Parameters</a></li>
<li><a href="#orgfb8bd07">1.3. System Identification - Without Gravity</a></li>
<li><a href="#org73ffaa0">1.4. System Identification - With Gravity</a></li>
<li><a href="#org0a007cf">1.5. Analytical Model</a>
<li><a href="#orgac27a65">1.1. Introduction</a></li>
<li><a href="#org991b9ad">1.2. Simscape Model - Parameters</a></li>
<li><a href="#org7417c14">1.3. System Identification - Without Gravity</a></li>
<li><a href="#org3ac74c3">1.4. System Identification - With Gravity</a></li>
<li><a href="#org13de6f7">1.5. Analytical Model</a>
<ul>
<li><a href="#orgd57f9d7">1.5.1. Parameters</a></li>
<li><a href="#orgccfbd49">1.5.2. Generation of the State Space Model</a></li>
<li><a href="#org29be676">1.5.3. Comparison with the Simscape Model</a></li>
<li><a href="#orgb216c9b">1.5.4. Analysis</a></li>
<li><a href="#org09d1753">1.5.5. Control Section</a></li>
<li><a href="#org08382b3">1.5.6. Greshgorin radius</a></li>
<li><a href="#org9828644">1.5.7. Injecting ground motion in the system to have the output</a></li>
<li><a href="#orgef157da">1.5.1. Parameters</a></li>
<li><a href="#orgb72d17d">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="#org2f7cb8f">1.5.4. Analysis</a></li>
<li><a href="#org218243e">1.5.5. Control Section</a></li>
<li><a href="#orgad11a63">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>
</ul>
</li>
</ul>
</li>
<li><a href="#org5cf1b81">2. Gravimeter - Functions</a>
<li><a href="#org23fa18d">2. Gravimeter - Functions</a>
<ul>
<li><a href="#org865007e">2.1. <code>align</code></a></li>
<li><a href="#org1ec1be4">2.2. <code>pzmap_testCL</code></a></li>
<li><a href="#org81c3333">2.1. <code>align</code></a></li>
<li><a href="#org8b6878d">2.2. <code>pzmap_testCL</code></a></li>
</ul>
</li>
<li><a href="#orgae25787">3. Stewart Platform - Simscape Model</a>
<li><a href="#org50746f8">3. Stewart Platform - Simscape Model</a>
<ul>
<li><a href="#orgddc1d62">3.1. Simscape Model - Parameters</a></li>
<li><a href="#org4bf5131">3.2. Identification of the plant</a></li>
<li><a href="#org1d0f1f6">3.3. Obtained Dynamics</a></li>
<li><a href="#orge008cca">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#org8f626c6">3.5. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#orgcb6b23d">3.6. Decoupled Plant</a></li>
<li><a href="#org35975ff">3.7. Diagonal Controller</a></li>
<li><a href="#orgfc10dab">3.8. Closed-Loop system Performances</a></li>
<li><a href="#orga12724f">3.1. Simscape Model - Parameters</a></li>
<li><a href="#org820527f">3.2. Identification of the plant</a></li>
<li><a href="#orga58761b">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="#org2f2890a">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="#orgc23974f">3.7. Diagonal Controller</a></li>
<li><a href="#org6e4ced6">3.8. Closed-Loop system Performances</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org29eb71f" class="outline-2">
<h2 id="org29eb71f"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div id="outline-container-org40c86ca" class="outline-2">
<h2 id="org40c86ca"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org3d08142" class="outline-3">
<h3 id="org3d08142"><span class="section-number-3">1.1</span> Introduction</h3>
<div id="outline-container-orgac27a65" class="outline-3">
<h3 id="orgac27a65"><span class="section-number-3">1.1</span> Introduction</h3>
<div class="outline-text-3" id="text-1-1">
<div id="org405c3e2" class="figure">
<div id="orgfaa8196" class="figure">
<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
@ -92,8 +92,8 @@
</div>
</div>
<div id="outline-container-org0e81328" class="outline-3">
<h3 id="org0e81328"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
<div id="outline-container-org991b9ad" class="outline-3">
<h3 id="org991b9ad"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<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 id="outline-container-orgfb8bd07" class="outline-3">
<h3 id="orgfb8bd07"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
<div id="outline-container-org7417c14" class="outline-3">
<h3 id="org7417c14"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
<div class="outline-text-3" id="text-1-3">
<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>
@ -147,7 +147,7 @@ G.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">
</pre>
</div>
<pre class="example" id="org9b4cec2">
<pre class="example" id="orgefbf7cd">
pole(G)
ans =
-0.000473481142385795 + 21.7596190728632i
@ -172,7 +172,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
<div id="orge565e71" class="figure">
<div id="orgfe2be7d" class="figure">
<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
</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 id="outline-container-org73ffaa0" class="outline-3">
<h3 id="org73ffaa0"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
<div id="outline-container-org3ac74c3" class="outline-3">
<h3 id="org3ac74c3"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
<div class="outline-text-3" id="text-1-4">
<div class="org-src-container">
<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>
We can now see that the system is unstable due to gravity.
</p>
<pre class="example" id="orga648ec2">
<pre class="example" id="org9de3a30">
pole(Gg)
ans =
-10.9848275341252 + 0i
@ -210,7 +210,7 @@ ans =
</pre>
<div id="orgddeeca1" class="figure">
<div id="org295f713" class="figure">
<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
</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 id="outline-container-org0a007cf" class="outline-3">
<h3 id="org0a007cf"><span class="section-number-3">1.5</span> Analytical Model</h3>
<div id="outline-container-org13de6f7" class="outline-3">
<h3 id="org13de6f7"><span class="section-number-3">1.5</span> Analytical Model</h3>
<div class="outline-text-3" id="text-1-5">
</div>
<div id="outline-container-orgd57f9d7" class="outline-4">
<h4 id="orgd57f9d7"><span class="section-number-4">1.5.1</span> Parameters</h4>
<div id="outline-container-orgef157da" class="outline-4">
<h4 id="orgef157da"><span class="section-number-4">1.5.1</span> Parameters</h4>
<div class="outline-text-4" id="text-1-5-1">
<p>
Bode options.
@ -255,8 +255,8 @@ Frequency vector.
</div>
</div>
<div id="outline-container-orgccfbd49" class="outline-4">
<h4 id="orgccfbd49"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
<div id="outline-container-orgb72d17d" class="outline-4">
<h4 id="orgb72d17d"><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">
<p>
Mass matrix
@ -357,11 +357,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div>
</div>
<div id="outline-container-org29be676" class="outline-4">
<h4 id="org29be676"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
<div id="outline-container-org3b77585" class="outline-4">
<h4 id="org3b77585"><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 id="orgd9d1275" class="figure">
<div id="org8f52253" class="figure">
<p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" />
</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 id="outline-container-orgb216c9b" class="outline-4">
<h4 id="orgb216c9b"><span class="section-number-4">1.5.4</span> Analysis</h4>
<div id="outline-container-org2f7cb8f" class="outline-4">
<h4 id="org2f7cb8f"><span class="section-number-4">1.5.4</span> Analysis</h4>
<div class="outline-text-4" id="text-1-5-4">
<div class="org-src-container">
<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 id="outline-container-org09d1753" class="outline-4">
<h4 id="org09d1753"><span class="section-number-4">1.5.5</span> Control Section</h4>
<div id="outline-container-org218243e" class="outline-4">
<h4 id="org218243e"><span class="section-number-4">1.5.5</span> Control Section</h4>
<div class="outline-text-4" id="text-1-5-5">
<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);
@ -579,8 +579,8 @@ legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'D
</div>
</div>
<div id="outline-container-org08382b3" class="outline-4">
<h4 id="org08382b3"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
<div id="outline-container-orgad11a63" class="outline-4">
<h4 id="orgad11a63"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
<div class="outline-text-4" id="text-1-5-6">
<div class="org-src-container">
<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 id="outline-container-org9828644" class="outline-4">
<h4 id="org9828644"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
<div id="outline-container-orga23d907" 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>
<div class="outline-text-4" id="text-1-5-7">
<div class="org-src-container">
<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 id="outline-container-org5cf1b81" class="outline-2">
<h2 id="org5cf1b81"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
<div id="outline-container-org23fa18d" class="outline-2">
<h2 id="org23fa18d"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org865007e" class="outline-3">
<h3 id="org865007e"><span class="section-number-3">2.1</span> <code>align</code></h3>
<div id="outline-container-org81c3333" class="outline-3">
<h3 id="org81c3333"><span class="section-number-3">2.1</span> <code>align</code></h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org4775cee"></a>
<a id="org303d818"></a>
</p>
<p>
@ -720,11 +720,11 @@ This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
</div>
<div id="outline-container-org1ec1be4" class="outline-3">
<h3 id="org1ec1be4"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
<div id="outline-container-org8b6878d" class="outline-3">
<h3 id="org8b6878d"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="org3661b72"></a>
<a id="org7c6ecb8"></a>
</p>
<p>
@ -773,15 +773,15 @@ This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
</div>
</div>
<div id="outline-container-orgae25787" class="outline-2">
<h2 id="orgae25787"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
<div id="outline-container-org50746f8" class="outline-2">
<h2 id="org50746f8"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
<div class="outline-text-2" id="text-3">
<p>
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org602832a">5</a>.
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org9c6bf2d">5</a>.
</p>
<div id="org602832a" class="figure">
<div id="org9c6bf2d" class="figure">
<p><img src="figs/SP_assembly.png" alt="SP_assembly.png" />
</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:
</p>
<ul class="org-ul">
<li>Section <a href="#orgac7dad3">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
<li>Section <a href="#org9d2dc26">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="#orgcb0f631">3.3</a>: The identified Dynamics is shown</li>
<li>Section <a href="#org60cd341">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="#org2ab4886">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="#orgbe91cc3">3.6</a>: The dynamics of the decoupled plant is shown</li>
<li>Section <a href="#org71de783">3.7</a>: A diagonal controller is defined to control the decoupled plant</li>
<li>Section <a href="#org2d444fe">3.8</a>: Finally, the closed loop system properties are studied</li>
<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="#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="#orgb9c44bf">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="#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="#org6e20bec">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="#orgfaeace7">3.8</a>: Finally, the closed loop system properties are studied</li>
</ul>
</div>
<div id="outline-container-orgddc1d62" class="outline-3">
<h3 id="orgddc1d62"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
<div id="outline-container-orga12724f" class="outline-3">
<h3 id="orga12724f"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="orgac7dad3"></a>
<a id="org5932d29"></a>
</p>
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>);
@ -844,11 +844,11 @@ We load the Jacobian (previously computed from the geometry).
</div>
</div>
<div id="outline-container-org4bf5131" class="outline-3">
<h3 id="org4bf5131"><span class="section-number-3">3.2</span> Identification of the plant</h3>
<div id="outline-container-org820527f" class="outline-3">
<h3 id="org820527f"><span class="section-number-3">3.2</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="org9d2dc26"></a>
<a id="org7980ba7"></a>
</p>
<p>
@ -885,11 +885,11 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
<p>
The &ldquo;centralized&rdquo; plant \(\bm{G}_x\) is now computed (Figure <a href="#org71176d2">6</a>).
The &ldquo;centralized&rdquo; plant \(\bm{G}_x\) is now computed (Figure <a href="#org249f9cd">6</a>).
</p>
<div id="org71176d2" class="figure">
<div id="org249f9cd" class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Centralized control architecture</p>
@ -907,22 +907,22 @@ Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string"
</div>
</div>
<div id="outline-container-org1d0f1f6" class="outline-3">
<h3 id="org1d0f1f6"><span class="section-number-3">3.3</span> Obtained Dynamics</h3>
<div id="outline-container-orga58761b" class="outline-3">
<h3 id="orga58761b"><span class="section-number-3">3.3</span> Obtained Dynamics</h3>
<div class="outline-text-3" id="text-3-3">
<p>
<a id="orgcb0f631"></a>
<a id="orgb9c44bf"></a>
</p>
<div id="orgab61fa1" class="figure">
<div id="org6d21a96" class="figure">
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
</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 id="orgdeda0eb" class="figure">
<div id="orge724936" class="figure">
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
</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 +930,11 @@ Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string"
</div>
</div>
<div id="outline-container-orge008cca" class="outline-3">
<h3 id="orge008cca"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
<div id="outline-container-orgb3d55c6" class="outline-3">
<h3 id="orgb3d55c6"><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">
<p>
<a id="org60cd341"></a>
<a id="orgea1a70b"></a>
</p>
<p>
@ -1034,7 +1034,7 @@ H1 = inv(D<span class="org-type">*</span>real(H1<span class="org-type">'*</span>
<p>
Please not that the plant \(G\) at \(\omega_c\) is already an almost real matrix.
Note that the plant \(G\) at \(\omega_c\) is already an almost real matrix.
This can be seen on the Bode plots where the phase is close to 1.
This can be verified below where only the real value of \(G(\omega_c)\) is shown
</p>
@ -1114,11 +1114,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown
</div>
</div>
<div id="outline-container-org8f626c6" class="outline-3">
<h3 id="org8f626c6"><span class="section-number-3">3.5</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3>
<div id="outline-container-org2f2890a" 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>
<div class="outline-text-3" id="text-3-5">
<p>
<a id="org2ab4886"></a>
<a id="org0cd9585"></a>
</p>
<p>
@ -1187,7 +1187,7 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
</div>
<div id="org8e3dd15" class="figure">
<div id="org4e85b3b" class="figure">
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
@ -1195,11 +1195,11 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
</div>
</div>
<div id="outline-container-orgcb6b23d" class="outline-3">
<h3 id="orgcb6b23d"><span class="section-number-3">3.6</span> Decoupled Plant</h3>
<div id="outline-container-org70b5fa2" class="outline-3">
<h3 id="org70b5fa2"><span class="section-number-3">3.6</span> Decoupled Plant</h3>
<div class="outline-text-3" id="text-3-6">
<p>
<a id="orgbe91cc3"></a>
<a id="org6e20bec"></a>
</p>
<p>
@ -1208,14 +1208,14 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</p>
<div id="org3040990" class="figure">
<div id="org82227b9" class="figure">
<p><img src="figs/simscape_model_decoupled_plant_svd.png" alt="simscape_model_decoupled_plant_svd.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Decoupled Plant using SVD</p>
</div>
<div id="orgb0513ee" class="figure">
<div id="org2cf3f8e" class="figure">
<p><img src="figs/simscape_model_decoupled_plant_jacobian.png" alt="simscape_model_decoupled_plant_jacobian.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Decoupled Plant using the Jacobian</p>
@ -1223,11 +1223,11 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</div>
</div>
<div id="outline-container-org35975ff" class="outline-3">
<h3 id="org35975ff"><span class="section-number-3">3.7</span> Diagonal Controller</h3>
<div id="outline-container-orgc23974f" class="outline-3">
<h3 id="orgc23974f"><span class="section-number-3">3.7</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-3-7">
<p>
<a id="org71de783"></a>
<a id="org7c9ebe2"></a>
</p>
<p>
@ -1243,7 +1243,7 @@ K = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<sp
</div>
<p>
The control diagram for the centralized control is shown in Figure <a href="#org71176d2">6</a>.
The control diagram for the centralized control is shown in Figure <a href="#org249f9cd">6</a>.
</p>
<p>
@ -1252,7 +1252,7 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
</p>
<div id="orgfbe6ac1" class="figure">
<div id="org6e49f6b" class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Control Diagram for the Centralized control</p>
@ -1267,11 +1267,11 @@ The feedback system is computed as shown below.
</div>
<p>
The SVD control architecture is shown in Figure <a href="#org4df0074">13</a>.
The SVD control architecture is shown in Figure <a href="#org98507fe">13</a>.
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
</p>
<div id="org4df0074" class="figure">
<div id="org98507fe" class="figure">
<p><img src="figs/svd_control.png" alt="svd_control.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Control Diagram for the SVD control</p>
@ -1287,11 +1287,11 @@ The feedback system is computed as shown below.
</div>
</div>
<div id="outline-container-orgfc10dab" class="outline-3">
<h3 id="orgfc10dab"><span class="section-number-3">3.8</span> Closed-Loop system Performances</h3>
<div id="outline-container-org6e4ced6" class="outline-3">
<h3 id="org6e4ced6"><span class="section-number-3">3.8</span> Closed-Loop system Performances</h3>
<div class="outline-text-3" id="text-3-8">
<p>
<a id="org2d444fe"></a>
<a id="orgfaeace7"></a>
</p>
<p>
@ -1322,11 +1322,11 @@ ans =
<p>
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org0481186">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="#org500fc7e">14</a>.
</p>
<div id="org0481186" class="figure">
<div id="org500fc7e" class="figure">
<p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Obtained Transmissibility</p>
@ -1337,7 +1337,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-06 ven. 12:04</p>
<p class="date">Created: 2020-11-06 ven. 12:22</p>
</div>
</body>
</html>

73
index.org
View File

@ -945,7 +945,7 @@ The real approximation is computed as follows:
| 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 |
Please not that the plant $G$ at $\omega_c$ is already an almost real matrix.
Note that the plant $G$ at $\omega_c$ is already an almost real matrix.
This can be seen on the Bode plots where the phase is close to 1.
This can be verified below where only the real value of $G(\omega_c)$ is shown
@ -1031,10 +1031,11 @@ Gershgorin Radii for the decoupled plant using the Jacobian:
hold off;
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
legend('location', 'northeast');
ylim([1e-3, 1e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_gershgorin_radii.pdf', 'width', 'full', 'height', 'full');
exportFig('figs/simscape_model_gershgorin_radii.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:simscape_model_gershgorin_radii
@ -1066,11 +1067,12 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
legend('location', 'northwest');
ylim([1e-3, 1e5]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_decoupled_plant_svd.pdf', 'width', 'full', 'height', 'full');
exportFig('figs/simscape_model_decoupled_plant_svd.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:simscape_model_decoupled_plant_svd
@ -1096,11 +1098,12 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
legend('location', 'northwest');
ylim([1e-3, 1e6]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'width', 'full', 'height', 'full');
exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:simscape_model_decoupled_plant_jacobian
@ -1212,71 +1215,73 @@ Let's first verify the stability of the closed-loop systems:
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
#+begin_src matlab :exports results
freqs = logspace(-3, 3, 1000);
freqs = logspace(-2, 2, 1000);
figure
figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = subplot(2, 3, 1);
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]');
ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest');
ax2 = subplot(2, 3, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
% ax2 = nexttile;
% hold on;
% plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
% plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
% plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
% hold off;
% set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
% ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
ax3 = subplot(2, 3, 3);
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]');
ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
ax4 = subplot(2, 3, 4);
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
ax5 = subplot(2, 3, 5);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
% ax5 = nexttile;
% hold on;
% plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
% plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
% plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
% hold off;
% set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
% ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
ax6 = subplot(2, 3, 6);
ax6 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
ylabel('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
xlim([freqs(1), freqs(end)]);
ylim([1e-5, 1e2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'width', 1600, 'height', 'full');
exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:stewart_platform_simscape_cl_transmissibility