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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>
<!-- 2020-09-30 mer. 17:15 --> <!-- 2020-09-30 mer. 17:21 -->
<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" />
@ -35,55 +35,55 @@
<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="#org8605f4d">1. Gravimeter - Simscape Model</a> <li><a href="#org8abfad3">1. Gravimeter - Simscape Model</a>
<ul> <ul>
<li><a href="#orgc32c7f1">1.1. Simscape Model - Parameters</a></li> <li><a href="#orgbb05dfd">1.1. Simscape Model - Parameters</a></li>
<li><a href="#org4bab8c7">1.2. System Identification - Without Gravity</a></li> <li><a href="#org46847b0">1.2. System Identification - Without Gravity</a></li>
<li><a href="#org1af907b">1.3. System Identification - With Gravity</a></li> <li><a href="#org47fe384">1.3. System Identification - With Gravity</a></li>
<li><a href="#orgf23381e">1.4. Analytical Model</a> <li><a href="#org67eee73">1.4. Analytical Model</a>
<ul> <ul>
<li><a href="#org86e662a">1.4.1. Parameters</a></li> <li><a href="#org1545adc">1.4.1. Parameters</a></li>
<li><a href="#orgd9883dd">1.4.2. generation of the state space model</a></li> <li><a href="#org22e1e2b">1.4.2. generation of the state space model</a></li>
<li><a href="#orgf1cd403">1.4.3. Comparison with the Simscape Model</a></li> <li><a href="#orgee11542">1.4.3. Comparison with the Simscape Model</a></li>
<li><a href="#orgc44aa74">1.4.4. Analysis</a></li> <li><a href="#org9524774">1.4.4. Analysis</a></li>
<li><a href="#org56c5430">1.4.5. Control Section</a></li> <li><a href="#org39b3c10">1.4.5. Control Section</a></li>
<li><a href="#org2f4000c">1.4.6. Greshgorin radius</a></li> <li><a href="#orgfd02af1">1.4.6. Greshgorin radius</a></li>
<li><a href="#orgacfa62b">1.4.7. Injecting ground motion in the system to have the output</a></li> <li><a href="#org88c39eb">1.4.7. Injecting ground motion in the system to have the output</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgdbdd0b8">2. Gravimeter - Functions</a> <li><a href="#org5a87ff1">2. Gravimeter - Functions</a>
<ul> <ul>
<li><a href="#org72eb8fe">2.1. <code>align</code></a></li> <li><a href="#org755e595">2.1. <code>align</code></a></li>
<li><a href="#orgf7acd4e">2.2. <code>pzmap_testCL</code></a></li> <li><a href="#org55b8479">2.2. <code>pzmap_testCL</code></a></li>
</ul> </ul>
</li> </li>
<li><a href="#org20de0f0">3. Stewart Platform - Simscape Model</a> <li><a href="#orgb23d007">3. Stewart Platform - Simscape Model</a>
<ul> <ul>
<li><a href="#orgf01ffb3">3.1. Jacobian</a></li> <li><a href="#org636de2e">3.1. Jacobian</a></li>
<li><a href="#orgab52d7f">3.2. Simscape Model</a></li> <li><a href="#org59d2125">3.2. Simscape Model</a></li>
<li><a href="#orgde1eb2c">3.3. Identification of the plant</a></li> <li><a href="#org77015bb">3.3. Identification of the plant</a></li>
<li><a href="#orgef70e0f">3.4. Obtained Dynamics</a></li> <li><a href="#org21a398b">3.4. Obtained Dynamics</a></li>
<li><a href="#org97d95a3">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li> <li><a href="#org6cab60a">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#org2460007">3.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li> <li><a href="#orgc331fa5">3.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#orgc89a913">3.7. Decoupled Plant</a></li> <li><a href="#orge6123eb">3.7. Decoupled Plant</a></li>
<li><a href="#org3d3c34c">3.8. Diagonal Controller</a></li> <li><a href="#orge9ddb65">3.8. Diagonal Controller</a></li>
<li><a href="#orgaa29814">3.9. Centralized Control</a></li> <li><a href="#orga25eea8">3.9. Centralized Control</a></li>
<li><a href="#orge736e6a">3.10. SVD Control</a></li> <li><a href="#org4255891">3.10. SVD Control</a></li>
<li><a href="#orgd1cfb41">3.11. Results</a></li> <li><a href="#org535f13b">3.11. Results</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org8605f4d" class="outline-2"> <div id="outline-container-org8abfad3" class="outline-2">
<h2 id="org8605f4d"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2> <h2 id="org8abfad3"><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-orgc32c7f1" class="outline-3"> <div id="outline-container-orgbb05dfd" class="outline-3">
<h3 id="orgc32c7f1"><span class="section-number-3">1.1</span> Simscape Model - Parameters</h3> <h3 id="orgbb05dfd"><span class="section-number-3">1.1</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open('gravimeter.slx') <pre class="src src-matlab">open('gravimeter.slx')
@ -114,8 +114,8 @@ g = 0; % Gravity [m/s2]
</div> </div>
</div> </div>
<div id="outline-container-org4bab8c7" class="outline-3"> <div id="outline-container-org46847b0" class="outline-3">
<h3 id="org4bab8c7"><span class="section-number-3">1.2</span> System Identification - Without Gravity</h3> <h3 id="org46847b0"><span class="section-number-3">1.2</span> System Identification - Without Gravity</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">%% Name of the Simulink File <pre class="src src-matlab">%% Name of the Simulink File
@ -162,7 +162,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
<div id="orgb31b43a" class="figure"> <div id="org0a2d774" 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 1: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p> <p><span class="figure-number">Figure 1: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
@ -170,8 +170,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org1af907b" class="outline-3"> <div id="outline-container-org47fe384" class="outline-3">
<h3 id="org1af907b"><span class="section-number-3">1.3</span> System Identification - With Gravity</h3> <h3 id="org47fe384"><span class="section-number-3">1.3</span> System Identification - With 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">g = 9.80665; % Gravity [m/s2] <pre class="src src-matlab">g = 9.80665; % Gravity [m/s2]
@ -200,7 +200,7 @@ ans =
</pre> </pre>
<div id="org6f96795" class="figure"> <div id="orgc26ed4a" 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 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p> <p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
@ -208,12 +208,12 @@ ans =
</div> </div>
</div> </div>
<div id="outline-container-orgf23381e" class="outline-3"> <div id="outline-container-org67eee73" class="outline-3">
<h3 id="orgf23381e"><span class="section-number-3">1.4</span> Analytical Model</h3> <h3 id="org67eee73"><span class="section-number-3">1.4</span> Analytical Model</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
</div> </div>
<div id="outline-container-org86e662a" class="outline-4"> <div id="outline-container-org1545adc" class="outline-4">
<h4 id="org86e662a"><span class="section-number-4">1.4.1</span> Parameters</h4> <h4 id="org1545adc"><span class="section-number-4">1.4.1</span> Parameters</h4>
<div class="outline-text-4" id="text-1-4-1"> <div class="outline-text-4" id="text-1-4-1">
<p> <p>
Control parameters Control parameters
@ -228,18 +228,17 @@ g_svd = 1e5;
System parameters System parameters
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">w0 = 2*pi*.5; % MinusK BM1 tablle <pre class="src src-matlab">l = 0.5; % Length of the mass [m]
la = 0.5; % Position of Act. [m]
l = 0.8; % [m] h = 1.7; % Height of the mass [m]
la = l; % [m] ha = 1.7; % Position of Act. [m]
h = 1.7; % [m] m = 400; % Mass [kg]
ha = h; % [m] I = 115; % Inertia [kg m^2]
m = 70; % [kg] k = 15e3; % Actuator Stiffness [N/m]
c = 0.03; % Actuator Damping [N/(m/s)]
k = 3e3; % [N/m]
I = 10; % [kg m^2]
</pre> </pre>
</div> </div>
@ -274,8 +273,8 @@ Frequency vector.
</div> </div>
</div> </div>
<div id="outline-container-orgd9883dd" class="outline-4"> <div id="outline-container-org22e1e2b" class="outline-4">
<h4 id="orgd9883dd"><span class="section-number-4">1.4.2</span> generation of the state space model</h4> <h4 id="org22e1e2b"><span class="section-number-4">1.4.2</span> generation of the state space model</h4>
<div class="outline-text-4" id="text-1-4-2"> <div class="outline-text-4" id="text-1-4-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">M = [m 0 0 <pre class="src src-matlab">M = [m 0 0
@ -296,7 +295,7 @@ Ja = [1 0 ha/2 %Left horizontal actuator
0 1 la/2]; %Right vertical actuator 0 1 la/2]; %Right vertical actuator
Jta = Ja'; Jta = Ja';
K = k*Jta*Ja; K = k*Jta*Ja;
C = 0.06*k*Jta*Ja; C = c*Jta*Ja;
E = [1 0 0 E = [1 0 0
0 1 0 0 1 0
@ -344,11 +343,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-orgf1cd403" class="outline-4"> <div id="outline-container-orgee11542" class="outline-4">
<h4 id="orgf1cd403"><span class="section-number-4">1.4.3</span> Comparison with the Simscape Model</h4> <h4 id="orgee11542"><span class="section-number-4">1.4.3</span> Comparison with the Simscape Model</h4>
<div class="outline-text-4" id="text-1-4-3"> <div class="outline-text-4" id="text-1-4-3">
<div id="org8683b2b" class="figure"> <div id="org2557d08" 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 3: </span>Comparison of the analytical and the Simscape models</p> <p><span class="figure-number">Figure 3: </span>Comparison of the analytical and the Simscape models</p>
@ -356,8 +355,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-orgc44aa74" class="outline-4"> <div id="outline-container-org9524774" class="outline-4">
<h4 id="orgc44aa74"><span class="section-number-4">1.4.4</span> Analysis</h4> <h4 id="org9524774"><span class="section-number-4">1.4.4</span> Analysis</h4>
<div class="outline-text-4" id="text-1-4-4"> <div class="outline-text-4" id="text-1-4-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">% figure <pre class="src src-matlab">% figure
@ -425,8 +424,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org56c5430" class="outline-4"> <div id="outline-container-org39b3c10" class="outline-4">
<h4 id="org56c5430"><span class="section-number-4">1.4.5</span> Control Section</h4> <h4 id="org39b3c10"><span class="section-number-4">1.4.5</span> Control Section</h4>
<div class="outline-text-4" id="text-1-4-5"> <div class="outline-text-4" id="text-1-4-5">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2*pi*10); <pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2*pi*10);
@ -566,8 +565,8 @@ legend('Control OFF','Decentralized control','Centralized control','SVD control'
</div> </div>
</div> </div>
<div id="outline-container-org2f4000c" class="outline-4"> <div id="outline-container-orgfd02af1" class="outline-4">
<h4 id="org2f4000c"><span class="section-number-4">1.4.6</span> Greshgorin radius</h4> <h4 id="orgfd02af1"><span class="section-number-4">1.4.6</span> Greshgorin radius</h4>
<div class="outline-text-4" id="text-1-4-6"> <div class="outline-text-4" id="text-1-4-6">
<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);
@ -613,8 +612,8 @@ ylabel('Greshgorin radius [-]');
</div> </div>
</div> </div>
<div id="outline-container-orgacfa62b" class="outline-4"> <div id="outline-container-org88c39eb" class="outline-4">
<h4 id="orgacfa62b"><span class="section-number-4">1.4.7</span> Injecting ground motion in the system to have the output</h4> <h4 id="org88c39eb"><span class="section-number-4">1.4.7</span> Injecting ground motion in the system to have the output</h4>
<div class="outline-text-4" id="text-1-4-7"> <div class="outline-text-4" id="text-1-4-7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Fr = logspace(-2,3,1e3); <pre class="src src-matlab">Fr = logspace(-2,3,1e3);
@ -670,15 +669,15 @@ rot = PHI(:,11,11);
</div> </div>
</div> </div>
<div id="outline-container-orgdbdd0b8" class="outline-2"> <div id="outline-container-org5a87ff1" class="outline-2">
<h2 id="orgdbdd0b8"><span class="section-number-2">2</span> Gravimeter - Functions</h2> <h2 id="org5a87ff1"><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-org72eb8fe" class="outline-3"> <div id="outline-container-org755e595" class="outline-3">
<h3 id="org72eb8fe"><span class="section-number-3">2.1</span> <code>align</code></h3> <h3 id="org755e595"><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="orgdf12c8c"></a> <a id="orga2dd16f"></a>
</p> </p>
<p> <p>
@ -707,11 +706,11 @@ end
</div> </div>
<div id="outline-container-orgf7acd4e" class="outline-3"> <div id="outline-container-org55b8479" class="outline-3">
<h3 id="orgf7acd4e"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3> <h3 id="org55b8479"><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="org8d6e185"></a> <a id="org536d2a8"></a>
</p> </p>
<p> <p>
@ -760,12 +759,12 @@ end
</div> </div>
</div> </div>
<div id="outline-container-org20de0f0" class="outline-2"> <div id="outline-container-orgb23d007" class="outline-2">
<h2 id="org20de0f0"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2> <h2 id="orgb23d007"><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">
</div> </div>
<div id="outline-container-orgf01ffb3" class="outline-3"> <div id="outline-container-org636de2e" class="outline-3">
<h3 id="orgf01ffb3"><span class="section-number-3">3.1</span> Jacobian</h3> <h3 id="org636de2e"><span class="section-number-3">3.1</span> Jacobian</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
First, the position of the &ldquo;joints&rdquo; (points of force application) are estimated and the Jacobian computed. First, the position of the &ldquo;joints&rdquo; (points of force application) are estimated and the Jacobian computed.
@ -807,8 +806,8 @@ save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
</div> </div>
</div> </div>
<div id="outline-container-orgab52d7f" class="outline-3"> <div id="outline-container-org59d2125" class="outline-3">
<h3 id="orgab52d7f"><span class="section-number-3">3.2</span> Simscape Model</h3> <h3 id="org59d2125"><span class="section-number-3">3.2</span> Simscape Model</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open('stewart_platform/drone_platform.slx'); <pre class="src src-matlab">open('stewart_platform/drone_platform.slx');
@ -839,8 +838,8 @@ We load the Jacobian.
</div> </div>
</div> </div>
<div id="outline-container-orgde1eb2c" class="outline-3"> <div id="outline-container-org77015bb" class="outline-3">
<h3 id="orgde1eb2c"><span class="section-number-3">3.3</span> Identification of the plant</h3> <h3 id="org77015bb"><span class="section-number-3">3.3</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform. The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
@ -897,32 +896,32 @@ Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
</div> </div>
</div> </div>
<div id="outline-container-orgef70e0f" class="outline-3"> <div id="outline-container-org21a398b" class="outline-3">
<h3 id="orgef70e0f"><span class="section-number-3">3.4</span> Obtained Dynamics</h3> <h3 id="org21a398b"><span class="section-number-3">3.4</span> Obtained Dynamics</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<div id="orgc85aa05" class="figure"> <div id="orgb43e051" 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 4: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p> <p><span class="figure-number">Figure 4: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
</div> </div>
<div id="orgc6ee760" class="figure"> <div id="org3b3901c" 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 5: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p> <p><span class="figure-number">Figure 5: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
</div> </div>
<div id="orgf90a83f" class="figure"> <div id="org089a60b" class="figure">
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" /> <p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p> <p><span class="figure-number">Figure 6: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p>
</div> </div>
<div id="org2b2322b" class="figure"> <div id="org2fb9fc4" class="figure">
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" /> <p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Transmissibility</p> <p><span class="figure-number">Figure 7: </span>Transmissibility</p>
@ -930,8 +929,8 @@ Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
</div> </div>
</div> </div>
<div id="outline-container-org97d95a3" class="outline-3"> <div id="outline-container-org6cab60a" class="outline-3">
<h3 id="org97d95a3"><span class="section-number-3">3.5</span> Real Approximation of \(G\) at the decoupling frequency</h3> <h3 id="org6cab60a"><span class="section-number-3">3.5</span> Real Approximation of \(G\) at the decoupling frequency</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-3-5">
<p> <p>
Let&rsquo;s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_c(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\). Let&rsquo;s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_c(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
@ -957,8 +956,8 @@ H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
</div> </div>
</div> </div>
<div id="outline-container-org2460007" class="outline-3"> <div id="outline-container-orgc331fa5" class="outline-3">
<h3 id="org2460007"><span class="section-number-3">3.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3> <h3 id="orgc331fa5"><span class="section-number-3">3.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-3-6"> <div class="outline-text-3" id="text-3-6">
<p> <p>
First, the Singular Value Decomposition of \(H_1\) is performed: First, the Singular Value Decomposition of \(H_1\) is performed:
@ -1026,7 +1025,7 @@ end
</div> </div>
<div id="org12b2460" class="figure"> <div id="orgb8a303c" 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 8: </span>Gershgorin Radii of the Coupled and Decoupled plants</p> <p><span class="figure-number">Figure 8: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
@ -1034,8 +1033,8 @@ end
</div> </div>
</div> </div>
<div id="outline-container-orgc89a913" class="outline-3"> <div id="outline-container-orge6123eb" class="outline-3">
<h3 id="orgc89a913"><span class="section-number-3">3.7</span> Decoupled Plant</h3> <h3 id="orge6123eb"><span class="section-number-3">3.7</span> Decoupled Plant</h3>
<div class="outline-text-3" id="text-3-7"> <div class="outline-text-3" id="text-3-7">
<p> <p>
Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\). Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
@ -1043,14 +1042,14 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</p> </p>
<div id="org46c3f4b" class="figure"> <div id="orgd9cd319" 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 9: </span>Decoupled Plant using SVD</p> <p><span class="figure-number">Figure 9: </span>Decoupled Plant using SVD</p>
</div> </div>
<div id="org69887b2" class="figure"> <div id="orgc1ec52e" 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 10: </span>Decoupled Plant using the Jacobian</p> <p><span class="figure-number">Figure 10: </span>Decoupled Plant using the Jacobian</p>
@ -1058,8 +1057,8 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</div> </div>
</div> </div>
<div id="outline-container-org3d3c34c" class="outline-3"> <div id="outline-container-orge9ddb65" class="outline-3">
<h3 id="org3d3c34c"><span class="section-number-3">3.8</span> Diagonal Controller</h3> <h3 id="orge9ddb65"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-3-8"> <div class="outline-text-3" id="text-3-8">
<p> <p>
The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\). The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
@ -1075,8 +1074,8 @@ K = eye(6)*C_g/(s+wc);
</div> </div>
</div> </div>
<div id="outline-container-orgaa29814" class="outline-3"> <div id="outline-container-orga25eea8" class="outline-3">
<h3 id="orgaa29814"><span class="section-number-3">3.9</span> Centralized Control</h3> <h3 id="orga25eea8"><span class="section-number-3">3.9</span> Centralized Control</h3>
<div class="outline-text-3" id="text-3-9"> <div class="outline-text-3" id="text-3-9">
<p> <p>
The control diagram for the centralized control is shown below. The control diagram for the centralized control is shown below.
@ -1100,8 +1099,8 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
</div> </div>
</div> </div>
<div id="outline-container-orge736e6a" class="outline-3"> <div id="outline-container-org4255891" class="outline-3">
<h3 id="orge736e6a"><span class="section-number-3">3.10</span> SVD Control</h3> <h3 id="org4255891"><span class="section-number-3">3.10</span> SVD Control</h3>
<div class="outline-text-3" id="text-3-10"> <div class="outline-text-3" id="text-3-10">
<p> <p>
The SVD control architecture is shown below. The SVD control architecture is shown below.
@ -1124,8 +1123,8 @@ SVD Control
</div> </div>
</div> </div>
<div id="outline-container-orgd1cfb41" class="outline-3"> <div id="outline-container-org535f13b" class="outline-3">
<h3 id="orgd1cfb41"><span class="section-number-3">3.11</span> Results</h3> <h3 id="org535f13b"><span class="section-number-3">3.11</span> Results</h3>
<div class="outline-text-3" id="text-3-11"> <div class="outline-text-3" id="text-3-11">
<p> <p>
Let&rsquo;s first verify the stability of the closed-loop systems: Let&rsquo;s first verify the stability of the closed-loop systems:
@ -1155,11 +1154,11 @@ 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="#orgce89643">13</a>. The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org59b4530">13</a>.
</p> </p>
<div id="orgce89643" class="figure"> <div id="org59b4530" 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 13: </span>Obtained Transmissibility</p> <p><span class="figure-number">Figure 13: </span>Obtained Transmissibility</p>
@ -1170,7 +1169,7 @@ 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: 2020-09-30 mer. 17:15</p> <p class="date">Created: 2020-09-30 mer. 17:21</p>
</div> </div>
</body> </body>
</html> </html>

19
index.org
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@ -213,18 +213,17 @@ Control parameters
System parameters System parameters
#+begin_src matlab #+begin_src matlab
w0 = 2*pi*.5; % MinusK BM1 tablle l = 0.5; % Length of the mass [m]
la = 0.5; % Position of Act. [m]
l = 0.8; % [m] h = 1.7; % Height of the mass [m]
la = l; % [m] ha = 1.7; % Position of Act. [m]
h = 1.7; % [m] m = 400; % Mass [kg]
ha = h; % [m] I = 115; % Inertia [kg m^2]
m = 70; % [kg] k = 15e3; % Actuator Stiffness [N/m]
c = 0.03; % Actuator Damping [N/(m/s)]
k = 3e3; % [N/m]
I = 10; % [kg m^2]
#+end_src #+end_src
Bode options. Bode options.
@ -270,7 +269,7 @@ Frequency vector.
0 1 la/2]; %Right vertical actuator 0 1 la/2]; %Right vertical actuator
Jta = Ja'; Jta = Ja';
K = k*Jta*Ja; K = k*Jta*Ja;
C = 0.06*k*Jta*Ja; C = c*Jta*Ja;
E = [1 0 0 E = [1 0 0
0 1 0 0 1 0