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<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>
<!-- 2020-10-05 lun. 18:06 --> <!-- 2020-10-05 lun. 18:28 -->
<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" />
@ -35,59 +35,59 @@
<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="#org4f444af">1. Gravimeter - Simscape Model</a> <li><a href="#org67dc64e">1. Gravimeter - Simscape Model</a>
<ul> <ul>
<li><a href="#org9c89f1b">1.1. Introduction</a></li> <li><a href="#orgbc83858">1.1. Introduction</a></li>
<li><a href="#org5c18570">1.2. Simscape Model - Parameters</a></li> <li><a href="#org6a10d93">1.2. Simscape Model - Parameters</a></li>
<li><a href="#org890db47">1.3. System Identification - Without Gravity</a></li> <li><a href="#org0efee8e">1.3. System Identification - Without Gravity</a></li>
<li><a href="#org8166361">1.4. System Identification - With Gravity</a></li> <li><a href="#org98fd3fd">1.4. System Identification - With Gravity</a></li>
<li><a href="#org1769947">1.5. Analytical Model</a> <li><a href="#org6400b2e">1.5. Analytical Model</a>
<ul> <ul>
<li><a href="#org6553335">1.5.1. Parameters</a></li> <li><a href="#orgd401b7a">1.5.1. Parameters</a></li>
<li><a href="#orgb9d8709">1.5.2. generation of the state space model</a></li> <li><a href="#orgdc4cf04">1.5.2. Generation of the State Space Model</a></li>
<li><a href="#org756ddd7">1.5.3. Comparison with the Simscape Model</a></li> <li><a href="#org2f36845">1.5.3. Comparison with the Simscape Model</a></li>
<li><a href="#orgbf8fd91">1.5.4. Analysis</a></li> <li><a href="#org028f15a">1.5.4. Analysis</a></li>
<li><a href="#org8a51806">1.5.5. Control Section</a></li> <li><a href="#orgaf39b24">1.5.5. Control Section</a></li>
<li><a href="#org28606d1">1.5.6. Greshgorin radius</a></li> <li><a href="#orga450746">1.5.6. Greshgorin radius</a></li>
<li><a href="#org6605d22">1.5.7. Injecting ground motion in the system to have the output</a></li> <li><a href="#orgd41a3f6">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="#org569be4b">2. Gravimeter - Functions</a> <li><a href="#org866fa85">2. Gravimeter - Functions</a>
<ul> <ul>
<li><a href="#orgb5adbf9">2.1. <code>align</code></a></li> <li><a href="#orgcf775e2">2.1. <code>align</code></a></li>
<li><a href="#orgdc0b26f">2.2. <code>pzmap_testCL</code></a></li> <li><a href="#org78f2c7e">2.2. <code>pzmap_testCL</code></a></li>
</ul> </ul>
</li> </li>
<li><a href="#org6b34180">3. Stewart Platform - Simscape Model</a> <li><a href="#orgd5b9491">3. Stewart Platform - Simscape Model</a>
<ul> <ul>
<li><a href="#org3ce25f0">3.1. Jacobian</a></li> <li><a href="#org6d58a07">3.1. Jacobian</a></li>
<li><a href="#org7544d0c">3.2. Simscape Model</a></li> <li><a href="#org4f58a34">3.2. Simscape Model</a></li>
<li><a href="#orgf131c5c">3.3. Identification of the plant</a></li> <li><a href="#org51c99d1">3.3. Identification of the plant</a></li>
<li><a href="#org9c8d3f6">3.4. Obtained Dynamics</a></li> <li><a href="#org84418dd">3.4. Obtained Dynamics</a></li>
<li><a href="#org3cbc2c6">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li> <li><a href="#org315ca7e">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#org2ed1102">3.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li> <li><a href="#org91c0ed9">3.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#org89bbbb9">3.7. Decoupled Plant</a></li> <li><a href="#org0bd0b38">3.7. Decoupled Plant</a></li>
<li><a href="#org16afb20">3.8. Diagonal Controller</a></li> <li><a href="#org4b22e32">3.8. Diagonal Controller</a></li>
<li><a href="#orgae885a2">3.9. Centralized Control</a></li> <li><a href="#orgac4cf9b">3.9. Centralized Control</a></li>
<li><a href="#org31ce44b">3.10. SVD Control</a></li> <li><a href="#org4ae317c">3.10. SVD Control</a></li>
<li><a href="#org6d772a3">3.11. Results</a></li> <li><a href="#orgabc897d">3.11. Results</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org4f444af" class="outline-2"> <div id="outline-container-org67dc64e" class="outline-2">
<h2 id="org4f444af"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2> <h2 id="org67dc64e"><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-org9c89f1b" class="outline-3"> <div id="outline-container-orgbc83858" class="outline-3">
<h3 id="org9c89f1b"><span class="section-number-3">1.1</span> Introduction</h3> <h3 id="orgbc83858"><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="orgfe166e2" class="figure"> <div id="orge6f0a72" 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>
@ -95,8 +95,8 @@
</div> </div>
</div> </div>
<div id="outline-container-org5c18570" class="outline-3"> <div id="outline-container-org6a10d93" class="outline-3">
<h3 id="org5c18570"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3> <h3 id="org6a10d93"><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>)
@ -127,8 +127,8 @@ g = 0; <span class="org-comment">% Gravity [m/s2]</span>
</div> </div>
</div> </div>
<div id="outline-container-org890db47" class="outline-3"> <div id="outline-container-org0efee8e" class="outline-3">
<h3 id="org890db47"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3> <h3 id="org0efee8e"><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>
@ -175,7 +175,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
<div id="org87b0848" class="figure"> <div id="orgf223fb8" 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>
@ -183,8 +183,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org8166361" class="outline-3"> <div id="outline-container-org98fd3fd" class="outline-3">
<h3 id="org8166361"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3> <h3 id="org98fd3fd"><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>
@ -213,7 +213,7 @@ ans =
</pre> </pre>
<div id="org69c29e0" class="figure"> <div id="org4d66bba" 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>
@ -221,12 +221,12 @@ ans =
</div> </div>
</div> </div>
<div id="outline-container-org1769947" class="outline-3"> <div id="outline-container-org6400b2e" class="outline-3">
<h3 id="org1769947"><span class="section-number-3">1.5</span> Analytical Model</h3> <h3 id="org6400b2e"><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-org6553335" class="outline-4"> <div id="outline-container-orgd401b7a" class="outline-4">
<h4 id="org6553335"><span class="section-number-4">1.5.1</span> Parameters</h4> <h4 id="orgd401b7a"><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.
@ -245,7 +245,6 @@ P.TickLabel.FontSize = 12;
P.Xlim = [1e<span class="org-type">-</span>1,1e2]; P.Xlim = [1e<span class="org-type">-</span>1,1e2];
P.MagLowerLimMode = <span class="org-string">'manual'</span>; P.MagLowerLimMode = <span class="org-string">'manual'</span>;
P.MagLowerLim= 1e<span class="org-type">-</span>3; P.MagLowerLim= 1e<span class="org-type">-</span>3;
<span class="org-comment">%P.PhaseVisible = 'off';</span>
</pre> </pre>
</div> </div>
@ -259,30 +258,62 @@ Frequency vector.
</div> </div>
</div> </div>
<div id="outline-container-orgb9d8709" class="outline-4"> <div id="outline-container-orgdc4cf04" class="outline-4">
<h4 id="orgb9d8709"><span class="section-number-4">1.5.2</span> generation of the state space model</h4> <h4 id="orgdc4cf04"><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>
Mass matrix
</p>
<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
0 m 0 0 m 0
0 0 I]; 0 0 I];
</pre>
</div>
<span class="org-comment">%Jacobian of the bottom sensor</span> <p>
Js1 = [1 0 h<span class="org-type">/</span>2 Jacobian of the bottom sensor
</p>
<div class="org-src-container">
<pre class="src src-matlab">Js1 = [1 0 h<span class="org-type">/</span>2
0 1 <span class="org-type">-</span>l<span class="org-type">/</span>2]; 0 1 <span class="org-type">-</span>l<span class="org-type">/</span>2];
<span class="org-comment">%Jacobian of the top sensor</span> </pre>
Js2 = [1 0 <span class="org-type">-</span>h<span class="org-type">/</span>2 </div>
0 1 0];
<span class="org-comment">%Jacobian of the actuators</span> <p>
Ja = [1 0 ha <span class="org-comment">% Left horizontal actuator</span> Jacobian of the top sensor
</p>
<div class="org-src-container">
<pre class="src src-matlab">Js2 = [1 0 <span class="org-type">-</span>h<span class="org-type">/</span>2
0 1 0];
</pre>
</div>
<p>
Jacobian of the actuators
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ja = [1 0 ha <span class="org-comment">% Left horizontal actuator</span>
0 1 <span class="org-type">-</span>la <span class="org-comment">% Left vertical actuator</span> 0 1 <span class="org-type">-</span>la <span class="org-comment">% Left vertical actuator</span>
0 1 la]; <span class="org-comment">% Right vertical actuator</span> 0 1 la]; <span class="org-comment">% Right vertical actuator</span>
Jta = Ja<span class="org-type">'</span>; Jta = Ja<span class="org-type">'</span>;
K = k<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja; </pre>
C = c<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja; </div>
E = [1 0 0 <p>
Stiffness and Damping matrices
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = k<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja;
C = c<span class="org-type">*</span>Jta<span class="org-type">*</span>Ja;
</pre>
</div>
<p>
State Space Matrices
</p>
<div class="org-src-container">
<pre class="src src-matlab">E = [1 0 0
0 1 0 0 1 0
0 0 1]; <span class="org-comment">%projecting ground motion in the directions of the legs</span> 0 0 1]; <span class="org-comment">%projecting ground motion in the directions of the legs</span>
@ -292,12 +323,6 @@ AA = [zeros(3) eye(3)
BB = [zeros(3,6) BB = [zeros(3,6)
M<span class="org-type">\</span>Jta M<span class="org-type">\</span>(k<span class="org-type">*</span>Jta<span class="org-type">*</span>E)]; M<span class="org-type">\</span>Jta M<span class="org-type">\</span>(k<span class="org-type">*</span>Jta<span class="org-type">*</span>E)];
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">BB </span></span><span class="org-comment">= [zeros(3,3)</span>
<span class="org-comment">% M\Jta ];</span>
<span class="org-comment">%</span>
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">CC </span></span><span class="org-comment">= [Ja zeros(3)];</span>
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">DD </span></span><span class="org-comment">= zeros(3,3);</span>
CC = [[Js1;Js2] zeros(4,3); CC = [[Js1;Js2] zeros(4,3);
zeros(2,6) zeros(2,6)
(Js1<span class="org-type">+</span>Js2)<span class="org-type">./</span>2 zeros(2,3) (Js1<span class="org-type">+</span>Js2)<span class="org-type">./</span>2 zeros(2,3)
@ -307,16 +332,23 @@ CC = [[Js1;Js2] zeros(4,3);
DD = [zeros(4,6) DD = [zeros(4,6)
zeros<span class="org-type">(2,3) eye(2,3)</span> zeros<span class="org-type">(2,3) eye(2,3)</span>
zeros<span class="org-type">(6,6)];</span> zeros<span class="org-type">(6,6)];</span>
system_dec = ss(AA,BB,CC,DD);
</pre> </pre>
</div> </div>
<p>
State Space model:
</p>
<ul class="org-ul"> <ul class="org-ul">
<li>Input = three actuators and three ground motions</li> <li>Input = three actuators and three ground motions</li>
<li>Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation</li> <li>Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation</li>
</ul> </ul>
<div class="org-src-container">
<pre class="src src-matlab">system_dec = ss(AA,BB,CC,DD);
</pre>
</div>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">size(system_dec) <pre class="src src-matlab">size(system_dec)
</pre> </pre>
@ -328,11 +360,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org756ddd7" class="outline-4"> <div id="outline-container-org2f36845" class="outline-4">
<h4 id="org756ddd7"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4> <h4 id="org2f36845"><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="orgcea07c6" class="figure"> <div id="orgd96c232" 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>
@ -340,8 +372,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-orgbf8fd91" class="outline-4"> <div id="outline-container-org028f15a" class="outline-4">
<h4 id="orgbf8fd91"><span class="section-number-4">1.5.4</span> Analysis</h4> <h4 id="org028f15a"><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>
@ -409,8 +441,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org8a51806" class="outline-4"> <div id="outline-container-orgaf39b24" class="outline-4">
<h4 id="org8a51806"><span class="section-number-4">1.5.5</span> Control Section</h4> <h4 id="orgaf39b24"><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);
@ -550,8 +582,8 @@ legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'D
</div> </div>
</div> </div>
<div id="outline-container-org28606d1" class="outline-4"> <div id="outline-container-orga450746" class="outline-4">
<h4 id="org28606d1"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4> <h4 id="orga450746"><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);
@ -597,8 +629,8 @@ ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
</div> </div>
</div> </div>
<div id="outline-container-org6605d22" class="outline-4"> <div id="outline-container-orgd41a3f6" class="outline-4">
<h4 id="org6605d22"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4> <h4 id="orgd41a3f6"><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);
@ -654,15 +686,15 @@ rot = PHI(<span class="org-type">:</span>,11,11);
</div> </div>
</div> </div>
<div id="outline-container-org569be4b" class="outline-2"> <div id="outline-container-org866fa85" class="outline-2">
<h2 id="org569be4b"><span class="section-number-2">2</span> Gravimeter - Functions</h2> <h2 id="org866fa85"><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-orgb5adbf9" class="outline-3"> <div id="outline-container-orgcf775e2" class="outline-3">
<h3 id="orgb5adbf9"><span class="section-number-3">2.1</span> <code>align</code></h3> <h3 id="orgcf775e2"><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="org3b1a721"></a> <a id="orgbb32c31"></a>
</p> </p>
<p> <p>
@ -691,11 +723,11 @@ This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
</div> </div>
<div id="outline-container-orgdc0b26f" class="outline-3"> <div id="outline-container-org78f2c7e" class="outline-3">
<h3 id="orgdc0b26f"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3> <h3 id="org78f2c7e"><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="org18df03d"></a> <a id="org655412c"></a>
</p> </p>
<p> <p>
@ -744,12 +776,12 @@ This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org6b34180" class="outline-2"> <div id="outline-container-orgd5b9491" class="outline-2">
<h2 id="org6b34180"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2> <h2 id="orgd5b9491"><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-org3ce25f0" class="outline-3"> <div id="outline-container-org6d58a07" class="outline-3">
<h3 id="org3ce25f0"><span class="section-number-3">3.1</span> Jacobian</h3> <h3 id="org6d58a07"><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.
@ -791,8 +823,8 @@ save(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">
</div> </div>
</div> </div>
<div id="outline-container-org7544d0c" class="outline-3"> <div id="outline-container-org4f58a34" class="outline-3">
<h3 id="org7544d0c"><span class="section-number-3">3.2</span> Simscape Model</h3> <h3 id="org4f58a34"><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(<span class="org-string">'stewart_platform/drone_platform.slx'</span>); <pre class="src src-matlab">open(<span class="org-string">'stewart_platform/drone_platform.slx'</span>);
@ -823,8 +855,8 @@ We load the Jacobian.
</div> </div>
</div> </div>
<div id="outline-container-orgf131c5c" class="outline-3"> <div id="outline-container-org51c99d1" class="outline-3">
<h3 id="orgf131c5c"><span class="section-number-3">3.3</span> Identification of the plant</h3> <h3 id="org51c99d1"><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.
@ -881,32 +913,32 @@ Gl.OutputName = {<span class="org-string">'A1'</span>, <span class="org-string"
</div> </div>
</div> </div>
<div id="outline-container-org9c8d3f6" class="outline-3"> <div id="outline-container-org84418dd" class="outline-3">
<h3 id="org9c8d3f6"><span class="section-number-3">3.4</span> Obtained Dynamics</h3> <h3 id="org84418dd"><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="orgbe1afe0" class="figure"> <div id="org77aab4b" 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 5: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p> <p><span class="figure-number">Figure 5: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
</div> </div>
<div id="orgb7a99f5" class="figure"> <div id="org9222b17" 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 6: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p> <p><span class="figure-number">Figure 6: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
</div> </div>
<div id="orgf03b0e7" class="figure"> <div id="org9d77253" 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 7: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p> <p><span class="figure-number">Figure 7: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p>
</div> </div>
<div id="orga22b4e1" class="figure"> <div id="org4cce08b" 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 8: </span>Transmissibility</p> <p><span class="figure-number">Figure 8: </span>Transmissibility</p>
@ -914,8 +946,8 @@ Gl.OutputName = {<span class="org-string">'A1'</span>, <span class="org-string"
</div> </div>
</div> </div>
<div id="outline-container-org3cbc2c6" class="outline-3"> <div id="outline-container-org315ca7e" class="outline-3">
<h3 id="org3cbc2c6"><span class="section-number-3">3.5</span> Real Approximation of \(G\) at the decoupling frequency</h3> <h3 id="org315ca7e"><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 \(\omega_c\).
@ -941,8 +973,8 @@ H1 = inv(D<span class="org-type">*</span>real(H1<span class="org-type">'*</span>
</div> </div>
</div> </div>
<div id="outline-container-org2ed1102" class="outline-3"> <div id="outline-container-org91c0ed9" class="outline-3">
<h3 id="org2ed1102"><span class="section-number-3">3.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3> <h3 id="org91c0ed9"><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:
@ -1010,7 +1042,7 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
</div> </div>
<div id="org040d1c3" class="figure"> <div id="orgda863a3" 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>
@ -1018,8 +1050,8 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
</div> </div>
</div> </div>
<div id="outline-container-org89bbbb9" class="outline-3"> <div id="outline-container-org0bd0b38" class="outline-3">
<h3 id="org89bbbb9"><span class="section-number-3">3.7</span> Decoupled Plant</h3> <h3 id="org0bd0b38"><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 \(G_d(s)\).
@ -1027,14 +1059,14 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</p> </p>
<div id="orge51c3a9" class="figure"> <div id="org6ba4690" 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="orgf570a47" class="figure"> <div id="org5342ca6" 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>
@ -1042,8 +1074,8 @@ Let&rsquo;s see the bode plot of the decoupled plant \(G_d(s)\).
</div> </div>
</div> </div>
<div id="outline-container-org16afb20" class="outline-3"> <div id="outline-container-org4b22e32" class="outline-3">
<h3 id="org16afb20"><span class="section-number-3">3.8</span> Diagonal Controller</h3> <h3 id="org4b22e32"><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 \(C_g\).
@ -1059,8 +1091,8 @@ K = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<sp
</div> </div>
</div> </div>
<div id="outline-container-orgae885a2" class="outline-3"> <div id="outline-container-orgac4cf9b" class="outline-3">
<h3 id="orgae885a2"><span class="section-number-3">3.9</span> Centralized Control</h3> <h3 id="orgac4cf9b"><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.
@ -1084,8 +1116,8 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
</div> </div>
</div> </div>
<div id="outline-container-org31ce44b" class="outline-3"> <div id="outline-container-org4ae317c" class="outline-3">
<h3 id="org31ce44b"><span class="section-number-3">3.10</span> SVD Control</h3> <h3 id="org4ae317c"><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.
@ -1108,8 +1140,8 @@ SVD Control
</div> </div>
</div> </div>
<div id="outline-container-org6d772a3" class="outline-3"> <div id="outline-container-orgabc897d" class="outline-3">
<h3 id="org6d772a3"><span class="section-number-3">3.11</span> Results</h3> <h3 id="orgabc897d"><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:
@ -1139,11 +1171,11 @@ 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="#orgd6d0049">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="#org92a495c">14</a>.
</p> </p>
<div id="orgd6d0049" class="figure"> <div id="org92a495c" 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>
@ -1154,7 +1186,7 @@ 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: 2020-10-05 lun. 18:06</p> <p class="date">Created: 2020-10-05 lun. 18:28</p>
</div> </div>
</body> </body>
</html> </html>

38
index.org
View File

@ -226,7 +226,6 @@ Bode options.
P.Xlim = [1e-1,1e2]; P.Xlim = [1e-1,1e2];
P.MagLowerLimMode = 'manual'; P.MagLowerLimMode = 'manual';
P.MagLowerLim= 1e-3; P.MagLowerLim= 1e-3;
%P.PhaseVisible = 'off';
#+end_src #+end_src
Frequency vector. Frequency vector.
@ -234,27 +233,42 @@ Frequency vector.
w = 2*pi*logspace(-1,2,1000); % [rad/s] w = 2*pi*logspace(-1,2,1000); % [rad/s]
#+end_src #+end_src
*** generation of the state space model *** Generation of the State Space Model
Mass matrix
#+begin_src matlab #+begin_src matlab
M = [m 0 0 M = [m 0 0
0 m 0 0 m 0
0 0 I]; 0 0 I];
#+end_src
%Jacobian of the bottom sensor Jacobian of the bottom sensor
#+begin_src matlab
Js1 = [1 0 h/2 Js1 = [1 0 h/2
0 1 -l/2]; 0 1 -l/2];
%Jacobian of the top sensor #+end_src
Jacobian of the top sensor
#+begin_src matlab
Js2 = [1 0 -h/2 Js2 = [1 0 -h/2
0 1 0]; 0 1 0];
#+end_src
%Jacobian of the actuators Jacobian of the actuators
#+begin_src matlab
Ja = [1 0 ha % Left horizontal actuator Ja = [1 0 ha % Left horizontal actuator
0 1 -la % Left vertical actuator 0 1 -la % Left vertical actuator
0 1 la]; % Right vertical actuator 0 1 la]; % Right vertical actuator
Jta = Ja'; Jta = Ja';
#+end_src
Stiffness and Damping matrices
#+begin_src matlab
K = k*Jta*Ja; K = k*Jta*Ja;
C = c*Jta*Ja; C = c*Jta*Ja;
#+end_src
State Space Matrices
#+begin_src matlab
E = [1 0 0 E = [1 0 0
0 1 0 0 1 0
0 0 1]; %projecting ground motion in the directions of the legs 0 0 1]; %projecting ground motion in the directions of the legs
@ -265,12 +279,6 @@ Frequency vector.
BB = [zeros(3,6) BB = [zeros(3,6)
M\Jta M\(k*Jta*E)]; M\Jta M\(k*Jta*E)];
% BB = [zeros(3,3)
% M\Jta ];
%
% CC = [Ja zeros(3)];
% DD = zeros(3,3);
CC = [[Js1;Js2] zeros(4,3); CC = [[Js1;Js2] zeros(4,3);
zeros(2,6) zeros(2,6)
(Js1+Js2)./2 zeros(2,3) (Js1+Js2)./2 zeros(2,3)
@ -280,12 +288,16 @@ Frequency vector.
DD = [zeros(4,6) DD = [zeros(4,6)
zeros(2,3) eye(2,3) zeros(2,3) eye(2,3)
zeros(6,6)]; zeros(6,6)];
#+end_src
State Space model:
- Input = three actuators and three ground motions
- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation
#+begin_src matlab
system_dec = ss(AA,BB,CC,DD); system_dec = ss(AA,BB,CC,DD);
#+end_src #+end_src
- Input = three actuators and three ground motions
- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation
#+begin_src matlab :results output replace #+begin_src matlab :results output replace
size(system_dec) size(system_dec)