Remove Gravity for the Stewart platform model
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-10-09 ven. 16:21 -->
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<!-- 2020-10-13 mar. 14:53 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>SVD Control</title>
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<meta name="generator" content="Org mode" />
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@ -35,75 +35,75 @@
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org90d7008">1. Gravimeter - Simscape Model</a>
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<li><a href="#org6dd65c1">1. Gravimeter - Simscape Model</a>
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<ul>
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<li><a href="#org29b9308">1.1. Introduction</a></li>
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<li><a href="#orgd333b87">1.2. Simscape Model - Parameters</a></li>
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<li><a href="#org09b581d">1.3. System Identification - Without Gravity</a></li>
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<li><a href="#org4f091cc">1.4. System Identification - With Gravity</a></li>
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<li><a href="#org7c4effc">1.5. Analytical Model</a>
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<li><a href="#org85dbe5c">1.1. Introduction</a></li>
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||||
<li><a href="#org0b31481">1.2. Simscape Model - Parameters</a></li>
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<li><a href="#org949338c">1.3. System Identification - Without Gravity</a></li>
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<li><a href="#org3e8d708">1.4. System Identification - With Gravity</a></li>
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<li><a href="#org8263a33">1.5. Analytical Model</a>
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<ul>
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<li><a href="#org20ea2aa">1.5.1. Parameters</a></li>
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<li><a href="#org02cb447">1.5.2. Generation of the State Space Model</a></li>
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<li><a href="#org9417f40">1.5.3. Comparison with the Simscape Model</a></li>
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<li><a href="#org6c56e64">1.5.4. Analysis</a></li>
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<li><a href="#orgeb20c08">1.5.5. Control Section</a></li>
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<li><a href="#org931022f">1.5.6. Greshgorin radius</a></li>
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<li><a href="#org1d56ec4">1.5.7. Injecting ground motion in the system to have the output</a></li>
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<li><a href="#org5ce809b">1.5.1. Parameters</a></li>
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<li><a href="#org485b7e0">1.5.2. Generation of the State Space Model</a></li>
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<li><a href="#orgb77d12b">1.5.3. Comparison with the Simscape Model</a></li>
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<li><a href="#orgbede3a4">1.5.4. Analysis</a></li>
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<li><a href="#org00d06a7">1.5.5. Control Section</a></li>
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<li><a href="#org8d48657">1.5.6. Greshgorin radius</a></li>
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<li><a href="#org7348f99">1.5.7. Injecting ground motion in the system to have the output</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org36d6b85">2. Gravimeter - Functions</a>
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<li><a href="#org534f1d2">2. Gravimeter - Functions</a>
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<ul>
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<li><a href="#orgbb4529b">2.1. <code>align</code></a></li>
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<li><a href="#orge0ed8bf">2.2. <code>pzmap_testCL</code></a></li>
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<li><a href="#org8fd3468">2.1. <code>align</code></a></li>
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<li><a href="#org7fc9d1b">2.2. <code>pzmap_testCL</code></a></li>
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</ul>
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</li>
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<li><a href="#org5afd29d">3. Stewart Platform - Simscape Model</a>
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<li><a href="#orga726921">3. Stewart Platform - Simscape Model</a>
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<ul>
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<li><a href="#orgff944f3">3.1. Jacobian</a></li>
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<li><a href="#org07ffe6c">3.2. Simscape Model</a></li>
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<li><a href="#org9aaf0d3">3.3. Identification of the plant</a></li>
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<li><a href="#orgb0b01e3">3.4. Obtained Dynamics</a></li>
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<li><a href="#org1de55ce">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
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<li><a href="#org53d60e1">3.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
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<li><a href="#org40c1d24">3.7. Decoupled Plant</a></li>
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<li><a href="#orgdfcd158">3.8. Diagonal Controller</a></li>
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<li><a href="#org25e3b35">3.9. Centralized Control</a></li>
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<li><a href="#org4d83793">3.10. SVD Control</a></li>
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<li><a href="#org7cece79">3.11. Results</a></li>
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<li><a href="#org0f4c378">3.1. Jacobian</a></li>
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<li><a href="#org8e93915">3.2. Simscape Model</a></li>
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<li><a href="#orga80ad9d">3.3. Identification of the plant</a></li>
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<li><a href="#org820395d">3.4. Obtained Dynamics</a></li>
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<li><a href="#org531c180">3.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
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<li><a href="#org04886ad">3.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
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<li><a href="#org96683a8">3.7. Decoupled Plant</a></li>
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<li><a href="#orgde9fab8">3.8. Diagonal Controller</a></li>
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<li><a href="#org47bbca8">3.9. Centralized Control</a></li>
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<li><a href="#org2c1e3f7">3.10. SVD Control</a></li>
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<li><a href="#orgd6985da">3.11. Results</a></li>
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</ul>
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</li>
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<li><a href="#org8b11aba">4. Stewart Platform - Analytical Model</a>
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<li><a href="#org99c6262">4. Stewart Platform - Analytical Model</a>
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<ul>
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<li><a href="#org2a175f6">4.1. Characteristics</a></li>
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<li><a href="#org9efa4f4">4.2. Mass Matrix</a></li>
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<li><a href="#org97bc497">4.3. Jacobian Matrix</a></li>
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<li><a href="#org7c9679d">4.4. Stifnness matrix and Damping matrix</a></li>
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<li><a href="#org00e8691">4.5. State Space System</a></li>
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<li><a href="#org8a70996">4.6. Transmissibility</a></li>
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<li><a href="#org12c95c9">4.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li>
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<li><a href="#orgc58b81c">4.8. Coupled and Decoupled Plant “Gershgorin Radii”</a></li>
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<li><a href="#org2ba91f6">4.9. Decoupled Plant</a></li>
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<li><a href="#orgc73a283">4.10. Controller</a></li>
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<li><a href="#org9c82ee4">4.11. Closed Loop System</a></li>
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<li><a href="#org80cd406">4.12. Results</a></li>
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<li><a href="#org6e044dd">4.1. Characteristics</a></li>
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<li><a href="#org20b7c2e">4.2. Mass Matrix</a></li>
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<li><a href="#org2f016df">4.3. Jacobian Matrix</a></li>
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<li><a href="#org2c9ff6d">4.4. Stifnness and Damping matrices</a></li>
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<li><a href="#orgffba0a8">4.5. State Space System</a></li>
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<li><a href="#org42b1b07">4.6. Transmissibility</a></li>
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<li><a href="#org38c8159">4.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li>
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<li><a href="#org477b3ce">4.8. Coupled and Decoupled Plant “Gershgorin Radii”</a></li>
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<li><a href="#orgde4eec1">4.9. Decoupled Plant</a></li>
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<li><a href="#org11b0182">4.10. Controller</a></li>
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<li><a href="#org5c893a8">4.11. Closed Loop System</a></li>
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<li><a href="#orgb1c0711">4.12. Results</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org90d7008" class="outline-2">
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<h2 id="org90d7008"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
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<div id="outline-container-org6dd65c1" class="outline-2">
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<h2 id="org6dd65c1"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org29b9308" class="outline-3">
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<h3 id="org29b9308"><span class="section-number-3">1.1</span> Introduction</h3>
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<div id="outline-container-org85dbe5c" class="outline-3">
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<h3 id="org85dbe5c"><span class="section-number-3">1.1</span> Introduction</h3>
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<div class="outline-text-3" id="text-1-1">
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<div id="org7df72f4" class="figure">
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<div id="org02345c4" class="figure">
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<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
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@ -111,8 +111,8 @@
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</div>
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</div>
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<div id="outline-container-orgd333b87" class="outline-3">
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<h3 id="orgd333b87"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
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<div id="outline-container-org0b31481" class="outline-3">
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<h3 id="org0b31481"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>)
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@ -143,8 +143,8 @@ g = 0; <span class="org-comment">% Gravity [m/s2]</span>
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</div>
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</div>
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<div id="outline-container-org09b581d" class="outline-3">
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<h3 id="org09b581d"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
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<div id="outline-container-org949338c" class="outline-3">
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<h3 id="org949338c"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
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<div class="outline-text-3" id="text-1-3">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
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@ -191,7 +191,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
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<div id="orgdd275bb" class="figure">
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<div id="orga082635" class="figure">
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<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
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@ -199,8 +199,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
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</div>
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</div>
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<div id="outline-container-org4f091cc" class="outline-3">
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<h3 id="org4f091cc"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
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<div id="outline-container-org3e8d708" class="outline-3">
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<h3 id="org3e8d708"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
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<div class="outline-text-3" id="text-1-4">
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<div class="org-src-container">
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<pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span>
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@ -229,7 +229,7 @@ ans =
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</pre>
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<div id="org392bf82" class="figure">
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<div id="org1a94741" class="figure">
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<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
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@ -237,12 +237,12 @@ ans =
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</div>
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</div>
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<div id="outline-container-org7c4effc" class="outline-3">
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<h3 id="org7c4effc"><span class="section-number-3">1.5</span> Analytical Model</h3>
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<div id="outline-container-org8263a33" class="outline-3">
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||||
<h3 id="org8263a33"><span class="section-number-3">1.5</span> Analytical Model</h3>
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<div class="outline-text-3" id="text-1-5">
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</div>
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<div id="outline-container-org20ea2aa" class="outline-4">
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||||
<h4 id="org20ea2aa"><span class="section-number-4">1.5.1</span> Parameters</h4>
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<div id="outline-container-org5ce809b" class="outline-4">
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||||
<h4 id="org5ce809b"><span class="section-number-4">1.5.1</span> Parameters</h4>
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<div class="outline-text-4" id="text-1-5-1">
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<p>
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Bode options.
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@ -274,8 +274,8 @@ Frequency vector.
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</div>
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</div>
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<div id="outline-container-org02cb447" class="outline-4">
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<h4 id="org02cb447"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
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<div id="outline-container-org485b7e0" class="outline-4">
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<h4 id="org485b7e0"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
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<div class="outline-text-4" id="text-1-5-2">
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<p>
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Mass matrix
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@ -376,11 +376,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
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</div>
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</div>
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<div id="outline-container-org9417f40" class="outline-4">
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<h4 id="org9417f40"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
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<div id="outline-container-orgb77d12b" class="outline-4">
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<h4 id="orgb77d12b"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
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<div class="outline-text-4" id="text-1-5-3">
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|
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<div id="orga6f165d" class="figure">
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<div id="org7bbc6ef" class="figure">
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<p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p>
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@ -388,8 +388,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
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</div>
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</div>
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<div id="outline-container-org6c56e64" class="outline-4">
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<h4 id="org6c56e64"><span class="section-number-4">1.5.4</span> Analysis</h4>
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<div id="outline-container-orgbede3a4" class="outline-4">
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<h4 id="orgbede3a4"><span class="section-number-4">1.5.4</span> Analysis</h4>
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<div class="outline-text-4" id="text-1-5-4">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-comment">% figure</span>
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@ -457,8 +457,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
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</div>
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</div>
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||||
|
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<div id="outline-container-orgeb20c08" class="outline-4">
|
||||
<h4 id="orgeb20c08"><span class="section-number-4">1.5.5</span> Control Section</h4>
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||||
<div id="outline-container-org00d06a7" class="outline-4">
|
||||
<h4 id="org00d06a7"><span class="section-number-4">1.5.5</span> Control Section</h4>
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||||
<div class="outline-text-4" id="text-1-5-5">
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||||
<div class="org-src-container">
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||||
<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);
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@ -598,8 +598,8 @@ legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'D
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</div>
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</div>
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<div id="outline-container-org931022f" class="outline-4">
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<h4 id="org931022f"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
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<div id="outline-container-org8d48657" class="outline-4">
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<h4 id="org8d48657"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
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<div class="outline-text-4" id="text-1-5-6">
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<div class="org-src-container">
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<pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w);
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@ -645,8 +645,8 @@ ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
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</div>
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</div>
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<div id="outline-container-org1d56ec4" class="outline-4">
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<h4 id="org1d56ec4"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
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<div id="outline-container-org7348f99" class="outline-4">
|
||||
<h4 id="org7348f99"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
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<div class="outline-text-4" id="text-1-5-7">
|
||||
<div class="org-src-container">
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<pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3);
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@ -702,15 +702,15 @@ rot = PHI(<span class="org-type">:</span>,11,11);
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</div>
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</div>
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<div id="outline-container-org36d6b85" class="outline-2">
|
||||
<h2 id="org36d6b85"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
|
||||
<div id="outline-container-org534f1d2" class="outline-2">
|
||||
<h2 id="org534f1d2"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
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||||
<div class="outline-text-2" id="text-2">
|
||||
</div>
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||||
<div id="outline-container-orgbb4529b" class="outline-3">
|
||||
<h3 id="orgbb4529b"><span class="section-number-3">2.1</span> <code>align</code></h3>
|
||||
<div id="outline-container-org8fd3468" class="outline-3">
|
||||
<h3 id="org8fd3468"><span class="section-number-3">2.1</span> <code>align</code></h3>
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||||
<div class="outline-text-3" id="text-2-1">
|
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<p>
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<a id="orgf2b803a"></a>
|
||||
<a id="org95a25f3"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -739,11 +739,11 @@ This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orge0ed8bf" class="outline-3">
|
||||
<h3 id="orge0ed8bf"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
|
||||
<div id="outline-container-org7fc9d1b" class="outline-3">
|
||||
<h3 id="org7fc9d1b"><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="orgf08bacf"></a>
|
||||
<a id="orge776d7f"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -792,12 +792,12 @@ This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5afd29d" class="outline-2">
|
||||
<h2 id="org5afd29d"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
|
||||
<div id="outline-container-orga726921" class="outline-2">
|
||||
<h2 id="orga726921"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
</div>
|
||||
<div id="outline-container-orgff944f3" class="outline-3">
|
||||
<h3 id="orgff944f3"><span class="section-number-3">3.1</span> Jacobian</h3>
|
||||
<div id="outline-container-org0f4c378" class="outline-3">
|
||||
<h3 id="org0f4c378"><span class="section-number-3">3.1</span> Jacobian</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
First, the position of the “joints” (points of force application) are estimated and the Jacobian computed.
|
||||
@ -839,11 +839,11 @@ save(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org07ffe6c" class="outline-3">
|
||||
<h3 id="org07ffe6c"><span class="section-number-3">3.2</span> Simscape Model</h3>
|
||||
<div id="outline-container-org8e93915" class="outline-3">
|
||||
<h3 id="org8e93915"><span class="section-number-3">3.2</span> Simscape Model</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<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">'drone_platform.slx'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -851,9 +851,9 @@ save(<span class="org-string">'./jacobian.mat'</span>, <span class="org-string">
|
||||
Definition of spring parameters
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">kx = 50; <span class="org-comment">% [N/m]</span>
|
||||
ky = 50;
|
||||
kz = 50;
|
||||
<pre class="src src-matlab">kx = 0.5<span class="org-type">*</span>1e3<span class="org-type">/</span>3; <span class="org-comment">% [N/m]</span>
|
||||
ky = 0.5<span class="org-type">*</span>1e3<span class="org-type">/</span>3;
|
||||
kz = 1e3<span class="org-type">/</span>3;
|
||||
|
||||
cx = 0.025; <span class="org-comment">% [Nm/rad]</span>
|
||||
cy = 0.025;
|
||||
@ -871,8 +871,8 @@ We load the Jacobian.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9aaf0d3" class="outline-3">
|
||||
<h3 id="org9aaf0d3"><span class="section-number-3">3.3</span> Identification of the plant</h3>
|
||||
<div id="outline-container-orga80ad9d" class="outline-3">
|
||||
<h3 id="orga80ad9d"><span class="section-number-3">3.3</span> Identification of the plant</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
|
||||
@ -929,32 +929,32 @@ Gl.OutputName = {<span class="org-string">'A1'</span>, <span class="org-string"
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb0b01e3" class="outline-3">
|
||||
<h3 id="orgb0b01e3"><span class="section-number-3">3.4</span> Obtained Dynamics</h3>
|
||||
<div id="outline-container-org820395d" class="outline-3">
|
||||
<h3 id="org820395d"><span class="section-number-3">3.4</span> Obtained Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
|
||||
<div id="org15e1aeb" class="figure">
|
||||
<div id="orgf45efb1" class="figure">
|
||||
<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
|
||||
</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 id="org1a9b1c6" class="figure">
|
||||
<div id="org7a9f376" class="figure">
|
||||
<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
|
||||
</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 id="org2c0cea0" class="figure">
|
||||
<div id="org01ccd4c" class="figure">
|
||||
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
|
||||
</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 id="org46a471a" class="figure">
|
||||
<div id="org18ecae5" class="figure">
|
||||
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Transmissibility</p>
|
||||
@ -962,8 +962,8 @@ Gl.OutputName = {<span class="org-string">'A1'</span>, <span class="org-string"
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1de55ce" class="outline-3">
|
||||
<h3 id="org1de55ce"><span class="section-number-3">3.5</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
||||
<div id="outline-container-org531c180" class="outline-3">
|
||||
<h3 id="org531c180"><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">
|
||||
<p>
|
||||
Let’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 \(\omega_c\).
|
||||
@ -989,8 +989,8 @@ H1 = inv(D<span class="org-type">*</span>real(H1<span class="org-type">'*</span>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org53d60e1" class="outline-3">
|
||||
<h3 id="org53d60e1"><span class="section-number-3">3.6</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div id="outline-container-org04886ad" class="outline-3">
|
||||
<h3 id="org04886ad"><span class="section-number-3">3.6</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||
<div class="outline-text-3" id="text-3-6">
|
||||
<p>
|
||||
First, the Singular Value Decomposition of \(H_1\) is performed:
|
||||
@ -1058,7 +1058,7 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6065705" class="figure">
|
||||
<div id="org98d0c86" 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>
|
||||
@ -1066,8 +1066,8 @@ H = abs(squeeze(freqresp(Gj, freqs, <span class="org-string">'Hz'</span>)));
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org40c1d24" class="outline-3">
|
||||
<h3 id="org40c1d24"><span class="section-number-3">3.7</span> Decoupled Plant</h3>
|
||||
<div id="outline-container-org96683a8" class="outline-3">
|
||||
<h3 id="org96683a8"><span class="section-number-3">3.7</span> Decoupled Plant</h3>
|
||||
<div class="outline-text-3" id="text-3-7">
|
||||
<p>
|
||||
Let’s see the bode plot of the decoupled plant \(G_d(s)\).
|
||||
@ -1075,14 +1075,14 @@ Let’s see the bode plot of the decoupled plant \(G_d(s)\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgbfa07c9" class="figure">
|
||||
<div id="org2351e85" 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="org28978a4" class="figure">
|
||||
<div id="org6699d5a" 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>
|
||||
@ -1090,8 +1090,8 @@ Let’s see the bode plot of the decoupled plant \(G_d(s)\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdfcd158" class="outline-3">
|
||||
<h3 id="orgdfcd158"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
|
||||
<div id="outline-container-orgde9fab8" class="outline-3">
|
||||
<h3 id="orgde9fab8"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
|
||||
<div class="outline-text-3" id="text-3-8">
|
||||
<p>
|
||||
The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
|
||||
@ -1107,8 +1107,8 @@ K = eye(6)<span class="org-type">*</span>C_g<span class="org-type">/</span>(s<sp
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org25e3b35" class="outline-3">
|
||||
<h3 id="org25e3b35"><span class="section-number-3">3.9</span> Centralized Control</h3>
|
||||
<div id="outline-container-org47bbca8" class="outline-3">
|
||||
<h3 id="org47bbca8"><span class="section-number-3">3.9</span> Centralized Control</h3>
|
||||
<div class="outline-text-3" id="text-3-9">
|
||||
<p>
|
||||
The control diagram for the centralized control is shown below.
|
||||
@ -1132,8 +1132,8 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4d83793" class="outline-3">
|
||||
<h3 id="org4d83793"><span class="section-number-3">3.10</span> SVD Control</h3>
|
||||
<div id="outline-container-org2c1e3f7" class="outline-3">
|
||||
<h3 id="org2c1e3f7"><span class="section-number-3">3.10</span> SVD Control</h3>
|
||||
<div class="outline-text-3" id="text-3-10">
|
||||
<p>
|
||||
The SVD control architecture is shown below.
|
||||
@ -1156,8 +1156,8 @@ SVD Control
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7cece79" class="outline-3">
|
||||
<h3 id="org7cece79"><span class="section-number-3">3.11</span> Results</h3>
|
||||
<div id="outline-container-orgd6985da" class="outline-3">
|
||||
<h3 id="orgd6985da"><span class="section-number-3">3.11</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-3-11">
|
||||
<p>
|
||||
Let’s first verify the stability of the closed-loop systems:
|
||||
@ -1182,16 +1182,16 @@ ans =
|
||||
<pre class="example">
|
||||
ans =
|
||||
logical
|
||||
1
|
||||
0
|
||||
</pre>
|
||||
|
||||
|
||||
<p>
|
||||
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org62fae46">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="#org0856618">14</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org62fae46" class="figure">
|
||||
<div id="org0856618" 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>
|
||||
@ -1200,83 +1200,85 @@ The obtained transmissibility in Open-loop, for the centralized control as well
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8b11aba" class="outline-2">
|
||||
<h2 id="org8b11aba"><span class="section-number-2">4</span> Stewart Platform - Analytical Model</h2>
|
||||
<div id="outline-container-org99c6262" class="outline-2">
|
||||
<h2 id="org99c6262"><span class="section-number-2">4</span> Stewart Platform - Analytical Model</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
</div>
|
||||
<div id="outline-container-org2a175f6" class="outline-3">
|
||||
<h3 id="org2a175f6"><span class="section-number-3">4.1</span> Characteristics</h3>
|
||||
<div id="outline-container-org6e044dd" class="outline-3">
|
||||
<h3 id="org6e044dd"><span class="section-number-3">4.1</span> Characteristics</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">L = 0.055;
|
||||
Zc = 0;
|
||||
m = 0.2;
|
||||
k = 1e3;
|
||||
c = 2<span class="org-type">*</span>0.1<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m);
|
||||
<pre class="src src-matlab">L = 0.055; <span class="org-comment">% Leg length [m]</span>
|
||||
Zc = 0; <span class="org-comment">% ?</span>
|
||||
m = 0.2; <span class="org-comment">% Top platform mass [m]</span>
|
||||
k = 1e3; <span class="org-comment">% Total vertical stiffness [N/m]</span>
|
||||
c = 2<span class="org-type">*</span>0.1<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m); <span class="org-comment">% Damping ? [N/(m/s)]</span>
|
||||
|
||||
Rx = 0.04;
|
||||
Rz = 0.04;
|
||||
Ix = m<span class="org-type">*</span>Rx<span class="org-type">^</span>2;
|
||||
Iy = m<span class="org-type">*</span>Rx<span class="org-type">^</span>2;
|
||||
Iz = m<span class="org-type">*</span>Rz<span class="org-type">^</span>2;
|
||||
Rx = 0.04; <span class="org-comment">% ?</span>
|
||||
Rz = 0.04; <span class="org-comment">% ?</span>
|
||||
Ix = m<span class="org-type">*</span>Rx<span class="org-type">^</span>2; <span class="org-comment">% ?</span>
|
||||
Iy = m<span class="org-type">*</span>Rx<span class="org-type">^</span>2; <span class="org-comment">% ?</span>
|
||||
Iz = m<span class="org-type">*</span>Rz<span class="org-type">^</span>2; <span class="org-comment">% ?</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9efa4f4" class="outline-3">
|
||||
<h3 id="org9efa4f4"><span class="section-number-3">4.2</span> Mass Matrix</h3>
|
||||
<div id="outline-container-org20b7c2e" class="outline-3">
|
||||
<h3 id="org20b7c2e"><span class="section-number-3">4.2</span> Mass Matrix</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">M = m<span class="org-type">*</span>[1 0 0 0 Zc 0;
|
||||
0 1 0 <span class="org-type">-</span>Zc 0 0;
|
||||
0 0 1 0 0 0;
|
||||
0 <span class="org-type">-</span>Zc 0 Rx<span class="org-type">^</span>2<span class="org-type">+</span>Zc<span class="org-type">^</span>2 0 0;
|
||||
Zc 0 0 0 Rx<span class="org-type">^</span>2<span class="org-type">+</span>Zc<span class="org-type">^</span>2 0;
|
||||
0 0 0 0 0 Rz<span class="org-type">^</span>2];
|
||||
<pre class="src src-matlab">M = m<span class="org-type">*</span>[1 0 0 0 Zc 0;
|
||||
0 1 0 <span class="org-type">-</span>Zc 0 0;
|
||||
0 0 1 0 0 0;
|
||||
0 <span class="org-type">-</span>Zc 0 Rx<span class="org-type">^</span>2<span class="org-type">+</span>Zc<span class="org-type">^</span>2 0 0;
|
||||
Zc 0 0 0 Rx<span class="org-type">^</span>2<span class="org-type">+</span>Zc<span class="org-type">^</span>2 0;
|
||||
0 0 0 0 0 Rz<span class="org-type">^</span>2];
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org97bc497" class="outline-3">
|
||||
<h3 id="org97bc497"><span class="section-number-3">4.3</span> Jacobian Matrix</h3>
|
||||
<div id="outline-container-org2f016df" class="outline-3">
|
||||
<h3 id="org2f016df"><span class="section-number-3">4.3</span> Jacobian Matrix</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Bj=1<span class="org-type">/</span>sqrt(6)<span class="org-type">*</span>[ 1 1 <span class="org-type">-</span>2 1 1 <span class="org-type">-</span>2;
|
||||
sqrt<span class="org-type">(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;</span>
|
||||
sqrt<span class="org-type">(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);</span>
|
||||
0 0 L L <span class="org-type">-</span>L <span class="org-type">-</span>L;
|
||||
<span class="org-type">-</span>L<span class="org-type">*</span>2<span class="org-type">/</span>sqrt(3) <span class="org-type">-</span>L<span class="org-type">*</span>2<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3);
|
||||
L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2) L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2) L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2)];
|
||||
<pre class="src src-matlab">Bj=1<span class="org-type">/</span>sqrt(6)<span class="org-type">*</span>[ 1 1 <span class="org-type">-</span>2 1 1 <span class="org-type">-</span>2;
|
||||
sqrt<span class="org-type">(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;</span>
|
||||
sqrt<span class="org-type">(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);</span>
|
||||
0 0 L L <span class="org-type">-</span>L <span class="org-type">-</span>L;
|
||||
<span class="org-type">-</span>L<span class="org-type">*</span>2<span class="org-type">/</span>sqrt(3) <span class="org-type">-</span>L<span class="org-type">*</span>2<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3) L<span class="org-type">/</span>sqrt(3);
|
||||
L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2) L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2) L<span class="org-type">*</span>sqrt(2) <span class="org-type">-</span>L<span class="org-type">*</span>sqrt(2)];
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7c9679d" class="outline-3">
|
||||
<h3 id="org7c9679d"><span class="section-number-3">4.4</span> Stifnness matrix and Damping matrix</h3>
|
||||
<div id="outline-container-org2c9ff6d" class="outline-3">
|
||||
<h3 id="org2c9ff6d"><span class="section-number-3">4.4</span> Stifnness and Damping matrices</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">kv = k<span class="org-type">/</span>3; <span class="org-comment">% [N/m]</span>
|
||||
kh = 0.5<span class="org-type">*</span>k<span class="org-type">/</span>3; <span class="org-comment">% [N/m]</span>
|
||||
|
||||
K = diag([3<span class="org-type">*</span>kh,3<span class="org-type">*</span>kh,3<span class="org-type">*</span>kv,3<span class="org-type">*</span>kv<span class="org-type">*</span>Rx<span class="org-type">^</span>2<span class="org-type">/</span>2,3<span class="org-type">*</span>kv<span class="org-type">*</span>Rx<span class="org-type">^</span>2<span class="org-type">/</span>2,3<span class="org-type">*</span>kh<span class="org-type">*</span>Rx<span class="org-type">^</span>2]); <span class="org-comment">% Stiffness Matrix</span>
|
||||
<pre class="src src-matlab">kv = k<span class="org-type">/</span>3; <span class="org-comment">% Vertical Stiffness of the springs [N/m]</span>
|
||||
kh = 0.5<span class="org-type">*</span>k<span class="org-type">/</span>3; <span class="org-comment">% Horizontal Stiffness of the springs [N/m]</span>
|
||||
|
||||
K = diag([3<span class="org-type">*</span>kh, 3<span class="org-type">*</span>kh, 3<span class="org-type">*</span>kv, 3<span class="org-type">*</span>kv<span class="org-type">*</span>Rx<span class="org-type">^</span>2<span class="org-type">/</span>2, 3<span class="org-type">*</span>kv<span class="org-type">*</span>Rx<span class="org-type">^</span>2<span class="org-type">/</span>2, 3<span class="org-type">*</span>kh<span class="org-type">*</span>Rx<span class="org-type">^</span>2]); <span class="org-comment">% Stiffness Matrix</span>
|
||||
C = c<span class="org-type">*</span>K<span class="org-type">/</span>100000; <span class="org-comment">% Damping Matrix</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org00e8691" class="outline-3">
|
||||
<h3 id="org00e8691"><span class="section-number-3">4.5</span> State Space System</h3>
|
||||
<div id="outline-container-orgffba0a8" class="outline-3">
|
||||
<h3 id="orgffba0a8"><span class="section-number-3">4.5</span> State Space System</h3>
|
||||
<div class="outline-text-3" id="text-4-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">A = [zeros(6) eye(6); <span class="org-type">-</span>M<span class="org-type">\</span>K <span class="org-type">-</span>M<span class="org-type">\</span>C];
|
||||
<pre class="src src-matlab">A = [ zeros(6) eye(6); ...
|
||||
<span class="org-type">-</span>M<span class="org-type">\</span>K <span class="org-type">-</span>M<span class="org-type">\</span>C];
|
||||
Bw = [zeros(6); <span class="org-type">-</span>eye(6)];
|
||||
Bu = [zeros(6); M<span class="org-type">\</span>Bj];
|
||||
|
||||
Co = [<span class="org-type">-</span>M<span class="org-type">\</span>K <span class="org-type">-</span>M<span class="org-type">\</span>C];
|
||||
|
||||
D = [zeros(6) M<span class="org-type">\</span>Bj];
|
||||
|
||||
ST = ss(A,[Bw Bu],Co,D);
|
||||
@ -1291,16 +1293,18 @@ ST = ss(A,[Bw Bu],Co,D);
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">ST.StateName = {<span class="org-string">'x'</span>;<span class="org-string">'y'</span>;<span class="org-string">'z'</span>;<span class="org-string">'theta_x'</span>;<span class="org-string">'theta_y'</span>;<span class="org-string">'theta_z'</span>;...
|
||||
<span class="org-string">'dx'</span>;<span class="org-string">'dy'</span>;<span class="org-string">'dz'</span>;<span class="org-string">'dtheta_x'</span>;<span class="org-string">'dtheta_y'</span>;<span class="org-string">'dtheta_z'</span>};
|
||||
|
||||
ST.InputName = {<span class="org-string">'w1'</span>;<span class="org-string">'w2'</span>;<span class="org-string">'w3'</span>;<span class="org-string">'w4'</span>;<span class="org-string">'w5'</span>;<span class="org-string">'w6'</span>;...
|
||||
<span class="org-string">'u1'</span>;<span class="org-string">'u2'</span>;<span class="org-string">'u3'</span>;<span class="org-string">'u4'</span>;<span class="org-string">'u5'</span>;<span class="org-string">'u6'</span>};
|
||||
|
||||
ST.OutputName = {<span class="org-string">'ax'</span>;<span class="org-string">'ay'</span>;<span class="org-string">'az'</span>;<span class="org-string">'atheta_x'</span>;<span class="org-string">'atheta_y'</span>;<span class="org-string">'atheta_z'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8a70996" class="outline-3">
|
||||
<h3 id="org8a70996"><span class="section-number-3">4.6</span> Transmissibility</h3>
|
||||
<div id="outline-container-org42b1b07" class="outline-3">
|
||||
<h3 id="org42b1b07"><span class="section-number-3">4.6</span> Transmissibility</h3>
|
||||
<div class="outline-text-3" id="text-4-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">TR=ST<span class="org-type">*</span>[eye(6); zeros(6)];
|
||||
@ -1310,22 +1314,22 @@ ST.OutputName = {<span class="org-string">'ax'</span>;<span class="org-string">'
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>
|
||||
subplot(231)
|
||||
bodemag(TR(1,1),opts);
|
||||
bodemag(TR(1,1));
|
||||
subplot(232)
|
||||
bodemag(TR(2,2),opts);
|
||||
bodemag(TR(2,2));
|
||||
subplot(233)
|
||||
bodemag(TR(3,3),opts);
|
||||
bodemag(TR(3,3));
|
||||
subplot(234)
|
||||
bodemag(TR(4,4),opts);
|
||||
bodemag(TR(4,4));
|
||||
subplot(235)
|
||||
bodemag(TR(5,5),opts);
|
||||
bodemag(TR(5,5));
|
||||
subplot(236)
|
||||
bodemag(TR(6,6),opts);
|
||||
bodemag(TR(6,6));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6ce913c" class="figure">
|
||||
<div id="orgd9b6731" class="figure">
|
||||
<p><img src="figs/stewart_platform_analytical_transmissibility.png" alt="stewart_platform_analytical_transmissibility.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Transmissibility</p>
|
||||
@ -1333,8 +1337,8 @@ bodemag(TR(6,6),opts);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org12c95c9" class="outline-3">
|
||||
<h3 id="org12c95c9"><span class="section-number-3">4.7</span> Real approximation of \(G(j\omega)\) at decoupling frequency</h3>
|
||||
<div id="outline-container-org38c8159" class="outline-3">
|
||||
<h3 id="org38c8159"><span class="section-number-3">4.7</span> Real approximation of \(G(j\omega)\) at decoupling frequency</h3>
|
||||
<div class="outline-text-3" id="text-4-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">sys1 = ST<span class="org-type">*</span>[zeros(6); eye(6)]; <span class="org-comment">% take only the forces inputs</span>
|
||||
@ -1362,8 +1366,8 @@ wf = logspace(<span class="org-type">-</span>1,2,1000);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc58b81c" class="outline-3">
|
||||
<h3 id="orgc58b81c"><span class="section-number-3">4.8</span> Coupled and Decoupled Plant “Gershgorin Radii”</h3>
|
||||
<div id="outline-container-org477b3ce" class="outline-3">
|
||||
<h3 id="org477b3ce"><span class="section-number-3">4.8</span> Coupled and Decoupled Plant “Gershgorin Radii”</h3>
|
||||
<div class="outline-text-3" id="text-4-8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||
@ -1375,7 +1379,7 @@ xlabel(<span class="org-string">'Frequency (Hz)'</span>); ylabel(<span class="or
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org20fc2fd" class="figure">
|
||||
<div id="org1416731" class="figure">
|
||||
<p><img src="figs/gershorin_raddii_coupled_analytical.png" alt="gershorin_raddii_coupled_analytical.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Gershorin Raddi for the coupled plant</p>
|
||||
@ -1391,7 +1395,7 @@ xlabel(<span class="org-string">'Frequency (Hz)'</span>); ylabel(<span class="or
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org586d327" class="figure">
|
||||
<div id="orgdfe4880" class="figure">
|
||||
<p><img src="figs/gershorin_raddii_decoupled_analytical.png" alt="gershorin_raddii_decoupled_analytical.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Gershorin Raddi for the decoupled plant</p>
|
||||
@ -1399,8 +1403,8 @@ xlabel(<span class="org-string">'Frequency (Hz)'</span>); ylabel(<span class="or
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2ba91f6" class="outline-3">
|
||||
<h3 id="org2ba91f6"><span class="section-number-3">4.9</span> Decoupled Plant</h3>
|
||||
<div id="outline-container-orgde4eec1" class="outline-3">
|
||||
<h3 id="orgde4eec1"><span class="section-number-3">4.9</span> Decoupled Plant</h3>
|
||||
<div class="outline-text-3" id="text-4-9">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||
@ -1409,7 +1413,7 @@ bodemag(U<span class="org-type">'*</span>sys1<span class="org-type">*</span>V,op
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5cd203f" class="figure">
|
||||
<div id="orge835f28" class="figure">
|
||||
<p><img src="figs/stewart_platform_analytical_decoupled_plant.png" alt="stewart_platform_analytical_decoupled_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Decoupled Plant</p>
|
||||
@ -1417,8 +1421,8 @@ bodemag(U<span class="org-type">'*</span>sys1<span class="org-type">*</span>V,op
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc73a283" class="outline-3">
|
||||
<h3 id="orgc73a283"><span class="section-number-3">4.10</span> Controller</h3>
|
||||
<div id="outline-container-org11b0182" class="outline-3">
|
||||
<h3 id="org11b0182"><span class="section-number-3">4.10</span> Controller</h3>
|
||||
<div class="outline-text-3" id="text-4-10">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">fc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; <span class="org-comment">% Crossover Frequency [rad/s]</span>
|
||||
@ -1430,8 +1434,8 @@ cont = eye(6)<span class="org-type">*</span>c_gain<span class="org-type">/</span
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9c82ee4" class="outline-3">
|
||||
<h3 id="org9c82ee4"><span class="section-number-3">4.11</span> Closed Loop System</h3>
|
||||
<div id="outline-container-org5c893a8" class="outline-3">
|
||||
<h3 id="org5c893a8"><span class="section-number-3">4.11</span> Closed Loop System</h3>
|
||||
<div class="outline-text-3" id="text-4-11">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">FEEDIN = [7<span class="org-type">:</span>12]; <span class="org-comment">% Input of controller</span>
|
||||
@ -1459,8 +1463,8 @@ TRsvd = STsvd<span class="org-type">*</span>[eye(6); zeros(6)];
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org80cd406" class="outline-3">
|
||||
<h3 id="org80cd406"><span class="section-number-3">4.12</span> Results</h3>
|
||||
<div id="outline-container-orgb1c0711" class="outline-3">
|
||||
<h3 id="orgb1c0711"><span class="section-number-3">4.12</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-4-12">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>
|
||||
@ -1486,7 +1490,7 @@ legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralize
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgfadf6e5" class="figure">
|
||||
<div id="orgb680082" class="figure">
|
||||
<p><img src="figs/stewart_platform_analytical_svd_cen_comp.png" alt="stewart_platform_analytical_svd_cen_comp.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Comparison of the obtained transmissibility for the centralized control and the SVD control</p>
|
||||
@ -1497,7 +1501,7 @@ legend(<span class="org-string">'OL'</span>,<span class="org-string">'Centralize
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-10-09 ven. 16:21</p>
|
||||
<p class="date">Created: 2020-10-13 mar. 14:53</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
10
index.org
10
index.org
@ -686,6 +686,9 @@ This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]].
|
||||
#+end_src
|
||||
|
||||
* Stewart Platform - Simscape Model
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle stewart_platform/simscape_model.m
|
||||
:END:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
@ -700,6 +703,10 @@ This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]].
|
||||
addpath('stewart_platform/STEP');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :eval no
|
||||
addpath('STEP');
|
||||
#+end_src
|
||||
|
||||
** Jacobian
|
||||
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
|
||||
#+begin_src matlab
|
||||
@ -1292,6 +1299,9 @@ The obtained transmissibility in Open-loop, for the centralized control as well
|
||||
[[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
|
||||
|
||||
* Stewart Platform - Analytical Model
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle stewart_platform/analytical_model.m
|
||||
:END:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
|
||||
196
stewart_platform/analytical_model.m
Normal file
196
stewart_platform/analytical_model.m
Normal file
@ -0,0 +1,196 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Bode plot options
|
||||
opts = bodeoptions('cstprefs');
|
||||
opts.FreqUnits = 'Hz';
|
||||
opts.MagUnits = 'abs';
|
||||
opts.MagScale = 'log';
|
||||
opts.PhaseWrapping = 'on';
|
||||
opts.xlim = [1 1000];
|
||||
|
||||
% Characteristics
|
||||
|
||||
L = 0.055; % Leg length [m]
|
||||
Zc = 0; % ?
|
||||
m = 0.2; % Top platform mass [m]
|
||||
k = 1e3; % Total vertical stiffness [N/m]
|
||||
c = 2*0.1*sqrt(k*m); % Damping ? [N/(m/s)]
|
||||
|
||||
Rx = 0.04; % ?
|
||||
Rz = 0.04; % ?
|
||||
Ix = m*Rx^2; % ?
|
||||
Iy = m*Rx^2; % ?
|
||||
Iz = m*Rz^2; % ?
|
||||
|
||||
% Mass Matrix
|
||||
|
||||
M = m*[1 0 0 0 Zc 0;
|
||||
0 1 0 -Zc 0 0;
|
||||
0 0 1 0 0 0;
|
||||
0 -Zc 0 Rx^2+Zc^2 0 0;
|
||||
Zc 0 0 0 Rx^2+Zc^2 0;
|
||||
0 0 0 0 0 Rz^2];
|
||||
|
||||
% Jacobian Matrix
|
||||
|
||||
Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
|
||||
sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
|
||||
sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
|
||||
0 0 L L -L -L;
|
||||
-L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
|
||||
L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
|
||||
|
||||
% Stifnness and Damping matrices
|
||||
|
||||
kv = k/3; % Vertical Stiffness of the springs [N/m]
|
||||
kh = 0.5*k/3; % Horizontal Stiffness of the springs [N/m]
|
||||
|
||||
K = diag([3*kh, 3*kh, 3*kv, 3*kv*Rx^2/2, 3*kv*Rx^2/2, 3*kh*Rx^2]); % Stiffness Matrix
|
||||
C = c*K/100000; % Damping Matrix
|
||||
|
||||
% State Space System
|
||||
|
||||
A = [ zeros(6) eye(6); ...
|
||||
-M\K -M\C];
|
||||
Bw = [zeros(6); -eye(6)];
|
||||
Bu = [zeros(6); M\Bj];
|
||||
|
||||
Co = [-M\K -M\C];
|
||||
|
||||
D = [zeros(6) M\Bj];
|
||||
|
||||
ST = ss(A,[Bw Bu],Co,D);
|
||||
|
||||
|
||||
|
||||
% - OUT 1-6: 6 dof
|
||||
% - IN 1-6 : ground displacement in the directions of the legs
|
||||
% - IN 7-12: forces in the actuators.
|
||||
|
||||
ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
|
||||
'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
|
||||
|
||||
ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
|
||||
'u1';'u2';'u3';'u4';'u5';'u6'};
|
||||
|
||||
ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
|
||||
|
||||
% Transmissibility
|
||||
|
||||
TR=ST*[eye(6); zeros(6)];
|
||||
|
||||
figure
|
||||
subplot(231)
|
||||
bodemag(TR(1,1));
|
||||
subplot(232)
|
||||
bodemag(TR(2,2));
|
||||
subplot(233)
|
||||
bodemag(TR(3,3));
|
||||
subplot(234)
|
||||
bodemag(TR(4,4));
|
||||
subplot(235)
|
||||
bodemag(TR(5,5));
|
||||
subplot(236)
|
||||
bodemag(TR(6,6));
|
||||
|
||||
% Real approximation of $G(j\omega)$ at decoupling frequency
|
||||
|
||||
sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
|
||||
|
||||
dec_fr = 20;
|
||||
H1 = evalfr(sys1,j*2*pi*dec_fr);
|
||||
H2 = H1;
|
||||
D = pinv(real(H2'*H2));
|
||||
H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
|
||||
[U,S,V] = svd(H1);
|
||||
|
||||
wf = logspace(-1,2,1000);
|
||||
for i = 1:length(wf)
|
||||
H = abs(evalfr(sys1,j*2*pi*wf(i)));
|
||||
H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r1(i,j) = (sum(H(j,:))-H(j,j))/H(j,j);
|
||||
g_r2(i,j) = (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
|
||||
% keyboard
|
||||
end
|
||||
g_lim(i) = 0.5;
|
||||
end
|
||||
|
||||
% Coupled and Decoupled Plant "Gershgorin Radii"
|
||||
|
||||
figure;
|
||||
title('Coupled plant')
|
||||
loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
|
||||
legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
|
||||
|
||||
|
||||
% #+name: fig:gershorin_raddii_coupled_analytical
|
||||
% #+caption: Gershorin Raddi for the coupled plant
|
||||
% #+RESULTS:
|
||||
% [[file:figs/gershorin_raddii_coupled_analytical.png]]
|
||||
|
||||
|
||||
figure;
|
||||
title('Decoupled plant (10 Hz)')
|
||||
loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
|
||||
legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
|
||||
% Decoupled Plant
|
||||
|
||||
figure;
|
||||
bodemag(U'*sys1*V,opts)
|
||||
|
||||
% Controller
|
||||
|
||||
fc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||
c_gain = 50; %
|
||||
|
||||
cont = eye(6)*c_gain/(s+fc);
|
||||
|
||||
% Closed Loop System
|
||||
|
||||
FEEDIN = [7:12]; % Input of controller
|
||||
FEEDOUT = [1:6]; % Output of controller
|
||||
|
||||
|
||||
|
||||
% Centralized Control
|
||||
|
||||
STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
|
||||
TRcen = STcen*[eye(6); zeros(6)];
|
||||
|
||||
|
||||
|
||||
% SVD Control
|
||||
|
||||
STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
|
||||
TRsvd = STsvd*[eye(6); zeros(6)];
|
||||
|
||||
% Results
|
||||
|
||||
figure
|
||||
subplot(231)
|
||||
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(232)
|
||||
bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(233)
|
||||
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(234)
|
||||
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(235)
|
||||
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(236)
|
||||
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
Binary file not shown.
BIN
stewart_platform/drone_platform_R2017b.slx
Normal file
BIN
stewart_platform/drone_platform_R2017b.slx
Normal file
Binary file not shown.
499
stewart_platform/simscape_model.m
Normal file
499
stewart_platform/simscape_model.m
Normal file
@ -0,0 +1,499 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
addpath('STEP');
|
||||
|
||||
% Jacobian
|
||||
% First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
|
||||
|
||||
open('drone_platform_jacobian.slx');
|
||||
|
||||
sim('drone_platform_jacobian');
|
||||
|
||||
Aa = [a1.Data(1,:);
|
||||
a2.Data(1,:);
|
||||
a3.Data(1,:);
|
||||
a4.Data(1,:);
|
||||
a5.Data(1,:);
|
||||
a6.Data(1,:)]';
|
||||
|
||||
Ab = [b1.Data(1,:);
|
||||
b2.Data(1,:);
|
||||
b3.Data(1,:);
|
||||
b4.Data(1,:);
|
||||
b5.Data(1,:);
|
||||
b6.Data(1,:)]';
|
||||
|
||||
As = (Ab - Aa)./vecnorm(Ab - Aa);
|
||||
|
||||
l = vecnorm(Ab - Aa)';
|
||||
|
||||
J = [As' , cross(Ab, As)'];
|
||||
|
||||
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
|
||||
% Simscape Model
|
||||
|
||||
open('drone_platform.slx');
|
||||
|
||||
|
||||
|
||||
% Definition of spring parameters
|
||||
|
||||
kx = 0.5*1e3/3; % [N/m]
|
||||
ky = 0.5*1e3/3;
|
||||
kz = 1e3/3;
|
||||
|
||||
cx = 0.025; % [Nm/rad]
|
||||
cy = 0.025;
|
||||
cz = 0.025;
|
||||
|
||||
|
||||
|
||||
% We load the Jacobian.
|
||||
|
||||
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
|
||||
% Identification of the plant
|
||||
% The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'drone_platform';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
|
||||
|
||||
|
||||
|
||||
% There are 24 states (6dof for the bottom platform + 6dof for the top platform).
|
||||
|
||||
size(G)
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% : State-space model with 6 outputs, 12 inputs, and 24 states.
|
||||
|
||||
|
||||
% G = G*blkdiag(inv(J), eye(6));
|
||||
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
|
||||
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
|
||||
|
||||
|
||||
% Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
|
||||
|
||||
Gx = G*blkdiag(eye(6), inv(J'));
|
||||
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
|
||||
Gl = J*G;
|
||||
Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
|
||||
|
||||
% Obtained Dynamics
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:stewart_platform_translations
|
||||
% #+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
|
||||
% #+RESULTS:
|
||||
% [[file:figs/stewart_platform_translations.png]]
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:stewart_platform_rotations
|
||||
% #+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
|
||||
% #+RESULTS:
|
||||
% [[file:figs/stewart_platform_rotations.png]]
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for out_i = 1:5
|
||||
for in_i = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
end
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:stewart_platform_legs
|
||||
% #+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
|
||||
% #+RESULTS:
|
||||
% [[file:figs/stewart_platform_legs.png]]
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
% plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(TR(1,1), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
plot(freqs, abs(squeeze(freqresp(TR(2,2), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
plot(freqs, abs(squeeze(freqresp(TR(3,3), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
% plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
|
||||
% plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(TR(4,4), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
plot(freqs, abs(squeeze(freqresp(TR(5,5), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
plot(freqs, abs(squeeze(freqresp(TR(6,6), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Real Approximation of $G$ at the decoupling frequency
|
||||
% Let'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 $\omega_c$.
|
||||
|
||||
wc = 2*pi*20; % Decoupling frequency [rad/s]
|
||||
|
||||
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
|
||||
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
|
||||
|
||||
H1 = evalfr(Gc, j*wc);
|
||||
|
||||
|
||||
|
||||
% The real approximation is computed as follows:
|
||||
|
||||
D = pinv(real(H1'*H1));
|
||||
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
||||
|
||||
% Verification of the decoupling using the "Gershgorin Radii"
|
||||
% First, the Singular Value Decomposition of $H_1$ is performed:
|
||||
% \[ H_1 = U \Sigma V^H \]
|
||||
|
||||
|
||||
[U,S,V] = svd(H1);
|
||||
|
||||
|
||||
|
||||
% Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
|
||||
% \[ G_d(s) = U^T G_c(s) V \]
|
||||
|
||||
% This is computed over the following frequencies.
|
||||
|
||||
freqs = logspace(-2, 2, 1000); % [Hz]
|
||||
|
||||
|
||||
|
||||
% Gershgorin Radii for the coupled plant:
|
||||
|
||||
Gr_coupled = zeros(length(freqs), size(Gc,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gc,2)
|
||||
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using SVD:
|
||||
|
||||
Gd = U'*Gc*V;
|
||||
Gr_decoupled = zeros(length(freqs), size(Gd,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gd,2)
|
||||
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using the Jacobian:
|
||||
|
||||
Gj = Gc*inv(J');
|
||||
Gr_jacobian = zeros(length(freqs), size(Gj,2));
|
||||
|
||||
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
|
||||
|
||||
for out_i = 1:size(Gj,2)
|
||||
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
|
||||
plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
|
||||
plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
|
||||
for in_i = 2:6
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',3)
|
||||
plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
|
||||
end
|
||||
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
legend('location', 'northeast');
|
||||
|
||||
% Decoupled Plant
|
||||
% Let's see the bode plot of the decoupled plant $G_d(s)$.
|
||||
% \[ G_d(s) = U^T G_c(s) V \]
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
|
||||
end
|
||||
for in_i = 1:5
|
||||
for out_i = in_i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:simscape_model_decoupled_plant_svd
|
||||
% #+caption: Decoupled Plant using SVD
|
||||
% #+RESULTS:
|
||||
% [[file:figs/simscape_model_decoupled_plant_svd.png]]
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for ch_i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
|
||||
end
|
||||
for in_i = 1:5
|
||||
for out_i = in_i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
|
||||
% Diagonal Controller
|
||||
% The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
|
||||
|
||||
|
||||
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||
C_g = 50; % DC Gain
|
||||
|
||||
K = eye(6)*C_g/(s+wc);
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% [[file:figs/centralized_control.png]]
|
||||
|
||||
|
||||
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% [[file:figs/svd_control.png]]
|
||||
|
||||
% SVD Control
|
||||
|
||||
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
|
||||
|
||||
% Results
|
||||
% Let's first verify the stability of the closed-loop systems:
|
||||
|
||||
isstable(G_cen)
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% : ans =
|
||||
% : logical
|
||||
% : 1
|
||||
|
||||
|
||||
isstable(G_svd)
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% : ans =
|
||||
% : logical
|
||||
% : 0
|
||||
|
||||
% 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]].
|
||||
|
||||
|
||||
freqs = logspace(-3, 3, 1000);
|
||||
|
||||
figure
|
||||
|
||||
ax1 = subplot(2, 3, 1);
|
||||
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]');
|
||||
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]');
|
||||
|
||||
ax3 = subplot(2, 3, 3);
|
||||
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]');
|
||||
|
||||
ax4 = subplot(2, 3, 4);
|
||||
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]');
|
||||
|
||||
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]');
|
||||
|
||||
ax6 = subplot(2, 3, 6);
|
||||
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]');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
Loading…
Reference in New Issue
Block a user