Add analysis about cube size
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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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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<!-- 2020-03-02 lun. 17:57 -->
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<!-- 2020-03-12 jeu. 18:06 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Stewart Platform - Decentralized Active Damping</title>
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@ -249,25 +249,25 @@
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<li><a href="#orgd59c804">1. Inertial Control</a>
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<ul>
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<li><a href="#org5f749c8">1.1. Identification of the Dynamics</a></li>
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<li><a href="#orgd0f78f7">1.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
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<li><a href="#org3f64d96">1.3. Obtained Damping</a></li>
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<li><a href="#org8e1ece7">1.4. Conclusion</a></li>
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<li><a href="#org81b6713">1.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
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<li><a href="#orge328103">1.3. Obtained Damping</a></li>
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<li><a href="#org48c963f">1.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org74c7eb4">2. Integral Force Feedback</a>
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<ul>
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<li><a href="#orgcd99b62">2.1. Identification of the Dynamics with perfect Joints</a></li>
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<li><a href="#org1b7a953">2.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
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<li><a href="#org1d362f0">2.3. Obtained Damping</a></li>
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<li><a href="#org63f9110">2.4. Conclusion</a></li>
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<li><a href="#org5364f58">2.1. Identification of the Dynamics with perfect Joints</a></li>
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<li><a href="#org2656032">2.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
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<li><a href="#org55bd5fb">2.3. Obtained Damping</a></li>
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<li><a href="#org4e07d49">2.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org08917d6">3. Direct Velocity Feedback</a>
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<ul>
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<li><a href="#org5364f58">3.1. Identification of the Dynamics with perfect Joints</a></li>
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<li><a href="#org81b6713">3.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
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<li><a href="#orge328103">3.3. Obtained Damping</a></li>
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<li><a href="#org48c963f">3.4. Conclusion</a></li>
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<li><a href="#org693d6f2">3.1. Identification of the Dynamics with perfect Joints</a></li>
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<li><a href="#org07bfe55">3.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
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<li><a href="#org6f708ec">3.3. Obtained Damping</a></li>
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<li><a href="#org12f722c">3.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org183f3f2">4. Compliance and Transmissibility Comparison</a>
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@ -366,8 +366,8 @@ The transfer function from actuator forces to force sensors is shown in Figure <
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</div>
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</div>
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<div id="outline-container-orgd0f78f7" class="outline-3">
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<h3 id="orgd0f78f7"><span class="section-number-3">1.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
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<div id="outline-container-org81b6713" class="outline-3">
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<h3 id="org81b6713"><span class="section-number-3">1.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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@ -403,8 +403,8 @@ The new dynamics from force actuator to force sensor is shown in Figure <a href=
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</div>
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</div>
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<div id="outline-container-org3f64d96" class="outline-3">
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<h3 id="org3f64d96"><span class="section-number-3">1.3</span> Obtained Damping</h3>
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<div id="outline-container-orge328103" class="outline-3">
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<h3 id="orge328103"><span class="section-number-3">1.3</span> Obtained Damping</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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The control is a performed in a decentralized manner.
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@ -429,8 +429,8 @@ The root locus is shown in figure <a href="#org9af9e33">3</a>.
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</div>
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</div>
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<div id="outline-container-org8e1ece7" class="outline-3">
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<h3 id="org8e1ece7"><span class="section-number-3">1.4</span> Conclusion</h3>
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<div id="outline-container-org48c963f" class="outline-3">
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<h3 id="org48c963f"><span class="section-number-3">1.4</span> Conclusion</h3>
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<div class="outline-text-3" id="text-1-4">
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<div class="important">
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<p>
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@ -461,8 +461,8 @@ To run the script, open the Simulink Project, and type <code>run active_damping_
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</div>
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</div>
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<div id="outline-container-orgcd99b62" class="outline-3">
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<h3 id="orgcd99b62"><span class="section-number-3">2.1</span> Identification of the Dynamics with perfect Joints</h3>
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<div id="outline-container-org5364f58" class="outline-3">
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<h3 id="org5364f58"><span class="section-number-3">2.1</span> Identification of the Dynamics with perfect Joints</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We first initialize the Stewart platform without joint stiffness.
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@ -520,8 +520,8 @@ The transfer function from actuator forces to force sensors is shown in Figure <
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</div>
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</div>
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<div id="outline-container-org1b7a953" class="outline-3">
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<h3 id="org1b7a953"><span class="section-number-3">2.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
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<div id="outline-container-org2656032" class="outline-3">
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<h3 id="org2656032"><span class="section-number-3">2.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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@ -557,8 +557,8 @@ The new dynamics from force actuator to force sensor is shown in Figure <a href=
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</div>
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</div>
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<div id="outline-container-org1d362f0" class="outline-3">
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<h3 id="org1d362f0"><span class="section-number-3">2.3</span> Obtained Damping</h3>
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<div id="outline-container-org55bd5fb" class="outline-3">
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<h3 id="org55bd5fb"><span class="section-number-3">2.3</span> Obtained Damping</h3>
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<div class="outline-text-3" id="text-2-3">
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<p>
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The control is a performed in a decentralized manner.
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@ -590,8 +590,8 @@ The root locus is shown in figure <a href="#orge21bbea">6</a> and the obtained p
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</div>
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</div>
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<div id="outline-container-org63f9110" class="outline-3">
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<h3 id="org63f9110"><span class="section-number-3">2.4</span> Conclusion</h3>
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<div id="outline-container-org4e07d49" class="outline-3">
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<h3 id="org4e07d49"><span class="section-number-3">2.4</span> Conclusion</h3>
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<div class="outline-text-3" id="text-2-4">
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<div class="important">
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<p>
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@ -623,8 +623,8 @@ To run the script, open the Simulink Project, and type <code>run active_damping_
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</div>
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</div>
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<div id="outline-container-org5364f58" class="outline-3">
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<h3 id="org5364f58"><span class="section-number-3">3.1</span> Identification of the Dynamics with perfect Joints</h3>
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<div id="outline-container-org693d6f2" class="outline-3">
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<h3 id="org693d6f2"><span class="section-number-3">3.1</span> Identification of the Dynamics with perfect Joints</h3>
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<div class="outline-text-3" id="text-3-1">
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<p>
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We first initialize the Stewart platform without joint stiffness.
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@ -687,8 +687,8 @@ The transfer function from actuator forces to relative motion sensors is shown i
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</div>
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<div id="outline-container-org81b6713" class="outline-3">
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<h3 id="org81b6713"><span class="section-number-3">3.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
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<div id="outline-container-org07bfe55" class="outline-3">
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<h3 id="org07bfe55"><span class="section-number-3">3.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
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We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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@ -724,8 +724,8 @@ The new dynamics from force actuator to relative motion sensor is shown in Figur
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</div>
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</div>
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<div id="outline-container-orge328103" class="outline-3">
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<h3 id="orge328103"><span class="section-number-3">3.3</span> Obtained Damping</h3>
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<div id="outline-container-org6f708ec" class="outline-3">
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<h3 id="org6f708ec"><span class="section-number-3">3.3</span> Obtained Damping</h3>
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<div class="outline-text-3" id="text-3-3">
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<p>
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The control is a performed in a decentralized manner.
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@ -750,8 +750,8 @@ The root locus is shown in figure <a href="#org277d60d">10</a>.
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</div>
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</div>
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<div id="outline-container-org48c963f" class="outline-3">
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<h3 id="org48c963f"><span class="section-number-3">3.4</span> Conclusion</h3>
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<div id="outline-container-org12f722c" class="outline-3">
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<h3 id="org12f722c"><span class="section-number-3">3.4</span> Conclusion</h3>
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<div class="outline-text-3" id="text-3-4">
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<div class="important">
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<p>
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@ -762,6 +762,7 @@ Joint stiffness does increase the resonance frequencies of the system but does n
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</div>
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</div>
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</div>
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<div id="outline-container-org183f3f2" class="outline-2">
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<h2 id="org183f3f2"><span class="section-number-2">4</span> Compliance and Transmissibility Comparison</h2>
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<div class="outline-text-2" id="text-4">
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@ -867,7 +868,7 @@ K_dvf = 1e4<span class="org-type">*</span>s<span class="org-type">/</span>(1<spa
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-03-02 lun. 17:57</p>
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<p class="date">Created: 2020-03-12 jeu. 18:06</p>
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</div>
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</body>
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</html>
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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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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<!-- 2020-03-11 mer. 18:59 -->
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<!-- 2020-03-12 jeu. 18:06 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Stewart Platform - Vibration Isolation</title>
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@ -249,28 +249,28 @@
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<li><a href="#org272e7f8">1. HAC-LAC (Cascade) Control - Integral Control</a>
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<ul>
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<li><a href="#orga5c9b98">1.1. Introduction</a></li>
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<li><a href="#orgc3f9713">1.2. Initialization</a></li>
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<li><a href="#org7e03b59">1.3. Identification</a>
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<li><a href="#org8418e03">1.2. Initialization</a></li>
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<li><a href="#org827d3cd">1.3. Identification</a>
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<ul>
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<li><a href="#org33f5d7c">1.3.1. HAC - Without LAC</a></li>
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<li><a href="#org09a49c0">1.3.2. HAC - IFF</a></li>
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<li><a href="#org9420024">1.3.3. HAC - DVF</a></li>
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<li><a href="#org9a68faf">1.3.1. HAC - Without LAC</a></li>
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<li><a href="#org2abd2cc">1.3.2. HAC - IFF</a></li>
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<li><a href="#org65a7b31">1.3.3. HAC - DVF</a></li>
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</ul>
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</li>
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<li><a href="#org61a6098">1.4. Control Architecture</a></li>
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<li><a href="#orgdca8b1b">1.5. 6x6 Plant Comparison</a></li>
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<li><a href="#orgf1f6f9b">1.6. HAC - DVF</a>
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<li><a href="#org56a04ba">1.6. HAC - DVF</a>
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<ul>
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<li><a href="#org5d68208">1.6.1. Plant</a></li>
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<li><a href="#orge650cdd">1.6.2. Controller Design</a></li>
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<li><a href="#orge5a568c">1.6.3. Obtained Performance</a></li>
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<li><a href="#org9060c71">1.6.1. Plant</a></li>
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<li><a href="#org57d2db6">1.6.2. Controller Design</a></li>
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<li><a href="#orgc77ad88">1.6.3. Obtained Performance</a></li>
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</ul>
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</li>
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<li><a href="#org2abd2cc">1.7. HAC - IFF</a>
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<li><a href="#orga7519aa">1.7. HAC - IFF</a>
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<ul>
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<li><a href="#org9060c71">1.7.1. Plant</a></li>
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<li><a href="#org57d2db6">1.7.2. Controller Design</a></li>
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<li><a href="#orgc77ad88">1.7.3. Obtained Performance</a></li>
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<li><a href="#orgdcb3512">1.7.1. Plant</a></li>
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<li><a href="#org7775b79">1.7.2. Controller Design</a></li>
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<li><a href="#org37a736f">1.7.3. Obtained Performance</a></li>
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</ul>
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</li>
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<li><a href="#org9224c01">1.8. Comparison</a></li>
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@ -278,11 +278,11 @@
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</li>
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<li><a href="#orgde62390">2. MIMO Analysis</a>
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<ul>
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<li><a href="#orgb2025bf">2.1. Initialization</a></li>
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<li><a href="#orgbdf2213">2.2. Identification</a>
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<li><a href="#orgf2ef3bf">2.1. Initialization</a></li>
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<li><a href="#org169782d">2.2. Identification</a>
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<ul>
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<li><a href="#org9a68faf">2.2.1. HAC - Without LAC</a></li>
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<li><a href="#org65a7b31">2.2.2. HAC - DVF</a></li>
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<li><a href="#org39e10f2">2.2.1. HAC - Without LAC</a></li>
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<li><a href="#org0f4faf4">2.2.2. HAC - DVF</a></li>
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<li><a href="#orgf7913d5">2.2.3. Cartesian Frame</a></li>
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</ul>
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</li>
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</li>
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<li><a href="#orgebf6121">3. Diagonal Control based on the damped plant</a>
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<ul>
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<li><a href="#orgf7aefc3">3.1. Initialization</a></li>
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<li><a href="#org827d3cd">3.2. Identification</a></li>
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<li><a href="#org8c2f437">3.1. Initialization</a></li>
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<li><a href="#orge2b1c03">3.2. Identification</a></li>
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<li><a href="#orgab6bc6f">3.3. Steady State Decoupling</a>
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<ul>
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<li><a href="#orga589a4a">3.3.1. Pre-Compensator Design</a></li>
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<li><a href="#org9eaf88f">3.3.2. Diagonal Control Design</a></li>
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<li><a href="#orge195d88">3.3.3. Results</a></li>
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<li><a href="#org5d77351">3.3.3. Results</a></li>
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</ul>
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</li>
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<li><a href="#org7af13df">3.4. Decoupling at Crossover</a></li>
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@ -305,10 +305,10 @@
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</li>
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<li><a href="#orgde0f265">4. Time Domain Simulation</a>
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<ul>
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<li><a href="#org8418e03">4.1. Initialization</a></li>
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<li><a href="#org327858f">4.1. Initialization</a></li>
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<li><a href="#org27ed7aa">4.2. HAC IFF</a></li>
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<li><a href="#orgfd6afac">4.3. HAC-DVF</a></li>
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<li><a href="#org5d77351">4.4. Results</a></li>
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<li><a href="#orgd68da79">4.4. Results</a></li>
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</ul>
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</li>
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<li><a href="#org1ce6b23">5. Functions</a>
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@ -385,8 +385,8 @@ First, the LAC loop is closed (the LAC control is described <a href="active-damp
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</div>
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</div>
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<div id="outline-container-orgc3f9713" class="outline-3">
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<h3 id="orgc3f9713"><span class="section-number-3">1.2</span> Initialization</h3>
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<div id="outline-container-org8418e03" class="outline-3">
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<h3 id="org8418e03"><span class="section-number-3">1.2</span> Initialization</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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We first initialize the Stewart platform.
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@ -417,8 +417,8 @@ payload = initializePayload(<span class="org-string">'type'</span>, <span class=
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</div>
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</div>
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<div id="outline-container-org7e03b59" class="outline-3">
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<h3 id="org7e03b59"><span class="section-number-3">1.3</span> Identification</h3>
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<div id="outline-container-org827d3cd" class="outline-3">
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<h3 id="org827d3cd"><span class="section-number-3">1.3</span> Identification</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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We identify the transfer function from the actuator forces \(\bm{\tau}\) to the absolute displacement of the mobile platform \(\bm{\mathcal{X}}\) in three different cases:
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@ -430,8 +430,8 @@ We identify the transfer function from the actuator forces \(\bm{\tau}\) to the
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</ul>
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</div>
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|
||||
<div id="outline-container-org33f5d7c" class="outline-4">
|
||||
<h4 id="org33f5d7c"><span class="section-number-4">1.3.1</span> HAC - Without LAC</h4>
|
||||
<div id="outline-container-org9a68faf" class="outline-4">
|
||||
<h4 id="org9a68faf"><span class="section-number-4">1.3.1</span> HAC - Without LAC</h4>
|
||||
<div class="outline-text-4" id="text-1-3-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
@ -456,8 +456,8 @@ G_ol.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org09a49c0" class="outline-4">
|
||||
<h4 id="org09a49c0"><span class="section-number-4">1.3.2</span> HAC - IFF</h4>
|
||||
<div id="outline-container-org2abd2cc" class="outline-4">
|
||||
<h4 id="org2abd2cc"><span class="section-number-4">1.3.2</span> HAC - IFF</h4>
|
||||
<div class="outline-text-4" id="text-1-3-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'iff'</span>);
|
||||
@ -483,8 +483,8 @@ G_iff.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-strin
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9420024" class="outline-4">
|
||||
<h4 id="org9420024"><span class="section-number-4">1.3.3</span> HAC - DVF</h4>
|
||||
<div id="outline-container-org65a7b31" class="outline-4">
|
||||
<h4 id="org65a7b31"><span class="section-number-4">1.3.3</span> HAC - DVF</h4>
|
||||
<div class="outline-text-4" id="text-1-3-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'dvf'</span>);
|
||||
@ -548,12 +548,12 @@ We then design a controller based on the transfer functions from \(\bm{\mathcal{
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf1f6f9b" class="outline-3">
|
||||
<h3 id="orgf1f6f9b"><span class="section-number-3">1.6</span> HAC - DVF</h3>
|
||||
<div id="outline-container-org56a04ba" class="outline-3">
|
||||
<h3 id="org56a04ba"><span class="section-number-3">1.6</span> HAC - DVF</h3>
|
||||
<div class="outline-text-3" id="text-1-6">
|
||||
</div>
|
||||
<div id="outline-container-org5d68208" class="outline-4">
|
||||
<h4 id="org5d68208"><span class="section-number-4">1.6.1</span> Plant</h4>
|
||||
<div id="outline-container-org9060c71" class="outline-4">
|
||||
<h4 id="org9060c71"><span class="section-number-4">1.6.1</span> Plant</h4>
|
||||
<div class="outline-text-4" id="text-1-6-1">
|
||||
|
||||
<div id="orgbe936ef" class="figure">
|
||||
@ -564,8 +564,8 @@ We then design a controller based on the transfer functions from \(\bm{\mathcal{
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge650cdd" class="outline-4">
|
||||
<h4 id="orge650cdd"><span class="section-number-4">1.6.2</span> Controller Design</h4>
|
||||
<div id="outline-container-org57d2db6" class="outline-4">
|
||||
<h4 id="org57d2db6"><span class="section-number-4">1.6.2</span> Controller Design</h4>
|
||||
<div class="outline-text-4" id="text-1-6-2">
|
||||
<p>
|
||||
We design a diagonal controller with equal bandwidth for the 6 terms.
|
||||
@ -600,8 +600,8 @@ Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implem
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge5a568c" class="outline-4">
|
||||
<h4 id="orge5a568c"><span class="section-number-4">1.6.3</span> Obtained Performance</h4>
|
||||
<div id="outline-container-orgc77ad88" class="outline-4">
|
||||
<h4 id="orgc77ad88"><span class="section-number-4">1.6.3</span> Obtained Performance</h4>
|
||||
<div class="outline-text-4" id="text-1-6-3">
|
||||
<p>
|
||||
We identify the transmissibility and compliance of the system.
|
||||
@ -638,12 +638,12 @@ We identify the transmissibility and compliance of the system.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2abd2cc" class="outline-3">
|
||||
<h3 id="org2abd2cc"><span class="section-number-3">1.7</span> HAC - IFF</h3>
|
||||
<div id="outline-container-orga7519aa" class="outline-3">
|
||||
<h3 id="orga7519aa"><span class="section-number-3">1.7</span> HAC - IFF</h3>
|
||||
<div class="outline-text-3" id="text-1-7">
|
||||
</div>
|
||||
<div id="outline-container-org9060c71" class="outline-4">
|
||||
<h4 id="org9060c71"><span class="section-number-4">1.7.1</span> Plant</h4>
|
||||
<div id="outline-container-orgdcb3512" class="outline-4">
|
||||
<h4 id="orgdcb3512"><span class="section-number-4">1.7.1</span> Plant</h4>
|
||||
<div class="outline-text-4" id="text-1-7-1">
|
||||
|
||||
<div id="orgcb10b82" class="figure">
|
||||
@ -654,8 +654,8 @@ We identify the transmissibility and compliance of the system.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org57d2db6" class="outline-4">
|
||||
<h4 id="org57d2db6"><span class="section-number-4">1.7.2</span> Controller Design</h4>
|
||||
<div id="outline-container-org7775b79" class="outline-4">
|
||||
<h4 id="org7775b79"><span class="section-number-4">1.7.2</span> Controller Design</h4>
|
||||
<div class="outline-text-4" id="text-1-7-2">
|
||||
<p>
|
||||
We design a diagonal controller with equal bandwidth for the 6 terms.
|
||||
@ -690,8 +690,8 @@ Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implem
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc77ad88" class="outline-4">
|
||||
<h4 id="orgc77ad88"><span class="section-number-4">1.7.3</span> Obtained Performance</h4>
|
||||
<div id="outline-container-org37a736f" class="outline-4">
|
||||
<h4 id="org37a736f"><span class="section-number-4">1.7.3</span> Obtained Performance</h4>
|
||||
<div class="outline-text-4" id="text-1-7-3">
|
||||
<p>
|
||||
We identify the transmissibility and compliance of the system.
|
||||
@ -847,8 +847,8 @@ Let’s define the system as shown in figure <a href="#orgba6519a">13</a>.
|
||||
</table>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb2025bf" class="outline-3">
|
||||
<h3 id="orgb2025bf"><span class="section-number-3">2.1</span> Initialization</h3>
|
||||
<div id="outline-container-orgf2ef3bf" class="outline-3">
|
||||
<h3 id="orgf2ef3bf"><span class="section-number-3">2.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We first initialize the Stewart platform.
|
||||
@ -879,12 +879,12 @@ payload = initializePayload(<span class="org-string">'type'</span>, <span class=
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbdf2213" class="outline-3">
|
||||
<h3 id="orgbdf2213"><span class="section-number-3">2.2</span> Identification</h3>
|
||||
<div id="outline-container-org169782d" class="outline-3">
|
||||
<h3 id="org169782d"><span class="section-number-3">2.2</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
</div>
|
||||
<div id="outline-container-org9a68faf" class="outline-4">
|
||||
<h4 id="org9a68faf"><span class="section-number-4">2.2.1</span> HAC - Without LAC</h4>
|
||||
<div id="outline-container-org39e10f2" class="outline-4">
|
||||
<h4 id="org39e10f2"><span class="section-number-4">2.2.1</span> HAC - Without LAC</h4>
|
||||
<div class="outline-text-4" id="text-2-2-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
@ -909,8 +909,8 @@ G_ol.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org65a7b31" class="outline-4">
|
||||
<h4 id="org65a7b31"><span class="section-number-4">2.2.2</span> HAC - DVF</h4>
|
||||
<div id="outline-container-org0f4faf4" class="outline-4">
|
||||
<h4 id="org0f4faf4"><span class="section-number-4">2.2.2</span> HAC - DVF</h4>
|
||||
<div class="outline-text-4" id="text-2-2-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'dvf'</span>);
|
||||
@ -1008,8 +1008,8 @@ There are mainly three different cases:
|
||||
</ol>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf7aefc3" class="outline-3">
|
||||
<h3 id="orgf7aefc3"><span class="section-number-3">3.1</span> Initialization</h3>
|
||||
<div id="outline-container-org8c2f437" class="outline-3">
|
||||
<h3 id="org8c2f437"><span class="section-number-3">3.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
We first initialize the Stewart platform.
|
||||
@ -1040,8 +1040,8 @@ payload = initializePayload(<span class="org-string">'type'</span>, <span class=
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org827d3cd" class="outline-3">
|
||||
<h3 id="org827d3cd"><span class="section-number-3">3.2</span> Identification</h3>
|
||||
<div id="outline-container-orge2b1c03" class="outline-3">
|
||||
<h3 id="orge2b1c03"><span class="section-number-3">3.2</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'dvf'</span>);
|
||||
@ -1156,8 +1156,8 @@ The overall controller is then \(K(s) = W_1 K_s(s)\) as shown in Figure <a href=
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge195d88" class="outline-4">
|
||||
<h4 id="orge195d88"><span class="section-number-4">3.3.3</span> Results</h4>
|
||||
<div id="outline-container-org5d77351" class="outline-4">
|
||||
<h4 id="org5d77351"><span class="section-number-4">3.3.3</span> Results</h4>
|
||||
<div class="outline-text-4" id="text-3-3-3">
|
||||
<p>
|
||||
We identify the transmissibility and compliance of the Stewart platform under open-loop and closed-loop control.
|
||||
@ -1205,8 +1205,8 @@ The results are shown in figure
|
||||
<h2 id="orgde0f265"><span class="section-number-2">4</span> Time Domain Simulation</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
</div>
|
||||
<div id="outline-container-org8418e03" class="outline-3">
|
||||
<h3 id="org8418e03"><span class="section-number-3">4.1</span> Initialization</h3>
|
||||
<div id="outline-container-org327858f" class="outline-3">
|
||||
<h3 id="org327858f"><span class="section-number-3">4.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
We first initialize the Stewart platform.
|
||||
@ -1329,8 +1329,8 @@ K_hac_dvf = inv(stewart.kinematics.J<span class="org-type">'</span>)<span class=
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5d77351" class="outline-3">
|
||||
<h3 id="org5d77351"><span class="section-number-3">4.4</span> Results</h3>
|
||||
<div id="outline-container-orgd68da79" class="outline-3">
|
||||
<h3 id="orgd68da79"><span class="section-number-3">4.4</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||
@ -1391,7 +1391,7 @@ ylabel(<span class="org-string">'Orientation error [rad]'</span>);
|
||||
<div class="outline-text-4" id="text-orgb457316">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">arguments
|
||||
args.type char {mustBeMember(args.type, {<span class="org-string">'open-loop'</span>, <span class="org-string">'iff'</span>, <span class="org-string">'dvf'</span>, <span class="org-string">'hac-iff'</span>, <span class="org-string">'hac-dvf'</span>, <span class="org-string">'ref-track-L'</span>, <span class="org-string">'ref-track-X'</span>})} = <span class="org-string">'open-loop'</span>
|
||||
args.type char {mustBeMember(args.type, {<span class="org-string">'open-loop'</span>, <span class="org-string">'iff'</span>, <span class="org-string">'dvf'</span>, <span class="org-string">'hac-iff'</span>, <span class="org-string">'hac-dvf'</span>, <span class="org-string">'ref-track-L'</span>, <span class="org-string">'ref-track-X'</span>, <span class="org-string">'ref-track-hac-dvf'</span>})} = <span class="org-string">'open-loop'</span>
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
@ -1427,6 +1427,8 @@ ylabel(<span class="org-string">'Orientation error [rad]'</span>);
|
||||
controller.type = 5;
|
||||
<span class="org-keyword">case</span> <span class="org-string">'ref-track-X'</span>
|
||||
controller.type = 6;
|
||||
<span class="org-keyword">case</span> <span class="org-string">'ref-track-hac-dvf'</span>
|
||||
controller.type = 7;
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
@ -1437,7 +1439,7 @@ ylabel(<span class="org-string">'Orientation error [rad]'</span>);
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-03-11 mer. 18:59</p>
|
||||
<p class="date">Created: 2020-03-12 jeu. 18:06</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -4,7 +4,7 @@
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-03-11 mer. 19:01 -->
|
||||
<!-- 2020-03-12 jeu. 18:06 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
<title>Stewart Platform - Tracking Control</title>
|
||||
@ -248,42 +248,42 @@
|
||||
<ul>
|
||||
<li><a href="#orgd7b25e5">1. Decentralized Control Architecture using Strut Length</a>
|
||||
<ul>
|
||||
<li><a href="#orgf22ae1f">1.1. Control Schematic</a></li>
|
||||
<li><a href="#orgfbc962f">1.2. Initialize the Stewart platform</a></li>
|
||||
<li><a href="#org5b50e6c">1.3. Identification of the plant</a></li>
|
||||
<li><a href="#org127af6e">1.4. Plant Analysis</a></li>
|
||||
<li><a href="#org64fe247">1.5. Controller Design</a></li>
|
||||
<li><a href="#org30afd72">1.6. Simulation</a></li>
|
||||
<li><a href="#org2183826">1.1. Control Schematic</a></li>
|
||||
<li><a href="#orga001fab">1.2. Initialize the Stewart platform</a></li>
|
||||
<li><a href="#orgdff5afa">1.3. Identification of the plant</a></li>
|
||||
<li><a href="#org0328e3c">1.4. Plant Analysis</a></li>
|
||||
<li><a href="#orge8a14d5">1.5. Controller Design</a></li>
|
||||
<li><a href="#org42d3b74">1.6. Simulation</a></li>
|
||||
<li><a href="#org974b430">1.7. Results</a></li>
|
||||
<li><a href="#orge3e2e02">1.8. Conclusion</a></li>
|
||||
<li><a href="#org94c3e48">1.8. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orga519721">2. Centralized Control Architecture using Pose Measurement</a>
|
||||
<ul>
|
||||
<li><a href="#org3846e3e">2.1. Control Schematic</a></li>
|
||||
<li><a href="#orga001fab">2.2. Initialize the Stewart platform</a></li>
|
||||
<li><a href="#orgdff5afa">2.3. Identification of the plant</a></li>
|
||||
<li><a href="#org1cb5954">2.1. Control Schematic</a></li>
|
||||
<li><a href="#orgb748d0f">2.2. Initialize the Stewart platform</a></li>
|
||||
<li><a href="#org131ba62">2.3. Identification of the plant</a></li>
|
||||
<li><a href="#org2223469">2.4. Diagonal Control - Leg’s Frame</a>
|
||||
<ul>
|
||||
<li><a href="#org224a0bb">2.4.1. Control Architecture</a></li>
|
||||
<li><a href="#org7cd17a1">2.4.2. Plant Analysis</a></li>
|
||||
<li><a href="#orgfe68d27">2.4.3. Controller Design</a></li>
|
||||
<li><a href="#org2481134">2.4.4. Simulation</a></li>
|
||||
<li><a href="#org51231d3">2.4.1. Control Architecture</a></li>
|
||||
<li><a href="#org474cd8b">2.4.2. Plant Analysis</a></li>
|
||||
<li><a href="#org98d44a8">2.4.3. Controller Design</a></li>
|
||||
<li><a href="#orgb2c0d3f">2.4.4. Simulation</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org26a8857">2.5. Diagonal Control - Cartesian Frame</a>
|
||||
<ul>
|
||||
<li><a href="#org0831ba6">2.5.1. Control Architecture</a></li>
|
||||
<li><a href="#orga750816">2.5.2. Plant Analysis</a></li>
|
||||
<li><a href="#org925664d">2.5.3. Controller Design</a></li>
|
||||
<li><a href="#org42d3b74">2.5.4. Simulation</a></li>
|
||||
<li><a href="#org8ab259f">2.5.1. Control Architecture</a></li>
|
||||
<li><a href="#org59acfd9">2.5.2. Plant Analysis</a></li>
|
||||
<li><a href="#orgebfcbb3">2.5.3. Controller Design</a></li>
|
||||
<li><a href="#org48cd0ec">2.5.4. Simulation</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgad7bc54">2.6. Diagonal Control - Steady State Decoupling</a>
|
||||
<ul>
|
||||
<li><a href="#org51231d3">2.6.1. Control Architecture</a></li>
|
||||
<li><a href="#org0328e3c">2.6.2. Plant Analysis</a></li>
|
||||
<li><a href="#orge8a14d5">2.6.3. Controller Design</a></li>
|
||||
<li><a href="#org19de50e">2.6.1. Control Architecture</a></li>
|
||||
<li><a href="#org7eca1bc">2.6.2. Plant Analysis</a></li>
|
||||
<li><a href="#org177398f">2.6.3. Controller Design</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orga2eadeb">2.7. Comparison</a>
|
||||
@ -292,12 +292,29 @@
|
||||
<li><a href="#org23ae479">2.7.2. Simulation Results</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org94c3e48">2.8. Conclusion</a></li>
|
||||
<li><a href="#orgb643774">2.8. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org4b8c360">3. Hybrid Control Architecture - HAC-LAC with relative DVF</a>
|
||||
<ul>
|
||||
<li><a href="#org2183826">3.1. Control Schematic</a></li>
|
||||
<li><a href="#org0e138be">3.1. Control Schematic</a></li>
|
||||
<li><a href="#org627b63c">3.2. Initialize the Stewart platform</a></li>
|
||||
<li><a href="#org3274a98">3.3. First Control Loop - \(\bm{K}_\mathcal{L}\)</a>
|
||||
<ul>
|
||||
<li><a href="#org4d65047">3.3.1. Identification</a></li>
|
||||
<li><a href="#org0f5d7cd">3.3.2. Obtained Plant</a></li>
|
||||
<li><a href="#org4e073ac">3.3.3. Controller Design</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org8440c0b">3.4. Second Control Loop - \(\bm{K}_\mathcal{X}\)</a>
|
||||
<ul>
|
||||
<li><a href="#orge29b065">3.4.1. Identification</a></li>
|
||||
<li><a href="#org78e6e29">3.4.2. Obtained Plant</a></li>
|
||||
<li><a href="#org3a9b8c2">3.4.3. Controller Design</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org74d3dcd">3.5. Simulations</a></li>
|
||||
<li><a href="#org35a41e9">3.6. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org445f7a9">4. Position Error computation</a></li>
|
||||
@ -325,8 +342,8 @@ Depending of the measured quantity, different control architectures can be used:
|
||||
<a id="orgea7df6c"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgf22ae1f" class="outline-3">
|
||||
<h3 id="orgf22ae1f"><span class="section-number-3">1.1</span> Control Schematic</h3>
|
||||
<div id="outline-container-org2183826" class="outline-3">
|
||||
<h3 id="org2183826"><span class="section-number-3">1.1</span> Control Schematic</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
The control architecture is shown in Figure <a href="#org4f704a1">1</a>.
|
||||
@ -349,8 +366,8 @@ Then, a diagonal (decentralized) controller \(\bm{K}_\mathcal{L}\) is used such
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfbc962f" class="outline-3">
|
||||
<h3 id="orgfbc962f"><span class="section-number-3">1.2</span> Initialize the Stewart platform</h3>
|
||||
<div id="outline-container-orga001fab" class="outline-3">
|
||||
<h3 id="orga001fab"><span class="section-number-3">1.2</span> Initialize the Stewart platform</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
@ -382,11 +399,11 @@ references = initializeReferences(stewart);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5b50e6c" class="outline-3">
|
||||
<h3 id="org5b50e6c"><span class="section-number-3">1.3</span> Identification of the plant</h3>
|
||||
<div id="outline-container-orgdff5afa" class="outline-3">
|
||||
<h3 id="orgdff5afa"><span class="section-number-3">1.3</span> Identification of the plant</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\).
|
||||
Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\).
|
||||
</p>
|
||||
<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>
|
||||
@ -406,8 +423,8 @@ G.OutputName = {<span class="org-string">'L1'</span>, <span class="org-string">'
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org127af6e" class="outline-3">
|
||||
<h3 id="org127af6e"><span class="section-number-3">1.4</span> Plant Analysis</h3>
|
||||
<div id="outline-container-org0328e3c" class="outline-3">
|
||||
<h3 id="org0328e3c"><span class="section-number-3">1.4</span> Plant Analysis</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<p>
|
||||
The diagonal terms of the plant is shown in Figure <a href="#org8c82316">2</a>.
|
||||
@ -441,15 +458,14 @@ We see that the plant is decoupled at low frequency which indicate that decentra
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org64fe247" class="outline-3">
|
||||
<h3 id="org64fe247"><span class="section-number-3">1.5</span> Controller Design</h3>
|
||||
<div id="outline-container-orge8a14d5" class="outline-3">
|
||||
<h3 id="orge8a14d5"><span class="section-number-3">1.5</span> Controller Design</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<p>
|
||||
The controller consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>A pure integrator</li>
|
||||
<li>A lead around the crossover frequency to increase the phase margin</li>
|
||||
<li>A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin</li>
|
||||
</ul>
|
||||
|
||||
@ -472,8 +488,8 @@ Kl = diag(1<span class="org-type">./</span>diag(abs(freqresp(G, wc)))) <span cla
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org30afd72" class="outline-3">
|
||||
<h3 id="org30afd72"><span class="section-number-3">1.6</span> Simulation</h3>
|
||||
<div id="outline-container-org42d3b74" class="outline-3">
|
||||
<h3 id="org42d3b74"><span class="section-number-3">1.6</span> Simulation</h3>
|
||||
<div class="outline-text-3" id="text-1-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">t = linspace(0, 10, 1000);
|
||||
@ -518,8 +534,8 @@ simout_D = simout;
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge3e2e02" class="outline-3">
|
||||
<h3 id="orge3e2e02"><span class="section-number-3">1.8</span> Conclusion</h3>
|
||||
<div id="outline-container-org94c3e48" class="outline-3">
|
||||
<h3 id="org94c3e48"><span class="section-number-3">1.8</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-1-8">
|
||||
<p>
|
||||
Such control architecture is easy to implement and give good results.
|
||||
@ -540,8 +556,8 @@ However, as \(\mathcal{X}\) is not directly measured, it is possible that import
|
||||
<a id="org48604d1"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org3846e3e" class="outline-3">
|
||||
<h3 id="org3846e3e"><span class="section-number-3">2.1</span> Control Schematic</h3>
|
||||
<div id="outline-container-org1cb5954" class="outline-3">
|
||||
<h3 id="org1cb5954"><span class="section-number-3">2.1</span> Control Schematic</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
The centralized controller takes the position error \(\bm{\epsilon}_\mathcal{X}\) as an inputs and generate actuator forces \(\bm{\tau}\) (see Figure <a href="#org97ec686">7</a>).
|
||||
@ -579,8 +595,8 @@ It is indeed a more complex computation explained in section <a href="#org5f6154
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga001fab" class="outline-3">
|
||||
<h3 id="orga001fab"><span class="section-number-3">2.2</span> Initialize the Stewart platform</h3>
|
||||
<div id="outline-container-orgb748d0f" class="outline-3">
|
||||
<h3 id="orgb748d0f"><span class="section-number-3">2.2</span> Initialize the Stewart platform</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
@ -612,11 +628,11 @@ references = initializeReferences(stewart);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdff5afa" class="outline-3">
|
||||
<h3 id="orgdff5afa"><span class="section-number-3">2.3</span> Identification of the plant</h3>
|
||||
<div id="outline-container-org131ba62" class="outline-3">
|
||||
<h3 id="org131ba62"><span class="section-number-3">2.3</span> Identification of the plant</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\).
|
||||
Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{\mathcal{X}}\).
|
||||
</p>
|
||||
<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>
|
||||
@ -643,8 +659,8 @@ G.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'
|
||||
<a id="org31fd942"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org224a0bb" class="outline-4">
|
||||
<h4 id="org224a0bb"><span class="section-number-4">2.4.1</span> Control Architecture</h4>
|
||||
<div id="outline-container-org51231d3" class="outline-4">
|
||||
<h4 id="org51231d3"><span class="section-number-4">2.4.1</span> Control Architecture</h4>
|
||||
<div class="outline-text-4" id="text-2-4-1">
|
||||
<p>
|
||||
The pose error \(\bm{\epsilon}_\mathcal{X}\) is first converted in the frame of the leg by using the Jacobian matrix.
|
||||
@ -665,8 +681,8 @@ Note here that the transformation from the pose error \(\bm{\epsilon}_\mathcal{X
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7cd17a1" class="outline-4">
|
||||
<h4 id="org7cd17a1"><span class="section-number-4">2.4.2</span> Plant Analysis</h4>
|
||||
<div id="outline-container-org474cd8b" class="outline-4">
|
||||
<h4 id="org474cd8b"><span class="section-number-4">2.4.2</span> Plant Analysis</h4>
|
||||
<div class="outline-text-4" id="text-2-4-2">
|
||||
<p>
|
||||
We now multiply the plant by the Jacobian matrix as shown in Figure <a href="#orgb1f5ad2">8</a> to obtain a more diagonal plant.
|
||||
@ -716,15 +732,14 @@ Thus \(J \cdot G(\omega = 0) = J \cdot \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfe68d27" class="outline-4">
|
||||
<h4 id="orgfe68d27"><span class="section-number-4">2.4.3</span> Controller Design</h4>
|
||||
<div id="outline-container-org98d44a8" class="outline-4">
|
||||
<h4 id="org98d44a8"><span class="section-number-4">2.4.3</span> Controller Design</h4>
|
||||
<div class="outline-text-4" id="text-2-4-3">
|
||||
<p>
|
||||
The controller consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>A pure integrator</li>
|
||||
<li>A lead around the crossover frequency to increase the phase margin</li>
|
||||
<li>A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin</li>
|
||||
</ul>
|
||||
|
||||
@ -755,8 +770,8 @@ The controller \(\bm{K} = \bm{K}_\mathcal{L} \bm{J}\) is computed.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2481134" class="outline-4">
|
||||
<h4 id="org2481134"><span class="section-number-4">2.4.4</span> Simulation</h4>
|
||||
<div id="outline-container-orgb2c0d3f" class="outline-4">
|
||||
<h4 id="orgb2c0d3f"><span class="section-number-4">2.4.4</span> Simulation</h4>
|
||||
<div class="outline-text-4" id="text-2-4-4">
|
||||
<p>
|
||||
We specify the reference path to follow.
|
||||
@ -796,8 +811,8 @@ simout_L = simout;
|
||||
<a id="orgfd201c3"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org0831ba6" class="outline-4">
|
||||
<h4 id="org0831ba6"><span class="section-number-4">2.5.1</span> Control Architecture</h4>
|
||||
<div id="outline-container-org8ab259f" class="outline-4">
|
||||
<h4 id="org8ab259f"><span class="section-number-4">2.5.1</span> Control Architecture</h4>
|
||||
<div class="outline-text-4" id="text-2-5-1">
|
||||
<p>
|
||||
A diagonal controller \(\bm{K}_\mathcal{X}\) take the pose error \(\bm{\epsilon}_\mathcal{X}\) and generate cartesian forces \(\bm{\mathcal{F}}\) that are then converted to actuators forces using the Jacobian as shown in Figure e <a href="#org6b158db">12</a>.
|
||||
@ -816,8 +831,8 @@ The final implemented controller is \(\bm{K} = \bm{J}^{-T} \cdot \bm{K}_\mathcal
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga750816" class="outline-4">
|
||||
<h4 id="orga750816"><span class="section-number-4">2.5.2</span> Plant Analysis</h4>
|
||||
<div id="outline-container-org59acfd9" class="outline-4">
|
||||
<h4 id="org59acfd9"><span class="section-number-4">2.5.2</span> Plant Analysis</h4>
|
||||
<div class="outline-text-4" id="text-2-5-2">
|
||||
<p>
|
||||
We now multiply the plant by the Jacobian matrix as shown in Figure <a href="#org6b158db">12</a> to obtain a more diagonal plant.
|
||||
@ -939,15 +954,14 @@ This control architecture can also give a dynamically decoupled plant if the Cen
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org925664d" class="outline-4">
|
||||
<h4 id="org925664d"><span class="section-number-4">2.5.3</span> Controller Design</h4>
|
||||
<div id="outline-container-orgebfcbb3" class="outline-4">
|
||||
<h4 id="orgebfcbb3"><span class="section-number-4">2.5.3</span> Controller Design</h4>
|
||||
<div class="outline-text-4" id="text-2-5-3">
|
||||
<p>
|
||||
The controller consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>A pure integrator</li>
|
||||
<li>A lead around the crossover frequency to increase the phase margin</li>
|
||||
<li>A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin</li>
|
||||
</ul>
|
||||
|
||||
@ -978,8 +992,8 @@ The controller \(\bm{K} = \bm{J}^{-T} \bm{K}_\mathcal{X}\) is computed.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org42d3b74" class="outline-4">
|
||||
<h4 id="org42d3b74"><span class="section-number-4">2.5.4</span> Simulation</h4>
|
||||
<div id="outline-container-org48cd0ec" class="outline-4">
|
||||
<h4 id="org48cd0ec"><span class="section-number-4">2.5.4</span> Simulation</h4>
|
||||
<div class="outline-text-4" id="text-2-5-4">
|
||||
<p>
|
||||
We specify the reference path to follow.
|
||||
@ -1019,8 +1033,8 @@ simout_X = simout;
|
||||
<a id="org789ba4a"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org51231d3" class="outline-4">
|
||||
<h4 id="org51231d3"><span class="section-number-4">2.6.1</span> Control Architecture</h4>
|
||||
<div id="outline-container-org19de50e" class="outline-4">
|
||||
<h4 id="org19de50e"><span class="section-number-4">2.6.1</span> Control Architecture</h4>
|
||||
<div class="outline-text-4" id="text-2-6-1">
|
||||
<p>
|
||||
The plant \(\bm{G}\) is pre-multiply by \(\bm{G}^{-1}(\omega = 0)\) such that the “shaped plant” \(\bm{G}_0 = \bm{G} \bm{G}^{-1}(\omega = 0)\) is diagonal at low frequency.
|
||||
@ -1043,8 +1057,8 @@ The control architecture is shown in Figure <a href="#orgb226e44">16</a>.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0328e3c" class="outline-4">
|
||||
<h4 id="org0328e3c"><span class="section-number-4">2.6.2</span> Plant Analysis</h4>
|
||||
<div id="outline-container-org7eca1bc" class="outline-4">
|
||||
<h4 id="org7eca1bc"><span class="section-number-4">2.6.2</span> Plant Analysis</h4>
|
||||
<div class="outline-text-4" id="text-2-6-2">
|
||||
<p>
|
||||
The plant is pre-multiplied by \(\bm{G}^{-1}(\omega = 0)\).
|
||||
@ -1072,8 +1086,8 @@ The diagonal elements of the shaped plant are shown in Figure <a href="#orgc15aa
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge8a14d5" class="outline-4">
|
||||
<h4 id="orge8a14d5"><span class="section-number-4">2.6.3</span> Controller Design</h4>
|
||||
<div id="outline-container-org177398f" class="outline-4">
|
||||
<h4 id="org177398f"><span class="section-number-4">2.6.3</span> Controller Design</h4>
|
||||
<div class="outline-text-4" id="text-2-6-3">
|
||||
<p>
|
||||
We have that:
|
||||
@ -1147,8 +1161,8 @@ This error is much lower when using the diagonal control in the frame of the leg
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org94c3e48" class="outline-3">
|
||||
<h3 id="org94c3e48"><span class="section-number-3">2.8</span> Conclusion</h3>
|
||||
<div id="outline-container-orgb643774" class="outline-3">
|
||||
<h3 id="orgb643774"><span class="section-number-3">2.8</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-8">
|
||||
<p>
|
||||
Both control architecture gives similar results even tough the control in the Leg’s frame gives slightly better results.
|
||||
@ -1231,68 +1245,332 @@ Thus, this method should be quite robust against parameter variation (e.g. the p
|
||||
<a id="org14e3e5f"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org2183826" class="outline-3">
|
||||
<h3 id="org2183826"><span class="section-number-3">3.1</span> Control Schematic</h3>
|
||||
<div id="outline-container-org0e138be" class="outline-3">
|
||||
<h3 id="org0e138be"><span class="section-number-3">3.1</span> Control Schematic</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
Let’s consider the control schematic shown in Figure <a href="#org3a1b1db">22</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The first loop containing \(\bm{K}_\mathcal{L}\) is a Decentralized Direct (Relative) Velocity Feedback.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
A reference \(\bm{r}_\mathcal{L}\) is computed using the inverse kinematics and corresponds to the wanted motion of each leg.
|
||||
The actual length of each leg \(\bm{\mathcal{L}}\) is subtracted and then passed trough the controller \(\bm{K}_\mathcal{L}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The controller is a diagonal controller with pure derivative action on the diagonal.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The effect of this loop is:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>it adds damping to the system (the force applied in each actuator is proportional to the relative velocity of the strut)</li>
|
||||
<li>it however does not go “against” the reference path \(\bm{r}_\mathcal{X}\) thanks to the use of the inverse kinematics</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Then, the second loop containing \(\bm{K}_\mathcal{X}\) is designed based on the already damped plant (represented by the gray area).
|
||||
This second loop is responsible for the reference tracking.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org3a1b1db" class="figure">
|
||||
<p><img src="figs/hybrid_reference_tracking.png" alt="hybrid_reference_tracking.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 22: </span>Centralized Controller</p>
|
||||
<p><span class="figure-number">Figure 22: </span>Hybrid Control Architecture</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org627b63c" class="outline-3">
|
||||
<h3 id="org627b63c"><span class="section-number-3">3.2</span> Initialize the Stewart platform</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'accelerometer'</span>, <span class="org-string">'freq'</span>, 5e3);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">disturbances = initializeDisturbances();
|
||||
references = initializeReferences(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3274a98" class="outline-3">
|
||||
<h3 id="org3274a98"><span class="section-number-3">3.3</span> First Control Loop - \(\bm{K}_\mathcal{L}\)</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
</div>
|
||||
<div id="outline-container-org4d65047" class="outline-4">
|
||||
<h4 id="org4d65047"><span class="section-number-4">3.3.1</span> Identification</h4>
|
||||
<div class="outline-text-4" id="text-3-3-1">
|
||||
<p>
|
||||
Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\).
|
||||
</p>
|
||||
<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>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
Gl = linearize(mdl, io);
|
||||
Gl.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
Gl.OutputName = {<span class="org-string">'L1'</span>, <span class="org-string">'L2'</span>, <span class="org-string">'L3'</span>, <span class="org-string">'L4'</span>, <span class="org-string">'L5'</span>, <span class="org-string">'L6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0f5d7cd" class="outline-4">
|
||||
<h4 id="org0f5d7cd"><span class="section-number-4">3.3.2</span> Obtained Plant</h4>
|
||||
<div class="outline-text-4" id="text-3-3-2">
|
||||
<p>
|
||||
The diagonal elements of the plant are shown in Figure <a href="#org687a922">23</a> while the off diagonal terms are shown in Figure <a href="#orge568b8a">24</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org687a922" class="figure">
|
||||
<p><img src="figs/hybrid_control_Kl_plant_diagonal.png" alt="hybrid_control_Kl_plant_diagonal.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 23: </span>Diagonal elements of the plant for the design of \(\bm{K}_\mathcal{L}\) (<a href="./figs/hybrid_control_Kl_plant_diagonal.png">png</a>, <a href="./figs/hybrid_control_Kl_plant_diagonal.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orge568b8a" class="figure">
|
||||
<p><img src="figs/hybrid_control_Kl_plant_off_diagonal.png" alt="hybrid_control_Kl_plant_off_diagonal.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 24: </span>Off-diagonal elements of the plant for the design of \(\bm{K}_\mathcal{L}\) (<a href="./figs/hybrid_control_Kl_plant_off_diagonal.png">png</a>, <a href="./figs/hybrid_control_Kl_plant_off_diagonal.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4e073ac" class="outline-4">
|
||||
<h4 id="org4e073ac"><span class="section-number-4">3.3.3</span> Controller Design</h4>
|
||||
<div class="outline-text-4" id="text-3-3-3">
|
||||
<p>
|
||||
We apply a decentralized (diagonal) direct velocity feedback.
|
||||
Thus, we apply a pure derivative action.
|
||||
In order to make the controller realizable, we add a low pass filter at high frequency.
|
||||
The gain of the controller is chosen such that the resonances are critically damped.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The obtain loop gain is shown in Figure <a href="#orgb74befe">25</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kl = 1e4 <span class="org-type">*</span> s <span class="org-type">/</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1e4) <span class="org-type">*</span> eye(6);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb74befe" class="figure">
|
||||
<p><img src="figs/hybrid_control_Kl_loop_gain.png" alt="hybrid_control_Kl_loop_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 25: </span>Obtain Loop Gain for the DVF control loop (<a href="./figs/hybrid_control_Kl_loop_gain.png">png</a>, <a href="./figs/hybrid_control_Kl_loop_gain.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8440c0b" class="outline-3">
|
||||
<h3 id="org8440c0b"><span class="section-number-3">3.4</span> Second Control Loop - \(\bm{K}_\mathcal{X}\)</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
</div>
|
||||
<div id="outline-container-orge29b065" class="outline-4">
|
||||
<h4 id="orge29b065"><span class="section-number-4">3.4.1</span> Identification</h4>
|
||||
<div class="outline-text-4" id="text-3-4-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kx = tf(zeros(6));
|
||||
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'ref-track-hac-dvf'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<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>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org78e6e29" class="outline-4">
|
||||
<h4 id="org78e6e29"><span class="section-number-4">3.4.2</span> Obtained Plant</h4>
|
||||
<div class="outline-text-4" id="text-3-4-2">
|
||||
<p>
|
||||
We use the Jacobian matrix to apply forces in the cartesian frame.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Gx = G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
|
||||
Gx.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained plant is shown in Figure <a href="#org2517e3d">26</a>.
|
||||
</p>
|
||||
|
||||
<div id="org2517e3d" class="figure">
|
||||
<p><img src="figs/hybrid_control_Kx_plant.png" alt="hybrid_control_Kx_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 26: </span>Diagonal and Off-diagonal elements of the plant for the design of \(\bm{K}_\mathcal{L}\) (<a href="./figs/hybrid_control_Kx_plant.png">png</a>, <a href="./figs/hybrid_control_Kx_plant.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3a9b8c2" class="outline-4">
|
||||
<h4 id="org3a9b8c2"><span class="section-number-4">3.4.3</span> Controller Design</h4>
|
||||
<div class="outline-text-4" id="text-3-4-3">
|
||||
<p>
|
||||
The controller consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>A pure integrator</li>
|
||||
<li>A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin</li>
|
||||
</ul>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>200; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
|
||||
|
||||
h = 3; <span class="org-comment">% Lead parameter</span>
|
||||
|
||||
Kx = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> ((s<span class="org-type">/</span>wc<span class="org-type">/</span>2 <span class="org-type">+</span> 1)<span class="org-type">/</span>(s<span class="org-type">/</span>wc<span class="org-type">/</span>2));
|
||||
|
||||
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
|
||||
Kx = Kx<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kx, wc))));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org30ad867" class="figure">
|
||||
<p><img src="figs/hybrid_control_Kx_loop_gain.png" alt="hybrid_control_Kx_loop_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 27: </span>Obtained Loop Gain for the controller \(\bm{K}_\mathcal{X}\) (<a href="./figs/hybrid_control_Kx_loop_gain.png">png</a>, <a href="./figs/hybrid_control_Kx_loop_gain.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Then we include the Jacobian in the controller matrix.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kx = inv(stewart.kinematics.J<span class="org-type">'</span>)<span class="org-type">*</span>Kx;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org74d3dcd" class="outline-3">
|
||||
<h3 id="org74d3dcd"><span class="section-number-3">3.5</span> Simulations</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
We specify the reference path to follow.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">t = linspace(0, 10, 10000);
|
||||
|
||||
r = zeros(6, length(t));
|
||||
|
||||
r(1, <span class="org-type">:</span>) = 5e<span class="org-type">-</span>3<span class="org-type">*</span>sin(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>t);
|
||||
|
||||
references = initializeReferences(stewart, <span class="org-string">'t'</span>, t, <span class="org-string">'r'</span>, r);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We run the simulation and we save the results.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'stewart_platform_model'</span>)
|
||||
simout_H = simout;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained position error is shown in Figure <a href="#org19456cf">28</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org19456cf" class="figure">
|
||||
<p><img src="figs/hybrid_control_Ex.png" alt="hybrid_control_Ex.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 28: </span>Obtained position error \(\bm{\epsilon}_\mathcal{X}\) (<a href="./figs/hybrid_control_Ex.png">png</a>, <a href="./figs/hybrid_control_Ex.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org35a41e9" class="outline-3">
|
||||
<h3 id="org35a41e9"><span class="section-number-3">3.6</span> Conclusion</h3>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org445f7a9" class="outline-2">
|
||||
<h2 id="org445f7a9"><span class="section-number-2">4</span> Position Error computation</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org5f61540"></a>
|
||||
Let’s denote:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(\{W\}\) the initial fixed frame (base in which the measurement is done)</li>
|
||||
<li>\(\{R\}\) the reference frame corresponding to the wanted pose of the sample</li>
|
||||
<li>\(\{M\}\) the frame corresponding to the measured pose of the sample</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
We have then computed:
|
||||
Let’s note:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\({}^W\bm{T}_R\) which corresponds to the wanted pose of the sample with respect to the granite</li>
|
||||
<li>\({}^W\bm{T}_M\) which corresponds to the measured pose of the sample with respect to the granite</li>
|
||||
<li>\(\{W\}\) the fixed measurement frame (corresponding to the metrology frame / the frame where the wanted displacement are expressed).
|
||||
The center of the frame if \(O_W\)</li>
|
||||
<li>\(\{M\}\) is the frame fixed to the measured elements.
|
||||
\(O_M\) is the point where the pose of the element is measured</li>
|
||||
<li>\(\{R\}\) is a virtual frame corresponding to the wanted pose of the element.
|
||||
\(O_R\) is the origin of this frame where the we want to position the point \(O_M\) of the element.</li>
|
||||
<li>\(\{V\}\) is a frame which its axes are aligned with \(\{W\}\) and its origin \(O_V\) is coincident with the \(O_M\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Reference Position with respect to fixed frame {W}: \({}^WT_R\)
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Dx = 0;
|
||||
Dy = 0;
|
||||
Dz = 0.1;
|
||||
Rx = <span class="org-constant">pi</span>;
|
||||
Ry = 0;
|
||||
Rz = 0;
|
||||
|
||||
WTr = zeros(4,4);
|
||||
|
||||
R = [cos(Rz) <span class="org-type">-</span>sin(Rz) 0;
|
||||
sin(Rz) cos(Rz) 0;
|
||||
0 0 1] <span class="org-type">*</span> ...
|
||||
[cos(Ry) 0 sin(Ry);
|
||||
0 1 0;
|
||||
<span class="org-type">-</span>sin(Ry) 0 cos(Ry)] <span class="org-type">*</span> ...
|
||||
[1 0 0;
|
||||
0 cos(Rx) <span class="org-type">-</span>sin(Rx);
|
||||
0 sin(Rx) cos(Rx)];
|
||||
|
||||
|
||||
WTr(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3) = R;
|
||||
WTr(1<span class="org-type">:</span>4, 4) = [Dx ; Dy ; Dz; 1];
|
||||
<pre class="src src-matlab">Dxr = 0;
|
||||
Dyr = 0;
|
||||
Dzr = 0.1;
|
||||
Rxr = <span class="org-constant">pi</span>;
|
||||
Ryr = 0;
|
||||
Rzr = 0;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -1300,47 +1578,89 @@ WTr(1<span class="org-type">:</span>4, 4) = [Dx ; Dy ; Dz; 1];
|
||||
Measured Position with respect to fixed frame {W}: \({}^WT_M\)
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Dx = 0;
|
||||
Dy = 0;
|
||||
Dz = 0;
|
||||
Rx = <span class="org-constant">pi</span>;
|
||||
Ry = 0;
|
||||
Rz = 0;
|
||||
|
||||
WTm = zeros(4,4);
|
||||
|
||||
R = [cos(Rz) <span class="org-type">-</span>sin(Rz) 0;
|
||||
sin(Rz) cos(Rz) 0;
|
||||
0 0 1] <span class="org-type">*</span> ...
|
||||
[cos(Ry) 0 sin(Ry);
|
||||
0 1 0;
|
||||
<span class="org-type">-</span>sin(Ry) 0 cos(Ry)] <span class="org-type">*</span> ...
|
||||
[1 0 0;
|
||||
0 cos(Rx) <span class="org-type">-</span>sin(Rx);
|
||||
0 sin(Rx) cos(Rx)];
|
||||
|
||||
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3) = R;
|
||||
WTm(1<span class="org-type">:</span>4, 4) = [Dx ; Dy ; Dz; 1];
|
||||
<pre class="src src-matlab">Dxm = 0;
|
||||
Dym = 0;
|
||||
Dzm = 0;
|
||||
Rxm = <span class="org-constant">pi</span>;
|
||||
Rym = 0;
|
||||
Rzm = 0;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We would like to compute \({}^M\bm{T}_R\) which corresponds to the wanted pose of the sample expressed in a frame attached to the top platform of the nano-hexapod (frame \(\{M\}\)).
|
||||
We measure the position and orientation (pose) of the element represented by the frame \(\{M\}\) with respect to frame \(\{W\}\).
|
||||
Thus we can compute the Homogeneous transformation matrix \({}^WT_M\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Measured Pose</span></span>
|
||||
WTm = zeros(4,4);
|
||||
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3) = [cos(Rzm) <span class="org-type">-</span>sin(Rzm) 0;
|
||||
sin(Rzm) cos(Rzm) 0;
|
||||
0 0 1] <span class="org-type">*</span> ...
|
||||
[cos(Rym) 0 sin(Rym);
|
||||
0 1 0;
|
||||
<span class="org-type">-</span>sin(Rym) 0 cos(Rym)] <span class="org-type">*</span> ...
|
||||
[1 0 0;
|
||||
0 cos(Rxm) <span class="org-type">-</span>sin(Rxm);
|
||||
0 sin(Rxm) cos(Rxm)];
|
||||
WTm(1<span class="org-type">:</span>4, 4) = [Dxm ; Dym ; Dzm; 1];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We have:
|
||||
We can also compute the Homogeneous transformation matrix \({}^WT_R\) corresponding to the transformation required to go from fixed frame \(\{W\}\) to the wanted frame \(\{R\}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Reference Pose</span></span>
|
||||
WTr = zeros(4,4);
|
||||
|
||||
WTr(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3) = [cos(Rzr) <span class="org-type">-</span>sin(Rzr) 0;
|
||||
sin(Rzr) cos(Rzr) 0;
|
||||
0 0 1] <span class="org-type">*</span> ...
|
||||
[cos(Ryr) 0 sin(Ryr);
|
||||
0 1 0;
|
||||
<span class="org-type">-</span>sin(Ryr) 0 cos(Ryr)] <span class="org-type">*</span> ...
|
||||
[1 0 0;
|
||||
0 cos(Rxr) <span class="org-type">-</span>sin(Rxr);
|
||||
0 sin(Rxr) cos(Rxr)];
|
||||
WTr(1<span class="org-type">:</span>4, 4) = [Dxr ; Dyr ; Dzr; 1];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We can also compute \({}^WT_V\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">WTv = eye(4);
|
||||
WTv(1<span class="org-type">:</span>3, 4) = WTm(1<span class="org-type">:</span>3, 4);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Now we want to express \({}^MT_R\) which corresponds to the transformation required to go to wanted position expressed in the frame of the measured element.
|
||||
This homogeneous transformation can be computed from the previously computed matrices:
|
||||
\[ {}^MT_R = ({{}^WT_M}^{-1}) {}^WT_R \]
|
||||
</p>
|
||||
\begin{align}
|
||||
{}^M\bm{T}_R &= {}^M\bm{T}_W \cdot {}^W\bm{T}_R \\
|
||||
&= {}^W{\bm{T}_M}^{-1} \cdot {}^W\bm{T}_R
|
||||
\end{align}
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-comment">% Error with respect to the top platform</span>
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Wanted pose expressed in a frame corresponding to the actual measured pose</span></span>
|
||||
MTr = [WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3)<span class="org-type">'</span>, <span class="org-type">-</span>WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3,4) ; 0 0 0 1]<span class="org-type">*</span>WTr;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Now we want to express \({}^VT_R\):
|
||||
\[ {}^VT_R = ({{}^WT_V}^{-1}) {}^WT_R \]
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Wanted pose expressed in a frame coincident with the actual position but with no rotation</span></span>
|
||||
VTr = [WTv(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3)<span class="org-type">'</span>, <span class="org-type">-</span>WTv(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3)<span class="org-type">'*</span>WTv(1<span class="org-type">:</span>3,4) ; 0 0 0 1] <span class="org-type">*</span> WTr;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Extract Translations and Rotations from the Homogeneous Matrix</span></span>
|
||||
T = MTr;
|
||||
Edx = T(1, 4);
|
||||
Edy = T(2, 4);
|
||||
@ -1350,8 +1670,6 @@ Edz = T(3, 4);
|
||||
Ery = atan2( T(1, 3), sqrt(T(1, 1)<span class="org-type">^</span>2 <span class="org-type">+</span> T(1, 2)<span class="org-type">^</span>2));
|
||||
Erx = atan2(<span class="org-type">-</span>T(2, 3)<span class="org-type">/</span>cos(Ery), T(3, 3)<span class="org-type">/</span>cos(Ery));
|
||||
Erz = atan2(<span class="org-type">-</span>T(1, 2)<span class="org-type">/</span>cos(Ery), T(1, 1)<span class="org-type">/</span>cos(Ery));
|
||||
|
||||
[Edx, Edy, Edz, Erx, Ery, Erz]
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@ -1359,7 +1677,7 @@ Erz = atan2(<span class="org-type">-</span>T(1, 2)<span class="org-type">/</span
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-03-11 mer. 19:01</p>
|
||||
<p class="date">Created: 2020-03-12 jeu. 18:06</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -4,7 +4,7 @@
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-03-03 mar. 15:51 -->
|
||||
<!-- 2020-03-12 jeu. 18:06 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
<title>Cubic configuration for the Stewart Platform</title>
|
||||
@ -252,33 +252,34 @@
|
||||
<li><a href="#orga88e79a">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
|
||||
<li><a href="#orge02ec88">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
|
||||
<li><a href="#org43fd7e4">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
|
||||
<li><a href="#org64395b6">1.5. Conclusion</a></li>
|
||||
<li><a href="#org3356db5">1.5. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgd70418b">2. Configuration with the Cube’s center above the mobile platform</a>
|
||||
<ul>
|
||||
<li><a href="#org8afa645">2.1. Having Cube’s center above the top platform</a></li>
|
||||
<li><a href="#org8e09793">2.2. Conclusion</a></li>
|
||||
<li><a href="#org25d045b">2.2. Size of the platforms</a></li>
|
||||
<li><a href="#org9477b7a">2.3. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgcc4ecce">3. Cubic size analysis</a>
|
||||
<ul>
|
||||
<li><a href="#org0029d8c">3.1. Analysis</a></li>
|
||||
<li><a href="#orgc5a2e1f">3.2. Conclusion</a></li>
|
||||
<li><a href="#org46632b3">3.2. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgf09da67">4. Dynamic Coupling in the Cartesian Frame</a>
|
||||
<ul>
|
||||
<li><a href="#org5fe01ec">4.1. Cube’s center at the Center of Mass of the mobile platform</a></li>
|
||||
<li><a href="#org4cb2a36">4.2. Cube’s center not coincident with the Mass of the Mobile platform</a></li>
|
||||
<li><a href="#org24cd25e">4.3. Conclusion</a></li>
|
||||
<li><a href="#orgc1b6a36">4.3. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org8f26dc0">5. Dynamic Coupling between actuators and sensors of each strut</a>
|
||||
<ul>
|
||||
<li><a href="#org6e391c9">5.1. Coupling between the actuators and sensors - Cubic Architecture</a></li>
|
||||
<li><a href="#orgafd808d">5.2. Coupling between the actuators and sensors - Non-Cubic Architecture</a></li>
|
||||
<li><a href="#org3356db5">5.3. Conclusion</a></li>
|
||||
<li><a href="#org87716af">5.3. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org3044455">6. Functions</a>
|
||||
@ -826,8 +827,8 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org64395b6" class="outline-3">
|
||||
<h3 id="org64395b6"><span class="section-number-3">1.5</span> Conclusion</h3>
|
||||
<div id="outline-container-org3356db5" class="outline-3">
|
||||
<h3 id="org3356db5"><span class="section-number-3">1.5</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<div class="important">
|
||||
<p>
|
||||
@ -1164,9 +1165,100 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8e09793" class="outline-3">
|
||||
<h3 id="org8e09793"><span class="section-number-3">2.2</span> Conclusion</h3>
|
||||
<div id="outline-container-org25d045b" class="outline-3">
|
||||
<h3 id="org25d045b"><span class="section-number-3">2.2</span> Size of the platforms</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
The minimum size of the platforms depends on the cube’s size and the height between the platform and the cube’s center.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Let’s denote:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(H\) the height between the cube’s center and the considered platform</li>
|
||||
<li>\(D\) the size of the cube’s edges</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Let’s denote by \(a\) and \(b\) the points of both ends of one of the cube’s edge.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Initially, we have:
|
||||
</p>
|
||||
\begin{align}
|
||||
a &= \frac{D}{2} \begin{bmatrix}-1 \\ -1 \\ 1\end{bmatrix} \\
|
||||
b &= \frac{D}{2} \begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
We rotate the cube around its center (origin of the rotated frame) such that one of its diagonal is vertical.
|
||||
\[ R = \begin{bmatrix}
|
||||
\frac{2}{\sqrt{6}} & 0 & \frac{1}{\sqrt{3}} \\
|
||||
\frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\
|
||||
\frac{-1}{\sqrt{6}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{3}}
|
||||
\end{bmatrix} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
After rotation, the points \(a\) and \(b\) become:
|
||||
</p>
|
||||
\begin{align}
|
||||
a &= \frac{D}{2} \begin{bmatrix}-\frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ -\frac{1}{\sqrt{3}}\end{bmatrix} \\
|
||||
b &= \frac{D}{2} \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ \frac{1}{\sqrt{3}}\end{bmatrix}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
Points \(a\) and \(b\) define a vector \(u = b - a\) that gives the orientation of one of the Stewart platform strut:
|
||||
\[ u = \frac{D}{\sqrt{3}} \begin{bmatrix} -\sqrt{2} \\ 0 \\ -1\end{bmatrix} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Then we want to find the intersection between the line that defines the strut with the plane defined by the height \(H\) from the cube’s center.
|
||||
To do so, we first find \(g\) such that:
|
||||
\[ a_z + g u_z = -H \]
|
||||
We obtain:
|
||||
</p>
|
||||
\begin{align}
|
||||
g &= - \frac{H + a_z}{u_z} \\
|
||||
&= \sqrt{3} \frac{H}{D} - \frac{1}{2}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
Then, the intersection point \(P\) is given by:
|
||||
</p>
|
||||
\begin{align}
|
||||
P &= a + g u \\
|
||||
&= \begin{bmatrix}
|
||||
H \sqrt{2} \\
|
||||
D \frac{1}{\sqrt{2}} \\
|
||||
H
|
||||
\end{bmatrix}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
Finally, the circle can contains the intersection point has a radius \(r\):
|
||||
</p>
|
||||
\begin{align}
|
||||
r &= \sqrt{P_x^2 + P_y^2} \\
|
||||
&= \sqrt{2 H^2 + \frac{1}{2}D^2}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
By symmetry, we can show that all the other intersection points will also be on the circle with a radius \(r\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
For a small cube:
|
||||
\[ r \approx \sqrt{2} H \]
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9477b7a" class="outline-3">
|
||||
<h3 id="org9477b7a"><span class="section-number-3">2.3</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<div class="important">
|
||||
<p>
|
||||
We found that we can have a diagonal stiffness matrix using the cubic architecture when \(\{A\}\) and \(\{B\}\) are located above the top platform.
|
||||
@ -1283,8 +1375,8 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc5a2e1f" class="outline-3">
|
||||
<h3 id="orgc5a2e1f"><span class="section-number-3">3.2</span> Conclusion</h3>
|
||||
<div id="outline-container-org46632b3" class="outline-3">
|
||||
<h3 id="org46632b3"><span class="section-number-3">3.2</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
|
||||
@ -1653,8 +1745,8 @@ This was expected as the mass matrix is not diagonal (the Center of Mass of the
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org24cd25e" class="outline-3">
|
||||
<h3 id="org24cd25e"><span class="section-number-3">4.3</span> Conclusion</h3>
|
||||
<div id="outline-container-orgc1b6a36" class="outline-3">
|
||||
<h3 id="orgc1b6a36"><span class="section-number-3">4.3</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div class="important">
|
||||
<p>
|
||||
@ -1743,6 +1835,12 @@ controller = initializeController(<span class="org-string">'type'</span>, <span
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">disturbances = initializeDisturbances();
|
||||
references = initializeReferences(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org67d7284" class="figure">
|
||||
<p><img src="figs/stewart_architecture_coupling_struts_cubic.png" alt="stewart_architecture_coupling_struts_cubic.png" />
|
||||
@ -1838,8 +1936,8 @@ And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relati
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3356db5" class="outline-3">
|
||||
<h3 id="org3356db5"><span class="section-number-3">5.3</span> Conclusion</h3>
|
||||
<div id="outline-container-org87716af" class="outline-3">
|
||||
<h3 id="org87716af"><span class="section-number-3">5.3</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<div class="important">
|
||||
<p>
|
||||
@ -1942,8 +2040,7 @@ H = stewart.geometry.H;
|
||||
<div class="outline-text-4" id="text-org771c630">
|
||||
<p>
|
||||
We define the useful points of the cube with respect to the Cube’s center.
|
||||
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
|
||||
located at the center of the cube and aligned with {F} and {M}.
|
||||
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is located at the center of the cube and aligned with {F} and {M}.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -2019,7 +2116,7 @@ stewart.platform_M.Mb = Mb;
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-03-03 mar. 15:51</p>
|
||||
<p class="date">Created: 2020-03-12 jeu. 18:06</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Binary file not shown.
@ -484,6 +484,67 @@ However, the rotational stiffnesses are increasing with the cube's size but the
|
||||
| -8e-17 | 0 | -3e-17 | -6.1e-19 | 0.094 | 0 |
|
||||
| -6.2e-18 | 7.2e-17 | 5.6e-17 | 2.3e-17 | 0 | 0.37 |
|
||||
|
||||
** Size of the platforms
|
||||
The minimum size of the platforms depends on the cube's size and the height between the platform and the cube's center.
|
||||
|
||||
Let's denote:
|
||||
- $H$ the height between the cube's center and the considered platform
|
||||
- $D$ the size of the cube's edges
|
||||
|
||||
Let's denote by $a$ and $b$ the points of both ends of one of the cube's edge.
|
||||
|
||||
Initially, we have:
|
||||
\begin{align}
|
||||
a &= \frac{D}{2} \begin{bmatrix}-1 \\ -1 \\ 1\end{bmatrix} \\
|
||||
b &= \frac{D}{2} \begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix}
|
||||
\end{align}
|
||||
|
||||
We rotate the cube around its center (origin of the rotated frame) such that one of its diagonal is vertical.
|
||||
\[ R = \begin{bmatrix}
|
||||
\frac{2}{\sqrt{6}} & 0 & \frac{1}{\sqrt{3}} \\
|
||||
\frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\
|
||||
\frac{-1}{\sqrt{6}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{3}}
|
||||
\end{bmatrix} \]
|
||||
|
||||
After rotation, the points $a$ and $b$ become:
|
||||
\begin{align}
|
||||
a &= \frac{D}{2} \begin{bmatrix}-\frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ -\frac{1}{\sqrt{3}}\end{bmatrix} \\
|
||||
b &= \frac{D}{2} \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ \frac{1}{\sqrt{3}}\end{bmatrix}
|
||||
\end{align}
|
||||
|
||||
Points $a$ and $b$ define a vector $u = b - a$ that gives the orientation of one of the Stewart platform strut:
|
||||
\[ u = \frac{D}{\sqrt{3}} \begin{bmatrix} -\sqrt{2} \\ 0 \\ -1\end{bmatrix} \]
|
||||
|
||||
Then we want to find the intersection between the line that defines the strut with the plane defined by the height $H$ from the cube's center.
|
||||
To do so, we first find $g$ such that:
|
||||
\[ a_z + g u_z = -H \]
|
||||
We obtain:
|
||||
\begin{align}
|
||||
g &= - \frac{H + a_z}{u_z} \\
|
||||
&= \sqrt{3} \frac{H}{D} - \frac{1}{2}
|
||||
\end{align}
|
||||
|
||||
Then, the intersection point $P$ is given by:
|
||||
\begin{align}
|
||||
P &= a + g u \\
|
||||
&= \begin{bmatrix}
|
||||
H \sqrt{2} \\
|
||||
D \frac{1}{\sqrt{2}} \\
|
||||
H
|
||||
\end{bmatrix}
|
||||
\end{align}
|
||||
|
||||
Finally, the circle can contains the intersection point has a radius $r$:
|
||||
\begin{align}
|
||||
r &= \sqrt{P_x^2 + P_y^2} \\
|
||||
&= \sqrt{2 H^2 + \frac{1}{2}D^2}
|
||||
\end{align}
|
||||
|
||||
By symmetry, we can show that all the other intersection points will also be on the circle with a radius $r$.
|
||||
|
||||
For a small cube:
|
||||
\[ r \approx \sqrt{2} H \]
|
||||
|
||||
** Conclusion
|
||||
#+begin_important
|
||||
We found that we can have a diagonal stiffness matrix using the cubic architecture when $\{A\}$ and $\{B\}$ are located above the top platform.
|
||||
@ -1119,6 +1180,11 @@ No flexibility below the Stewart platform and no payload.
|
||||
controller = initializeController('type', 'open-loop');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
disturbances = initializeDisturbances();
|
||||
references = initializeReferences(stewart);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
displayArchitecture(stewart, 'labels', false, 'view', 'all');
|
||||
#+end_src
|
||||
@ -1509,8 +1575,7 @@ This Matlab function is accessible [[file:../src/generateCubicConfiguration.m][h
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
We define the useful points of the cube with respect to the Cube's center.
|
||||
${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is
|
||||
located at the center of the cube and aligned with {F} and {M}.
|
||||
${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is located at the center of the cube and aligned with {F} and {M}.
|
||||
|
||||
#+begin_src matlab
|
||||
sx = [ 2; -1; -1];
|
||||
|
||||
Loading…
Reference in New Issue
Block a user