Add study of active damping techniques
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<title>Stewart Platform - Active Damping</title>
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<title>Stewart Platform - Decentralized Active Damping</title>
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<meta name="author" content="Dehaeze Thomas" />
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Stewart Platform - Active Damping</h1>
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<h1 class="title">Stewart Platform - Decentralized Active Damping</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgbab9da7">1. Inertial Control</a>
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<li><a href="#orgfba33d4">1. Inertial Control</a>
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<ul>
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<li><a href="#orgd9e8d20">1.1. Simscape Model</a></li>
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<li><a href="#orge6285ea">1.2. Initialize the Stewart model</a></li>
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<li><a href="#org471f4bb">1.3. Identification of the Dynamics</a></li>
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<li><a href="#org30b2e11">1.4. Analysis</a></li>
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<li><a href="#org7f07dca">1.5. Conclusion</a></li>
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<li><a href="#org0ea4bd4">1.1. Identification of the Dynamics</a></li>
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<li><a href="#org5a29480">1.2. Effect of the Flexible Joint stiffness on the Dynamics</a></li>
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<li><a href="#orga92be75">1.3. Obtained Damping</a></li>
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<li><a href="#orgb29f377">1.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org5fde56d">2. Integral Force Feedback</a>
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<ul>
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<li><a href="#org8823e64">2.1. Identification of the Dynamics with perfect Joints</a></li>
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<li><a href="#org2aff899">2.2. Effect of the Flexible Joint stiffness on the Dynamics</a></li>
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<li><a href="#org40dffdd">2.3. Obtained Damping</a></li>
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<li><a href="#org2ae5aaf">2.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org9425768">3. Direct Velocity Feedback</a>
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<ul>
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<li><a href="#org61043ac">3.1. Identification of the Dynamics with perfect Joints</a></li>
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<li><a href="#org8f71141">3.2. Effect of the Flexible Joint stiffness on the Dynamics</a></li>
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<li><a href="#org87c6911">3.3. Obtained Damping</a></li>
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<li><a href="#org516fed1">3.4. Conclusion</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgbab9da7" class="outline-2">
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<h2 id="orgbab9da7"><span class="section-number-2">1</span> Inertial Control</h2>
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<p>
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The following decentralized active damping techniques are briefly studied:
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</p>
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<ul class="org-ul">
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<li>Inertial Control (proportional feedback of the absolute velocity): Section <a href="#org3c68d9e">1</a></li>
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<li>Integral Force Feedback: Section <a href="#org62cd19c">2</a></li>
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<li>Direct feedback of the relative velocity of each strut: Section <a href="#org587277a">3</a></li>
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</ul>
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<div id="outline-container-orgfba33d4" class="outline-2">
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<h2 id="orgfba33d4"><span class="section-number-2">1</span> Inertial Control</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org3c68d9e"></a>
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</p>
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</div>
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<div id="outline-container-orgd9e8d20" class="outline-3">
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<h3 id="orgd9e8d20"><span class="section-number-3">1.1</span> Simscape Model</h3>
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<div id="outline-container-org0ea4bd4" class="outline-3">
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<h3 id="org0ea4bd4"><span class="section-number-3">1.1</span> Identification of the Dynamics</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="org-src-container">
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<pre class="src src-matlab">open(<span class="org-string">'simulink/stewart_active_damping.slx'</span>)
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orge6285ea" class="outline-3">
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<h3 id="orge6285ea"><span class="section-number-3">1.2</span> Initialize the Stewart model</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab">stewart = initializeFramesPositions(<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);
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stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 40e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 45e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 5e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 5e<span class="org-type">-</span>3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones(6,1));
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org471f4bb" class="outline-3">
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<h3 id="org471f4bb"><span class="section-number-3">1.3</span> Identification of the Dynamics</h3>
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<div class="outline-text-3" id="text-1-3">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
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options = linearizeOptions;
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@ -313,59 +339,373 @@ mdl = <span class="org-string">'stewart_active_damping'</span>;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
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clear io; io_i = 1;
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io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/WVB'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Dm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
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io(io_i) = linio([mdl, <span class="org-string">'/F'</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>
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io(io_i) = linio([mdl, <span class="org-string">'/Vm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute velocity of each leg [m/s]</span>
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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G = linearize(mdl, io, options);
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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>};
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G.OutputName = {<span class="org-string">'Vx'</span>, <span class="org-string">'Vy'</span>, <span class="org-string">'Vz'</span>, <span class="org-string">'Wx'</span>, <span class="org-string">'Wy'</span>, <span class="org-string">'Wz'</span>, ...
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<span class="org-string">'D1'</span>, <span class="org-string">'D2'</span>, <span class="org-string">'D3'</span>, <span class="org-string">'D4'</span>, <span class="org-string">'D5'</span>, <span class="org-string">'D6'</span>};
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org30b2e11" class="outline-3">
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<h3 id="org30b2e11"><span class="section-number-3">1.4</span> Analysis</h3>
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<div class="outline-text-3" id="text-1-4">
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<div class="org-src-container">
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<pre class="src src-matlab">freqs = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>logspace(1, 4, 1000);
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<span class="org-type">figure</span>;
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bode(G({<span class="org-string">'D1'</span>, <span class="org-string">'D2'</span>, <span class="org-string">'D3'</span>, <span class="org-string">'D4'</span>, <span class="org-string">'D5'</span>, <span class="org-string">'D6'</span>}, {<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>}), freqs)
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<span class="org-type">figure</span>;
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bode(stewart.J<span class="org-type">*</span>G({<span class="org-string">'Vx'</span>, <span class="org-string">'Vy'</span>, <span class="org-string">'Vz'</span>, <span class="org-string">'Wx'</span>, <span class="org-string">'Wy'</span>, <span class="org-string">'Wz'</span>}, {<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>}), freqs)
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G.OutputName = {<span class="org-string">'Vm1'</span>, <span class="org-string">'Vm2'</span>, <span class="org-string">'Vm3'</span>, <span class="org-string">'Vm4'</span>, <span class="org-string">'Vm5'</span>, <span class="org-string">'Vm6'</span>};
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-type">figure</span>;
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bode(G({<span class="org-string">'D1'</span>, <span class="org-string">'D2'</span>, <span class="org-string">'D3'</span>, <span class="org-string">'D4'</span>, <span class="org-string">'D5'</span>, <span class="org-string">'D6'</span>}, {<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>}), stewart.J<span class="org-type">*</span>G({<span class="org-string">'Vx'</span>, <span class="org-string">'Vy'</span>, <span class="org-string">'Vz'</span>, <span class="org-string">'Wx'</span>, <span class="org-string">'Wy'</span>, <span class="org-string">'Wz'</span>}, {<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>}), freqs)
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org7f07dca" class="outline-3">
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<h3 id="org7f07dca"><span class="section-number-3">1.5</span> Conclusion</h3>
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<div class="outline-text-3" id="text-1-5">
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<p>
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It is similar to use:
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The transfer function from actuator forces to force sensors is shown in Figure <a href="#orgfc5367b">1</a>.
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</p>
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<ul class="org-ul">
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<li>one 6dof inertial sensor and the Jacobian the have the velocity of each lim</li>
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<li>6 1dof inertial sensor in each top part of the limbs</li>
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</ul>
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<div id="orgfc5367b" class="figure">
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<p><img src="figs/inertial_plant_coupling.png" alt="inertial_plant_coupling.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Transfer function from the Actuator force \(F_{i}\) to the absolute velocity of the same leg \(v_{m,i}\) and to the absolute velocity of the other legs \(v_{m,j}\) with \(i \neq j\) in grey (<a href="./figs/inertial_plant_coupling.png">png</a>, <a href="./figs/inertial_plant_coupling.pdf">pdf</a>)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org5a29480" class="outline-3">
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<h3 id="org5a29480"><span class="section-number-3">1.2</span> Effect of the Flexible Joint stiffness 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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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
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Gf = linearize(mdl, io, options);
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Gf.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>};
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Gf.OutputName = {<span class="org-string">'Vm1'</span>, <span class="org-string">'Vm2'</span>, <span class="org-string">'Vm3'</span>, <span class="org-string">'Vm4'</span>, <span class="org-string">'Vm5'</span>, <span class="org-string">'Vm6'</span>};
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</pre>
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</div>
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<p>
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The new dynamics from force actuator to force sensor is shown in Figure <a href="#org2ee5d65">2</a>.
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</p>
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<div id="org2ee5d65" class="figure">
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<p><img src="figs/inertial_plant_flexible_joint_decentralized.png" alt="inertial_plant_flexible_joint_decentralized.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Transfer function from the Actuator force \(F_{i}\) to the absolute velocity sensor \(v_{m,i}\) (<a href="./figs/inertial_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/inertial_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orga92be75" class="outline-3">
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<h3 id="orga92be75"><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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The \(6 \times 6\) control is a diagonal matrix with pure proportional action on the diagonal:
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\[ K(s) = g
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\begin{bmatrix}
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1 & & 0 \\
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& \ddots & \\
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0 & & 1
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\end{bmatrix} \]
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</p>
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<p>
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The root locus is shown in figure <a href="#org78a599c">3</a> and the obtained pole damping function of the control gain is shown in figure <a href="#org0b6bb28">4</a>.
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</p>
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<div id="org78a599c" class="figure">
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<p><img src="figs/root_locus_inertial_rot_stiffness.png" alt="root_locus_inertial_rot_stiffness.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints (<a href="./figs/root_locus_inertial_rot_stiffness.png">png</a>, <a href="./figs/root_locus_inertial_rot_stiffness.pdf">pdf</a>)</p>
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</div>
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<div id="org0b6bb28" class="figure">
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<p><img src="figs/pole_damping_gain_inertial_rot_stiffness.png" alt="pole_damping_gain_inertial_rot_stiffness.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Damping of the poles with respect to the gain of the Decentralized Inertial Control when considering the stiffness of flexible joints (<a href="./figs/pole_damping_gain_inertial_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_inertial_rot_stiffness.pdf">pdf</a>)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgb29f377" class="outline-3">
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<h3 id="orgb29f377"><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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Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
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</p>
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</div>
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</div>
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</div>
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</div>
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||||
<div id="outline-container-org5fde56d" class="outline-2">
|
||||
<h2 id="org5fde56d"><span class="section-number-2">2</span> Integral Force Feedback</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="org62cd19c"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8823e64" class="outline-3">
|
||||
<h3 id="org8823e64"><span class="section-number-3">2.1</span> Identification of the Dynamics with perfect Joints</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We first initialize the Stewart platform without joint stiffness.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeFramesPositions(<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">'disable'</span>, <span class="org-constant">true</span>);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we identify the dynamics from force actuators to force sensors.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_active_damping'</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">'/F'</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">'/Fm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Force Sensor Outputs [N]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
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">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The transfer function from actuator forces to force sensors is shown in Figure <a href="#orgae4e327">5</a>.
|
||||
</p>
|
||||
|
||||
<div id="orgae4e327" class="figure">
|
||||
<p><img src="figs/iff_plant_coupling.png" alt="iff_plant_coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Transfer function from the Actuator force \(F_{i}\) to the Force sensor of the same leg \(F_{m,i}\) and to the force sensor of the other legs \(F_{m,j}\) with \(i \neq j\) in grey (<a href="./figs/iff_plant_coupling.png">png</a>, <a href="./figs/iff_plant_coupling.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2aff899" class="outline-3">
|
||||
<h3 id="org2aff899"><span class="section-number-3">2.2</span> Effect of the Flexible Joint stiffness on the Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.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>};
|
||||
Gf.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The new dynamics from force actuator to force sensor is shown in Figure <a href="#orgd21a8a8">6</a>.
|
||||
</p>
|
||||
|
||||
<div id="orgd21a8a8" class="figure">
|
||||
<p><img src="figs/iff_plant_flexible_joint_decentralized.png" alt="iff_plant_flexible_joint_decentralized.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Transfer function from the Actuator force \(F_{i}\) to the force sensor \(F_{m,i}\) (<a href="./figs/iff_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/iff_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org40dffdd" class="outline-3">
|
||||
<h3 id="org40dffdd"><span class="section-number-3">2.3</span> Obtained Damping</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
The control is a performed in a decentralized manner.
|
||||
The \(6 \times 6\) control is a diagonal matrix with pure integration action on the diagonal:
|
||||
\[ K(s) = g
|
||||
\begin{bmatrix}
|
||||
\frac{1}{s} & & 0 \\
|
||||
& \ddots & \\
|
||||
0 & & \frac{1}{s}
|
||||
\end{bmatrix} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The root locus is shown in figure <a href="#org2cdbf69">7</a> and the obtained pole damping function of the control gain is shown in figure <a href="#orge344229">8</a>.
|
||||
</p>
|
||||
|
||||
<div id="org2cdbf69" class="figure">
|
||||
<p><img src="figs/root_locus_iff_rot_stiffness.png" alt="root_locus_iff_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints (<a href="./figs/root_locus_iff_rot_stiffness.png">png</a>, <a href="./figs/root_locus_iff_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orge344229" class="figure">
|
||||
<p><img src="figs/pole_damping_gain_iff_rot_stiffness.png" alt="pole_damping_gain_iff_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints (<a href="./figs/pole_damping_gain_iff_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_iff_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2ae5aaf" class="outline-3">
|
||||
<h3 id="org2ae5aaf"><span class="section-number-3">2.4</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<div class="important">
|
||||
<p>
|
||||
The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
|
||||
Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9425768" class="outline-2">
|
||||
<h2 id="org9425768"><span class="section-number-2">3</span> Direct Velocity Feedback</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org587277a"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org61043ac" class="outline-3">
|
||||
<h3 id="org61043ac"><span class="section-number-3">3.1</span> Identification of the Dynamics with perfect Joints</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
We first initialize the Stewart platform without joint stiffness.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeFramesPositions(<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">'disable'</span>, <span class="org-constant">true</span>);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we identify the dynamics from force actuators to force sensors.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_active_damping'</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">'/F'</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">'/Dm'</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 [N]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
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">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The transfer function from actuator forces to relative motion sensors is shown in Figure <a href="#orgd8d51db">9</a>.
|
||||
</p>
|
||||
|
||||
<div id="orgd8d51db" class="figure">
|
||||
<p><img src="figs/dvf_plant_coupling.png" alt="dvf_plant_coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Transfer function from the Actuator force \(F_{i}\) to the Relative Motion Sensor \(D_{m,j}\) with \(i \neq j\) (<a href="./figs/dvf_plant_coupling.png">png</a>, <a href="./figs/dvf_plant_coupling.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org8f71141" class="outline-3">
|
||||
<h3 id="org8f71141"><span class="section-number-3">3.2</span> Effect of the Flexible Joint stiffness on the Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.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>};
|
||||
Gf.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The new dynamics from force actuator to relative motion sensor is shown in Figure <a href="#orgb18f950">10</a>.
|
||||
</p>
|
||||
|
||||
<div id="orgb18f950" class="figure">
|
||||
<p><img src="figs/dvf_plant_flexible_joint_decentralized.png" alt="dvf_plant_flexible_joint_decentralized.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Transfer function from the Actuator force \(F_{i}\) to the relative displacement sensor \(D_{m,i}\) (<a href="./figs/dvf_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/dvf_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org87c6911" class="outline-3">
|
||||
<h3 id="org87c6911"><span class="section-number-3">3.3</span> Obtained Damping</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
The control is a performed in a decentralized manner.
|
||||
The \(6 \times 6\) control is a diagonal matrix with pure derivative action on the diagonal:
|
||||
\[ K(s) = g
|
||||
\begin{bmatrix}
|
||||
s & & \\
|
||||
& \ddots & \\
|
||||
& & s
|
||||
\end{bmatrix} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The root locus is shown in figure <a href="#org5cb31c8">11</a> and the obtained pole damping function of the control gain is shown in figure <a href="#org4618492">12</a>.
|
||||
</p>
|
||||
|
||||
<div id="org5cb31c8" class="figure">
|
||||
<p><img src="figs/root_locus_dvf_rot_stiffness.png" alt="root_locus_dvf_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints (<a href="./figs/root_locus_dvf_rot_stiffness.png">png</a>, <a href="./figs/root_locus_dvf_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org4618492" class="figure">
|
||||
<p><img src="figs/pole_damping_gain_dvf_rot_stiffness.png" alt="pole_damping_gain_dvf_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Damping of the poles with respect to the gain of the Direct Velocity Feedback when considering the Stiffness of flexible joints (<a href="./figs/pole_damping_gain_dvf_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_dvf_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org516fed1" class="outline-3">
|
||||
<h3 id="org516fed1"><span class="section-number-3">3.4</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<div class="important">
|
||||
<p>
|
||||
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-01-22 mer. 16:31</p>
|
||||
<p class="date">Created: 2020-02-06 jeu. 15:39</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -1,4 +1,4 @@
|
||||
#+TITLE: Stewart Platform - Active Damping
|
||||
#+TITLE: Stewart Platform - Decentralized Active Damping
|
||||
:DRAWER:
|
||||
#+HTML_LINK_HOME: ./index.html
|
||||
#+HTML_LINK_UP: ./index.html
|
||||
@ -20,7 +20,249 @@
|
||||
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||
:END:
|
||||
|
||||
* Introduction :ignore:
|
||||
|
||||
The following decentralized active damping techniques are briefly studied:
|
||||
- Inertial Control (proportional feedback of the absolute velocity): Section [[sec:active_damping_inertial]]
|
||||
- Integral Force Feedback: Section [[sec:active_damping_iff]]
|
||||
- Direct feedback of the relative velocity of each strut: Section [[sec:active_damping_dvf]]
|
||||
|
||||
* Inertial Control
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/active_damping_inertial.m
|
||||
:header-args:matlab+: :comments org :mkdirp yes
|
||||
:END:
|
||||
<<sec:active_damping_inertial>>
|
||||
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
simulinkproject('./');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
open('simulink/stewart_active_damping.slx')
|
||||
#+end_src
|
||||
|
||||
** Identification of the Dynamics
|
||||
#+begin_src matlab
|
||||
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, 'disable', true);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_active_damping';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Vm'], 1, 'openoutput'); io_i = io_i + 1; % Absolute velocity of each leg [m/s]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
|
||||
#+end_src
|
||||
|
||||
The transfer function from actuator forces to force sensors is shown in Figure [[fig:inertial_plant_coupling]].
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/inertial_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:inertial_plant_coupling
|
||||
#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity of the same leg $v_{m,i}$ and to the absolute velocity of the other legs $v_{m,j}$ with $i \neq j$ in grey ([[./figs/inertial_plant_coupling.png][png]], [[./figs/inertial_plant_coupling.pdf][pdf]])
|
||||
[[file:figs/inertial_plant_coupling.png]]
|
||||
|
||||
** Effect of the Flexible Joint stiffness on the Dynamics
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
#+begin_src matlab
|
||||
stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
Gf.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
|
||||
#+end_src
|
||||
|
||||
The new dynamics from force actuator to force sensor is shown in Figure [[fig:inertial_plant_flexible_joint_decentralized]].
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend('location', 'southwest')
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/inertial_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:inertial_plant_flexible_joint_decentralized
|
||||
#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity sensor $v_{m,i}$ ([[./figs/inertial_plant_flexible_joint_decentralized.png][png]], [[./figs/inertial_plant_flexible_joint_decentralized.pdf][pdf]])
|
||||
[[file:figs/inertial_plant_flexible_joint_decentralized.png]]
|
||||
|
||||
** Obtained Damping
|
||||
The control is a performed in a decentralized manner.
|
||||
The $6 \times 6$ control is a diagonal matrix with pure proportional action on the diagonal:
|
||||
\[ K(s) = g
|
||||
\begin{bmatrix}
|
||||
1 & & 0 \\
|
||||
& \ddots & \\
|
||||
0 & & 1
|
||||
\end{bmatrix} \]
|
||||
|
||||
The root locus is shown in figure [[fig:root_locus_inertial_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_inertial_rot_stiffness]].
|
||||
#+begin_src matlab :exports none
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(pole(G)), imag(pole(G)), 'x');
|
||||
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
||||
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
||||
for i = 1:length(gains)
|
||||
cl_poles = pole(feedback(G, gains(i)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
ylim([0,2000]);
|
||||
xlim([-2000,0]);
|
||||
xlabel('Real Part')
|
||||
ylabel('Imaginary Part')
|
||||
axis square
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/root_locus_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:root_locus_inertial_rot_stiffness
|
||||
#+caption: Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/root_locus_inertial_rot_stiffness.png][png]], [[./figs/root_locus_inertial_rot_stiffness.pdf][pdf]])
|
||||
[[file:figs/root_locus_inertial_rot_stiffness.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
cl_poles = pole(feedback(G, gains(i)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
end
|
||||
xlabel('Control Gain');
|
||||
ylabel('Damping of the Poles');
|
||||
set(gca, 'XScale', 'log');
|
||||
ylim([0,pi/2]);
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/pole_damping_gain_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:pole_damping_gain_inertial_rot_stiffness
|
||||
#+caption: Damping of the poles with respect to the gain of the Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_inertial_rot_stiffness.png][png]], [[./figs/pole_damping_gain_inertial_rot_stiffness.pdf][pdf]])
|
||||
[[file:figs/pole_damping_gain_inertial_rot_stiffness.png]]
|
||||
|
||||
** Conclusion
|
||||
#+begin_important
|
||||
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
||||
#+end_important
|
||||
|
||||
* Integral Force Feedback
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/active_damping_iff.m
|
||||
:header-args:matlab+: :comments org :mkdirp yes
|
||||
:END:
|
||||
<<sec:active_damping_iff>>
|
||||
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
@ -34,24 +276,25 @@
|
||||
simulinkproject('./');
|
||||
#+end_src
|
||||
|
||||
** Simscape Model
|
||||
#+begin_src matlab
|
||||
open('simulink/stewart_active_damping.slx')
|
||||
#+end_src
|
||||
|
||||
** Initialize the Stewart model
|
||||
** Identification of the Dynamics with perfect Joints
|
||||
We first initialize the Stewart platform without joint stiffness.
|
||||
#+begin_src matlab
|
||||
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
|
||||
stewart = generateCubicConfiguration(stewart, 'Hc', 40e-3, 'FOc', 45e-3, 'FHa', 5e-3, 'MHb', 5e-3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1));
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, 'disable', true);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
#+end_src
|
||||
|
||||
** Identification of the Dynamics
|
||||
And we identify the dynamics from force actuators to force sensors.
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
@ -62,34 +305,414 @@
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/WVB'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor Outputs [N]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Vx', 'Vy', 'Vz', 'Wx', 'Wy', 'Wz', ...
|
||||
'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
|
||||
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
||||
#+end_src
|
||||
|
||||
** Analysis
|
||||
The transfer function from actuator forces to force sensors is shown in Figure [[fig:iff_plant_coupling]].
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/iff_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:iff_plant_coupling
|
||||
#+caption: Transfer function from the Actuator force $F_{i}$ to the Force sensor of the same leg $F_{m,i}$ and to the force sensor of the other legs $F_{m,j}$ with $i \neq j$ in grey ([[./figs/iff_plant_coupling.png][png]], [[./figs/iff_plant_coupling.pdf][pdf]])
|
||||
[[file:figs/iff_plant_coupling.png]]
|
||||
|
||||
** Effect of the Flexible Joint stiffness on the Dynamics
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
#+begin_src matlab
|
||||
freqs = 2*pi*logspace(1, 4, 1000);
|
||||
|
||||
figure;
|
||||
bode(G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), freqs)
|
||||
|
||||
figure;
|
||||
bode(stewart.J*G({'Vx', 'Vy', 'Vz', 'Wx', 'Wy', 'Wz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), freqs)
|
||||
stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
The new dynamics from force actuator to force sensor is shown in Figure [[fig:iff_plant_flexible_joint_decentralized]].
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
bode(G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), stewart.J*G({'Vx', 'Vy', 'Vz', 'Wx', 'Wy', 'Wz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), freqs)
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend('location', 'southwest')
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/iff_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:iff_plant_flexible_joint_decentralized
|
||||
#+caption: Transfer function from the Actuator force $F_{i}$ to the force sensor $F_{m,i}$ ([[./figs/iff_plant_flexible_joint_decentralized.png][png]], [[./figs/iff_plant_flexible_joint_decentralized.pdf][pdf]])
|
||||
[[file:figs/iff_plant_flexible_joint_decentralized.png]]
|
||||
|
||||
** Obtained Damping
|
||||
The control is a performed in a decentralized manner.
|
||||
The $6 \times 6$ control is a diagonal matrix with pure integration action on the diagonal:
|
||||
\[ K(s) = g
|
||||
\begin{bmatrix}
|
||||
\frac{1}{s} & & 0 \\
|
||||
& \ddots & \\
|
||||
0 & & \frac{1}{s}
|
||||
\end{bmatrix} \]
|
||||
|
||||
The root locus is shown in figure [[fig:root_locus_iff_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_iff_rot_stiffness]].
|
||||
#+begin_src matlab :exports none
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(pole(G)), imag(pole(G)), 'x');
|
||||
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
||||
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
||||
for i = 1:length(gains)
|
||||
cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
ylim([0,inf]);
|
||||
xlim([-3000,0]);
|
||||
xlabel('Real Part')
|
||||
ylabel('Imaginary Part')
|
||||
axis square
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/root_locus_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:root_locus_iff_rot_stiffness
|
||||
#+caption: Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/root_locus_iff_rot_stiffness.png][png]], [[./figs/root_locus_iff_rot_stiffness.pdf][pdf]])
|
||||
[[file:figs/root_locus_iff_rot_stiffness.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
end
|
||||
xlabel('Control Gain');
|
||||
ylabel('Damping of the Poles');
|
||||
set(gca, 'XScale', 'log');
|
||||
ylim([0,pi/2]);
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/pole_damping_gain_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:pole_damping_gain_iff_rot_stiffness
|
||||
#+caption: Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_iff_rot_stiffness.png][png]], [[./figs/pole_damping_gain_iff_rot_stiffness.pdf][pdf]])
|
||||
[[file:figs/pole_damping_gain_iff_rot_stiffness.png]]
|
||||
|
||||
** Conclusion
|
||||
It is similar to use:
|
||||
- one 6dof inertial sensor and the Jacobian the have the velocity of each lim
|
||||
- 6 1dof inertial sensor in each top part of the limbs
|
||||
#+begin_important
|
||||
The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
|
||||
Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
|
||||
#+end_important
|
||||
|
||||
* Direct Velocity Feedback
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/active_damping_dvf.m
|
||||
:header-args:matlab+: :comments org :mkdirp yes
|
||||
:END:
|
||||
<<sec:active_damping_dvf>>
|
||||
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
simulinkproject('./');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
open('simulink/stewart_active_damping.slx')
|
||||
#+end_src
|
||||
|
||||
** Identification of the Dynamics with perfect Joints
|
||||
We first initialize the Stewart platform without joint stiffness.
|
||||
#+begin_src matlab
|
||||
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, 'disable', true);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
#+end_src
|
||||
|
||||
And we identify the dynamics from force actuators to force sensors.
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_active_damping';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [N]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||||
#+end_src
|
||||
|
||||
The transfer function from actuator forces to relative motion sensors is shown in Figure [[fig:dvf_plant_coupling]].
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$D_{m,i}/F_i$', '$D_{m,j}/F_i$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/dvf_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:dvf_plant_coupling
|
||||
#+caption: Transfer function from the Actuator force $F_{i}$ to the Relative Motion Sensor $D_{m,j}$ with $i \neq j$ ([[./figs/dvf_plant_coupling.png][png]], [[./figs/dvf_plant_coupling.pdf][pdf]])
|
||||
[[file:figs/dvf_plant_coupling.png]]
|
||||
|
||||
|
||||
** Effect of the Flexible Joint stiffness on the Dynamics
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
#+begin_src matlab
|
||||
stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||||
#+end_src
|
||||
|
||||
The new dynamics from force actuator to relative motion sensor is shown in Figure [[fig:dvf_plant_flexible_joint_decentralized]].
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/dvf_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:dvf_plant_flexible_joint_decentralized
|
||||
#+caption: Transfer function from the Actuator force $F_{i}$ to the relative displacement sensor $D_{m,i}$ ([[./figs/dvf_plant_flexible_joint_decentralized.png][png]], [[./figs/dvf_plant_flexible_joint_decentralized.pdf][pdf]])
|
||||
[[file:figs/dvf_plant_flexible_joint_decentralized.png]]
|
||||
|
||||
** Obtained Damping
|
||||
The control is a performed in a decentralized manner.
|
||||
The $6 \times 6$ control is a diagonal matrix with pure derivative action on the diagonal:
|
||||
\[ K(s) = g
|
||||
\begin{bmatrix}
|
||||
s & & \\
|
||||
& \ddots & \\
|
||||
& & s
|
||||
\end{bmatrix} \]
|
||||
|
||||
The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_dvf_rot_stiffness]].
|
||||
#+begin_src matlab :exports none
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(pole(G)), imag(pole(G)), 'x');
|
||||
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
||||
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
||||
for i = 1:length(gains)
|
||||
cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
ylim([0,inf]);
|
||||
xlim([-3000,0]);
|
||||
xlabel('Real Part')
|
||||
ylabel('Imaginary Part')
|
||||
axis square
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/root_locus_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:root_locus_dvf_rot_stiffness
|
||||
#+caption: Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/root_locus_dvf_rot_stiffness.png][png]], [[./figs/root_locus_dvf_rot_stiffness.pdf][pdf]])
|
||||
[[file:figs/root_locus_dvf_rot_stiffness.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
end
|
||||
xlabel('Control Gain');
|
||||
ylabel('Damping of the Poles');
|
||||
set(gca, 'XScale', 'log');
|
||||
ylim([0,pi/2]);
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/pole_damping_gain_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:pole_damping_gain_dvf_rot_stiffness
|
||||
#+caption: Damping of the poles with respect to the gain of the Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/pole_damping_gain_dvf_rot_stiffness.png][png]], [[./figs/pole_damping_gain_dvf_rot_stiffness.pdf][pdf]])
|
||||
[[file:figs/pole_damping_gain_dvf_rot_stiffness.png]]
|
||||
|
||||
** Conclusion
|
||||
#+begin_important
|
||||
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
||||
#+end_important
|
||||
|
||||
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figs/dvf_plant_coupling.pdf
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|
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figs/dvf_plant_flexible_joint_decentralized.pdf
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figs/dvf_plant_flexible_joint_decentralized.pdf
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figs/iff_plant_coupling.pdf
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figs/iff_plant_flexible_joint_decentralized.pdf
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figs/iff_plant_flexible_joint_decentralized.pdf
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figs/iff_plant_flexible_joint_decentralized.png
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figs/iff_plant_flexible_joint_decentralized.png
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figs/inertial_plant_coupling.pdf
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figs/inertial_plant_flexible_joint_decentralized.pdf
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figs/inertial_plant_flexible_joint_decentralized.pdf
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figs/inertial_plant_flexible_joint_decentralized.png
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figs/inertial_plant_flexible_joint_decentralized.png
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|
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figs/pole_damping_gain_dvf_rot_stiffness.pdf
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figs/pole_damping_gain_dvf_rot_stiffness.pdf
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figs/pole_damping_gain_dvf_rot_stiffness.png
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178
matlab/active_damping_dvf.m
Normal file
178
matlab/active_damping_dvf.m
Normal file
@ -0,0 +1,178 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
simulinkproject('./');
|
||||
|
||||
open('simulink/stewart_active_damping.slx')
|
||||
|
||||
% Identification of the Dynamics with perfect Joints
|
||||
% We first initialize the Stewart platform without joint stiffness.
|
||||
|
||||
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, 'disable', true);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
|
||||
|
||||
|
||||
% And we identify the dynamics from force actuators to force sensors.
|
||||
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_active_damping';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [N]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||||
|
||||
|
||||
|
||||
% The transfer function from actuator forces to relative motion sensors is shown in Figure [[fig:dvf_plant_coupling]].
|
||||
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$D_{m,i}/F_i$', '$D_{m,j}/F_i$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Effect of the Flexible Joint stiffness on the Dynamics
|
||||
% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
|
||||
stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
||||
|
||||
|
||||
|
||||
% The new dynamics from force actuator to relative motion sensor is shown in Figure [[fig:dvf_plant_flexible_joint_decentralized]].
|
||||
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Obtained Damping
|
||||
% The control is a performed in a decentralized manner.
|
||||
% The $6 \times 6$ control is a diagonal matrix with pure derivative action on the diagonal:
|
||||
% \[ K(s) = g
|
||||
% \begin{bmatrix}
|
||||
% s & & \\
|
||||
% & \ddots & \\
|
||||
% & & s
|
||||
% \end{bmatrix} \]
|
||||
|
||||
% The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_dvf_rot_stiffness]].
|
||||
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(pole(G)), imag(pole(G)), 'x');
|
||||
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
||||
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
||||
for i = 1:length(gains)
|
||||
cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
ylim([0,inf]);
|
||||
xlim([-3000,0]);
|
||||
xlabel('Real Part')
|
||||
ylabel('Imaginary Part')
|
||||
axis square
|
||||
|
||||
|
||||
|
||||
% #+name: fig:root_locus_dvf_rot_stiffness
|
||||
% #+caption: Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/root_locus_dvf_rot_stiffness.png][png]], [[./figs/root_locus_dvf_rot_stiffness.pdf][pdf]])
|
||||
% [[file:figs/root_locus_dvf_rot_stiffness.png]]
|
||||
|
||||
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
end
|
||||
xlabel('Control Gain');
|
||||
ylabel('Damping of the Poles');
|
||||
set(gca, 'XScale', 'log');
|
||||
ylim([0,pi/2]);
|
||||
178
matlab/active_damping_iff.m
Normal file
178
matlab/active_damping_iff.m
Normal file
@ -0,0 +1,178 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
simulinkproject('./');
|
||||
|
||||
open('simulink/stewart_active_damping.slx')
|
||||
|
||||
% Identification of the Dynamics with perfect Joints
|
||||
% We first initialize the Stewart platform without joint stiffness.
|
||||
|
||||
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, 'disable', true);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
|
||||
|
||||
|
||||
% And we identify the dynamics from force actuators to force sensors.
|
||||
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_active_damping';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor Outputs [N]
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
||||
|
||||
|
||||
|
||||
% The transfer function from actuator forces to force sensors is shown in Figure [[fig:iff_plant_coupling]].
|
||||
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Effect of the Flexible Joint stiffness on the Dynamics
|
||||
% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
|
||||
stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
|
||||
|
||||
|
||||
|
||||
% The new dynamics from force actuator to force sensor is shown in Figure [[fig:iff_plant_flexible_joint_decentralized]].
|
||||
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend('location', 'southwest')
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Obtained Damping
|
||||
% The control is a performed in a decentralized manner.
|
||||
% The $6 \times 6$ control is a diagonal matrix with pure integration action on the diagonal:
|
||||
% \[ K(s) = g
|
||||
% \begin{bmatrix}
|
||||
% \frac{1}{s} & & 0 \\
|
||||
% & \ddots & \\
|
||||
% 0 & & \frac{1}{s}
|
||||
% \end{bmatrix} \]
|
||||
|
||||
% The root locus is shown in figure [[fig:root_locus_iff_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_iff_rot_stiffness]].
|
||||
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(pole(G)), imag(pole(G)), 'x');
|
||||
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
||||
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
||||
for i = 1:length(gains)
|
||||
cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
ylim([0,inf]);
|
||||
xlim([-3000,0]);
|
||||
xlabel('Real Part')
|
||||
ylabel('Imaginary Part')
|
||||
axis square
|
||||
|
||||
|
||||
|
||||
% #+name: fig:root_locus_iff_rot_stiffness
|
||||
% #+caption: Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/root_locus_iff_rot_stiffness.png][png]], [[./figs/root_locus_iff_rot_stiffness.pdf][pdf]])
|
||||
% [[file:figs/root_locus_iff_rot_stiffness.png]]
|
||||
|
||||
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
end
|
||||
xlabel('Control Gain');
|
||||
ylabel('Damping of the Poles');
|
||||
set(gca, 'XScale', 'log');
|
||||
ylim([0,pi/2]);
|
||||
173
matlab/active_damping_inertial.m
Normal file
173
matlab/active_damping_inertial.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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simulinkproject('./');
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open('simulink/stewart_active_damping.slx')
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% Identification of the Dynamics
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stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'disable', true);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_active_damping';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Vm'], 1, 'openoutput'); io_i = io_i + 1; % Absolute velocity of each leg [m/s]
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
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% The transfer function from actuator forces to force sensors is shown in Figure [[fig:inertial_plant_coupling]].
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
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linkaxes([ax1,ax2],'x');
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% Effect of the Flexible Joint stiffness on the Dynamics
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% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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stewart = initializeJointDynamics(stewart);
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Gf = linearize(mdl, io, options);
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Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Gf.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
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% The new dynamics from force actuator to force sensor is shown in Figure [[fig:inertial_plant_flexible_joint_decentralized]].
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend('location', 'southwest')
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linkaxes([ax1,ax2],'x');
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% Obtained Damping
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% The control is a performed in a decentralized manner.
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% The $6 \times 6$ control is a diagonal matrix with pure proportional action on the diagonal:
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% \[ K(s) = g
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% \begin{bmatrix}
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% 1 & & 0 \\
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||||
% & \ddots & \\
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% 0 & & 1
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% \end{bmatrix} \]
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% The root locus is shown in figure [[fig:root_locus_inertial_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_inertial_rot_stiffness]].
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gains = logspace(0, 5, 1000);
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figure;
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hold on;
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plot(real(pole(G)), imag(pole(G)), 'x');
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plot(real(pole(Gf)), imag(pole(Gf)), 'x');
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set(gca,'ColorOrderIndex',1);
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plot(real(tzero(G)), imag(tzero(G)), 'o');
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plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
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for i = 1:length(gains)
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cl_poles = pole(feedback(G, gains(i)*eye(6)));
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set(gca,'ColorOrderIndex',1);
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plot(real(cl_poles), imag(cl_poles), '.');
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cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
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set(gca,'ColorOrderIndex',2);
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plot(real(cl_poles), imag(cl_poles), '.');
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end
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ylim([0,2000]);
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xlim([-2000,0]);
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xlabel('Real Part')
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ylabel('Imaginary Part')
|
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axis square
|
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|
||||
|
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% #+name: fig:root_locus_inertial_rot_stiffness
|
||||
% #+caption: Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/root_locus_inertial_rot_stiffness.png][png]], [[./figs/root_locus_inertial_rot_stiffness.pdf][pdf]])
|
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% [[file:figs/root_locus_inertial_rot_stiffness.png]]
|
||||
|
||||
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',1);
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||||
cl_poles = pole(feedback(G, gains(i)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
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set(gca,'ColorOrderIndex',2);
|
||||
cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
end
|
||||
xlabel('Control Gain');
|
||||
ylabel('Damping of the Poles');
|
||||
set(gca, 'XScale', 'log');
|
||||
ylim([0,pi/2]);
|
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