Analysis + Control of Simscape Model
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>SVD Control</title>
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<title>SVD Control</title>
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<h2>Table of Contents</h2>
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<div id="text-table-of-contents">
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<ul>
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<ul>
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<li><a href="#org09b41c5">1. Simscape Model - Gravimeter</a>
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<li><a href="#orge03ef95">1. Gravimeter - Simscape Model</a>
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<ul>
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<ul>
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<li><a href="#orgaf12c1d">1.1. Simulink</a></li>
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<li><a href="#org94cda63">1.1. Simulink</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org84efeb7">2. Simscape Model - Stewart Platform</a>
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<li><a href="#org01f2bcf">2. Stewart Platform - Simscape Model</a>
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<ul>
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<ul>
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<li><a href="#org157458d">2.1. Jacobian</a></li>
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<li><a href="#org5d90a14">2.1. Jacobian</a></li>
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<li><a href="#org8947fec">2.2. Simulink</a></li>
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<li><a href="#org7bbb169">2.2. Simscape Model</a></li>
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<li><a href="#org2a265c4">2.3. Identification of the plant</a></li>
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<li><a href="#orgfa83a84">2.4. Obtained Dynamics</a></li>
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<li><a href="#org92dd977">2.5. Real Approximation of \(G\) at the decoupling frequency</a></li>
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<li><a href="#orgebf7751">2.6. Verification of the decoupling using the “Gershgorin Radii”</a></li>
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<li><a href="#orge21a525">2.7. Decoupled Plant</a></li>
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<li><a href="#org4c1f528">2.8. Diagonal Controller</a></li>
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<li><a href="#org4f88748">2.9. Centralized Control</a></li>
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<li><a href="#org6eac181">2.10. SVD Control</a></li>
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<li><a href="#org89ccc9f">2.11. Results</a></li>
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</ul>
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</li>
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<li><a href="#orgdcb6e90">3. Stewart Platform - Analytical Model</a>
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<ul>
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<li><a href="#orgeb4b14b">3.1. Characteristics</a></li>
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<li><a href="#orgeff797b">3.2. Mass Matrix</a></li>
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<li><a href="#org7027995">3.3. Jacobian Matrix</a></li>
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<li><a href="#org51bab7b">3.4. Stifnness matrix and Damping matrix</a></li>
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<li><a href="#orga9e6cf5">3.5. State Space System</a></li>
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<li><a href="#org769c38a">3.6. Transmissibility</a></li>
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<li><a href="#org24eb81f">3.7. Real approximation of \(G(j\omega)\) at decoupling frequency</a></li>
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<li><a href="#org824e380">3.8. Coupled and Decoupled Plant “Gershgorin Radii”</a></li>
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<li><a href="#org8e5d2c7">3.9. Decoupled Plant</a></li>
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<li><a href="#org102382b">3.10. Controller</a></li>
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<li><a href="#org27bf3be">3.11. Closed Loop System</a></li>
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<li><a href="#org419f877">3.12. Results</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</div>
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</div>
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</div>
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<div id="outline-container-org09b41c5" class="outline-2">
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<div id="outline-container-orge03ef95" class="outline-2">
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<h2 id="org09b41c5"><span class="section-number-2">1</span> Simscape Model - Gravimeter</h2>
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<h2 id="orge03ef95"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
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<div class="outline-text-2" id="text-1">
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<div class="outline-text-2" id="text-1">
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<div id="outline-container-orgaf12c1d" class="outline-3">
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<div id="outline-container-org94cda63" class="outline-3">
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<h3 id="orgaf12c1d"><span class="section-number-3">1.1</span> Simulink</h3>
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<h3 id="org94cda63"><span class="section-number-3">1.1</span> Simulink</h3>
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<div class="outline-text-3" id="text-1-1">
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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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<div class="org-src-container">
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<pre class="src src-matlab">open('gravimeter.slx')
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<pre class="src src-matlab">open('gravimeter.slx')
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@ -89,7 +122,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
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<div id="org1c9b0ec" class="figure">
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<div id="org57d8f45" class="figure">
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<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
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<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
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</p>
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</p>
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<p><span class="figure-number">Figure 1: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
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<p><span class="figure-number">Figure 1: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
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@ -98,17 +131,16 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
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</div>
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<div id="outline-container-org84efeb7" class="outline-2">
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<div id="outline-container-org01f2bcf" class="outline-2">
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<h2 id="org84efeb7"><span class="section-number-2">2</span> Simscape Model - Stewart Platform</h2>
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<h2 id="org01f2bcf"><span class="section-number-2">2</span> Stewart Platform - Simscape Model</h2>
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<div class="outline-text-2" id="text-2">
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<div id="outline-container-org157458d" class="outline-3">
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<div id="outline-container-org5d90a14" class="outline-3">
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<h3 id="org157458d"><span class="section-number-3">2.1</span> Jacobian</h3>
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<h3 id="org5d90a14"><span class="section-number-3">2.1</span> Jacobian</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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<p>
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First, the position of the “joints” (points of force application) are estimated and the Jacobian computed.
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First, the position of the “joints” (points of force application) are estimated and the Jacobian computed.
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</p>
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</p>
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<div class="org-src-container">
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<div class="org-src-container">
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<pre class="src src-matlab">open('stewart_platform/drone_platform_jacobian.slx');
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<pre class="src src-matlab">open('stewart_platform/drone_platform_jacobian.slx');
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</pre>
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</pre>
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@ -146,8 +178,8 @@ save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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<div id="outline-container-org7bbb169" class="outline-3">
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<h3 id="org8947fec"><span class="section-number-3">2.2</span> Simulink</h3>
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<h3 id="org7bbb169"><span class="section-number-3">2.2</span> Simscape Model</h3>
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<div class="outline-text-3" id="text-2-2">
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<div class="outline-text-3" id="text-2-2">
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<div class="org-src-container">
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<div class="org-src-container">
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<pre class="src src-matlab">open('stewart_platform/drone_platform.slx');
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<pre class="src src-matlab">open('stewart_platform/drone_platform.slx');
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@ -175,8 +207,12 @@ We load the Jacobian.
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<pre class="src src-matlab">load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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<pre class="src src-matlab">load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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</pre>
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<div id="outline-container-org2a265c4" class="outline-3">
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<h3 id="org2a265c4"><span class="section-number-3">2.3</span> Identification of the plant</h3>
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<div class="outline-text-3" id="text-2-3">
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<p>
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<p>
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The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
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The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
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</p>
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</p>
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@ -186,64 +222,554 @@ mdl = 'drone_platform';
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%% Input/Output definition
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%% Input/Output definition
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clear io; io_i = 1;
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
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'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
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G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
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</pre>
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</pre>
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<p>
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There are 24 states (6dof for the bottom platform + 6dof for the top platform).
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</p>
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<div class="org-src-container">
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<div class="org-src-container">
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<pre class="src src-matlab">size(G)
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<pre class="src src-matlab">size(G)
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</pre>
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</pre>
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<pre class="example">
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<pre class="example">
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State-space model with 6 outputs, 6 inputs, and 12 states.
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State-space model with 6 outputs, 12 inputs, and 24 states.
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</pre>
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</pre>
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<div class="org-src-container">
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<pre class="src src-matlab">% G = G*blkdiag(inv(J), eye(6));
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% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
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% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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</pre>
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<p>
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<p>
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Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
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Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
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</p>
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</p>
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<div class="org-src-container">
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<div class="org-src-container">
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<pre class="src src-matlab">Gx = -G*inv(J');
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<pre class="src src-matlab">Gx = G*blkdiag(eye(6), inv(J'));
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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
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'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gl = -J*G;
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Gl = J*G;
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Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
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Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
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</pre>
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</pre>
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<h3 id="orgfa83a84"><span class="section-number-3">2.4</span> Obtained Dynamics</h3>
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<div class="outline-text-3" id="text-2-4">
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<div id="orgc94fa6a" class="figure">
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<div id="orga7d2bfa" class="figure">
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<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
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<p><img src="figs/stewart_platform_translations.png" alt="stewart_platform_translations.png" />
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</p>
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</p>
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<p><span class="figure-number">Figure 2: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
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<p><span class="figure-number">Figure 2: </span>Stewart Platform Plant from forces applied by the legs to the acceleration of the platform</p>
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<div id="org5e7bd8e" class="figure">
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<div id="orge8ecc72" class="figure">
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<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
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<p><img src="figs/stewart_platform_rotations.png" alt="stewart_platform_rotations.png" />
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</p>
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</p>
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<p><span class="figure-number">Figure 3: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
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<p><span class="figure-number">Figure 3: </span>Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform</p>
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</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgce0e5a7" class="figure">
|
<div id="orga068faf" class="figure">
|
||||||
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
|
<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 4: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p>
|
<p><span class="figure-number">Figure 4: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="orgf48c4d4" class="figure">
|
||||||
|
<p><img src="figs/stewart_platform_transmissibility.png" alt="stewart_platform_transmissibility.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 5: </span>Transmissibility</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org92dd977" class="outline-3">
|
||||||
|
<h3 id="org92dd977"><span class="section-number-3">2.5</span> Real Approximation of \(G\) at the decoupling frequency</h3>
|
||||||
|
<div class="outline-text-3" id="text-2-5">
|
||||||
|
<p>
|
||||||
|
Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_c(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">wc = 2*pi*20; % Decoupling frequency [rad/s]
|
||||||
|
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">H1 = evalfr(Gc, j*wc);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The real approximation is computed as follows:
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">D = pinv(real(H1'*H1));
|
||||||
|
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgebf7751" class="outline-3">
|
||||||
|
<h3 id="orgebf7751"><span class="section-number-3">2.6</span> Verification of the decoupling using the “Gershgorin Radii”</h3>
|
||||||
|
<div class="outline-text-3" id="text-2-6">
|
||||||
|
<p>
|
||||||
|
First, the Singular Value Decomposition of \(H_1\) is performed:
|
||||||
|
\[ H_1 = U \Sigma V^H \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">[U,S,V] = svd(H1);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Then, the “Gershgorin Radii” is computed for the plant \(G_c(s)\) and the “SVD Decoupled Plant” \(G_d(s)\):
|
||||||
|
\[ G_d(s) = U^T G_c(s) V \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
It is done over the following frequencies.
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">freqs = logspace(-1,2,1000); % [Hz]
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">for i = 1:length(freqs)
|
||||||
|
H = abs(evalfr(Gc, j*2*pi*freqs(i)));
|
||||||
|
for j = 1:size(H,2)
|
||||||
|
g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">Gd = U'*Gc*V;
|
||||||
|
|
||||||
|
for i = 1:length(freqs)
|
||||||
|
H_dec = abs(evalfr(Gd, j*2*pi*freqs(i)));
|
||||||
|
for j = 1:size(H,2)
|
||||||
|
g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orge21a525" class="outline-3">
|
||||||
|
<h3 id="orge21a525"><span class="section-number-3">2.7</span> Decoupled Plant</h3>
|
||||||
|
<div class="outline-text-3" id="text-2-7">
|
||||||
|
<p>
|
||||||
|
Let’s see the bode plot of the decoupled plant \(G_d(s)\).
|
||||||
|
\[ G_d(s) = U^T G_c(s) V \]
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org4c1f528" class="outline-3">
|
||||||
|
<h3 id="org4c1f528"><span class="section-number-3">2.8</span> Diagonal Controller</h3>
|
||||||
|
<div class="outline-text-3" id="text-2-8">
|
||||||
|
<p>
|
||||||
|
The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">wc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||||
|
C_g = 50; % DC Gain
|
||||||
|
|
||||||
|
K = eye(6)*C_g/(s+wc);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org4f88748" class="outline-3">
|
||||||
|
<h3 id="org4f88748"><span class="section-number-3">2.9</span> Centralized Control</h3>
|
||||||
|
<div class="outline-text-3" id="text-2-9">
|
||||||
|
<p>
|
||||||
|
The control diagram for the centralized control is shown below.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The controller \(K_c\) is “working” in an cartesian frame.
|
||||||
|
The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
|
||||||
|
<div class="figure">
|
||||||
|
<p><img src="figs/centralized_control.png" alt="centralized_control.png" />
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org6eac181" class="outline-3">
|
||||||
|
<h3 id="org6eac181"><span class="section-number-3">2.10</span> SVD Control</h3>
|
||||||
|
<div class="outline-text-3" id="text-2-10">
|
||||||
|
<p>
|
||||||
|
The SVD control architecture is shown below.
|
||||||
|
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
|
||||||
|
<div class="figure">
|
||||||
|
<p><img src="figs/svd_control.png" alt="svd_control.png" />
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
SVD Control
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org89ccc9f" class="outline-3">
|
||||||
|
<h3 id="org89ccc9f"><span class="section-number-3">2.11</span> Results</h3>
|
||||||
|
<div class="outline-text-3" id="text-2-11">
|
||||||
|
<p>
|
||||||
|
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#orgfaedd1c">8</a>.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="orgfaedd1c" class="figure">
|
||||||
|
<p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 8: </span>Obtained Transmissibility</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgdcb6e90" class="outline-2">
|
||||||
|
<h2 id="orgdcb6e90"><span class="section-number-2">3</span> Stewart Platform - Analytical Model</h2>
|
||||||
|
<div class="outline-text-2" id="text-3">
|
||||||
|
</div>
|
||||||
|
<div id="outline-container-orgeb4b14b" class="outline-3">
|
||||||
|
<h3 id="orgeb4b14b"><span class="section-number-3">3.1</span> Characteristics</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-1">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">L = 0.055;
|
||||||
|
Zc = 0;
|
||||||
|
m = 0.2;
|
||||||
|
k = 1e3;
|
||||||
|
c = 2*0.1*sqrt(k*m);
|
||||||
|
|
||||||
|
Rx = 0.04;
|
||||||
|
Rz = 0.04;
|
||||||
|
Ix = m*Rx^2;
|
||||||
|
Iy = m*Rx^2;
|
||||||
|
Iz = m*Rz^2;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgeff797b" class="outline-3">
|
||||||
|
<h3 id="orgeff797b"><span class="section-number-3">3.2</span> Mass Matrix</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-2">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">M = m*[1 0 0 0 Zc 0;
|
||||||
|
0 1 0 -Zc 0 0;
|
||||||
|
0 0 1 0 0 0;
|
||||||
|
0 -Zc 0 Rx^2+Zc^2 0 0;
|
||||||
|
Zc 0 0 0 Rx^2+Zc^2 0;
|
||||||
|
0 0 0 0 0 Rz^2];
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org7027995" class="outline-3">
|
||||||
|
<h3 id="org7027995"><span class="section-number-3">3.3</span> Jacobian Matrix</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-3">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
|
||||||
|
sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
|
||||||
|
sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
|
||||||
|
0 0 L L -L -L;
|
||||||
|
-L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
|
||||||
|
L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org51bab7b" class="outline-3">
|
||||||
|
<h3 id="org51bab7b"><span class="section-number-3">3.4</span> Stifnness matrix and Damping matrix</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-4">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">kv = k/3; % [N/m]
|
||||||
|
kh = 0.5*k/3; % [N/m]
|
||||||
|
|
||||||
|
K = diag([3*kh,3*kh,3*kv,3*kv*Rx^2/2,3*kv*Rx^2/2,3*kh*Rx^2]); % Stiffness Matrix
|
||||||
|
|
||||||
|
C = c*K/100000; % Damping Matrix
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orga9e6cf5" class="outline-3">
|
||||||
|
<h3 id="orga9e6cf5"><span class="section-number-3">3.5</span> State Space System</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-5">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">A = [zeros(6) eye(6); -M\K -M\C];
|
||||||
|
Bw = [zeros(6); -eye(6)];
|
||||||
|
Bu = [zeros(6); M\Bj];
|
||||||
|
Co = [-M\K -M\C];
|
||||||
|
D = [zeros(6) M\Bj];
|
||||||
|
|
||||||
|
ST = ss(A,[Bw Bu],Co,D);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<ul class="org-ul">
|
||||||
|
<li>OUT 1-6: 6 dof</li>
|
||||||
|
<li>IN 1-6 : ground displacement in the directions of the legs</li>
|
||||||
|
<li>IN 7-12: forces in the actuators.</li>
|
||||||
|
</ul>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
|
||||||
|
'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
|
||||||
|
ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
|
||||||
|
'u1';'u2';'u3';'u4';'u5';'u6'};
|
||||||
|
ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org769c38a" class="outline-3">
|
||||||
|
<h3 id="org769c38a"><span class="section-number-3">3.6</span> Transmissibility</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-6">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">TR=ST*[eye(6); zeros(6)];
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">figure
|
||||||
|
subplot(231)
|
||||||
|
bodemag(TR(1,1),opts);
|
||||||
|
subplot(232)
|
||||||
|
bodemag(TR(2,2),opts);
|
||||||
|
subplot(233)
|
||||||
|
bodemag(TR(3,3),opts);
|
||||||
|
subplot(234)
|
||||||
|
bodemag(TR(4,4),opts);
|
||||||
|
subplot(235)
|
||||||
|
bodemag(TR(5,5),opts);
|
||||||
|
subplot(236)
|
||||||
|
bodemag(TR(6,6),opts);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="org55a5d25" class="figure">
|
||||||
|
<p><img src="figs/stewart_platform_analytical_transmissibility.png" alt="stewart_platform_analytical_transmissibility.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 9: </span>Transmissibility</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org24eb81f" class="outline-3">
|
||||||
|
<h3 id="org24eb81f"><span class="section-number-3">3.7</span> Real approximation of \(G(j\omega)\) at decoupling frequency</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-7">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
|
||||||
|
|
||||||
|
dec_fr = 20;
|
||||||
|
H1 = evalfr(sys1,j*2*pi*dec_fr);
|
||||||
|
H2 = H1;
|
||||||
|
D = pinv(real(H2'*H2));
|
||||||
|
H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
|
||||||
|
[U,S,V] = svd(H1);
|
||||||
|
|
||||||
|
wf = logspace(-1,2,1000);
|
||||||
|
for i = 1:length(wf)
|
||||||
|
H = abs(evalfr(sys1,j*2*pi*wf(i)));
|
||||||
|
H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
|
||||||
|
for j = 1:size(H,2)
|
||||||
|
g_r1(i,j) = (sum(H(j,:))-H(j,j))/H(j,j);
|
||||||
|
g_r2(i,j) = (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
|
||||||
|
% keyboard
|
||||||
|
end
|
||||||
|
g_lim(i) = 0.5;
|
||||||
|
end
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org824e380" class="outline-3">
|
||||||
|
<h3 id="org824e380"><span class="section-number-3">3.8</span> Coupled and Decoupled Plant “Gershgorin Radii”</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-8">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">figure;
|
||||||
|
title('Coupled plant')
|
||||||
|
loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
|
||||||
|
legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
|
||||||
|
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="org7d8bf66" class="figure">
|
||||||
|
<p><img src="figs/gershorin_raddii_coupled_analytical.png" alt="gershorin_raddii_coupled_analytical.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 10: </span>Gershorin Raddi for the coupled plant</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">figure;
|
||||||
|
title('Decoupled plant (10 Hz)')
|
||||||
|
loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
|
||||||
|
legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
|
||||||
|
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="org319f0f6" class="figure">
|
||||||
|
<p><img src="figs/gershorin_raddii_decoupled_analytical.png" alt="gershorin_raddii_decoupled_analytical.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 11: </span>Gershorin Raddi for the decoupled plant</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org8e5d2c7" class="outline-3">
|
||||||
|
<h3 id="org8e5d2c7"><span class="section-number-3">3.9</span> Decoupled Plant</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-9">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">figure;
|
||||||
|
bodemag(U'*sys1*V,opts)
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="org057e23e" class="figure">
|
||||||
|
<p><img src="figs/stewart_platform_analytical_decoupled_plant.png" alt="stewart_platform_analytical_decoupled_plant.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 12: </span>Decoupled Plant</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org102382b" class="outline-3">
|
||||||
|
<h3 id="org102382b"><span class="section-number-3">3.10</span> Controller</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-10">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">fc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||||
|
c_gain = 50; %
|
||||||
|
|
||||||
|
cont = eye(6)*c_gain/(s+fc);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org27bf3be" class="outline-3">
|
||||||
|
<h3 id="org27bf3be"><span class="section-number-3">3.11</span> Closed Loop System</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-11">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">FEEDIN = [7:12]; % Input of controller
|
||||||
|
FEEDOUT = [1:6]; % Output of controller
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Centralized Control
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
|
||||||
|
TRcen = STcen*[eye(6); zeros(6)];
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
SVD Control
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
|
||||||
|
TRsvd = STsvd*[eye(6); zeros(6)];
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org419f877" class="outline-3">
|
||||||
|
<h3 id="org419f877"><span class="section-number-3">3.12</span> Results</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-12">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">figure
|
||||||
|
subplot(231)
|
||||||
|
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(232)
|
||||||
|
bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(233)
|
||||||
|
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(234)
|
||||||
|
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(235)
|
||||||
|
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(236)
|
||||||
|
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="orgbde8c92" class="figure">
|
||||||
|
<p><img src="figs/stewart_platform_analytical_svd_cen_comp.png" alt="stewart_platform_analytical_svd_cen_comp.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 13: </span>Comparison of the obtained transmissibility for the centralized control and the SVD control</p>
|
||||||
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
<div id="postamble" class="status">
|
<div id="postamble" class="status">
|
||||||
<p class="author">Author: Dehaeze Thomas</p>
|
<p class="author">Author: Dehaeze Thomas</p>
|
||||||
<p class="date">Created: 2020-09-21 lun. 13:14</p>
|
<p class="date">Created: 2020-09-21 lun. 18:03</p>
|
||||||
</div>
|
</div>
|
||||||
</body>
|
</body>
|
||||||
</html>
|
</html>
|
||||||
|
|||||||
617
index.org
617
index.org
@ -34,14 +34,16 @@
|
|||||||
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
|
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
|
||||||
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
|
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
|
||||||
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
|
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
|
||||||
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
|
#+PROPERTY: header-args:latex+ :results file raw replace
|
||||||
|
#+PROPERTY: header-args:latex+ :buffer no
|
||||||
#+PROPERTY: header-args:latex+ :eval no-export
|
#+PROPERTY: header-args:latex+ :eval no-export
|
||||||
#+PROPERTY: header-args:latex+ :exports both
|
#+PROPERTY: header-args:latex+ :exports results
|
||||||
#+PROPERTY: header-args:latex+ :mkdirp yes
|
#+PROPERTY: header-args:latex+ :mkdirp yes
|
||||||
#+PROPERTY: header-args:latex+ :output-dir figs
|
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||||
|
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
|
||||||
:END:
|
:END:
|
||||||
|
|
||||||
* Simscape Model - Gravimeter
|
* Gravimeter - Simscape Model
|
||||||
** Matlab Init :noexport: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)
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
<<matlab-dir>>
|
<<matlab-dir>>
|
||||||
@ -380,7 +382,7 @@ The plant as 6 states as expected (2 translations + 1 rotation)
|
|||||||
rot = PHI(:,11,11);
|
rot = PHI(:,11,11);
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
* Simscape Model - Stewart Platform
|
* Stewart Platform - Simscape Model
|
||||||
** Matlab Init :noexport: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)
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
<<matlab-dir>>
|
<<matlab-dir>>
|
||||||
@ -392,7 +394,6 @@ The plant as 6 states as expected (2 translations + 1 rotation)
|
|||||||
|
|
||||||
** Jacobian
|
** Jacobian
|
||||||
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
|
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
|
||||||
|
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
open('stewart_platform/drone_platform_jacobian.slx');
|
open('stewart_platform/drone_platform_jacobian.slx');
|
||||||
#+end_src
|
#+end_src
|
||||||
@ -425,7 +426,7 @@ First, the position of the "joints" (points of force application) are estimated
|
|||||||
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
** Simulink
|
** Simscape Model
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
open('stewart_platform/drone_platform.slx');
|
open('stewart_platform/drone_platform.slx');
|
||||||
#+end_src
|
#+end_src
|
||||||
@ -446,7 +447,7 @@ We load the Jacobian.
|
|||||||
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
** Identification of the plant
|
||||||
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
|
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
%% Name of the Simulink File
|
%% Name of the Simulink File
|
||||||
@ -454,30 +455,41 @@ The dynamics is identified from forces applied by each legs to the measured acce
|
|||||||
|
|
||||||
%% Input/Output definition
|
%% Input/Output definition
|
||||||
clear io; io_i = 1;
|
clear io; io_i = 1;
|
||||||
|
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
|
||||||
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
|
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
|
||||||
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
|
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
|
||||||
|
|
||||||
G = linearize(mdl, io);
|
G = linearize(mdl, io);
|
||||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||||
|
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||||
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
|
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
|
||||||
#+begin_src matlab :results output replace
|
#+begin_src matlab :results output replace
|
||||||
size(G)
|
size(G)
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
: State-space model with 6 outputs, 6 inputs, and 12 states.
|
: State-space model with 6 outputs, 12 inputs, and 24 states.
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
% G = G*blkdiag(inv(J), eye(6));
|
||||||
|
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
|
||||||
|
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||||
|
#+end_src
|
||||||
|
|
||||||
Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
|
Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
Gx = -G*inv(J');
|
Gx = G*blkdiag(eye(6), inv(J'));
|
||||||
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||||
|
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||||
|
|
||||||
Gl = -J*G;
|
Gl = J*G;
|
||||||
Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
|
Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
** Obtained Dynamics
|
||||||
#+begin_src matlab :exports none
|
#+begin_src matlab :exports none
|
||||||
freqs = logspace(-1, 2, 1000);
|
freqs = logspace(-1, 2, 1000);
|
||||||
|
|
||||||
@ -485,9 +497,9 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
|||||||
|
|
||||||
ax1 = subplot(2, 1, 1);
|
ax1 = subplot(2, 1, 1);
|
||||||
hold on;
|
hold on;
|
||||||
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
|
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
|
||||||
plot(freqs, abs(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
|
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
|
||||||
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
|
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
|
||||||
hold off;
|
hold off;
|
||||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
@ -495,13 +507,13 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
|||||||
|
|
||||||
ax2 = subplot(2, 1, 2);
|
ax2 = subplot(2, 1, 2);
|
||||||
hold on;
|
hold on;
|
||||||
for i = 1:3
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
|
||||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz')))));
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
|
||||||
end
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
|
||||||
hold off;
|
hold off;
|
||||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
ylim([-270, 90]);
|
ylim([-180, 180]);
|
||||||
yticks([-360:90:360]);
|
yticks([-360:90:360]);
|
||||||
|
|
||||||
linkaxes([ax1,ax2],'x');
|
linkaxes([ax1,ax2],'x');
|
||||||
@ -523,9 +535,9 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
|||||||
|
|
||||||
ax1 = subplot(2, 1, 1);
|
ax1 = subplot(2, 1, 1);
|
||||||
hold on;
|
hold on;
|
||||||
plot(freqs, abs(squeeze(freqresp(Gx(4, 4), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
|
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
|
||||||
plot(freqs, abs(squeeze(freqresp(Gx(5, 5), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
|
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
|
||||||
plot(freqs, abs(squeeze(freqresp(Gx(6, 6), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
|
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
|
||||||
hold off;
|
hold off;
|
||||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
|
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
|
||||||
@ -533,13 +545,13 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
|||||||
|
|
||||||
ax2 = subplot(2, 1, 2);
|
ax2 = subplot(2, 1, 2);
|
||||||
hold on;
|
hold on;
|
||||||
for i = 4:6
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
|
||||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz')))));
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
|
||||||
end
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
|
||||||
hold off;
|
hold off;
|
||||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
ylim([-270, 90]);
|
ylim([-180, 180]);
|
||||||
yticks([-360:90:360]);
|
yticks([-360:90:360]);
|
||||||
|
|
||||||
linkaxes([ax1,ax2],'x');
|
linkaxes([ax1,ax2],'x');
|
||||||
@ -562,11 +574,11 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
|||||||
ax1 = subplot(2, 1, 1);
|
ax1 = subplot(2, 1, 1);
|
||||||
hold on;
|
hold on;
|
||||||
for i = 1:6
|
for i = 1:6
|
||||||
plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
|
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', i)), freqs, 'Hz'))));
|
||||||
end
|
end
|
||||||
for i = 1:5
|
for i = 1:5
|
||||||
for j = i+1:6
|
for j = i+1:6
|
||||||
plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', j)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
hold off;
|
hold off;
|
||||||
@ -576,12 +588,12 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
|||||||
ax2 = subplot(2, 1, 2);
|
ax2 = subplot(2, 1, 2);
|
||||||
hold on;
|
hold on;
|
||||||
for i = 1:6
|
for i = 1:6
|
||||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz')))));
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', i)), freqs, 'Hz'))));
|
||||||
end
|
end
|
||||||
hold off;
|
hold off;
|
||||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
ylim([-270, 90]);
|
ylim([-180, 180]);
|
||||||
yticks([-360:90:360]);
|
yticks([-360:90:360]);
|
||||||
|
|
||||||
linkaxes([ax1,ax2],'x');
|
linkaxes([ax1,ax2],'x');
|
||||||
@ -595,3 +607,548 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
|||||||
#+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
|
#+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
[[file:figs/stewart_platform_legs.png]]
|
[[file:figs/stewart_platform_legs.png]]
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(-1, 2, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
|
||||||
|
legend('location', 'northeast');
|
||||||
|
|
||||||
|
ax2 = subplot(2, 1, 2);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
|
||||||
|
legend('location', 'northeast');
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/stewart_platform_transmissibility.pdf', 'width', 'full', 'height', 'full');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:stewart_platform_transmissibility
|
||||||
|
#+caption: Transmissibility
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/stewart_platform_transmissibility.png]]
|
||||||
|
|
||||||
|
** Real Approximation of $G$ at the decoupling frequency
|
||||||
|
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
|
||||||
|
#+begin_src matlab
|
||||||
|
wc = 2*pi*20; % Decoupling frequency [rad/s]
|
||||||
|
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
H1 = evalfr(Gc, j*wc);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
The real approximation is computed as follows:
|
||||||
|
#+begin_src matlab
|
||||||
|
D = pinv(real(H1'*H1));
|
||||||
|
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Verification of the decoupling using the "Gershgorin Radii"
|
||||||
|
First, the Singular Value Decomposition of $H_1$ is performed:
|
||||||
|
\[ H_1 = U \Sigma V^H \]
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
[U,S,V] = svd(H1);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
|
||||||
|
\[ G_d(s) = U^T G_c(s) V \]
|
||||||
|
|
||||||
|
It is done over the following frequencies.
|
||||||
|
#+begin_src matlab
|
||||||
|
freqs = logspace(-1,2,1000); % [Hz]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
for i = 1:length(freqs)
|
||||||
|
H = abs(evalfr(Gc, j*2*pi*freqs(i)));
|
||||||
|
for j = 1:size(H,2)
|
||||||
|
g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
Gd = U'*Gc*V;
|
||||||
|
|
||||||
|
for i = 1:length(freqs)
|
||||||
|
H_dec = abs(evalfr(Gd, j*2*pi*freqs(i)));
|
||||||
|
for j = 1:size(H,2)
|
||||||
|
g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports results
|
||||||
|
figure;
|
||||||
|
hold on;
|
||||||
|
plot(freqs, g_r1(:,1), 'DisplayName', '$a_x$')
|
||||||
|
plot(freqs, g_r1(:,2), 'DisplayName', '$a_y$')
|
||||||
|
plot(freqs, g_r1(:,3), 'DisplayName', '$a_z$')
|
||||||
|
plot(freqs, g_r1(:,4), 'DisplayName', '$a_{R_x}$')
|
||||||
|
plot(freqs, g_r1(:,5), 'DisplayName', '$a_{R_y}$')
|
||||||
|
plot(freqs, g_r1(:,6), 'DisplayName', '$a_{R_z}$')
|
||||||
|
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
hold off;
|
||||||
|
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports results
|
||||||
|
figure;
|
||||||
|
hold on;
|
||||||
|
plot(freqs, g_r2(:,1), 'DisplayName', '$a_x$')
|
||||||
|
plot(freqs, g_r2(:,2), 'DisplayName', '$a_y$')
|
||||||
|
plot(freqs, g_r2(:,3), 'DisplayName', '$a_z$')
|
||||||
|
plot(freqs, g_r2(:,4), 'DisplayName', '$a_{R_x}$')
|
||||||
|
plot(freqs, g_r2(:,5), 'DisplayName', '$a_{R_y}$')
|
||||||
|
plot(freqs, g_r2(:,6), 'DisplayName', '$a_{R_z}$')
|
||||||
|
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Decoupled Plant
|
||||||
|
Let's see the bode plot of the decoupled plant $G_d(s)$.
|
||||||
|
\[ G_d(s) = U^T G_c(s) V \]
|
||||||
|
|
||||||
|
#+begin_src matlab :exports results
|
||||||
|
freqs = logspace(-1, 2, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gd(i, i), freqs, 'Hz'))), ...
|
||||||
|
'DisplayName', sprintf('$G(%i, %i)$', i, i));
|
||||||
|
end
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||||
|
'HandleVisibility', 'off');
|
||||||
|
end
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
||||||
|
legend('location', 'southeast');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Diagonal Controller
|
||||||
|
The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||||
|
C_g = 50; % DC Gain
|
||||||
|
|
||||||
|
K = eye(6)*C_g/(s+wc);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Centralized Control
|
||||||
|
The control diagram for the centralized control is shown below.
|
||||||
|
|
||||||
|
The controller $K_c$ is "working" in an cartesian frame.
|
||||||
|
The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.
|
||||||
|
|
||||||
|
#+begin_src latex :file centralized_control.pdf
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node[block={2cm}{1.5cm}] (G) {$G$};
|
||||||
|
\node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
|
||||||
|
\node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$};
|
||||||
|
|
||||||
|
% Inputs of the controllers
|
||||||
|
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
|
||||||
|
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
|
||||||
|
|
||||||
|
% Connections and labels
|
||||||
|
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
|
||||||
|
\draw[->] (G.east) -- ++(2.0, 0) node[above left]{$a$};
|
||||||
|
\draw[->] ($(G.east)+(1.4, 0)$)node[branch]{} |- (K.east);
|
||||||
|
\draw[->] (K.west) -- (J.east) node[above right]{$\mathcal{F}$};
|
||||||
|
\draw[->] (J.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/centralized_control.png]]
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** SVD Control
|
||||||
|
The SVD control architecture is shown below.
|
||||||
|
The matrices $U$ and $V$ are used to decoupled the plant $G$.
|
||||||
|
|
||||||
|
#+begin_src latex :file svd_control.pdf
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node[block={2cm}{1.5cm}] (G) {$G$};
|
||||||
|
\node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$};
|
||||||
|
\node[block, below=0.6 of G] (K) {$K_{\text{SVD}}$};
|
||||||
|
\node[block, below left= 0.6 and 0 of G] (V) {$V^{-T}$};
|
||||||
|
|
||||||
|
% Inputs of the controllers
|
||||||
|
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
|
||||||
|
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
|
||||||
|
|
||||||
|
% Connections and labels
|
||||||
|
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
|
||||||
|
\draw[->] (G.east) -- ++(2.5, 0) node[above left]{$a$};
|
||||||
|
\draw[->] ($(G.east)+(2.0, 0)$) node[branch]{} |- (U.east);
|
||||||
|
\draw[->] (U.west) -- (K.east);
|
||||||
|
\draw[->] (K.west) -- (V.east);
|
||||||
|
\draw[->] (V.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/svd_control.png]]
|
||||||
|
|
||||||
|
SVD Control
|
||||||
|
#+begin_src matlab
|
||||||
|
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Results
|
||||||
|
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
|
||||||
|
|
||||||
|
#+begin_src matlab :exports results
|
||||||
|
freqs = logspace(-3, 3, 1000);
|
||||||
|
|
||||||
|
figure
|
||||||
|
|
||||||
|
ax1 = subplot(2, 3, 1);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]');
|
||||||
|
legend('location', 'southwest');
|
||||||
|
|
||||||
|
ax2 = subplot(2, 3, 2);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
|
||||||
|
|
||||||
|
ax3 = subplot(2, 3, 3);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]');
|
||||||
|
|
||||||
|
ax4 = subplot(2, 3, 4);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
|
||||||
|
|
||||||
|
ax5 = subplot(2, 3, 5);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
|
||||||
|
|
||||||
|
ax6 = subplot(2, 3, 6);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||||
|
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
||||||
|
xlim([freqs(1), freqs(end)]);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'width', 1600, 'height', 'full');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:stewart_platform_simscape_cl_transmissibility
|
||||||
|
#+caption: Obtained Transmissibility
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
|
||||||
|
|
||||||
|
* Stewart Platform - Analytical Model
|
||||||
|
** 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
|
||||||
|
%% Bode plot options
|
||||||
|
opts = bodeoptions('cstprefs');
|
||||||
|
opts.FreqUnits = 'Hz';
|
||||||
|
opts.MagUnits = 'abs';
|
||||||
|
opts.MagScale = 'log';
|
||||||
|
opts.PhaseWrapping = 'on';
|
||||||
|
opts.xlim = [1 1000];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Characteristics
|
||||||
|
#+begin_src matlab
|
||||||
|
L = 0.055;
|
||||||
|
Zc = 0;
|
||||||
|
m = 0.2;
|
||||||
|
k = 1e3;
|
||||||
|
c = 2*0.1*sqrt(k*m);
|
||||||
|
|
||||||
|
Rx = 0.04;
|
||||||
|
Rz = 0.04;
|
||||||
|
Ix = m*Rx^2;
|
||||||
|
Iy = m*Rx^2;
|
||||||
|
Iz = m*Rz^2;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Mass Matrix
|
||||||
|
#+begin_src matlab
|
||||||
|
M = m*[1 0 0 0 Zc 0;
|
||||||
|
0 1 0 -Zc 0 0;
|
||||||
|
0 0 1 0 0 0;
|
||||||
|
0 -Zc 0 Rx^2+Zc^2 0 0;
|
||||||
|
Zc 0 0 0 Rx^2+Zc^2 0;
|
||||||
|
0 0 0 0 0 Rz^2];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Jacobian Matrix
|
||||||
|
#+begin_src matlab
|
||||||
|
Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
|
||||||
|
sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
|
||||||
|
sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
|
||||||
|
0 0 L L -L -L;
|
||||||
|
-L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
|
||||||
|
L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Stifnness matrix and Damping matrix
|
||||||
|
#+begin_src matlab
|
||||||
|
kv = k/3; % [N/m]
|
||||||
|
kh = 0.5*k/3; % [N/m]
|
||||||
|
|
||||||
|
K = diag([3*kh,3*kh,3*kv,3*kv*Rx^2/2,3*kv*Rx^2/2,3*kh*Rx^2]); % Stiffness Matrix
|
||||||
|
|
||||||
|
C = c*K/100000; % Damping Matrix
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** State Space System
|
||||||
|
#+begin_src matlab
|
||||||
|
A = [zeros(6) eye(6); -M\K -M\C];
|
||||||
|
Bw = [zeros(6); -eye(6)];
|
||||||
|
Bu = [zeros(6); M\Bj];
|
||||||
|
Co = [-M\K -M\C];
|
||||||
|
D = [zeros(6) M\Bj];
|
||||||
|
|
||||||
|
ST = ss(A,[Bw Bu],Co,D);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
- OUT 1-6: 6 dof
|
||||||
|
- IN 1-6 : ground displacement in the directions of the legs
|
||||||
|
- IN 7-12: forces in the actuators.
|
||||||
|
#+begin_src matlab
|
||||||
|
ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
|
||||||
|
'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
|
||||||
|
ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
|
||||||
|
'u1';'u2';'u3';'u4';'u5';'u6'};
|
||||||
|
ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Transmissibility
|
||||||
|
#+begin_src matlab
|
||||||
|
TR=ST*[eye(6); zeros(6)];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
figure
|
||||||
|
subplot(231)
|
||||||
|
bodemag(TR(1,1),opts);
|
||||||
|
subplot(232)
|
||||||
|
bodemag(TR(2,2),opts);
|
||||||
|
subplot(233)
|
||||||
|
bodemag(TR(3,3),opts);
|
||||||
|
subplot(234)
|
||||||
|
bodemag(TR(4,4),opts);
|
||||||
|
subplot(235)
|
||||||
|
bodemag(TR(5,5),opts);
|
||||||
|
subplot(236)
|
||||||
|
bodemag(TR(6,6),opts);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/stewart_platform_analytical_transmissibility.pdf', 'width', 'full', 'height', 'full');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:stewart_platform_analytical_transmissibility
|
||||||
|
#+caption: Transmissibility
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/stewart_platform_analytical_transmissibility.png]]
|
||||||
|
|
||||||
|
** Real approximation of $G(j\omega)$ at decoupling frequency
|
||||||
|
#+begin_src matlab
|
||||||
|
sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
|
||||||
|
|
||||||
|
dec_fr = 20;
|
||||||
|
H1 = evalfr(sys1,j*2*pi*dec_fr);
|
||||||
|
H2 = H1;
|
||||||
|
D = pinv(real(H2'*H2));
|
||||||
|
H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
|
||||||
|
[U,S,V] = svd(H1);
|
||||||
|
|
||||||
|
wf = logspace(-1,2,1000);
|
||||||
|
for i = 1:length(wf)
|
||||||
|
H = abs(evalfr(sys1,j*2*pi*wf(i)));
|
||||||
|
H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
|
||||||
|
for j = 1:size(H,2)
|
||||||
|
g_r1(i,j) = (sum(H(j,:))-H(j,j))/H(j,j);
|
||||||
|
g_r2(i,j) = (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
|
||||||
|
% keyboard
|
||||||
|
end
|
||||||
|
g_lim(i) = 0.5;
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Coupled and Decoupled Plant "Gershgorin Radii"
|
||||||
|
#+begin_src matlab
|
||||||
|
figure;
|
||||||
|
title('Coupled plant')
|
||||||
|
loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
|
||||||
|
legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
|
||||||
|
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/gershorin_raddii_coupled_analytical.pdf', 'width', 'full', 'height', 'full');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:gershorin_raddii_coupled_analytical
|
||||||
|
#+caption: Gershorin Raddi for the coupled plant
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/gershorin_raddii_coupled_analytical.png]]
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
figure;
|
||||||
|
title('Decoupled plant (10 Hz)')
|
||||||
|
loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
|
||||||
|
legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
|
||||||
|
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/gershorin_raddii_decoupled_analytical.pdf', 'width', 'full', 'height', 'full');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:gershorin_raddii_decoupled_analytical
|
||||||
|
#+caption: Gershorin Raddi for the decoupled plant
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/gershorin_raddii_decoupled_analytical.png]]
|
||||||
|
|
||||||
|
** Decoupled Plant
|
||||||
|
#+begin_src matlab
|
||||||
|
figure;
|
||||||
|
bodemag(U'*sys1*V,opts)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/stewart_platform_analytical_decoupled_plant.pdf', 'width', 'full', 'height', 'full');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:stewart_platform_analytical_decoupled_plant
|
||||||
|
#+caption: Decoupled Plant
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/stewart_platform_analytical_decoupled_plant.png]]
|
||||||
|
|
||||||
|
** Controller
|
||||||
|
#+begin_src matlab
|
||||||
|
fc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||||
|
c_gain = 50; %
|
||||||
|
|
||||||
|
cont = eye(6)*c_gain/(s+fc);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Closed Loop System
|
||||||
|
#+begin_src matlab
|
||||||
|
FEEDIN = [7:12]; % Input of controller
|
||||||
|
FEEDOUT = [1:6]; % Output of controller
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Centralized Control
|
||||||
|
#+begin_src matlab
|
||||||
|
STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
|
||||||
|
TRcen = STcen*[eye(6); zeros(6)];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
SVD Control
|
||||||
|
#+begin_src matlab
|
||||||
|
STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
|
||||||
|
TRsvd = STsvd*[eye(6); zeros(6)];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Results
|
||||||
|
#+begin_src matlab
|
||||||
|
figure
|
||||||
|
subplot(231)
|
||||||
|
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(232)
|
||||||
|
bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(233)
|
||||||
|
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(234)
|
||||||
|
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(235)
|
||||||
|
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
subplot(236)
|
||||||
|
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
|
||||||
|
legend('OL','Centralized','SVD')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/stewart_platform_analytical_svd_cen_comp.pdf', 'width', 'full', 'height', 'full');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:stewart_platform_analytical_svd_cen_comp
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#+caption: Comparison of the obtained transmissibility for the centralized control and the SVD control
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#+RESULTS:
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[[file:figs/stewart_platform_analytical_svd_cen_comp.png]]
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