2020-09-21 13:08:36 +02:00
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2020-09-21 13:14:25 +02:00
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<!-- 2020-09-21 lun. 13:14 -->
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2020-09-21 13:08:36 +02:00
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>SVD Control</title>
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<meta name="author" content="Dehaeze Thomas" />
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</head>
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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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">SVD Control</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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2020-09-21 13:14:25 +02:00
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<li><a href="#org09b41c5">1. Simscape Model - Gravimeter</a>
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<ul>
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<li><a href="#orgaf12c1d">1.1. Simulink</a></li>
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</ul>
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</li>
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<li><a href="#org84efeb7">2. Simscape Model - Stewart Platform</a>
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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="#org8947fec">2.2. Simulink</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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2020-09-21 13:14:25 +02:00
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<div id="outline-container-org09b41c5" 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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<div class="outline-text-2" id="text-1">
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</div>
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2020-09-21 13:14:25 +02:00
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<div id="outline-container-orgaf12c1d" class="outline-3">
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<h3 id="orgaf12c1d"><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="org-src-container">
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<pre class="src src-matlab">open('gravimeter.slx')
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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">%% Name of the Simulink File
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mdl = 'gravimeter';
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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, '/F1'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3'};
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G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
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</pre>
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</div>
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<p>
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The plant as 6 states as expected (2 translations + 1 rotation)
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</p>
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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>
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</div>
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<pre class="example">
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State-space model with 4 outputs, 3 inputs, and 6 states.
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</pre>
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2020-09-21 13:14:25 +02:00
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<div id="org1c9b0ec" class="figure">
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<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</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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2020-09-21 13:14:25 +02:00
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<div id="outline-container-org84efeb7" 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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<div class="outline-text-2" id="text-2">
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</div>
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<div id="outline-container-org157458d" class="outline-3">
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<h3 id="org157458d"><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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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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<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>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">sim('drone_platform_jacobian');
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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">Aa = [a1.Data(1,:);
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a2.Data(1,:);
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a3.Data(1,:);
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a4.Data(1,:);
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a5.Data(1,:);
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a6.Data(1,:)]';
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Ab = [b1.Data(1,:);
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b2.Data(1,:);
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b3.Data(1,:);
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b4.Data(1,:);
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b5.Data(1,:);
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b6.Data(1,:)]';
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As = (Ab - Aa)./vecnorm(Ab - Aa);
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l = vecnorm(Ab - Aa)';
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J = [As' , cross(Ab, As)'];
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save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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</pre>
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</div>
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</div>
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</div>
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2020-09-21 13:14:25 +02:00
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<div id="outline-container-org8947fec" class="outline-3">
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<h3 id="org8947fec"><span class="section-number-3">2.2</span> Simulink</h3>
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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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<pre class="src src-matlab">open('stewart_platform/drone_platform.slx');
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</pre>
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</div>
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<p>
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Definition of spring parameters
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">kx = 50; % [N/m]
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ky = 50;
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kz = 50;
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cx = 0.025; % [Nm/rad]
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cy = 0.025;
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cz = 0.025;
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</pre>
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</div>
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<p>
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We load the Jacobian.
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</p>
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<div class="org-src-container">
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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>
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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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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">%% Name of the Simulink File
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mdl = 'drone_platform';
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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, '/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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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
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</pre>
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</div>
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2020-09-21 13:14:25 +02:00
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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>
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</div>
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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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</pre>
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2020-09-21 13:08:36 +02:00
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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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</p>
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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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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gl = -J*G;
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Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
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</pre>
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</div>
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2020-09-21 13:14:25 +02:00
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<div id="orgc94fa6a" class="figure">
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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><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>
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2020-09-21 13:14:25 +02:00
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<div id="org5e7bd8e" class="figure">
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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><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>
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2020-09-21 13:14:25 +02:00
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<div id="orgce0e5a7" class="figure">
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<p><img src="figs/stewart_platform_legs.png" alt="stewart_platform_legs.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Stewart Platform Plant from forces applied by the legs to displacement of the legs</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>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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2020-09-21 13:14:25 +02:00
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<p class="date">Created: 2020-09-21 lun. 13:14</p>
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2020-09-21 13:08:36 +02:00
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</div>
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</body>
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