587 lines
20 KiB
HTML
587 lines
20 KiB
HTML
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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">Identification of the Stewart Platform using Simscape</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgcb2f4c2">1. Modal Analysis of the Stewart Platform</a>
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<ul>
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<li><a href="#org66d09e9">1.1. Initialize the Stewart Platform</a></li>
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<li><a href="#org8b1c587">1.2. Identification</a></li>
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<li><a href="#orge68adea">1.3. Coordinate transformation</a></li>
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<li><a href="#org4973ae1">1.4. Analysis</a></li>
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<li><a href="#orge7b97c8">1.5. Visualizing the modes</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgcb2f4c2" class="outline-2">
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<h2 id="orgcb2f4c2"><span class="section-number-2">1</span> Modal Analysis of the Stewart Platform</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org66d09e9" class="outline-3">
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<h3 id="org66d09e9"><span class="section-number-3">1.1</span> Initialize the Stewart Platform</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">stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart);
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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">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org8b1c587" class="outline-3">
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<h3 id="org8b1c587"><span class="section-number-3">1.2</span> Identification</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
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options = linearizeOptions;
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options.SampleTime = 0;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
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mdl = <span class="org-string">'stewart_platform_model'</span>;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
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clear io; io_i = 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
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io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
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io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Velocity of {B} w.r.t. {A}</span>
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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G = linearize(mdl, io);
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<span class="org-comment">% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};</span>
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<span class="org-comment">% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};</span>
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</pre>
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</div>
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<p>
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Let’s check the size of <code>G</code>:
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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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size(G)
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State-space model with 12 outputs, 6 inputs, and 18 states.
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'org_babel_eoe'
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ans =
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'org_babel_eoe'
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</pre>
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<p>
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We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Gm = minreal(G);
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</pre>
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</div>
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<pre class="example">
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Gm = minreal(G);
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6 states removed.
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</pre>
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<p>
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And indeed, we obtain 12 states.
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</p>
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</div>
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</div>
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<div id="outline-container-orge68adea" class="outline-3">
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<h3 id="orge68adea"><span class="section-number-3">1.3</span> Coordinate transformation</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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We can perform the following transformation using the <code>ss2ss</code> command.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Gt = ss2ss(Gm, Gm.C);
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</pre>
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</div>
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<p>
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Then, the <code>C</code> matrix of <code>Gt</code> is the unity matrix which means that the states of the state space model are equal to the measurements \(\bm{Y}\).
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</p>
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<p>
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The measurements are the 6 displacement and 6 velocities of mobile platform with respect to \(\{B\}\).
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</p>
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<p>
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We could perform the transformation by hand:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">At = Gm.C<span class="org-type">*</span>Gm.A<span class="org-type">*</span>pinv(Gm.C);
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Bt = Gm.C<span class="org-type">*</span>Gm.B;
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Ct = eye(12);
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Dt = zeros(12, 6);
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|
|
|
Gt = ss(At, Bt, Ct, Dt);
|
|
</pre>
|
|
</div>
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|
</div>
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|
</div>
|
|
|
|
<div id="outline-container-org4973ae1" class="outline-3">
|
|
<h3 id="org4973ae1"><span class="section-number-3">1.4</span> Analysis</h3>
|
|
<div class="outline-text-3" id="text-1-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">[V,D] = eig(Gt.A);
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-right">Mode Number</th>
|
|
<th scope="col" class="org-right">Resonance Frequency [Hz]</th>
|
|
<th scope="col" class="org-right">Damping Ratio [%]</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">1.0</td>
|
|
<td class="org-right">780.6</td>
|
|
<td class="org-right">0.4</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2.0</td>
|
|
<td class="org-right">780.6</td>
|
|
<td class="org-right">0.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.0</td>
|
|
<td class="org-right">903.9</td>
|
|
<td class="org-right">0.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.0</td>
|
|
<td class="org-right">1061.4</td>
|
|
<td class="org-right">0.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5.0</td>
|
|
<td class="org-right">1061.4</td>
|
|
<td class="org-right">0.2</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6.0</td>
|
|
<td class="org-right">1269.6</td>
|
|
<td class="org-right">0.2</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge7b97c8" class="outline-3">
|
|
<h3 id="orge7b97c8"><span class="section-number-3">1.5</span> Visualizing the modes</h3>
|
|
<div class="outline-text-3" id="text-1-5">
|
|
<p>
|
|
To visualize the i’th mode, we may excite the system using the inputs \(U_i\) such that \(B U_i\) is co-linear to \(\xi_i\) (the mode we want to excite).
|
|
</p>
|
|
|
|
<p>
|
|
\[ U(t) = e^{\alpha t} ( ) \]
|
|
</p>
|
|
|
|
<p>
|
|
Let’s first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ws = imag(diag(D));
|
|
[ws,I] = sort(ws)
|
|
ws = ws(7<span class="org-type">:</span>end); I = I(7<span class="org-type">:</span>end);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(ws)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">i_mode = I(<span class="org-constant">i</span>); <span class="org-comment">% the argument is the i'th mode</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">lambda_i = D(i_mode, i_mode);
|
|
xi_i = V(<span class="org-type">:</span>,i_mode);
|
|
|
|
a_i = real(lambda_i);
|
|
b_i = imag(lambda_i);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let do 10 periods of the mode.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">t = linspace(0, 10<span class="org-type">/</span>(imag(lambda_i)<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span>), 1000);
|
|
U_i = pinv(Gt.B) <span class="org-type">*</span> real(xi_i <span class="org-type">*</span> lambda_i <span class="org-type">*</span> (cos(b_i <span class="org-type">*</span> t) <span class="org-type">+</span> 1<span class="org-constant">i</span><span class="org-type">*</span>sin(b_i <span class="org-type">*</span> t)));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">U = timeseries(U_i, t);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Simulation:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, num2str(t(<span class="org-variable-name">end</span>)));
|
|
<span class="org-matlab-simulink-keyword">sim</span>(mdl);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Save the movie of the mode shape.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">smwritevideo(mdl, sprintf(<span class="org-string">'figs/mode%i'</span>, <span class="org-constant">i</span>), ...
|
|
<span class="org-string">'PlaybackSpeedRatio'</span>, 1<span class="org-type">/</span>(b_i<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span>), ...
|
|
<span class="org-string">'FrameRate'</span>, 30, ...
|
|
<span class="org-string">'FrameSize'</span>, [800, 400]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgb15855a" class="figure">
|
|
<p><img src="figs/mode1.gif" alt="mode1.gif" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Identified mode - 1</p>
|
|
</div>
|
|
|
|
|
|
<div id="org1816e59" class="figure">
|
|
<p><img src="figs/mode3.gif" alt="mode3.gif" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Identified mode - 3</p>
|
|
</div>
|
|
|
|
|
|
<div id="org01c8dca" class="figure">
|
|
<p><img src="figs/mode5.gif" alt="mode5.gif" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Identified mode - 5</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-02-13 jeu. 15:44</p>
|
|
</div>
|
|
</body>
|
|
</html>
|