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<h1 class="title">Identification of the Stewart Platform using Simscape</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<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>
<li><a href="#org8b1c587">1.2. Identification</a></li>
<li><a href="#orge68adea">1.3. Coordinate transformation</a></li>
<li><a href="#org4973ae1">1.4. Analysis</a></li>
<li><a href="#orge7b97c8">1.5. Visualizing the modes</a></li>
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</ul>
</li>
<li><a href="#org2891722">2. Transmissibility Analysis</a>
<ul>
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<li><a href="#orgc00b850">2.1. Initialize the Stewart platform</a></li>
<li><a href="#org5338f20">2.2. Transmissibility</a></li>
</ul>
</li>
<li><a href="#orgc94edbd">3. Compliance Analysis</a>
<ul>
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<li><a href="#orge13761a">3.1. Initialize the Stewart platform</a></li>
<li><a href="#org1177029">3.2. Compliance</a></li>
</ul>
</li>
<li><a href="#org68ca336">4. Functions</a>
<ul>
<li><a href="#org487c4d4">4.1. Compute the Transmissibility</a>
<ul>
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<li><a href="#orgd5cf0cf">Function description</a></li>
<li><a href="#orgdce5d62">Optional Parameters</a></li>
<li><a href="#org4629501">Identification of the Transmissibility Matrix</a></li>
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<li><a href="#orge202ae7">Computation of the Frobenius norm</a></li>
</ul>
</li>
<li><a href="#org50e35a6">4.2. Compute the Compliance</a>
<ul>
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<li><a href="#org4630aae">Function description</a></li>
<li><a href="#orgc2d7cfd">Optional Parameters</a></li>
<li><a href="#orgef06b63">Identification of the Compliance Matrix</a></li>
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<li><a href="#org45205c2">Computation of the Frobenius norm</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
<p>
In this document, we discuss the various methods to identify the behavior of the Stewart platform.
</p>
<ul class="org-ul">
<li><a href="#org7981e88">1</a></li>
<li><a href="#orga989615">2</a></li>
<li><a href="#org4579374">3</a></li>
</ul>
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<div id="outline-container-orgcb2f4c2" class="outline-2">
<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">
<p>
<a id="org7981e88"></a>
</p>
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</div>
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<div id="outline-container-org66d09e9" class="outline-3">
<h3 id="org66d09e9"><span class="section-number-3">1.1</span> Initialize the Stewart Platform</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
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<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
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stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart);
</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
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</pre>
</div>
</div>
</div>
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<div id="outline-container-org8b1c587" class="outline-3">
<h3 id="org8b1c587"><span class="section-number-3">1.2</span> Identification</h3>
<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">%% Options for Linearized
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options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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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, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 2, 'openoutput'); io_i = io_i + 1; % Velocity of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io);
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% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
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</pre>
</div>
<p>
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Let&rsquo;s check the size of <code>G</code>:
</p>
<div class="org-src-container">
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<pre class="src src-matlab">size(G)
</pre>
</div>
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<pre class="example">
size(G)
State-space model with 12 outputs, 6 inputs, and 18 states.
'org_babel_eoe'
ans =
'org_babel_eoe'
</pre>
<p>
We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
</p>
<div class="org-src-container">
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<pre class="src src-matlab">Gm = minreal(G);
</pre>
</div>
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<pre class="example">
Gm = minreal(G);
6 states removed.
</pre>
<p>
And indeed, we obtain 12 states.
</p>
</div>
</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>
<div class="outline-text-3" id="text-1-3">
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<p>
We can perform the following transformation using the <code>ss2ss</code> command.
</p>
<div class="org-src-container">
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<pre class="src src-matlab">Gt = ss2ss(Gm, Gm.C);
</pre>
</div>
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<p>
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 \(\bm{Y}\).
</p>
<p>
The measurements are the 6 displacement and 6 velocities of mobile platform with respect to \(\{B\}\).
</p>
<p>
We could perform the transformation by hand:
</p>
<div class="org-src-container">
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<pre class="src src-matlab">At = Gm.C*Gm.A*pinv(Gm.C);
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Bt = Gm.C*Gm.B;
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Ct = eye(12);
Dt = zeros(12, 6);
Gt = ss(At, Bt, Ct, Dt);
</pre>
</div>
</div>
</div>
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<div id="outline-container-org4973ae1" class="outline-3">
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<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">
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<pre class="src src-matlab">[V,D] = eig(Gt.A);
</pre>
</div>
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<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>
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<td class="org-right">780.6</td>
<td class="org-right">0.4</td>
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</tr>
<tr>
<td class="org-right">2.0</td>
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<td class="org-right">780.6</td>
<td class="org-right">0.3</td>
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</tr>
<tr>
<td class="org-right">3.0</td>
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<td class="org-right">903.9</td>
<td class="org-right">0.3</td>
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</tr>
<tr>
<td class="org-right">4.0</td>
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<td class="org-right">1061.4</td>
<td class="org-right">0.3</td>
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</tr>
<tr>
<td class="org-right">5.0</td>
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<td class="org-right">1061.4</td>
<td class="org-right">0.2</td>
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</tr>
<tr>
<td class="org-right">6.0</td>
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<td class="org-right">1269.6</td>
<td class="org-right">0.2</td>
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</tr>
</tbody>
</table>
</div>
</div>
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<div id="outline-container-orge7b97c8" class="outline-3">
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<h3 id="orge7b97c8"><span class="section-number-3">1.5</span> Visualizing the modes</h3>
<div class="outline-text-3" id="text-1-5">
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<p>
To visualize the i&rsquo;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&rsquo;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">
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<pre class="src src-matlab">ws = imag(diag(D));
[ws,I] = sort(ws)
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ws = ws(7:end); I = I(7:end);
</pre>
</div>
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<div class="org-src-container">
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<pre class="src src-matlab">for i = 1:length(ws)
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</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">i_mode = I(i); % the argument is the i'th mode
</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">lambda_i = D(i_mode, i_mode);
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xi_i = V(:,i_mode);
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a_i = real(lambda_i);
b_i = imag(lambda_i);
</pre>
</div>
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<p>
Let do 10 periods of the mode.
</p>
<div class="org-src-container">
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<pre class="src src-matlab">t = linspace(0, 10/(imag(lambda_i)/2/pi), 1000);
U_i = pinv(Gt.B) * real(xi_i * lambda_i * (cos(b_i * t) + 1i*sin(b_i * t)));
</pre>
</div>
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<div class="org-src-container">
<pre class="src src-matlab">U = timeseries(U_i, t);
</pre>
</div>
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<p>
Simulation:
</p>
<div class="org-src-container">
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<pre class="src src-matlab">load('mat/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', num2str(t(end)));
sim(mdl);
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</pre>
</div>
<p>
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Save the movie of the mode shape.
</p>
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<div class="org-src-container">
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<pre class="src src-matlab">smwritevideo(mdl, sprintf('figs/mode%i', i), ...
'PlaybackSpeedRatio', 1/(b_i/2/pi), ...
'FrameRate', 30, ...
'FrameSize', [800, 400]);
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</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">end
</pre>
</div>
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<div id="orgb15855a" class="figure">
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<p><img src="figs/mode1.gif" alt="mode1.gif" />
</p>
<p><span class="figure-number">Figure 1: </span>Identified mode - 1</p>
</div>
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<div id="org1816e59" class="figure">
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<p><img src="figs/mode3.gif" alt="mode3.gif" />
</p>
<p><span class="figure-number">Figure 2: </span>Identified mode - 3</p>
</div>
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<div id="org01c8dca" class="figure">
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<p><img src="figs/mode5.gif" alt="mode5.gif" />
</p>
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<p><span class="figure-number">Figure 3: </span>Identified mode - 5</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org2891722" class="outline-2">
<h2 id="org2891722"><span class="section-number-2">2</span> Transmissibility Analysis</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orga989615"></a>
</p>
</div>
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<div id="outline-container-orgc00b850" class="outline-3">
<h3 id="orgc00b850"><span class="section-number-3">2.1</span> Initialize the Stewart platform</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
</pre>
</div>
<p>
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
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<pre class="src src-matlab">ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'rigid');
controller = initializeController('type', 'open-loop');
</pre>
</div>
</div>
</div>
<div id="outline-container-org5338f20" class="outline-3">
<h3 id="org5338f20"><span class="section-number-3">2.2</span> Transmissibility</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
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<pre class="src src-matlab">%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
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%% Run the linearization
T = linearize(mdl, io, options);
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T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
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figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
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plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-5, 10]);
xlim([freqs(1), freqs(end)]);
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if ix &lt; 6
xticklabels({});
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end
if iy &gt; 1
yticklabels({});
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end
end
end
</pre>
</div>
<p>
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From (<a href="#citeproc_bib_item_1">Preumont et al. 2007</a>), one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
</p>
\begin{align*}
\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
T_norm = zeros(length(freqs), 1);
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for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
</pre>
</div>
<p>
And we normalize by a factor \(\sqrt{6}\) to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
</p>
<div class="org-src-container">
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<pre class="src src-matlab">Gamma = T_norm/sqrt(6);
</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">figure;
plot(freqs, Gamma)
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgc94edbd" class="outline-2">
<h2 id="orgc94edbd"><span class="section-number-2">3</span> Compliance Analysis</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org4579374"></a>
</p>
</div>
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<div id="outline-container-orge13761a" class="outline-3">
<h3 id="orge13761a"><span class="section-number-3">3.1</span> Initialize the Stewart platform</h3>
<div class="outline-text-3" id="text-3-1">
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
</pre>
</div>
<p>
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
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<pre class="src src-matlab">ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid');
controller = initializeController('type', 'open-loop');
</pre>
</div>
</div>
</div>
<div id="outline-container-org1177029" class="outline-3">
<h3 id="org1177029"><span class="section-number-3">3.2</span> Compliance</h3>
<div class="outline-text-3" id="text-3-2">
<div class="org-src-container">
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<pre class="src src-matlab">%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
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%% Run the linearization
C = linearize(mdl, io, options);
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C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
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figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
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plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-10, 1e-3]);
xlim([freqs(1), freqs(end)]);
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if ix &lt; 6
xticklabels({});
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end
if iy &gt; 1
yticklabels({});
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end
end
end
</pre>
</div>
<p>
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
C_norm = zeros(length(freqs), 1);
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for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">figure;
plot(freqs, C_norm)
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org68ca336" class="outline-2">
<h2 id="org68ca336"><span class="section-number-2">4</span> Functions</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org487c4d4" class="outline-3">
<h3 id="org487c4d4"><span class="section-number-3">4.1</span> Compute the Transmissibility</h3>
<div class="outline-text-3" id="text-4-1">
<p>
<a id="orgbca579c"></a>
</p>
</div>
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<div id="outline-container-orgd5cf0cf" class="outline-4">
<h4 id="orgd5cf0cf">Function description</h4>
<div class="outline-text-4" id="text-orgd5cf0cf">
<div class="org-src-container">
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<pre class="src src-matlab">function [T, T_norm, freqs] = computeTransmissibility(args)
% computeTransmissibility -
%
% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - T [6x6 ss] - Transmissibility matrix
% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
</pre>
</div>
</div>
</div>
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<div id="outline-container-orgdce5d62" class="outline-4">
<h4 id="orgdce5d62">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgdce5d62">
<div class="org-src-container">
<pre class="src src-matlab">arguments
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args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
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end
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = args.freqs;
</pre>
</div>
</div>
</div>
<div id="outline-container-org4629501" class="outline-4">
<h4 id="org4629501">Identification of the Transmissibility Matrix</h4>
<div class="outline-text-4" id="text-org4629501">
<div class="org-src-container">
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<pre class="src src-matlab">%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
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%% Run the linearization
T = linearize(mdl, io, options);
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T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
</pre>
</div>
<p>
If wanted, the 6x6 transmissibility matrix is plotted.
</p>
<div class="org-src-container">
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<pre class="src src-matlab">p_handle = zeros(6*6,1);
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if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
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plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix &lt; 6
xticklabels({});
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end
if iy &gt; 1
yticklabels({});
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end
end
end
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linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
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ylim([1e-5, 1e2]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Transmissibility [m/m]');
end
</pre>
</div>
</div>
</div>
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<div id="outline-container-orge202ae7" class="outline-4">
<h4 id="orge202ae7">Computation of the Frobenius norm</h4>
<div class="outline-text-4" id="text-orge202ae7">
<div class="org-src-container">
<pre class="src src-matlab">T_norm = zeros(length(freqs), 1);
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for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">T_norm = T_norm/sqrt(6);
</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">if args.plots
figure;
plot(freqs, T_norm)
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Transmissibility - Frobenius Norm');
end
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org50e35a6" class="outline-3">
<h3 id="org50e35a6"><span class="section-number-3">4.2</span> Compute the Compliance</h3>
<div class="outline-text-3" id="text-4-2">
<p>
<a id="org0a73574"></a>
</p>
</div>
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<div id="outline-container-org4630aae" class="outline-4">
<h4 id="org4630aae">Function description</h4>
<div class="outline-text-4" id="text-org4630aae">
<div class="org-src-container">
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<pre class="src src-matlab">function [C, C_norm, freqs] = computeCompliance(args)
% computeCompliance -
%
% Syntax: [C, C_norm, freqs] = computeCompliance(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - C [6x6 ss] - Compliance matrix
% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
</pre>
</div>
</div>
</div>
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<div id="outline-container-orgc2d7cfd" class="outline-4">
<h4 id="orgc2d7cfd">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgc2d7cfd">
<div class="org-src-container">
<pre class="src src-matlab">arguments
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args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
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end
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = args.freqs;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgef06b63" class="outline-4">
<h4 id="orgef06b63">Identification of the Compliance Matrix</h4>
<div class="outline-text-4" id="text-orgef06b63">
<div class="org-src-container">
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<pre class="src src-matlab">%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, N*m]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
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%% Run the linearization
C = linearize(mdl, io, options);
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C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
</pre>
</div>
<p>
If wanted, the 6x6 transmissibility matrix is plotted.
</p>
<div class="org-src-container">
2020-08-05 13:28:14 +02:00
<pre class="src src-matlab">p_handle = zeros(6*6,1);
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if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
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plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix &lt; 6
xticklabels({});
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end
if iy &gt; 1
yticklabels({});
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end
end
end
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linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
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han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Compliance [m/N, rad/(N*m)]');
end
</pre>
</div>
</div>
</div>
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<div id="outline-container-org45205c2" class="outline-4">
<h4 id="org45205c2">Computation of the Frobenius norm</h4>
<div class="outline-text-4" id="text-org45205c2">
<div class="org-src-container">
<pre class="src src-matlab">freqs = args.freqs;
C_norm = zeros(length(freqs), 1);
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for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
</pre>
</div>
<div class="org-src-container">
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<pre class="src src-matlab">if args.plots
figure;
plot(freqs, C_norm)
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Compliance - Frobenius Norm');
end
</pre>
</div>
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
<div class="csl-bib-body">
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” <i>Journal of Sound and Vibration</i> 300 (3-5):64461. <a href="https://doi.org/10.1016/j.jsv.2006.07.050">https://doi.org/10.1016/j.jsv.2006.07.050</a>.</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
2020-01-29 17:52:04 +01:00
<p class="author">Author: Dehaeze Thomas</p>
2020-08-05 13:28:14 +02:00
<p class="date">Created: 2020-08-05 mer. 13:27</p>
</div>
</body>
</html>