Transmissibility and Compliance comp IFF/DVF/OL
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@@ -277,13 +255,58 @@ for the JavaScript code in this tag.
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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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<li><a href="#org2891722">2. Transmissibility Analysis</a>
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<ul>
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<li><a href="#org8c667e9">2.1. Initialize the Stewart platform</a></li>
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||||
<li><a href="#org5338f20">2.2. Transmissibility</a></li>
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</ul>
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</li>
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<li><a href="#orgc94edbd">3. Compliance Analysis</a>
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<ul>
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<li><a href="#orgc8e1f51">3.1. Initialize the Stewart platform</a></li>
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<li><a href="#org1177029">3.2. Compliance</a></li>
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</ul>
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</li>
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<li><a href="#org68ca336">4. Functions</a>
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<ul>
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<li><a href="#org487c4d4">4.1. Compute the Transmissibility</a>
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<ul>
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<li><a href="#org851f84d">Function description</a></li>
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<li><a href="#orgf5e24cd">Optional Parameters</a></li>
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<li><a href="#org4629501">Identification of the Transmissibility Matrix</a></li>
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<li><a href="#org989379a">Computation of the Frobenius norm</a></li>
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</ul>
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</li>
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<li><a href="#org50e35a6">4.2. Compute the Compliance</a>
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<ul>
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<li><a href="#org64fc1e2">Function description</a></li>
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<li><a href="#org54cab00">Optional Parameters</a></li>
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<li><a href="#orgef06b63">Identification of the Compliance Matrix</a></li>
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<li><a href="#org6f63d37">Computation of the Frobenius norm</a></li>
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</ul>
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</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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<p>
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In this document, we discuss the various methods to identify the behavior of the Stewart platform.
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</p>
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<ul class="org-ul">
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<li><a href="#org7981e88">1</a></li>
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<li><a href="#orga989615">2</a></li>
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<li><a href="#org4579374">3</a></li>
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</ul>
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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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<p>
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<a id="org7981e88"></a>
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</p>
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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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@@ -577,10 +600,521 @@ Save the movie of the mode shape.
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</div>
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</div>
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</div>
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<div id="outline-container-org2891722" class="outline-2">
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<h2 id="org2891722"><span class="section-number-2">2</span> Transmissibility Analysis</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orga989615"></a>
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</p>
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</div>
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<div id="outline-container-org8c667e9" class="outline-3">
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<h3 id="org8c667e9"><span class="section-number-3">2.1</span> Initialize the Stewart platform</h3>
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<div class="outline-text-3" id="text-2-1">
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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, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
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stewart = 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, <span class="org-string">'type'</span>, <span class="org-string">'accelerometer'</span>, <span class="org-string">'freq'</span>, 5e3);
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</pre>
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</div>
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<p>
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We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
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</p>
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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">'rigid'</span>, <span class="org-string">'rot_point'</span>, stewart.platform_F.FO_A);
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payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</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-org5338f20" class="outline-3">
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<h3 id="org5338f20"><span class="section-number-3">2.2</span> Transmissibility</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"><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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||||
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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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||||
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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">'/Disturbances/D_w'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
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||||
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
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||||
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||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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||||
T = linearize(mdl, io, options);
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||||
T.InputName = {<span class="org-string">'Wdx'</span>, <span class="org-string">'Wdy'</span>, <span class="org-string">'Wdz'</span>, <span class="org-string">'Wrx'</span>, <span class="org-string">'Wry'</span>, <span class="org-string">'Wrz'</span>};
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||||
T.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
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||||
</pre>
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||||
</div>
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||||
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<div class="org-src-container">
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||||
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
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||||
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<span class="org-type">figure</span>;
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||||
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
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||||
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
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||||
subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
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||||
hold on;
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||||
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
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||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
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||||
ylim([1e<span class="org-type">-</span>5, 10]);
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||||
xlim([freqs(1), freqs(end)]);
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||||
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
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xticklabels({});
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||||
<span class="org-keyword">end</span>
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||||
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
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yticklabels({});
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<span class="org-keyword">end</span>
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||||
<span class="org-keyword">end</span>
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||||
<span class="org-keyword">end</span>
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||||
</pre>
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||||
</div>
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||||
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||||
<p>
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||||
From <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>, one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
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||||
</p>
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||||
\begin{align*}
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||||
\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
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||||
&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
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||||
\end{align*}
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||||
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||||
<div class="org-src-container">
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||||
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
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||||
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||||
T_norm = zeros(length(freqs), 1);
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||||
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||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
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||||
T_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
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||||
<span class="org-keyword">end</span>
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||||
</pre>
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||||
</div>
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||||
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||||
<p>
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||||
And we normalize by a factor \(\sqrt{6}\) to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
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||||
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
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||||
</p>
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||||
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||||
<div class="org-src-container">
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||||
<pre class="src src-matlab">Gamma = T_norm<span class="org-type">/</span>sqrt(6);
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||||
</pre>
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||||
</div>
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||||
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||||
<div class="org-src-container">
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||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
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||||
plot(freqs, Gamma)
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||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</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>
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||||
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||||
<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>
|
||||
<div id="outline-container-orgc8e1f51" class="outline-3">
|
||||
<h3 id="orgc8e1f51"><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();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'accelerometer'</span>, <span class="org-string">'freq'</span>, 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">
|
||||
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
</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">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances/F_ext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
C = linearize(mdl, io, options);
|
||||
C.InputName = {<span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
|
||||
C.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
|
||||
subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||||
ylim([1e<span class="org-type">-</span>10, 1e<span class="org-type">-</span>3]);
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
|
||||
xticklabels({});
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
|
||||
yticklabels({});
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
</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);
|
||||
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
|
||||
C_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||
plot(freqs, C_norm)
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||||
</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>
|
||||
|
||||
<div id="outline-container-org851f84d" class="outline-4">
|
||||
<h4 id="org851f84d">Function description</h4>
|
||||
<div class="outline-text-4" id="text-org851f84d">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[T, T_norm, freqs]</span> = <span class="org-function-name">computeTransmissibility</span>(<span class="org-variable-name">args</span>)
|
||||
<span class="org-comment">% computeTransmissibility -</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Inputs:</span>
|
||||
<span class="org-comment">% - args - Structure with the following fields:</span>
|
||||
<span class="org-comment">% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm</span>
|
||||
<span class="org-comment">% - freqs [] - Frequency vector to estimate the Frobenius norm</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Outputs:</span>
|
||||
<span class="org-comment">% - T [6x6 ss] - Transmissibility matrix</span>
|
||||
<span class="org-comment">% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix</span>
|
||||
<span class="org-comment">% - freqs [length(freqs)x1] - Frequency vector in [Hz]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf5e24cd" class="outline-4">
|
||||
<h4 id="orgf5e24cd">Optional Parameters</h4>
|
||||
<div class="outline-text-4" id="text-orgf5e24cd">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">arguments
|
||||
args.plots logical {mustBeNumericOrLogical} = <span class="org-constant">false</span>
|
||||
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
|
||||
<span class="org-keyword">end</span>
|
||||
</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">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances/D_w'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
T = linearize(mdl, io, options);
|
||||
T.InputName = {<span class="org-string">'Wdx'</span>, <span class="org-string">'Wdy'</span>, <span class="org-string">'Wdz'</span>, <span class="org-string">'Wrx'</span>, <span class="org-string">'Wry'</span>, <span class="org-string">'Wrz'</span>};
|
||||
T.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
If wanted, the 6x6 transmissibility matrix is plotted.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">p_handle = zeros(6<span class="org-type">*</span>6,1);
|
||||
|
||||
<span class="org-keyword">if</span> args.plots
|
||||
fig = <span class="org-type">figure</span>;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
|
||||
p_handle((ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy) = subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||||
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
|
||||
xticklabels({});
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
|
||||
yticklabels({});
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
|
||||
linkaxes(p_handle, <span class="org-string">'xy'</span>)
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e<span class="org-type">-</span>5, 1e2]);
|
||||
|
||||
han = <span class="org-type">axes</span>(fig, <span class="org-string">'visible'</span>, <span class="org-string">'off'</span>);
|
||||
han.XLabel.Visible = <span class="org-string">'on'</span>;
|
||||
han.YLabel.Visible = <span class="org-string">'on'</span>;
|
||||
ylabel(han, <span class="org-string">'Frequency [Hz]'</span>);
|
||||
xlabel(han, <span class="org-string">'Transmissibility [m/m]'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org989379a" class="outline-4">
|
||||
<h4 id="org989379a">Computation of the Frobenius norm</h4>
|
||||
<div class="outline-text-4" id="text-org989379a">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">T_norm = zeros(length(freqs), 1);
|
||||
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
|
||||
T_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">T_norm = T_norm<span class="org-type">/</span>sqrt(6);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">if</span> args.plots
|
||||
<span class="org-type">figure</span>;
|
||||
plot(freqs, T_norm)
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
|
||||
ylabel(<span class="org-string">'Transmissibility - Frobenius Norm'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</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>
|
||||
|
||||
<div id="outline-container-org64fc1e2" class="outline-4">
|
||||
<h4 id="org64fc1e2">Function description</h4>
|
||||
<div class="outline-text-4" id="text-org64fc1e2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[C, C_norm, freqs]</span> = <span class="org-function-name">computeCompliance</span>(<span class="org-variable-name">args</span>)
|
||||
<span class="org-comment">% computeCompliance -</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Syntax: [C, C_norm, freqs] = computeCompliance(args)</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Inputs:</span>
|
||||
<span class="org-comment">% - args - Structure with the following fields:</span>
|
||||
<span class="org-comment">% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm</span>
|
||||
<span class="org-comment">% - freqs [] - Frequency vector to estimate the Frobenius norm</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Outputs:</span>
|
||||
<span class="org-comment">% - C [6x6 ss] - Compliance matrix</span>
|
||||
<span class="org-comment">% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix</span>
|
||||
<span class="org-comment">% - freqs [length(freqs)x1] - Frequency vector in [Hz]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org54cab00" class="outline-4">
|
||||
<h4 id="org54cab00">Optional Parameters</h4>
|
||||
<div class="outline-text-4" id="text-org54cab00">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">arguments
|
||||
args.plots logical {mustBeNumericOrLogical} = <span class="org-constant">false</span>
|
||||
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
|
||||
<span class="org-keyword">end</span>
|
||||
</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">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances/F_ext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces [N, N*m]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
C = linearize(mdl, io, options);
|
||||
C.InputName = {<span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
|
||||
C.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
If wanted, the 6x6 transmissibility matrix is plotted.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">p_handle = zeros(6<span class="org-type">*</span>6,1);
|
||||
|
||||
<span class="org-keyword">if</span> args.plots
|
||||
fig = <span class="org-type">figure</span>;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
|
||||
p_handle((ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy) = subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||||
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
|
||||
xticklabels({});
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
|
||||
yticklabels({});
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
|
||||
linkaxes(p_handle, <span class="org-string">'xy'</span>)
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
han = <span class="org-type">axes</span>(fig, <span class="org-string">'visible'</span>, <span class="org-string">'off'</span>);
|
||||
han.XLabel.Visible = <span class="org-string">'on'</span>;
|
||||
han.YLabel.Visible = <span class="org-string">'on'</span>;
|
||||
xlabel(han, <span class="org-string">'Frequency [Hz]'</span>);
|
||||
ylabel(han, <span class="org-string">'Compliance [m/N, rad/(N*m)]'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6f63d37" class="outline-4">
|
||||
<h4 id="org6f63d37">Computation of the Frobenius norm</h4>
|
||||
<div class="outline-text-4" id="text-org6f63d37">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = args.freqs;
|
||||
|
||||
C_norm = zeros(length(freqs), 1);
|
||||
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
|
||||
C_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">if</span> args.plots
|
||||
<span class="org-type">figure</span>;
|
||||
plot(freqs, C_norm)
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
|
||||
ylabel(<span class="org-string">'Compliance - Frobenius Norm'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</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>
|
||||
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user