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Better Matlab notations

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Thomas Dehaeze
2020-11-09 14:37:04 +01:00
parent 292ba73fb1
commit e97a3d58ab
25 changed files with 1786 additions and 1666 deletions
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<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org8480cb8">1. Gravimeter - Simscape Model</a> <li><a href="#org4262bdd">1. Gravimeter - Simscape Model</a>
<ul> <ul>
<li><a href="#org669566a">1.1. Introduction</a></li> <li><a href="#org46fc636">1.1. Introduction</a></li>
<li><a href="#org513c970">1.2. Simscape Model - Parameters</a></li> <li><a href="#org137a1ed">1.2. Simscape Model - Parameters</a></li>
<li><a href="#org7e0371b">1.3. System Identification - Without Gravity</a></li> <li><a href="#org08acfbd">1.3. System Identification - Without Gravity</a></li>
<li><a href="#org9dfc541">1.4. System Identification - With Gravity</a></li> <li><a href="#orge4c219d">1.4. System Identification - With Gravity</a></li>
<li><a href="#org06067ff">1.5. Analytical Model</a> <li><a href="#org744c6c9">1.5. Analytical Model</a>
<ul> <ul>
<li><a href="#org063c200">1.5.1. Parameters</a></li> <li><a href="#orga42f590">1.5.1. Parameters</a></li>
<li><a href="#orgec24c80">1.5.2. Generation of the State Space Model</a></li> <li><a href="#org288ddf0">1.5.2. Generation of the State Space Model</a></li>
<li><a href="#org1590891">1.5.3. Comparison with the Simscape Model</a></li> <li><a href="#orgcd68a21">1.5.3. Comparison with the Simscape Model</a></li>
<li><a href="#orgb615e54">1.5.4. Analysis</a></li> <li><a href="#orga3239b9">1.5.4. Analysis</a></li>
<li><a href="#org2243155">1.5.5. Control Section</a></li> <li><a href="#orgda0f1ad">1.5.5. Control Section</a></li>
<li><a href="#orgd28ecdb">1.5.6. Greshgorin radius</a></li> <li><a href="#org7ffae54">1.5.6. Greshgorin radius</a></li>
<li><a href="#org24f83eb">1.5.7. Injecting ground motion in the system to have the output</a></li> <li><a href="#org72dd1a0">1.5.7. Injecting ground motion in the system to have the output</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgc1560cf">2. Gravimeter - Functions</a> <li><a href="#org01bdedf">2. Gravimeter - Functions</a>
<ul> <ul>
<li><a href="#org814c22c">2.1. <code>align</code></a></li> <li><a href="#org4647e37">2.1. <code>align</code></a></li>
<li><a href="#orga936ee3">2.2. <code>pzmap_testCL</code></a></li> <li><a href="#orga0981c0">2.2. <code>pzmap_testCL</code></a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgf783e5e">3. Stewart Platform - Simscape Model</a> <li><a href="#orgd6f892a">3. Stewart Platform - Simscape Model</a>
<ul> <ul>
<li><a href="#org698a574">3.1. Simscape Model - Parameters</a></li> <li><a href="#org98f27a1">3.1. Simscape Model - Parameters</a></li>
<li><a href="#orgdfc6136">3.2. Identification of the plant</a></li> <li><a href="#orgfc4057f">3.2. Identification of the plant</a></li>
<li><a href="#orgadaff5c">3.3. Physical Decoupling using the Jacobian</a></li> <li><a href="#org06bff3b">3.3. Physical Decoupling using the Jacobian</a></li>
<li><a href="#org6ba1c1a">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li> <li><a href="#org7208fcb">3.4. Real Approximation of \(G\) at the decoupling frequency</a></li>
<li><a href="#org7e2a42e">3.5. SVD Decoupling</a></li> <li><a href="#orgdcfefc4">3.5. SVD Decoupling</a></li>
<li><a href="#orgc6f3016">3.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li> <li><a href="#orgeedb4ac">3.6. Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</a></li>
<li><a href="#orgedf5c94">3.7. Obtained Decoupled Plants</a></li> <li><a href="#orga3edea8">3.7. Obtained Decoupled Plants</a></li>
<li><a href="#orgff44b51">3.8. Diagonal Controller</a></li> <li><a href="#orgb371cb1">3.8. Diagonal Controller</a></li>
<li><a href="#org949d9ca">3.9. Closed-Loop system Performances</a></li> <li><a href="#orgb6d90eb">3.9. Closed-Loop system Performances</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org8480cb8" class="outline-2"> <div id="outline-container-org4262bdd" class="outline-2">
<h2 id="org8480cb8"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2> <h2 id="org4262bdd"><span class="section-number-2">1</span> Gravimeter - Simscape Model</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org669566a" class="outline-3"> <div id="outline-container-org46fc636" class="outline-3">
<h3 id="org669566a"><span class="section-number-3">1.1</span> Introduction</h3> <h3 id="org46fc636"><span class="section-number-3">1.1</span> Introduction</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<div id="org1f9eedf" class="figure"> <div id="orgca5b956" class="figure">
<p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" /> <p><img src="figs/gravimeter_model.png" alt="gravimeter_model.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p> <p><span class="figure-number">Figure 1: </span>Model of the gravimeter</p>
@@ -93,8 +93,8 @@
</div> </div>
</div> </div>
<div id="outline-container-org513c970" class="outline-3"> <div id="outline-container-org137a1ed" class="outline-3">
<h3 id="org513c970"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3> <h3 id="org137a1ed"><span class="section-number-3">1.2</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>) <pre class="src src-matlab">open(<span class="org-string">'gravimeter.slx'</span>)
@@ -125,8 +125,8 @@ g = 0; <span class="org-comment">% Gravity [m/s2]</span>
</div> </div>
</div> </div>
<div id="outline-container-org7e0371b" class="outline-3"> <div id="outline-container-org08acfbd" class="outline-3">
<h3 id="org7e0371b"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3> <h3 id="org08acfbd"><span class="section-number-3">1.3</span> System Identification - Without Gravity</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
@@ -148,7 +148,7 @@ G.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string">
</pre> </pre>
</div> </div>
<pre class="example" id="org6eb6401"> <pre class="example" id="org2c5c71d">
pole(G) pole(G)
ans = ans =
-0.000473481142385795 + 21.7596190728632i -0.000473481142385795 + 21.7596190728632i
@@ -173,7 +173,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
<div id="org3874001" class="figure"> <div id="orgddb1793" class="figure">
<p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" /> <p><img src="figs/open_loop_tf.png" alt="open_loop_tf.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p> <p><span class="figure-number">Figure 2: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers</p>
@@ -181,8 +181,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org9dfc541" class="outline-3"> <div id="outline-container-orge4c219d" class="outline-3">
<h3 id="org9dfc541"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3> <h3 id="orge4c219d"><span class="section-number-3">1.4</span> System Identification - With Gravity</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span> <pre class="src src-matlab">g = 9.80665; <span class="org-comment">% Gravity [m/s2]</span>
@@ -199,7 +199,7 @@ Gg.OutputName = {<span class="org-string">'Ax1'</span>, <span class="org-string"
<p> <p>
We can now see that the system is unstable due to gravity. We can now see that the system is unstable due to gravity.
</p> </p>
<pre class="example" id="orga5f4271"> <pre class="example" id="org78beae2">
pole(Gg) pole(Gg)
ans = ans =
-10.9848275341252 + 0i -10.9848275341252 + 0i
@@ -211,7 +211,7 @@ ans =
</pre> </pre>
<div id="org5e8aee0" class="figure"> <div id="org70961c1" class="figure">
<p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" /> <p><img src="figs/open_loop_tf_g.png" alt="open_loop_tf_g.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p> <p><span class="figure-number">Figure 3: </span>Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity</p>
@@ -219,12 +219,12 @@ ans =
</div> </div>
</div> </div>
<div id="outline-container-org06067ff" class="outline-3"> <div id="outline-container-org744c6c9" class="outline-3">
<h3 id="org06067ff"><span class="section-number-3">1.5</span> Analytical Model</h3> <h3 id="org744c6c9"><span class="section-number-3">1.5</span> Analytical Model</h3>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-3" id="text-1-5">
</div> </div>
<div id="outline-container-org063c200" class="outline-4"> <div id="outline-container-orga42f590" class="outline-4">
<h4 id="org063c200"><span class="section-number-4">1.5.1</span> Parameters</h4> <h4 id="orga42f590"><span class="section-number-4">1.5.1</span> Parameters</h4>
<div class="outline-text-4" id="text-1-5-1"> <div class="outline-text-4" id="text-1-5-1">
<p> <p>
Bode options. Bode options.
@@ -256,8 +256,8 @@ Frequency vector.
</div> </div>
</div> </div>
<div id="outline-container-orgec24c80" class="outline-4"> <div id="outline-container-org288ddf0" class="outline-4">
<h4 id="orgec24c80"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4> <h4 id="org288ddf0"><span class="section-number-4">1.5.2</span> Generation of the State Space Model</h4>
<div class="outline-text-4" id="text-1-5-2"> <div class="outline-text-4" id="text-1-5-2">
<p> <p>
Mass matrix Mass matrix
@@ -358,11 +358,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org1590891" class="outline-4"> <div id="outline-container-orgcd68a21" class="outline-4">
<h4 id="org1590891"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4> <h4 id="orgcd68a21"><span class="section-number-4">1.5.3</span> Comparison with the Simscape Model</h4>
<div class="outline-text-4" id="text-1-5-3"> <div class="outline-text-4" id="text-1-5-3">
<div id="orgfa66619" class="figure"> <div id="orgacf77cc" class="figure">
<p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" /> <p><img src="figs/gravimeter_analytical_system_open_loop_models.png" alt="gravimeter_analytical_system_open_loop_models.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p> <p><span class="figure-number">Figure 4: </span>Comparison of the analytical and the Simscape models</p>
@@ -370,8 +370,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-orgb615e54" class="outline-4"> <div id="outline-container-orga3239b9" class="outline-4">
<h4 id="orgb615e54"><span class="section-number-4">1.5.4</span> Analysis</h4> <h4 id="orga3239b9"><span class="section-number-4">1.5.4</span> Analysis</h4>
<div class="outline-text-4" id="text-1-5-4"> <div class="outline-text-4" id="text-1-5-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% figure</span> <pre class="src src-matlab"><span class="org-comment">% figure</span>
@@ -439,8 +439,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
</div> </div>
</div> </div>
<div id="outline-container-org2243155" class="outline-4"> <div id="outline-container-orgda0f1ad" class="outline-4">
<h4 id="org2243155"><span class="section-number-4">1.5.5</span> Control Section</h4> <h4 id="orgda0f1ad"><span class="section-number-4">1.5.5</span> Control Section</h4>
<div class="outline-text-4" id="text-1-5-5"> <div class="outline-text-4" id="text-1-5-5">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10); <pre class="src src-matlab">system_dec_10Hz = freqresp(system_dec,2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10);
@@ -580,8 +580,8 @@ legend(<span class="org-string">'Control OFF'</span>,<span class="org-string">'D
</div> </div>
</div> </div>
<div id="outline-container-orgd28ecdb" class="outline-4"> <div id="outline-container-org7ffae54" class="outline-4">
<h4 id="orgd28ecdb"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4> <h4 id="org7ffae54"><span class="section-number-4">1.5.6</span> Greshgorin radius</h4>
<div class="outline-text-4" id="text-1-5-6"> <div class="outline-text-4" id="text-1-5-6">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w); <pre class="src src-matlab">system_dec_freq = freqresp(system_dec,w);
@@ -627,8 +627,8 @@ ylabel(<span class="org-string">'Greshgorin radius [-]'</span>);
</div> </div>
</div> </div>
<div id="outline-container-org24f83eb" class="outline-4"> <div id="outline-container-org72dd1a0" class="outline-4">
<h4 id="org24f83eb"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4> <h4 id="org72dd1a0"><span class="section-number-4">1.5.7</span> Injecting ground motion in the system to have the output</h4>
<div class="outline-text-4" id="text-1-5-7"> <div class="outline-text-4" id="text-1-5-7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3); <pre class="src src-matlab">Fr = logspace(<span class="org-type">-</span>2,3,1e3);
@@ -684,15 +684,15 @@ rot = PHI(<span class="org-type">:</span>,11,11);
</div> </div>
</div> </div>
<div id="outline-container-orgc1560cf" class="outline-2"> <div id="outline-container-org01bdedf" class="outline-2">
<h2 id="orgc1560cf"><span class="section-number-2">2</span> Gravimeter - Functions</h2> <h2 id="org01bdedf"><span class="section-number-2">2</span> Gravimeter - Functions</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
</div> </div>
<div id="outline-container-org814c22c" class="outline-3"> <div id="outline-container-org4647e37" class="outline-3">
<h3 id="org814c22c"><span class="section-number-3">2.1</span> <code>align</code></h3> <h3 id="org4647e37"><span class="section-number-3">2.1</span> <code>align</code></h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
<a id="org3643797"></a> <a id="org787b0b4"></a>
</p> </p>
<p> <p>
@@ -721,11 +721,11 @@ This Matlab function is accessible <a href="gravimeter/align.m">here</a>.
</div> </div>
<div id="outline-container-orga936ee3" class="outline-3"> <div id="outline-container-orga0981c0" class="outline-3">
<h3 id="orga936ee3"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3> <h3 id="orga0981c0"><span class="section-number-3">2.2</span> <code>pzmap_testCL</code></h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
<a id="org7c6bace"></a> <a id="org6adb39c"></a>
</p> </p>
<p> <p>
@@ -774,11 +774,11 @@ This Matlab function is accessible <a href="gravimeter/pzmap_testCL.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-orgf783e5e" class="outline-2"> <div id="outline-container-orgd6f892a" class="outline-2">
<h2 id="orgf783e5e"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2> <h2 id="orgd6f892a"><span class="section-number-2">3</span> Stewart Platform - Simscape Model</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org325e3af">5</a>. In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure <a href="#org2113119">5</a>.
</p> </p>
<p> <p>
@@ -791,7 +791,7 @@ Some notes about the system:
</ul> </ul>
<div id="org325e3af" class="figure"> <div id="org2113119" class="figure">
<p><img src="figs/SP_assembly.png" alt="SP_assembly.png" /> <p><img src="figs/SP_assembly.png" alt="SP_assembly.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Stewart Platform CAD View</p> <p><span class="figure-number">Figure 5: </span>Stewart Platform CAD View</p>
@@ -801,22 +801,22 @@ Some notes about the system:
The analysis of the SVD control applied to the Stewart platform is performed in the following sections: The analysis of the SVD control applied to the Stewart platform is performed in the following sections:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org46c7682">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li> <li>Section <a href="#org9eff470">3.1</a>: The parameters of the Simscape model of the Stewart platform are defined</li>
<li>Section <a href="#orgacf8e97">3.2</a>: The plant is identified from the Simscape model and the system coupling is shown</li> <li>Section <a href="#orgb8efc36">3.2</a>: The plant is identified from the Simscape model and the system coupling is shown</li>
<li>Section <a href="#orgf5489c1">3.3</a>: The plant is first decoupled using the Jacobian</li> <li>Section <a href="#org9d45510">3.3</a>: The plant is first decoupled using the Jacobian</li>
<li>Section <a href="#orgccd7599">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li> <li>Section <a href="#orgbe757a9">3.4</a>: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)</li>
<li>Section <a href="#org5abe937">3.5</a>: The decoupling is performed thanks to the SVD</li> <li>Section <a href="#orgb593bce">3.5</a>: The decoupling is performed thanks to the SVD</li>
<li>Section <a href="#org7691f01">3.6</a>: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii</li> <li>Section <a href="#org9c68bed">3.6</a>: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii</li>
<li>Section <a href="#orgabc21ab">3.7</a>: The dynamics of the decoupled plants are shown</li> <li>Section <a href="#orgc065295">3.7</a>: The dynamics of the decoupled plants are shown</li>
<li>Section <a href="#org9620c1c">3.8</a>: A diagonal controller is defined to control the decoupled plant</li> <li>Section <a href="#orgaf53d60">3.8</a>: A diagonal controller is defined to control the decoupled plant</li>
<li>Section <a href="#org823e1cb">3.9</a>: Finally, the closed loop system properties are studied</li> <li>Section <a href="#org60a86ad">3.9</a>: Finally, the closed loop system properties are studied</li>
</ul> </ul>
</div> </div>
<div id="outline-container-org698a574" class="outline-3"> <div id="outline-container-org98f27a1" class="outline-3">
<h3 id="org698a574"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3> <h3 id="org98f27a1"><span class="section-number-3">3.1</span> Simscape Model - Parameters</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
<a id="org46c7682"></a> <a id="org9eff470"></a>
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>); <pre class="src src-matlab">open(<span class="org-string">'drone_platform.slx'</span>);
@@ -864,14 +864,14 @@ Kc = tf(zeros(6));
</div> </div>
<div id="org48cc1aa" class="figure"> <div id="orgf541900" class="figure">
<p><img src="figs/stewart_simscape.png" alt="stewart_simscape.png" /> <p><img src="figs/stewart_simscape.png" alt="stewart_simscape.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>General view of the Simscape Model</p> <p><span class="figure-number">Figure 6: </span>General view of the Simscape Model</p>
</div> </div>
<div id="orgd93f514" class="figure"> <div id="orge4629ec" class="figure">
<p><img src="figs/stewart_platform_details.png" alt="stewart_platform_details.png" /> <p><img src="figs/stewart_platform_details.png" alt="stewart_platform_details.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Simscape model of the Stewart platform</p> <p><span class="figure-number">Figure 7: </span>Simscape model of the Stewart platform</p>
@@ -879,15 +879,15 @@ Kc = tf(zeros(6));
</div> </div>
</div> </div>
<div id="outline-container-orgdfc6136" class="outline-3"> <div id="outline-container-orgfc4057f" class="outline-3">
<h3 id="orgdfc6136"><span class="section-number-3">3.2</span> Identification of the plant</h3> <h3 id="orgfc4057f"><span class="section-number-3">3.2</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
<a id="orgacf8e97"></a> <a id="orgb8efc36"></a>
</p> </p>
<p> <p>
The plant shown in Figure <a href="#org6611cbe">8</a> is identified from the Simscape model. The plant shown in Figure <a href="#orge3a32c6">8</a> is identified from the Simscape model.
</p> </p>
<p> <p>
@@ -903,10 +903,10 @@ The outputs are the 6 accelerations measured by the inertial unit.
</p> </p>
<div id="org6611cbe" class="figure"> <div id="orge3a32c6" class="figure">
<p><img src="figs/stewart_platform_plant.png" alt="stewart_platform_plant.png" /> <p><img src="figs/stewart_platform_plant.png" alt="stewart_platform_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform</p> <p><span class="figure-number">Figure 8: </span>Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform</p>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
@@ -923,6 +923,11 @@ G = linearize(mdl, io);
G.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>, ... G.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>, ...
<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}; <span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
G.OutputName = {<span class="org-string">'Ax'</span>, <span class="org-string">'Ay'</span>, <span class="org-string">'Az'</span>, <span class="org-string">'Arx'</span>, <span class="org-string">'Ary'</span>, <span class="org-string">'Arz'</span>}; G.OutputName = {<span class="org-string">'Ax'</span>, <span class="org-string">'Ay'</span>, <span class="org-string">'Az'</span>, <span class="org-string">'Arx'</span>, <span class="org-string">'Ary'</span>, <span class="org-string">'Arz'</span>};
<span class="org-comment">% Plant</span>
Gu = G(<span class="org-type">:</span>, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>});
<span class="org-comment">% Disturbance dynamics</span>
Gd = G(<span class="org-type">:</span>, {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>});
</pre> </pre>
</div> </div>
@@ -940,7 +945,7 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
<p> <p>
The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure <a href="#org3e2e269">9</a>. The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure <a href="#org602aa13">9</a>.
</p> </p>
<p> <p>
@@ -948,25 +953,25 @@ One can easily see that the system is strongly coupled.
</p> </p>
<div id="org3e2e269" class="figure"> <div id="org602aa13" class="figure">
<p><img src="figs/stewart_platform_coupled_plant.png" alt="stewart_platform_coupled_plant.png" /> <p><img src="figs/stewart_platform_coupled_plant.png" alt="stewart_platform_coupled_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Magnitude of all 36 elements of the transfer function matrix \(\bm{G}\)</p> <p><span class="figure-number">Figure 9: </span>Magnitude of all 36 elements of the transfer function matrix \(G_u\)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgadaff5c" class="outline-3"> <div id="outline-container-org06bff3b" class="outline-3">
<h3 id="orgadaff5c"><span class="section-number-3">3.3</span> Physical Decoupling using the Jacobian</h3> <h3 id="org06bff3b"><span class="section-number-3">3.3</span> Physical Decoupling using the Jacobian</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
<a id="orgf5489c1"></a> <a id="org9d45510"></a>
Consider the control architecture shown in Figure <a href="#orgeef0f77">10</a>. Consider the control architecture shown in Figure <a href="#org1c673db">10</a>.
The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.
</p> </p>
<div id="orgeef0f77" class="figure"> <div id="org1c673db" class="figure">
<p><img src="figs/plant_decouple_jacobian.png" alt="plant_decouple_jacobian.png" /> <p><img src="figs/plant_decouple_jacobian.png" alt="plant_decouple_jacobian.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)</p> <p><span class="figure-number">Figure 10: </span>Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)</p>
@@ -982,31 +987,27 @@ We define a new plant:
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Gx = G<span class="org-type">*</span>blkdiag(eye(6), inv(J<span class="org-type">'</span>)); <pre class="src src-matlab">Gx = Gu<span class="org-type">*</span>inv(J<span class="org-type">'</span>);
Gx.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Rwx'</span>, <span class="org-string">'Rwy'</span>, <span class="org-string">'Rwz'</span>, ... Gx.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
</pre> </pre>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org6ba1c1a" class="outline-3"> <div id="outline-container-org7208fcb" class="outline-3">
<h3 id="org6ba1c1a"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3> <h3 id="org7208fcb"><span class="section-number-3">3.4</span> Real Approximation of \(G\) at the decoupling frequency</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
<a id="orgccd7599"></a> <a id="orgbe757a9"></a>
</p> </p>
<p> <p>
Let&rsquo;s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\). Let&rsquo;s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_u(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30; <span class="org-comment">% Decoupling frequency [rad/s]</span> <pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30; <span class="org-comment">% Decoupling frequency [rad/s]</span>
Gc = G({<span class="org-string">'Ax'</span>, <span class="org-string">'Ay'</span>, <span class="org-string">'Az'</span>, <span class="org-string">'Arx'</span>, <span class="org-string">'Ary'</span>, <span class="org-string">'Arz'</span>}, ... H1 = evalfr(Gu, <span class="org-constant">j</span><span class="org-type">*</span>wc);
{<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}); <span class="org-comment">% Transfer function to find a real approximation</span>
H1 = evalfr(Gc, <span class="org-constant">j</span><span class="org-type">*</span>wc);
</pre> </pre>
</div> </div>
@@ -1094,9 +1095,9 @@ H1 = inv(D<span class="org-type">*</span>real(H1<span class="org-type">'*</span>
<p> <p>
Note that the plant \(G\) at \(\omega_c\) is already an almost real matrix. Note that the plant \(G_u\) at \(\omega_c\) is already an almost real matrix.
This can be seen on the Bode plots where the phase is close to 1. This can be seen on the Bode plots where the phase is close to 1.
This can be verified below where only the real value of \(G(\omega_c)\) is shown This can be verified below where only the real value of \(G_u(\omega_c)\) is shown
</p> </p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@@ -1174,11 +1175,11 @@ This can be verified below where only the real value of \(G(\omega_c)\) is shown
</div> </div>
</div> </div>
<div id="outline-container-org7e2a42e" class="outline-3"> <div id="outline-container-orgdcfefc4" class="outline-3">
<h3 id="org7e2a42e"><span class="section-number-3">3.5</span> SVD Decoupling</h3> <h3 id="orgdcfefc4"><span class="section-number-3">3.5</span> SVD Decoupling</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-3-5">
<p> <p>
<a id="org5abe937"></a> <a id="orgb593bce"></a>
</p> </p>
<p> <p>
@@ -1187,16 +1188,16 @@ First, the Singular Value Decomposition of \(H_1\) is performed:
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">[U,S,V] = svd(H1); <pre class="src src-matlab">[U,<span class="org-type">~</span>,V] = svd(H1);
</pre> </pre>
</div> </div>
<p> <p>
The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure <a href="#orga151aa3">11</a>. The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure <a href="#orgfbe015c">11</a>.
</p> </p>
<div id="orga151aa3" class="figure"> <div id="orgfbe015c" class="figure">
<p><img src="figs/plant_decouple_svd.png" alt="plant_decouple_svd.png" /> <p><img src="figs/plant_decouple_svd.png" alt="plant_decouple_svd.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition</p> <p><span class="figure-number">Figure 11: </span>Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition</p>
@@ -1204,22 +1205,32 @@ The obtained matrices \(U\) and \(V\) are used to decouple the system as shown i
<p> <p>
The decoupled plant is then: The decoupled plant is then:
\[ G_{SVD}(s) = U^{-1} G(s) V^{-H} \] \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
</p> </p>
<div class="org-src-container">
<pre class="src src-matlab">Gsvd = inv(U)<span class="org-type">*</span>Gu<span class="org-type">*</span>inv(V<span class="org-type">'</span>);
</pre>
</div>
</div> </div>
</div> </div>
<div id="outline-container-orgc6f3016" class="outline-3"> <div id="outline-container-orgeedb4ac" class="outline-3">
<h3 id="orgc6f3016"><span class="section-number-3">3.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3> <h3 id="orgeedb4ac"><span class="section-number-3">3.6</span> Verification of the decoupling using the &ldquo;Gershgorin Radii&rdquo;</h3>
<div class="outline-text-3" id="text-3-6"> <div class="outline-text-3" id="text-3-6">
<p> <p>
<a id="org7691f01"></a> <a id="org9c68bed"></a>
</p> </p>
<p> <p>
The &ldquo;Gershgorin Radii&rdquo; is computed for the coupled plant \(G(s)\), for the &ldquo;Jacobian plant&rdquo; \(G_x(s)\) and the &ldquo;SVD Decoupled Plant&rdquo; \(G_{SVD}(s)\): The &ldquo;Gershgorin Radii&rdquo; is computed for the coupled plant \(G(s)\), for the &ldquo;Jacobian plant&rdquo; \(G_x(s)\) and the &ldquo;SVD Decoupled Plant&rdquo; \(G_{SVD}(s)\):
</p> </p>
<p>
The &ldquo;Gershgorin Radii&rdquo; of a matrix \(S\) is defined by:
\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
</p>
<p> <p>
This is computed over the following frequencies. This is computed over the following frequencies.
</p> </p>
@@ -1229,7 +1240,7 @@ This is computed over the following frequencies.
</div> </div>
<div id="orgea46431" class="figure"> <div id="org0864583" class="figure">
<p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" /> <p><img src="figs/simscape_model_gershgorin_radii.png" alt="simscape_model_gershgorin_radii.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Gershgorin Radii of the Coupled and Decoupled plants</p> <p><span class="figure-number">Figure 12: </span>Gershgorin Radii of the Coupled and Decoupled plants</p>
@@ -1237,30 +1248,30 @@ This is computed over the following frequencies.
</div> </div>
</div> </div>
<div id="outline-container-orgedf5c94" class="outline-3"> <div id="outline-container-orga3edea8" class="outline-3">
<h3 id="orgedf5c94"><span class="section-number-3">3.7</span> Obtained Decoupled Plants</h3> <h3 id="orga3edea8"><span class="section-number-3">3.7</span> Obtained Decoupled Plants</h3>
<div class="outline-text-3" id="text-3-7"> <div class="outline-text-3" id="text-3-7">
<p> <p>
<a id="orgabc21ab"></a> <a id="orgc065295"></a>
</p> </p>
<p> <p>
The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure <a href="#org0947ccc">13</a>. The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure <a href="#org57489d0">13</a>.
</p> </p>
<div id="org0947ccc" class="figure"> <div id="org57489d0" class="figure">
<p><img src="figs/simscape_model_decoupled_plant_svd.png" alt="simscape_model_decoupled_plant_svd.png" /> <p><img src="figs/simscape_model_decoupled_plant_svd.png" alt="simscape_model_decoupled_plant_svd.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Decoupled Plant using SVD</p> <p><span class="figure-number">Figure 13: </span>Decoupled Plant using SVD</p>
</div> </div>
<p> <p>
Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure <a href="#org14484c8">14</a>. Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure <a href="#orga4b3cd1">14</a>.
</p> </p>
<div id="org14484c8" class="figure"> <div id="orga4b3cd1" class="figure">
<p><img src="figs/simscape_model_decoupled_plant_jacobian.png" alt="simscape_model_decoupled_plant_jacobian.png" /> <p><img src="figs/simscape_model_decoupled_plant_jacobian.png" alt="simscape_model_decoupled_plant_jacobian.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)</p> <p><span class="figure-number">Figure 14: </span>Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)</p>
@@ -1268,15 +1279,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
</div> </div>
</div> </div>
<div id="outline-container-orgff44b51" class="outline-3"> <div id="outline-container-orgb371cb1" class="outline-3">
<h3 id="orgff44b51"><span class="section-number-3">3.8</span> Diagonal Controller</h3> <h3 id="orgb371cb1"><span class="section-number-3">3.8</span> Diagonal Controller</h3>
<div class="outline-text-3" id="text-3-8"> <div class="outline-text-3" id="text-3-8">
<p> <p>
<a id="org9620c1c"></a> <a id="orgaf53d60"></a>
</p> The control diagram for the centralized control is shown in Figure <a href="#org457c7b6">15</a>.
<p>
The control diagram for the centralized control is shown in Figure <a href="#org656626f">15</a>.
</p> </p>
<p> <p>
@@ -1285,19 +1293,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
</p> </p>
<div id="org656626f" class="figure"> <div id="org457c7b6" class="figure">
<p><img src="figs/centralized_control.png" alt="centralized_control.png" /> <p><img src="figs/centralized_control.png" alt="centralized_control.png" />
</p> </p>
<p><span class="figure-number">Figure 15: </span>Control Diagram for the Centralized control</p> <p><span class="figure-number">Figure 15: </span>Control Diagram for the Centralized control</p>
</div> </div>
<p> <p>
The SVD control architecture is shown in Figure <a href="#org0f3cfd0">16</a>. The SVD control architecture is shown in Figure <a href="#org84af546">16</a>.
The matrices \(U\) and \(V\) are used to decoupled the plant \(G\). The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
</p> </p>
<div id="org0f3cfd0" class="figure"> <div id="org84af546" class="figure">
<p><img src="figs/svd_control.png" alt="svd_control.png" /> <p><img src="figs/svd_control.png" alt="svd_control.png" />
</p> </p>
<p><span class="figure-number">Figure 16: </span>Control Diagram for the SVD control</p> <p><span class="figure-number">Figure 16: </span>Control Diagram for the SVD control</p>
@@ -1314,31 +1322,31 @@ We choose the controller to be a low pass filter:
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>80; <pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>80; <span class="org-comment">% Crossover Frequency [rad/s]</span>
w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>0.1; <span class="org-comment">% Controller Pole [rad/s]</span>
</pre> </pre>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">K_cen = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gx(1<span class="org-type">:</span>6, 7<span class="org-type">:</span>12), <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0); <pre class="src src-matlab">K_cen = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gx, <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0);
L_cen = K_cen<span class="org-type">*</span>Gx(1<span class="org-type">:</span>6, 7<span class="org-type">:</span>12); L_cen = K_cen<span class="org-type">*</span>Gx;
G_cen = feedback(G, pinv(J<span class="org-type">'</span>)<span class="org-type">*</span>K_cen, [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]); G_cen = feedback(G, pinv(J<span class="org-type">'</span>)<span class="org-type">*</span>K_cen, [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
</pre> </pre>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">K_svd = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gd, <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0); <pre class="src src-matlab">K_svd = diag(1<span class="org-type">./</span>diag(abs(evalfr(Gsvd, <span class="org-constant">j</span><span class="org-type">*</span>wc))))<span class="org-type">*</span>(1<span class="org-type">/</span>abs(evalfr(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0), <span class="org-constant">j</span><span class="org-type">*</span>wc)))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>w0);
L_svd = K_svd<span class="org-type">*</span>Gd; L_svd = K_svd<span class="org-type">*</span>Gsvd;
G_svd = feedback(G, inv(V<span class="org-type">'</span>)<span class="org-type">*</span>K_svd<span class="org-type">*</span>inv(U), [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]); G_svd = feedback(G, inv(V<span class="org-type">'</span>)<span class="org-type">*</span>K_svd<span class="org-type">*</span>inv(U), [7<span class="org-type">:</span>12], [1<span class="org-type">:</span>6]);
</pre> </pre>
</div> </div>
<p> <p>
The obtained diagonal elements of the loop gains are shown in Figure <a href="#orgc313067">17</a>. The obtained diagonal elements of the loop gains are shown in Figure <a href="#org51e7e94">17</a>.
</p> </p>
<div id="orgc313067" class="figure"> <div id="org51e7e94" class="figure">
<p><img src="figs/stewart_comp_loop_gain_diagonal.png" alt="stewart_comp_loop_gain_diagonal.png" /> <p><img src="figs/stewart_comp_loop_gain_diagonal.png" alt="stewart_comp_loop_gain_diagonal.png" />
</p> </p>
<p><span class="figure-number">Figure 17: </span>Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one</p> <p><span class="figure-number">Figure 17: </span>Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one</p>
@@ -1346,11 +1354,11 @@ The obtained diagonal elements of the loop gains are shown in Figure <a href="#o
</div> </div>
</div> </div>
<div id="outline-container-org949d9ca" class="outline-3"> <div id="outline-container-orgb6d90eb" class="outline-3">
<h3 id="org949d9ca"><span class="section-number-3">3.9</span> Closed-Loop system Performances</h3> <h3 id="orgb6d90eb"><span class="section-number-3">3.9</span> Closed-Loop system Performances</h3>
<div class="outline-text-3" id="text-3-9"> <div class="outline-text-3" id="text-3-9">
<p> <p>
<a id="org823e1cb"></a> <a id="org60a86ad"></a>
</p> </p>
<p> <p>
@@ -1381,11 +1389,11 @@ ans =
<p> <p>
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org32234b0">18</a>. The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure <a href="#org5c79b0b">18</a>.
</p> </p>
<div id="org32234b0" class="figure"> <div id="org5c79b0b" class="figure">
<p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" /> <p><img src="figs/stewart_platform_simscape_cl_transmissibility.png" alt="stewart_platform_simscape_cl_transmissibility.png" />
</p> </p>
<p><span class="figure-number">Figure 18: </span>Obtained Transmissibility</p> <p><span class="figure-number">Figure 18: </span>Obtained Transmissibility</p>
@@ -1396,7 +1404,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-09 lun. 10:54</p> <p class="date">Created: 2020-11-09 lun. 14:36</p>
</div> </div>
</body> </body>
</html> </html>

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@@ -820,7 +820,7 @@ The outputs are the 6 accelerations measured by the inertial unit.
#+begin_src latex :file stewart_platform_plant.pdf :tangle no :exports results #+begin_src latex :file stewart_platform_plant.pdf :tangle no :exports results
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$}; \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
\node[above] at (G.north) {$\bm{G}$}; \node[above] at (G.north) {$\bm{G}$};
% Inputs of the controllers % Inputs of the controllers
@@ -835,7 +835,7 @@ The outputs are the 6 accelerations measured by the inertial unit.
#+end_src #+end_src
#+name: fig:stewart_platform_plant #+name: fig:stewart_platform_plant
#+caption: Considered plant $\bm{G} = \begin{bmatrix}G_d\\G\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform #+caption: Considered plant $\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform
#+RESULTS: #+RESULTS:
[[file:figs/stewart_platform_plant.png]] [[file:figs/stewart_platform_plant.png]]
@@ -853,6 +853,11 @@ The outputs are the 6 accelerations measured by the inertial unit.
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ... G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}; G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
% Plant
Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
% Disturbance dynamics
Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
#+end_src #+end_src
There are 24 states (6dof for the bottom platform + 6dof for the top platform). There are 24 states (6dof for the bottom platform + 6dof for the top platform).
@@ -876,15 +881,15 @@ One can easily see that the system is strongly coupled.
hold on; hold on;
for i_in = 1:6 for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6] for i_out = [1:i_in-1, i_in+1:6]
plot(freqs, abs(squeeze(freqresp(G(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
plot(freqs, abs(squeeze(freqresp(G(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G(i,j)\ i \neq j$'); 'DisplayName', '$G_u(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1) set(gca,'ColorOrderIndex',1)
for i_in_out = 1:6 for i_in_out = 1:6
plot(freqs, abs(squeeze(freqresp(G(i_in_out, 6+i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G(%d,%d)$', i_in_out, i_in_out)); plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -898,7 +903,7 @@ One can easily see that the system is strongly coupled.
#+end_src #+end_src
#+name: fig:stewart_platform_coupled_plant #+name: fig:stewart_platform_coupled_plant
#+caption: Magnitude of all 36 elements of the transfer function matrix $\bm{G}$ #+caption: Magnitude of all 36 elements of the transfer function matrix $G_u$
#+RESULTS: #+RESULTS:
[[file:figs/stewart_platform_coupled_plant.png]] [[file:figs/stewart_platform_coupled_plant.png]]
@@ -909,22 +914,16 @@ The Jacobian matrix is used to transform forces/torques applied on the top platf
#+begin_src latex :file plant_decouple_jacobian.pdf :tangle no :exports results #+begin_src latex :file plant_decouple_jacobian.pdf :tangle no :exports results
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$}; \node[block] (G) {$G_u$};
\node[block, left=0.6 of G] (J) {$J^{-T}$};
% Inputs of the controllers
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
\node[block, left=0.6 of inputu] (J) {$J^{-T}$};
% Connections and labels % Connections and labels
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$}; \draw[<-] (J.west) -- ++(-1.0, 0) node[above right]{$\mathcal{F}$};
\draw[->] (G.east) -- ++( 0.8, 0) node[above left]{$a$}; \draw[->] (J.east) -- (G.west) node[above left]{$\tau$};
\draw[->] (J.east) -- (inputu) node[above left]{$\tau$}; \draw[->] (G.east) -- ++( 1.0, 0) node[above left]{$a$};
\draw[<-] (J.west) -- ++(-0.8, 0) node[above right]{$\mathcal{F}$};
\begin{scope}[on background layer] \begin{scope}[on background layer]
\node[fit={(J.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=8pt] (Gx) {}; \node[fit={(J.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
\node[below right] at (Gx.north west) {$\bm{G}_x$}; \node[below right] at (Gx.north west) {$\bm{G}_x$};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
@@ -941,22 +940,18 @@ We define a new plant:
$G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform. $G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
#+begin_src matlab #+begin_src matlab
Gx = G*blkdiag(eye(6), inv(J')); Gx = Gu*inv(J');
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ... Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
#+end_src #+end_src
** Real Approximation of $G$ at the decoupling frequency ** Real Approximation of $G$ at the decoupling frequency
<<sec:stewart_real_approx>> <<sec:stewart_real_approx>>
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$. Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
#+begin_src matlab #+begin_src matlab
wc = 2*pi*30; % Decoupling frequency [rad/s] wc = 2*pi*30; % Decoupling frequency [rad/s]
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ... H1 = evalfr(Gu, j*wc);
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
H1 = evalfr(Gc, j*wc);
#+end_src #+end_src
The real approximation is computed as follows: The real approximation is computed as follows:
@@ -979,12 +974,12 @@ The real approximation is computed as follows:
| 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 | | 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 |
Note that the plant $G$ at $\omega_c$ is already an almost real matrix. Note that the plant $G_u$ at $\omega_c$ is already an almost real matrix.
This can be seen on the Bode plots where the phase is close to 1. This can be seen on the Bode plots where the phase is close to 1.
This can be verified below where only the real value of $G(\omega_c)$ is shown This can be verified below where only the real value of $G_u(\omega_c)$ is shown
#+begin_src matlab :exports results :results value table replace :tangle no #+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(real(evalfr(Gc, j*wc)), {}, {}, ' %.1f '); data2orgtable(real(evalfr(Gu, j*wc)), {}, {}, ' %.1f ');
#+end_src #+end_src
#+RESULTS: #+RESULTS:
@@ -1002,31 +997,26 @@ First, the Singular Value Decomposition of $H_1$ is performed:
\[ H_1 = U \Sigma V^H \] \[ H_1 = U \Sigma V^H \]
#+begin_src matlab #+begin_src matlab
[U,S,V] = svd(H1); [U,~,V] = svd(H1);
#+end_src #+end_src
The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure [[fig:plant_decouple_svd]]. The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure [[fig:plant_decouple_svd]].
#+begin_src latex :file plant_decouple_svd.pdf :tangle no :exports results #+begin_src latex :file plant_decouple_svd.pdf :tangle no :exports results
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$}; \node[block] (G) {$G_u$};
% Inputs of the controllers \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
\node[block, left=0.6 of inputu] (V) {$V^{-T}$};
\node[block, right=0.6 of G.east] (U) {$U^{-1}$}; \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
% Connections and labels % Connections and labels
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$}; \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$};
\draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
\draw[->] (G.east) -- (U.west) node[above left]{$a$}; \draw[->] (G.east) -- (U.west) node[above left]{$a$};
\draw[->] (U.east) -- ++( 0.8, 0) node[above left]{$y$}; \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$};
\draw[->] (V.east) -- (inputu) node[above left]{$\tau$};
\draw[<-] (V.west) -- ++(-0.8, 0) node[above right]{$u$};
\begin{scope}[on background layer] \begin{scope}[on background layer]
\node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=8pt] (Gsvd) {}; \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
\node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$}; \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
@@ -1038,13 +1028,20 @@ The obtained matrices $U$ and $V$ are used to decouple the system as shown in Fi
[[file:figs/plant_decouple_svd.png]] [[file:figs/plant_decouple_svd.png]]
The decoupled plant is then: The decoupled plant is then:
\[ G_{SVD}(s) = U^{-1} G(s) V^{-H} \] \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
#+begin_src matlab
Gsvd = inv(U)*Gu*inv(V');
#+end_src
** Verification of the decoupling using the "Gershgorin Radii" ** Verification of the decoupling using the "Gershgorin Radii"
<<sec:comp_decoupling>> <<sec:comp_decoupling>>
The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$: The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
The "Gershgorin Radii" of a matrix $S$ is defined by:
\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
This is computed over the following frequencies. This is computed over the following frequencies.
#+begin_src matlab #+begin_src matlab
freqs = logspace(-2, 2, 1000); % [Hz] freqs = logspace(-2, 2, 1000); % [Hz]
@@ -1052,29 +1049,23 @@ This is computed over the following frequencies.
#+begin_src matlab :exports none #+begin_src matlab :exports none
% Gershgorin Radii for the coupled plant: % Gershgorin Radii for the coupled plant:
Gr_coupled = zeros(length(freqs), size(Gc,2)); Gr_coupled = zeros(length(freqs), size(Gu,2));
H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
H = abs(squeeze(freqresp(Gc, freqs, 'Hz'))); for out_i = 1:size(Gu,2)
for out_i = 1:size(Gc,2)
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :)); Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
% Gershgorin Radii for the decoupled plant using SVD: % Gershgorin Radii for the decoupled plant using SVD:
Gd = inv(U)*Gc*inv(V'); Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
Gr_decoupled = zeros(length(freqs), size(Gd,2)); H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
for out_i = 1:size(Gsvd,2)
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
for out_i = 1:size(Gd,2)
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :)); Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
% Gershgorin Radii for the decoupled plant using the Jacobian: % Gershgorin Radii for the decoupled plant using the Jacobian:
Gj = Gc*inv(J'); Gr_jacobian = zeros(length(freqs), size(Gx,2));
Gr_jacobian = zeros(length(freqs), size(Gj,2)); H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
for out_i = 1:size(Gx,2)
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
for out_i = 1:size(Gj,2)
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :)); Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
#+end_src #+end_src
@@ -1126,15 +1117,15 @@ The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown i
hold on; hold on;
for i_in = 1:6 for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6] for i_out = [1:i_in-1, i_in+1:6]
plot(freqs, abs(squeeze(freqresp(Gd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
plot(freqs, abs(squeeze(freqresp(Gd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ... plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
'DisplayName', '$G_{SVD}(i,j),\ i \neq j$'); 'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1) set(gca,'ColorOrderIndex',1)
for ch_i = 1:6 for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ... plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i)); 'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
end end
hold off; hold off;
@@ -1147,7 +1138,7 @@ The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown i
ax2 = nexttile; ax2 = nexttile;
hold on; hold on;
for ch_i = 1:6 for ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
@@ -1180,7 +1171,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
hold on; hold on;
for i_in = 1:6 for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6] for i_out = [1:i_in-1, i_in+1:6]
plot(freqs, abs(squeeze(freqresp(Gx(i_out, 6+i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
@@ -1228,14 +1219,6 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
** Diagonal Controller ** Diagonal Controller
<<sec:stewart_diagonal_control>> <<sec:stewart_diagonal_control>>
#+begin_src matlab :exports none :tangle no
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
C_g = 50; % DC Gain
Kc = eye(6)*C_g/(s+wc);
#+end_src
The control diagram for the centralized control is shown in Figure [[fig:centralized_control]]. The control diagram for the centralized control is shown in Figure [[fig:centralized_control]].
The controller $K_c$ is "working" in an cartesian frame. The controller $K_c$ is "working" in an cartesian frame.
@@ -1243,7 +1226,7 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
#+begin_src latex :file centralized_control.pdf :tangle no :exports results #+begin_src latex :file centralized_control.pdf :tangle no :exports results
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G\end{bmatrix}$}; \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
\node[above] at (G.north) {$\bm{G}$}; \node[above] at (G.north) {$\bm{G}$};
\node[block, below right=0.6 and -0.5 of G] (K) {$K_c$}; \node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
\node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$}; \node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$};
@@ -1271,7 +1254,8 @@ The matrices $U$ and $V$ are used to decoupled the plant $G$.
#+begin_src latex :file svd_control.pdf :tangle no :exports results #+begin_src latex :file svd_control.pdf :tangle no :exports results
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$G$}; \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
\node[above] at (G.north) {$\bm{G}$};
\node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$}; \node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$};
\node[block, below=0.6 of G] (K) {$K_{\text{SVD}}$}; \node[block, below=0.6 of G] (K) {$K_{\text{SVD}}$};
\node[block, below left= 0.6 and 0 of G] (V) {$V^{-T}$}; \node[block, below left= 0.6 and 0 of G] (V) {$V^{-T}$};
@@ -1302,19 +1286,19 @@ We choose the controller to be a low pass filter:
$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$ $G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
#+begin_src matlab #+begin_src matlab
wc = 2*pi*80; wc = 2*pi*80; % Crossover Frequency [rad/s]
w0 = 2*pi*0.1; w0 = 2*pi*0.1; % Controller Pole [rad/s]
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
K_cen = diag(1./diag(abs(evalfr(Gx(1:6, 7:12), j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_cen = K_cen*Gx(1:6, 7:12); L_cen = K_cen*Gx;
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]); G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
K_svd = diag(1./diag(abs(evalfr(Gd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_svd = K_svd*Gd; L_svd = K_svd*Gsvd;
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]); G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
#+end_src #+end_src

550
stewart_platform/simscape_model.m
View File

@@ -6,42 +6,14 @@ s = zpk('s');
addpath('STEP'); addpath('STEP');
% Jacobian % Simscape Model - Parameters
% First, the position of the "joints" (points of force application) are estimated and the Jacobian computed. % <<sec:stewart_simscape>>
open('drone_platform_jacobian.slx');
sim('drone_platform_jacobian');
Aa = [a1.Data(1,:);
a2.Data(1,:);
a3.Data(1,:);
a4.Data(1,:);
a5.Data(1,:);
a6.Data(1,:)]';
Ab = [b1.Data(1,:);
b2.Data(1,:);
b3.Data(1,:);
b4.Data(1,:);
b5.Data(1,:);
b6.Data(1,:)]';
As = (Ab - Aa)./vecnorm(Ab - Aa);
l = vecnorm(Ab - Aa)';
J = [As' , cross(Ab, As)'];
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
% Simscape Model
open('drone_platform.slx'); open('drone_platform.slx');
% Definition of spring parameters % Definition of spring parameters:
kx = 0.5*1e3/3; % [N/m] kx = 0.5*1e3/3; % [N/m]
ky = 0.5*1e3/3; ky = 0.5*1e3/3;
@@ -51,31 +23,53 @@ cx = 0.025; % [Nm/rad]
cy = 0.025; cy = 0.025;
cz = 0.025; cz = 0.025;
% Gravity:
g = 0; g = 0;
% We load the Jacobian. % We load the Jacobian (previously computed from the geometry):
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J'); load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
% Identification of the plant
% The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
% We initialize other parameters:
U = eye(6);
V = eye(6);
Kc = tf(zeros(6));
% #+name: fig:stewart_platform_plant
% #+caption: Considered plant $\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}$. $D_w$ is the translation/rotation of the support, $\tau$ the actuator forces, $a$ the acceleration/angular acceleration of the top platform
% #+RESULTS:
% [[file:figs/stewart_platform_plant.png]]
%% Name of the Simulink File %% Name of the Simulink File
mdl = 'drone_platform'; mdl = 'drone_platform';
%% Input/Output definition %% Input/Output definition
clear io; io_i = 1; clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
G = linearize(mdl, io); G = linearize(mdl, io);
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ... G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}; G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
% Plant
Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
% Disturbance dynamics
Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
% There are 24 states (6dof for the bottom platform + 6dof for the top platform). % There are 24 states (6dof for the bottom platform + 6dof for the top platform).
@@ -87,176 +81,59 @@ size(G)
% #+RESULTS: % #+RESULTS:
% : State-space model with 6 outputs, 12 inputs, and 24 states. % : State-space model with 6 outputs, 12 inputs, and 24 states.
% The elements of the transfer matrix $\bm{G}$ corresponding to the transfer function from actuator forces $\tau$ to the measured acceleration $a$ are shown in Figure [[fig:stewart_platform_coupled_plant]].
% G = G*blkdiag(inv(J), eye(6)); % One can easily see that the system is strongly coupled.
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
% Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
Gx = G*blkdiag(eye(6), inv(J'));
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
% Gl = J*G;
% Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
% Obtained Dynamics
freqs = logspace(-1, 2, 1000); freqs = logspace(-1, 2, 1000);
figure; figure;
ax1 = subplot(2, 1, 1); % Magnitude
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$'); for i_in = 1:6
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$'); for i_out = [1:i_in-1, i_in+1:6]
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$'); plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_u(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:6
plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast'); ylim([1e-2, 1e5]);
legend('location', 'northwest');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
% #+name: fig:stewart_platform_translations % #+name: fig:plant_decouple_jacobian
% #+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform % #+caption: Decoupled plant $\bm{G}_x$ using the Jacobian matrix $J$
% #+RESULTS: % #+RESULTS:
% [[file:figs/stewart_platform_translations.png]] % [[file:figs/plant_decouple_jacobian.png]]
% We define a new plant:
% \[ G_x(s) = G(s) J^{-T} \]
% $G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
freqs = logspace(-1, 2, 1000); Gx = Gu*inv(J');
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
% #+name: fig:stewart_platform_rotations
% #+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
% #+RESULTS:
% [[file:figs/stewart_platform_rotations.png]]
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for out_i = 1:5
for in_i = i+1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
% #+name: fig:stewart_platform_legs
% #+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
% #+RESULTS:
% [[file:figs/stewart_platform_legs.png]]
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
% set(gca,'ColorOrderIndex',1)
% plot(freqs, abs(squeeze(freqresp(TR(1,1), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(2,2), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(3,3), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
% set(gca,'ColorOrderIndex',1)
% plot(freqs, abs(squeeze(freqresp(TR(4,4), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(5,5), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(6,6), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
linkaxes([ax1,ax2],'x');
% Real Approximation of $G$ at the decoupling frequency % Real Approximation of $G$ at the decoupling frequency
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$. % <<sec:stewart_real_approx>>
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
wc = 2*pi*30; % Decoupling frequency [rad/s] wc = 2*pi*30; % Decoupling frequency [rad/s]
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ... H1 = evalfr(Gu, j*wc);
{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
H1 = evalfr(Gc, j*wc);
@@ -265,55 +142,58 @@ H1 = evalfr(Gc, j*wc);
D = pinv(real(H1'*H1)); D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
% Verification of the decoupling using the "Gershgorin Radii" % SVD Decoupling
% <<sec:stewart_svd_decoupling>>
% First, the Singular Value Decomposition of $H_1$ is performed: % First, the Singular Value Decomposition of $H_1$ is performed:
% \[ H_1 = U \Sigma V^H \] % \[ H_1 = U \Sigma V^H \]
[U,S,V] = svd(H1); [U,~,V] = svd(H1);
% Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$: % #+name: fig:plant_decouple_svd
% \[ G_d(s) = U^T G_c(s) V \] % #+caption: Decoupled plant $\bm{G}_{SVD}$ using the Singular Value Decomposition
% #+RESULTS:
% [[file:figs/plant_decouple_svd.png]]
% The decoupled plant is then:
% \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
Gsvd = inv(U)*Gu*inv(V');
% Verification of the decoupling using the "Gershgorin Radii"
% <<sec:comp_decoupling>>
% The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
% The "Gershgorin Radii" of a matrix $S$ is defined by:
% \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
% This is computed over the following frequencies. % This is computed over the following frequencies.
freqs = logspace(-2, 2, 1000); % [Hz] freqs = logspace(-2, 2, 1000); % [Hz]
% Gershgorin Radii for the coupled plant: % Gershgorin Radii for the coupled plant:
Gr_coupled = zeros(length(freqs), size(Gu,2));
Gr_coupled = zeros(length(freqs), size(Gc,2)); H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
for out_i = 1:size(Gu,2)
H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
for out_i = 1:size(Gc,2)
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :)); Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
% Gershgorin Radii for the decoupled plant using SVD: % Gershgorin Radii for the decoupled plant using SVD:
Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
Gd = U'*Gc*V; H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
Gr_decoupled = zeros(length(freqs), size(Gd,2)); for out_i = 1:size(Gsvd,2)
H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
for out_i = 1:size(Gd,2)
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :)); Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
% Gershgorin Radii for the decoupled plant using the Jacobian: % Gershgorin Radii for the decoupled plant using the Jacobian:
Gr_jacobian = zeros(length(freqs), size(Gx,2));
Gj = Gc*inv(J'); H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
Gr_jacobian = zeros(length(freqs), size(Gj,2)); for out_i = 1:size(Gx,2)
H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
for out_i = 1:size(Gj,2)
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :)); Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end end
@@ -334,31 +214,55 @@ plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
hold off; hold off;
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii') xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
legend('location', 'northeast'); legend('location', 'northwest');
ylim([1e-3, 1e3]);
% Decoupled Plant % Obtained Decoupled Plants
% Let's see the bode plot of the decoupled plant $G_d(s)$. % <<sec:stewart_decoupled_plant>>
% \[ G_d(s) = U^T G_c(s) V \]
% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
freqs = logspace(-1, 2, 1000); freqs = logspace(-1, 2, 1000);
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on; hold on;
for ch_i = 1:6 for i_in = 1:6
plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ... for i_out = [1:i_in-1, i_in+1:6]
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i)); plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
end
for in_i = 1:5
for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast'); legend('location', 'northwest');
ylim([1e-1, 1e5])
% Phase
ax2 = nexttile;
hold on;
for ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180:90:360]);
linkaxes([ax1,ax2],'x');
@@ -367,53 +271,135 @@ legend('location', 'southeast');
% #+RESULTS: % #+RESULTS:
% [[file:figs/simscape_model_decoupled_plant_svd.png]] % [[file:figs/simscape_model_decoupled_plant_svd.png]]
% Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:simscape_model_decoupled_plant_jacobian]].
freqs = logspace(-1, 2, 1000); freqs = logspace(-1, 2, 1000);
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on; hold on;
for ch_i = 1:6 for i_in = 1:6
plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ... for i_out = [1:i_in-1, i_in+1:6]
'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i)); plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
end
for in_i = 1:5
for out_i = in_i+1:6
plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
'DisplayName', '$G_x(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast'); legend('location', 'northwest');
ylim([1e-2, 2e6])
% Diagonal Controller % Phase
% The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$. ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([0, 180]);
yticks([0:45:360]);
linkaxes([ax1,ax2],'x');
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
C_g = 50; % DC Gain
K = eye(6)*C_g/(s+wc);
% #+RESULTS:
% [[file:figs/centralized_control.png]]
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
% #+name: fig:svd_control
% #+caption: Control Diagram for the SVD control
% #+RESULTS: % #+RESULTS:
% [[file:figs/svd_control.png]] % [[file:figs/svd_control.png]]
% SVD Control
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]); % We choose the controller to be a low pass filter:
% \[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]
% $G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
wc = 2*pi*80; % Crossover Frequency [rad/s]
w0 = 2*pi*0.1; % Controller Pole [rad/s]
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_cen = K_cen*Gx;
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_svd = K_svd*Gsvd;
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
% The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
for i_in_out = 2:6
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
'DisplayName', '$L_{J}(i,i)$');
for i_in_out = 2:6
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest');
ylim([5e-2, 2e3])
% Phase
ax2 = nexttile;
hold on;
for i_in_out = 1:6
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
end
set(gca,'ColorOrderIndex',2)
for i_in_out = 1:6
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180:90:360]);
linkaxes([ax1,ax2],'x');
% Closed-Loop system Performances
% <<sec:stewart_closed_loop_results>>
% Results
% Let's first verify the stability of the closed-loop systems: % Let's first verify the stability of the closed-loop systems:
isstable(G_cen) isstable(G_cen)
@@ -433,69 +419,61 @@ isstable(G_svd)
% #+RESULTS: % #+RESULTS:
% : ans = % : ans =
% : logical % : logical
% : 0 % : 1
% The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]]. % The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
freqs = logspace(-3, 3, 1000); freqs = logspace(-2, 2, 1000);
figure figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = subplot(2, 3, 1); ax1 = nexttile;
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop'); plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized'); plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD'); plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]'); ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest'); legend('location', 'southwest');
ax2 = subplot(2, 3, 2); ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
ax3 = subplot(2, 3, 3);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))), '--');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]'); ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
ax4 = subplot(2, 3, 4); ax3 = nexttile;
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))), '--');
hold off; set(gca,'ColorOrderIndex',1)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
ax5 = subplot(2, 3, 5);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))), '--');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]'); ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
ax6 = subplot(2, 3, 6); ax4 = nexttile;
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))), '--');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]'); ylabel('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); linkaxes([ax1,ax2,ax3,ax4],'xy');
xlim([freqs(1), freqs(end)]); xlim([freqs(1), freqs(end)]);
ylim([1e-3, 1e2]);