Add new file about optimal stiffness / control
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docs/figs/opt_stiff_iff_root_locus.pdf
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docs/figs/opt_stiff_iff_root_locus.pdf
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docs/figs/opt_stiff_iff_root_locus.png
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docs/figs/opt_stiff_iff_root_locus.png.png
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docs/optimal_stiffness_control.html
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docs/optimal_stiffness_control.html
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<?xml version="1.0" encoding="utf-8"?>
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<title>Control of the NASS with optimal stiffness</title>
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<meta name="author" content="Dehaeze Thomas" />
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#org-div-home-and-up
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/*]]>*/-->
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// @license magnet:?xt=urn:btih:1f739d935676111cfff4b4693e3816e664797050&dn=gpl-3.0.txt GPL-v3-or-Later
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="./index.html"> UP </a>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Control of the NASS with optimal stiffness</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgb09af41">1. Low Authority Control - Decentralized Integral Force Feedback</a>
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<ul>
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<li><a href="#orgf3f8aed">1.1. Initialization</a></li>
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<li><a href="#org93676a4">1.2. Identification</a></li>
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<li><a href="#org1f2e125">1.3. Controller Design</a></li>
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</ul>
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</li>
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<li><a href="#orgf825c29">2. Primary Control</a>
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<ul>
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<li><a href="#org611a860">2.1. Identification</a></li>
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<li><a href="#orgfef1a3f">2.2. Controller Design</a></li>
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</ul>
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</li>
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<li><a href="#org84f68cc">3. Simulations</a></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgb09af41" class="outline-2">
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<h2 id="orgb09af41"><span class="section-number-2">1</span> Low Authority Control - Decentralized Integral Force Feedback</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-orgf3f8aed" class="outline-3">
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<h3 id="orgf3f8aed"><span class="section-number-3">1.1</span> Initialization</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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We initialize all the stages with the default parameters.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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</pre>
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</div>
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<p>
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We set the references that corresponds to a tomography experiment.
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<div class="org-src-container">
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<pre class="src src-matlab">initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 1);
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initializeSimscapeConfiguration();
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initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
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initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-iff'</span>);
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</pre>
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<div id="outline-container-org93676a4" class="outline-3">
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<h3 id="org93676a4"><span class="section-number-3">1.2</span> Identification</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab">Kx = tf(zeros(6));
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Kiff = tf(zeros(6));
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</pre>
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</div>
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<pre class="src src-matlab">Ms = [1, 10, 50];
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Gm_iff = {zeros(length(Ms), 1)};
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</pre>
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||||||
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</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'k'</span>, 1e5, <span class="org-string">'c'</span>, 2e2);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org1f2e125" class="outline-3">
|
||||||
|
<h3 id="org1f2e125"><span class="section-number-3">1.3</span> Controller Design</h3>
|
||||||
|
<div class="outline-text-3" id="text-1-3">
|
||||||
|
<p>
|
||||||
|
Root Locus
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div id="orgf657dfe" class="figure">
|
||||||
|
<p><img src="figs/opt_stiff_iff_root_locus.png" alt="opt_stiff_iff_root_locus.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 1: </span>Root Locus for the</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Damping as function of the gain
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<div id="outline-container-orgf825c29" class="outline-2">
|
||||||
|
<h2 id="orgf825c29"><span class="section-number-2">2</span> Primary Control</h2>
|
||||||
|
<div class="outline-text-2" id="text-2">
|
||||||
|
</div>
|
||||||
|
<div id="outline-container-org611a860" class="outline-3">
|
||||||
|
<h3 id="org611a860"><span class="section-number-3">2.1</span> Identification</h3>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgfef1a3f" class="outline-3">
|
||||||
|
<h3 id="orgfef1a3f"><span class="section-number-3">2.2</span> Controller Design</h3>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org84f68cc" class="outline-2">
|
||||||
|
<h2 id="org84f68cc"><span class="section-number-2">3</span> Simulations</h2>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<div id="postamble" class="status">
|
||||||
|
<p class="author">Author: Dehaeze Thomas</p>
|
||||||
|
<p class="date">Created: 2020-04-08 mer. 17:22</p>
|
||||||
|
</div>
|
||||||
|
</body>
|
||||||
|
</html>
|
||||||
@ -4,7 +4,7 @@
|
|||||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||||
<head>
|
<head>
|
||||||
<!-- 2020-04-08 mer. 12:17 -->
|
<!-- 2020-04-08 mer. 17:20 -->
|
||||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||||
<title>Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</title>
|
<title>Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</title>
|
||||||
@ -257,7 +257,7 @@
|
|||||||
<li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
|
<li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
|
||||||
<li><a href="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
|
<li><a href="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
|
||||||
<li><a href="#org0448746">2.5. Direct Forces (Compliance).</a></li>
|
<li><a href="#org0448746">2.5. Direct Forces (Compliance).</a></li>
|
||||||
<li><a href="#org6791692">2.6. Conclusion</a></li>
|
<li><a href="#org08f24cd">2.6. Conclusion</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org6527e58">3. Effect of granite stiffness</a>
|
<li><a href="#org6527e58">3. Effect of granite stiffness</a>
|
||||||
@ -270,7 +270,7 @@
|
|||||||
</li>
|
</li>
|
||||||
<li><a href="#org9215f81">3.2. Soft Granite</a></li>
|
<li><a href="#org9215f81">3.2. Soft Granite</a></li>
|
||||||
<li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
|
<li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
|
||||||
<li><a href="#orga001da4">3.4. Conclusion</a></li>
|
<li><a href="#orgbc8931e">3.4. Conclusion</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
|
<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
|
||||||
@ -278,7 +278,7 @@
|
|||||||
<li><a href="#org6bd588f">4.1. Noise Budgeting - Theory</a></li>
|
<li><a href="#org6bd588f">4.1. Noise Budgeting - Theory</a></li>
|
||||||
<li><a href="#orgcc86f59">4.2. Power Spectral Densities</a></li>
|
<li><a href="#orgcc86f59">4.2. Power Spectral Densities</a></li>
|
||||||
<li><a href="#orgef96b89">4.3. Cumulative Amplitude Spectrum</a></li>
|
<li><a href="#orgef96b89">4.3. Cumulative Amplitude Spectrum</a></li>
|
||||||
<li><a href="#org4352c0d">4.4. Conclusion</a></li>
|
<li><a href="#org18bab44">4.4. Conclusion</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org34c0f38">5. Closed Loop Budget Error</a>
|
<li><a href="#org34c0f38">5. Closed Loop Budget Error</a>
|
||||||
@ -287,7 +287,7 @@
|
|||||||
<li><a href="#orgf2d36a1">5.2. Reduction thanks to feedback - Required bandwidth</a></li>
|
<li><a href="#orgf2d36a1">5.2. Reduction thanks to feedback - Required bandwidth</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org08f24cd">6. Conclusion</a></li>
|
<li><a href="#orgd9198a1">6. Conclusion</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
@ -497,8 +497,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6791692" class="outline-3">
|
<div id="outline-container-org08f24cd" class="outline-3">
|
||||||
<h3 id="org6791692"><span class="section-number-3">2.6</span> Conclusion</h3>
|
<h3 id="org08f24cd"><span class="section-number-3">2.6</span> Conclusion</h3>
|
||||||
<div class="outline-text-3" id="text-2-6">
|
<div class="outline-text-3" id="text-2-6">
|
||||||
<div class="important">
|
<div class="important">
|
||||||
<p>
|
<p>
|
||||||
@ -678,8 +678,8 @@ From Figures <a href="#orgc4c14fb">11</a> and <a href="#org533cc4b">12</a>, we s
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga001da4" class="outline-3">
|
<div id="outline-container-orgbc8931e" class="outline-3">
|
||||||
<h3 id="orga001da4"><span class="section-number-3">3.4</span> Conclusion</h3>
|
<h3 id="orgbc8931e"><span class="section-number-3">3.4</span> Conclusion</h3>
|
||||||
<div class="outline-text-3" id="text-3-4">
|
<div class="outline-text-3" id="text-3-4">
|
||||||
<div class="important">
|
<div class="important">
|
||||||
<p>
|
<p>
|
||||||
@ -848,8 +848,8 @@ The black dashed line corresponds to the performance objective of a sample vibra
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org4352c0d" class="outline-3">
|
<div id="outline-container-org18bab44" class="outline-3">
|
||||||
<h3 id="org4352c0d"><span class="section-number-3">4.4</span> Conclusion</h3>
|
<h3 id="org18bab44"><span class="section-number-3">4.4</span> Conclusion</h3>
|
||||||
<div class="outline-text-3" id="text-4-4">
|
<div class="outline-text-3" id="text-4-4">
|
||||||
<div class="important">
|
<div class="important">
|
||||||
<p>
|
<p>
|
||||||
@ -1012,8 +1012,8 @@ The obtained required bandwidth (approximate upper and lower bounds) to obtained
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org08f24cd" class="outline-2">
|
<div id="outline-container-orgd9198a1" class="outline-2">
|
||||||
<h2 id="org08f24cd"><span class="section-number-2">6</span> Conclusion</h2>
|
<h2 id="orgd9198a1"><span class="section-number-2">6</span> Conclusion</h2>
|
||||||
<div class="outline-text-2" id="text-6">
|
<div class="outline-text-2" id="text-6">
|
||||||
<div class="important">
|
<div class="important">
|
||||||
<p>
|
<p>
|
||||||
@ -1021,7 +1021,7 @@ From Figure <a href="#orgd677910">23</a> and Table <a href="#org5ab4860">1</a>,
|
|||||||
</p>
|
</p>
|
||||||
<ul class="org-ul">
|
<ul class="org-ul">
|
||||||
<li>For a soft nano-hexapod (\(k < 10^4\ [N/m]\)), the required bandwidth is \(\omega_c \approx 50-100\ Hz\)</li>
|
<li>For a soft nano-hexapod (\(k < 10^4\ [N/m]\)), the required bandwidth is \(\omega_c \approx 50-100\ Hz\)</li>
|
||||||
<li>For a nano-hexapods with \(10^5 < k < 10^6\ [N/m]\)), the required bandwidth is \(\omega_c \approx 150-300\ Hz\)</li>
|
<li>For a nano-hexapods with \(10^5 < k < 10^6\ [N/m]\), the required bandwidth is \(\omega_c \approx 150-300\ Hz\)</li>
|
||||||
<li>For a stiff nano-hexapods (\(k > 10^7\ [N/m]\)), the required bandwidth is \(\omega_c \approx 250-500\ Hz\)</li>
|
<li>For a stiff nano-hexapods (\(k > 10^7\ [N/m]\)), the required bandwidth is \(\omega_c \approx 250-500\ Hz\)</li>
|
||||||
</ul>
|
</ul>
|
||||||
|
|
||||||
@ -1031,7 +1031,7 @@ From Figure <a href="#orgd677910">23</a> and Table <a href="#org5ab4860">1</a>,
|
|||||||
</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: 2020-04-08 mer. 12:17</p>
|
<p class="date">Created: 2020-04-08 mer. 17:20</p>
|
||||||
</div>
|
</div>
|
||||||
</body>
|
</body>
|
||||||
</html>
|
</html>
|
||||||
|
|||||||
BIN
mat/conf_log.mat
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mat/conf_log.mat
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mat/stages.mat
BIN
mat/stages.mat
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779
org/optimal_stiffness_control.org
Normal file
779
org/optimal_stiffness_control.org
Normal file
@ -0,0 +1,779 @@
|
|||||||
|
#+TITLE: Control of the NASS with optimal stiffness
|
||||||
|
:DRAWER:
|
||||||
|
#+STARTUP: overview
|
||||||
|
|
||||||
|
#+LANGUAGE: en
|
||||||
|
#+EMAIL: dehaeze.thomas@gmail.com
|
||||||
|
#+AUTHOR: Dehaeze Thomas
|
||||||
|
|
||||||
|
#+HTML_LINK_HOME: ./index.html
|
||||||
|
#+HTML_LINK_UP: ./index.html
|
||||||
|
|
||||||
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
||||||
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
||||||
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
|
||||||
|
|
||||||
|
#+HTML_MATHJAX: align: center tagside: right font: TeX
|
||||||
|
|
||||||
|
#+PROPERTY: header-args:matlab :session *MATLAB*
|
||||||
|
#+PROPERTY: header-args:matlab+ :comments org
|
||||||
|
#+PROPERTY: header-args:matlab+ :results none
|
||||||
|
#+PROPERTY: header-args:matlab+ :exports both
|
||||||
|
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||||
|
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||||
|
#+PROPERTY: header-args:matlab+ :tangle no
|
||||||
|
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
||||||
|
|
||||||
|
#+PROPERTY: header-args:shell :eval no-export
|
||||||
|
|
||||||
|
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/org/}{config.tex}")
|
||||||
|
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
|
||||||
|
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
|
||||||
|
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
|
||||||
|
#+PROPERTY: header-args:latex+ :results file raw replace
|
||||||
|
#+PROPERTY: header-args:latex+ :buffer no
|
||||||
|
#+PROPERTY: header-args:latex+ :eval no-export
|
||||||
|
#+PROPERTY: header-args:latex+ :exports results
|
||||||
|
#+PROPERTY: header-args:latex+ :mkdirp yes
|
||||||
|
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||||
|
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
|
||||||
|
:END:
|
||||||
|
|
||||||
|
* Introduction :ignore:
|
||||||
|
|
||||||
|
* Low Authority Control - Decentralized Integral Force Feedback
|
||||||
|
** Introduction :ignore:
|
||||||
|
|
||||||
|
** Matlab Init :noexport:ignore:
|
||||||
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
|
<<matlab-dir>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none :results silent :noweb yes
|
||||||
|
<<matlab-init>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no
|
||||||
|
simulinkproject('../');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
load('mat/conf_simulink.mat');
|
||||||
|
open('nass_model.slx')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Initialization
|
||||||
|
We initialize all the stages with the default parameters.
|
||||||
|
#+begin_src matlab
|
||||||
|
initializeGround();
|
||||||
|
initializeGranite();
|
||||||
|
initializeTy();
|
||||||
|
initializeRy();
|
||||||
|
initializeRz();
|
||||||
|
initializeMicroHexapod();
|
||||||
|
initializeAxisc();
|
||||||
|
initializeMirror();
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
We set the references that corresponds to a tomography experiment.
|
||||||
|
#+begin_src matlab
|
||||||
|
initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 1);
|
||||||
|
initializeSimscapeConfiguration();
|
||||||
|
initializeDisturbances('enable', false);
|
||||||
|
initializeLoggingConfiguration('log', 'none');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
initializeController('type', 'hac-iff');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Identification
|
||||||
|
#+begin_src matlab
|
||||||
|
Kx = tf(zeros(6));
|
||||||
|
Kiff = tf(zeros(6));
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
Ms = [1, 10, 50];
|
||||||
|
|
||||||
|
Gm_iff = {zeros(length(Ms), 1)};
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
initializeNanoHexapod('k', 1e5, 'c', 2e2);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
%% Name of the Simulink File
|
||||||
|
mdl = 'nass_model';
|
||||||
|
|
||||||
|
%% Input/Output definition
|
||||||
|
clear io; io_i = 1;
|
||||||
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
|
||||||
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
initializeSample('mass', Ms(i), 'freq', 200*ones(6,1));
|
||||||
|
|
||||||
|
%% Run the linearization
|
||||||
|
G_iff = linearize(mdl, io);
|
||||||
|
G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
||||||
|
G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
|
||||||
|
Gm_iff(i) = {G_iff};
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Controller Design
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(-1, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax2 = subplot(2, 1, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz')))), ...
|
||||||
|
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
legend('location', 'northeast');
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Root Locus
|
||||||
|
#+begin_src matlab :exports none :post
|
||||||
|
figure;
|
||||||
|
|
||||||
|
gains = logspace(0, 3, 300);
|
||||||
|
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(real(pole(Gm_iff{i})), imag(pole(Gm_iff{i})), 'x', ...
|
||||||
|
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(real(tzero(Gm_iff{i})), imag(tzero(Gm_iff{i})), 'o', ...
|
||||||
|
'HandleVisibility', 'off');
|
||||||
|
for k = 1:length(gains)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
|
||||||
|
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||||
|
'HandleVisibility', 'off');
|
||||||
|
end
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
axis square;
|
||||||
|
xlim([-140, 10]); ylim([0, 150]);
|
||||||
|
|
||||||
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||||
|
legend('location', 'northwest');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/opt_stiff_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:opt_stiff_iff_root_locus
|
||||||
|
#+caption: Root Locus for the
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/opt_stiff_iff_root_locus.png]]
|
||||||
|
|
||||||
|
Damping as function of the gain
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
c1 = [ 0 0.4470 0.7410]; % Blue
|
||||||
|
c2 = [0.8500 0.3250 0.0980]; % Orange
|
||||||
|
c3 = [0.9290 0.6940 0.1250]; % Yellow
|
||||||
|
c4 = [0.4940 0.1840 0.5560]; % Purple
|
||||||
|
c5 = [0.4660 0.6740 0.1880]; % Green
|
||||||
|
c6 = [0.3010 0.7450 0.9330]; % Light Blue
|
||||||
|
c7 = [0.6350 0.0780 0.1840]; % Red
|
||||||
|
colors = [c1; c2; c3; c4; c5; c6; c7];
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
gains = logspace(0, 3, 100);
|
||||||
|
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
for k = 1:length(gains)
|
||||||
|
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(gains(k), sin(-pi/2 + angle(cl_poles)), '.', 'color', colors(i, :));
|
||||||
|
end
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
xlabel('IFF Gain'); ylabel('Modal Damping');
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylim([0, 1]);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
Kiff = -200/s*eye(6);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
* Primary Control
|
||||||
|
** Introduction :ignore:
|
||||||
|
|
||||||
|
|
||||||
|
** Matlab Init :noexport:ignore:
|
||||||
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
|
<<matlab-dir>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none :results silent :noweb yes
|
||||||
|
<<matlab-init>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Identification
|
||||||
|
#+begin_src matlab
|
||||||
|
Gm_x = {zeros(length(Ms), 1)};
|
||||||
|
Gm_l = {zeros(length(Ms), 1)};
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
load('mat/stages.mat', 'nano_hexapod');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
%% Name of the Simulink File
|
||||||
|
mdl = 'nass_model';
|
||||||
|
|
||||||
|
%% Input/Output definition
|
||||||
|
clear io; io_i = 1;
|
||||||
|
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
|
||||||
|
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
initializeSample('mass', Ms(i), 'freq', 200*ones(6,1));
|
||||||
|
|
||||||
|
%% Run the linearization
|
||||||
|
G = linearize(mdl, io);
|
||||||
|
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
||||||
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
||||||
|
|
||||||
|
Gx = -G*inv(nano_hexapod.J');
|
||||||
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||||
|
Gm_x(i) = {Gx};
|
||||||
|
|
||||||
|
Gl = -nano_hexapod.J*G;
|
||||||
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
||||||
|
Gm_l(i) = {Gl};
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Controller in the task space
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'};
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 2, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax2 = subplot(2, 2, 3);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))), 'DisplayName', labels{i});
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
legend();
|
||||||
|
|
||||||
|
ax3 = subplot(2, 2, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Off-Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax4 = subplot(2, 2, 4);
|
||||||
|
hold on;
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Translation
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 2, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(1, 1), freqs, 'Hz'))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(2, 2), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')
|
||||||
|
|
||||||
|
ax2 = subplot(2, 2, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(3, 3), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_z/\mathcal{F}_z$')
|
||||||
|
|
||||||
|
ax3 = subplot(2, 2, 3);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(1, 1), freqs, 'Hz')))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(2, 2), freqs, 'Hz')))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
|
||||||
|
ax4 = subplot(2, 2, 4);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(3, 3), freqs, 'Hz')))), ...
|
||||||
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
legend('location', 'southwest');
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
Kx = tf(zeros(6));
|
||||||
|
|
||||||
|
h = 1.5;
|
||||||
|
Kx(1,1) = 2e6 * ...
|
||||||
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||||
|
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||||
|
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||||
|
|
||||||
|
Kx(2,2) = Kx(1,1);
|
||||||
|
|
||||||
|
h = 1.5;
|
||||||
|
Kx(3,3) = 1e7 * ...
|
||||||
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||||
|
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||||
|
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 2, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(1, 1)*Kx(1,1), freqs, 'Hz'))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(2, 2)*Kx(2,2), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')
|
||||||
|
|
||||||
|
ax2 = subplot(2, 2, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(3, 3)*Kx(3,3), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_z/\mathcal{F}_z$')
|
||||||
|
|
||||||
|
ax3 = subplot(2, 2, 3);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(1, 1)*Kx(1,1), freqs, 'Hz')))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(2, 2)*Kx(2,2), freqs, 'Hz')))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
|
||||||
|
ax4 = subplot(2, 2, 4);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(3, 3)*Kx(3,3), freqs, 'Hz')))), ...
|
||||||
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
legend('location', 'southwest');
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Rotations
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 2, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(4, 4), freqs, 'Hz'))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(5, 5), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_{R_x}/\mathcal{M}_x$, $\mathcal{X}_{R_y}/\mathcal{M}_y$')
|
||||||
|
|
||||||
|
ax2 = subplot(2, 2, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(6, 6), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_{R_z}/\mathcal{M}_z$')
|
||||||
|
|
||||||
|
ax3 = subplot(2, 2, 3);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(4, 4), freqs, 'Hz')))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(5, 5), freqs, 'Hz')))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
|
||||||
|
ax4 = subplot(2, 2, 4);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(6, 6), freqs, 'Hz')))), ...
|
||||||
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
legend('location', 'southwest');
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
h = 1.5;
|
||||||
|
Kx(4,4) = 1e5 * ...
|
||||||
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||||
|
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||||
|
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||||
|
|
||||||
|
Kx(5,5) = Kx(4,4);
|
||||||
|
|
||||||
|
h = 1.5;
|
||||||
|
Kx(6,6) = 2e5 * ...
|
||||||
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||||
|
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||||
|
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 2, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(4, 4)*Kx(4,4), freqs, 'Hz'))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(5, 5)*Kx(5,5), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_{R_x}/\mathcal{M}_x$, $\mathcal{X}_{R_y}/\mathcal{M}_y$')
|
||||||
|
|
||||||
|
ax2 = subplot(2, 2, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(6, 6)*Kx(6,6), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('$\mathcal{X}_{R_z}/\mathcal{M}_z$')
|
||||||
|
|
||||||
|
ax3 = subplot(2, 2, 3);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(4, 4)*Kx(4,4), freqs, 'Hz')))));
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(5, 5)*Kx(5,5), freqs, 'Hz')))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
|
||||||
|
ax4 = subplot(2, 2, 4);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(6, 6)*Kx(6,6), freqs, 'Hz')))), ...
|
||||||
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-270, 90]);
|
||||||
|
yticks([-360:90:360]);
|
||||||
|
legend('location', 'southwest');
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Stability
|
||||||
|
#+begin_src matlab
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
isstable(feedback(Gm_x{i}*Kx, eye(6), -1))
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Simulation
|
||||||
|
|
||||||
|
** Control in the leg space
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 2, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax2 = subplot(2, 2, 3);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))), ...
|
||||||
|
'DisplayName', sprintf('$d\\mathcal{L}_%i / \\tau_%i$', i, i));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
legend();
|
||||||
|
|
||||||
|
ax3 = subplot(2, 2, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Off-Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax4 = subplot(2, 2, 4);
|
||||||
|
hold on;
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
for j = 1:6
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax2 = subplot(2, 1, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
for j = 1:6
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
h = 1.5;
|
||||||
|
Kl = 5e6 * eye(6) * ...
|
||||||
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||||
|
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||||
|
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
isstable(feedback(Gm_l{i}(1,1)*Kl(1,1), 1, -1))
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:length(Ms)
|
||||||
|
for j = 1:6
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
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 i = 1:length(Ms)
|
||||||
|
for j = 1:6
|
||||||
|
set(gca,'ColorOrderIndex',i);
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
|
||||||
|
* Simulations
|
||||||
@ -1240,6 +1240,6 @@ data2orgtable([wb1'; wb2'], {'Required wc with L1 [Hz]', 'Required wc with L2 [H
|
|||||||
#+begin_important
|
#+begin_important
|
||||||
From Figure [[fig:opt_stiff_req_bandwidth_K1_K2]] and Table [[tab:approx_required_wc_10nm]], we can clearly see three different results depending on the nano-hexapod stiffness:
|
From Figure [[fig:opt_stiff_req_bandwidth_K1_K2]] and Table [[tab:approx_required_wc_10nm]], we can clearly see three different results depending on the nano-hexapod stiffness:
|
||||||
- For a soft nano-hexapod ($k < 10^4\ [N/m]$), the required bandwidth is $\omega_c \approx 50-100\ Hz$
|
- For a soft nano-hexapod ($k < 10^4\ [N/m]$), the required bandwidth is $\omega_c \approx 50-100\ Hz$
|
||||||
- For a nano-hexapods with $10^5 < k < 10^6\ [N/m]$), the required bandwidth is $\omega_c \approx 150-300\ Hz$
|
- For a nano-hexapods with $10^5 < k < 10^6\ [N/m]$, the required bandwidth is $\omega_c \approx 150-300\ Hz$
|
||||||
- For a stiff nano-hexapods ($k > 10^7\ [N/m]$), the required bandwidth is $\omega_c \approx 250-500\ Hz$
|
- For a stiff nano-hexapods ($k > 10^7\ [N/m]$), the required bandwidth is $\omega_c \approx 250-500\ Hz$
|
||||||
#+end_important
|
#+end_important
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user