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Update files for new definition of hexapods

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Thomas Dehaeze
2020-05-05 11:38:52 +02:00
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docs/uncertainty_support.html
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@@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-04-17 ven. 09:35 -->
<!-- 2020-05-05 mar. 10:33 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Effect of Uncertainty on the support&rsquo;s dynamics on the isolation platform dynamics</title>
<meta name="generator" content="Org mode" />
@@ -37,17 +37,17 @@
<ul>
<li><a href="#orgbe6e0b8">1. Simple Introductory Example</a>
<ul>
<li><a href="#orgf4562a5">1.1. Equations of motion</a></li>
<li><a href="#org3d4902a">1.1. Equations of motion</a></li>
<li><a href="#org8bd2a4a">1.2. Initialization of the support dynamics</a></li>
<li><a href="#orgefb9b71">1.3. Initialization of the isolation platform</a></li>
<li><a href="#org3bc4ad1">1.4. Comparison</a></li>
<li><a href="#org2a9bf99">1.5. Conclusion</a></li>
<li><a href="#org999e1c5">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#orge1d3484">2. Generalization to arbitrary dynamics</a>
<ul>
<li><a href="#org3948d1f">2.1. Introduction</a></li>
<li><a href="#org3d4902a">2.2. Equations of motion</a></li>
<li><a href="#org18c1c3f">2.2. Equations of motion</a></li>
<li><a href="#orgc20cabb">2.3. Compliance of the Support</a></li>
<li><a href="#org67810a4">2.4. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#orge950395">2.5. Effect of the Isolation platform Stiffness</a></li>
@@ -58,7 +58,7 @@
<li><a href="#orgd2fc303">2.6.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul>
</li>
<li><a href="#org999e1c5">2.7. Conclusion</a></li>
<li><a href="#orgde3616e">2.7. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -112,8 +112,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the suppor
</div>
</div>
<div id="outline-container-orgf4562a5" class="outline-3">
<h3 id="orgf4562a5"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-org3d4902a" class="outline-3">
<h3 id="org3d4902a"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div class="outline-text-3" id="text-1-1">
<p>
If we write the equation of motion of the system in Figure <a href="#org41bc770">1</a>, we obtain:
@@ -156,9 +156,9 @@ kpi = 1e8;
Let&rsquo;s also consider some uncertainty in those parameters:
</p>
<div class="org-src-container">
<pre class="src src-matlab">mp = ureal(<span class="org-string">'m'</span>, mpi, <span class="org-string">'Percentage'</span>, 30);
cp = ureal(<span class="org-string">'c'</span>, cpi, <span class="org-string">'Percentage'</span>, 30);
kp = ureal(<span class="org-string">'k'</span>, kpi, <span class="org-string">'Percentage'</span>, 30);
<pre class="src src-matlab">mp = ureal('m', mpi, 'Percentage', 30);
cp = ureal('c', cpi, 'Percentage', 30);
kp = ureal('k', kpi, 'Percentage', 30);
</pre>
</div>
@@ -217,8 +217,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-org2a9bf99" class="outline-3">
<h3 id="org2a9bf99"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org999e1c5" class="outline-3">
<h3 id="org999e1c5"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@@ -263,8 +263,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org3d4902a" class="outline-3">
<h3 id="org3d4902a"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org18c1c3f" class="outline-3">
<h3 id="org18c1c3f"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We have to following equations of motion:
@@ -294,7 +294,7 @@ In order to verify that the formula is correct, let&rsquo;s take the same mass-s
<p>
And we obtain
\[ \frac{x}{F} = \frac{m^\prime s^2 + c^\prime s + k^\prime}{(ms^2 + cs + k)(m^\prime s^2 + c^\prime s + k^\prime) + ms^2(cs + k)} \]
Which is the same transfer function that was obtained in section <a href="#org232d01f">1</a> (Eq. \eqref{org2d73355}).
Which is the same transfer function that was obtained in section <a href="#org232d01f">1</a> (Eq. \eqref{eq:plant_simple_system}).
</p>
</div>
</div>
@@ -316,7 +316,7 @@ The main resonance of the support is then \(\omega^\prime = \sqrt{\frac{m^\prime
c0 = 5e4;
k0 = 1e8;
Gp0 = 1<span class="org-type">/</span>(m0<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c0<span class="org-type">*</span>s <span class="org-type">+</span> k0);
Gp0 = 1/(m0*s^2 + c0*s + k0);
</pre>
</div>
@@ -347,10 +347,10 @@ The parameters are defined below.
</p>
<div class="org-src-container">
<pre class="src src-matlab">r0 = 0.5;
tau = 1<span class="org-type">/</span>(100<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>);
tau = 1/(100*2*pi);
rinf = 10;
wI = (tau<span class="org-type">*</span>s <span class="org-type">+</span> r0)<span class="org-type">/</span>((tau<span class="org-type">/</span>rinf)<span class="org-type">*</span>s <span class="org-type">+</span> 1);
wI = (tau*s + r0)/((tau/rinf)*s + 1);
</pre>
</div>
@@ -358,7 +358,7 @@ wI = (tau<span class="org-type">*</span>s <span class="org-type">+</span> r0)<sp
We then generate a complex \(\Delta\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">DeltaI = ucomplex(<span class="org-string">'A'</span>,0);
<pre class="src src-matlab">DeltaI = ucomplex('A',0);
</pre>
</div>
@@ -366,7 +366,7 @@ We then generate a complex \(\Delta\).
We generate the uncertain plant \(G^\prime(s)\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gp = Gp0<span class="org-type">*</span>(1<span class="org-type">+</span>wI<span class="org-type">*</span>DeltaI);
<pre class="src src-matlab">Gp = Gp0*(1+wI*DeltaI);
</pre>
</div>
@@ -445,12 +445,12 @@ And we generate three isolation platforms:
Soft Isolation Platform:
</p>
<div class="org-src-container">
<pre class="src src-matlab">k_soft = m<span class="org-type">*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5)<span class="org-type">^</span>2;
c_soft = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_soft);
<pre class="src src-matlab">k_soft = m*(2*pi*5)^2;
c_soft = 0.1*sqrt(m*k_soft);
G_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp);
G0_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp0);
wiI_soft = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>G0_soft<span class="org-type">*</span>wI;
G_soft = 1/(m*s^2 + c_soft*s + k_soft + m*s^2*(c_soft*s + k_soft)*Gp);
G0_soft = 1/(m*s^2 + c_soft*s + k_soft + m*s^2*(c_soft*s + k_soft)*Gp0);
wiI_soft = Gp0*m*s^2*(c_soft*s + k_soft)*G0_soft*wI;
</pre>
</div>
@@ -458,12 +458,12 @@ wiI_soft = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<s
Mid Isolation Platform
</p>
<div class="org-src-container">
<pre class="src src-matlab">k_mid = m<span class="org-type">*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50)<span class="org-type">^</span>2;
c_mid = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_mid);
<pre class="src src-matlab">k_mid = m*(2*pi*50)^2;
c_mid = 0.1*sqrt(m*k_mid);
G_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp);
G0_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp0);
wiI_mid = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>G0_mid<span class="org-type">*</span>wI;
G_mid = 1/(m*s^2 + c_mid*s + k_mid + m*s^2*(c_mid*s + k_mid)*Gp);
G0_mid = 1/(m*s^2 + c_mid*s + k_mid + m*s^2*(c_mid*s + k_mid)*Gp0);
wiI_mid = Gp0*m*s^2*(c_mid*s + k_mid)*G0_mid*wI;
</pre>
</div>
@@ -471,12 +471,12 @@ wiI_mid = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<sp
Stiff Isolation Platform
</p>
<div class="org-src-container">
<pre class="src src-matlab">k_stiff = m<span class="org-type">*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>500)<span class="org-type">^</span>2;
c_stiff = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_stiff);
<pre class="src src-matlab">k_stiff = m*(2*pi*500)^2;
c_stiff = 0.1*sqrt(m*k_stiff);
G_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp);
G0_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp0);
wiI_stiff = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>G0_stiff<span class="org-type">*</span>wI;
G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + m*s^2*(c_stiff*s + k_stiff)*Gp);
G0_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + m*s^2*(c_stiff*s + k_stiff)*Gp0);
wiI_stiff = Gp0*m*s^2*(c_stiff*s + k_stiff)*G0_stiff*wI;
</pre>
</div>
@@ -600,8 +600,8 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
</div>
</div>
<div id="outline-container-org999e1c5" class="outline-3">
<h3 id="org999e1c5"><span class="section-number-3">2.7</span> Conclusion</h3>
<div id="outline-container-orgde3616e" class="outline-3">
<h3 id="orgde3616e"><span class="section-number-3">2.7</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-7">
<div class="important">
<p>
@@ -629,7 +629,7 @@ Thus, if a stiff isolation platform is used, the recommendation is to have the l
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-17 ven. 09:35</p>
<p class="date">Created: 2020-05-05 mar. 10:33</p>
</div>
</body>
</html>