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Thomas Dehaeze
2020-04-01 16:17:26 +02:00
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@@ -1,11 +1,10 @@
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-03-26 jeu. 17:25 -->
<!-- 2020-04-01 mer. 16:14 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Effect of Uncertainty on the payload&rsquo;s dynamics on the isolation platform dynamics</title>
@@ -203,50 +202,28 @@
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<script type="text/javascript" src="./js/readtheorg.js"></script>
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@@ -272,22 +249,22 @@ for the JavaScript code in this tag.
<ul>
<li><a href="#orgcc5f0ec">1. Simple Introductory Example</a>
<ul>
<li><a href="#org00d3412">1.1. Equations of motion</a></li>
<li><a href="#org6264842">1.1. Equations of motion</a></li>
<li><a href="#org4efccbf">1.2. Initialization of the payload dynamics</a></li>
<li><a href="#orgb400ca3">1.3. Initialization of the isolation platform</a></li>
<li><a href="#orgd0dd88b">1.4. Comparison</a></li>
<li><a href="#org3bef024">1.5. Conclusion</a></li>
<li><a href="#org1b051ce">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org1f8e63e">2. Generalization to arbitrary dynamics</a>
<ul>
<li><a href="#orgc4fa63e">2.1. Introduction</a></li>
<li><a href="#org6264842">2.2. Equations of motion</a></li>
<li><a href="#org35ac80d">2.2. Equations of motion</a></li>
<li><a href="#orge217a33">2.3. Impedance \(G^\prime(s)\) of a mass-spring-damper payload</a></li>
<li><a href="#org0ee44da">2.4. First Analytical analysis</a></li>
<li><a href="#orgfe81c1c">2.5. Impedance of the Payload and Dynamical Uncertainty</a></li>
<li><a href="#orgcc0c290">2.6. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org5e0366d">2.7. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#org5e0366d">2.6. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#orgcc0c290">2.7. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org1466bd9">2.8. Reduce the Uncertainty on the plant</a>
<ul>
<li><a href="#org4be463f">2.8.1. Effect of the platform&rsquo;s stiffness \(k\)</a></li>
@@ -295,7 +272,7 @@ for the JavaScript code in this tag.
<li><a href="#org9086831">2.8.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul>
</li>
<li><a href="#org1b051ce">2.9. Conclusion</a></li>
<li><a href="#org43f33dc">2.9. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -349,8 +326,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the payloa
</div>
</div>
<div id="outline-container-org00d3412" class="outline-3">
<h3 id="org00d3412"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-org6264842" class="outline-3">
<h3 id="org6264842"><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="#orgaa77a57">1</a>, we obtain:
@@ -418,6 +395,7 @@ One can see that the payload has a resonance frequency of \(\omega_0^\prime = 25
</div>
</div>
</div>
<div id="outline-container-orgb400ca3" class="outline-3">
<h3 id="orgb400ca3"><span class="section-number-3">1.3</span> Initialization of the isolation platform</h3>
<div class="outline-text-3" id="text-1-3">
@@ -456,8 +434,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-org3bef024" class="outline-3">
<h3 id="org3bef024"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org1b051ce" class="outline-3">
<h3 id="org1b051ce"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@@ -509,8 +487,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org6264842" class="outline-3">
<h3 id="org6264842"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org35ac80d" class="outline-3">
<h3 id="org35ac80d"><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:
@@ -757,10 +735,47 @@ A set of uncertainty payload&rsquo;s impedance transfer functions is shown in Fi
</div>
</div>
<div id="outline-container-orgcc0c290" class="outline-3">
<h3 id="orgcc0c290"><span class="section-number-3">2.6</span> Effect of the Isolation platform Stiffness</h3>
<div id="outline-container-org5e0366d" class="outline-3">
<h3 id="org5e0366d"><span class="section-number-3">2.6</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-6">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + G_0^\prime(s)} \cdot \frac{1}{1 + \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#org90e7eef">9</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} = G_0(s) G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="org90e7eef" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Inverse Multiplicative Uncertainty</p>
</div>
</div>
</div>
<div id="outline-container-orgcc0c290" class="outline-3">
<h3 id="orgcc0c290"><span class="section-number-3">2.7</span> Effect of the Isolation platform Stiffness</h3>
<div class="outline-text-3" id="text-2-7">
<p>
Let&rsquo;s first fix the mass of the isolation platform:
</p>
<div class="org-src-container">
@@ -785,6 +800,8 @@ Soft Isolation Platform:
c_soft = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>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> 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> Gp0);
wiI_soft = Gp0<span class="org-type">*</span>G0_soft<span class="org-type">*</span>wI;
</pre>
</div>
@@ -796,6 +813,8 @@ Mid Isolation Platform
c_mid = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>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> 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> Gp0);
wiI_mid = Gp0<span class="org-type">*</span>G0_mid<span class="org-type">*</span>wI;
</pre>
</div>
@@ -807,55 +826,20 @@ Stiff Isolation Platform
c_stiff = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>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> 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> Gp0);
wiI_stiff = Gp0<span class="org-type">*</span>G0_stiff<span class="org-type">*</span>wI;
</pre>
</div>
<p>
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org864ba65">9</a>.
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org864ba65">10</a>.
</p>
<div id="org864ba65" class="figure">
<p><img src="figs/plant_uncertainty_impedance_payload.png" alt="plant_uncertainty_impedance_payload.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_impedance_payload.png">png</a>, <a href="./figs/plant_uncertainty_impedance_payload.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org5e0366d" class="outline-3">
<h3 id="org5e0366d"><span class="section-number-3">2.7</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-7">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + G_0^\prime(s)} \cdot \frac{1}{1 + \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#org90e7eef">10</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} = G_0(s) G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="org90e7eef" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Inverse Multiplicative Uncertainty</p>
<p><span class="figure-number">Figure 10: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_impedance_payload.png">png</a>, <a href="./figs/plant_uncertainty_impedance_payload.pdf">pdf</a>)</p>
</div>
</div>
</div>
@@ -961,8 +945,8 @@ 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-org1b051ce" class="outline-3">
<h3 id="org1b051ce"><span class="section-number-3">2.9</span> Conclusion</h3>
<div id="outline-container-org43f33dc" class="outline-3">
<h3 id="org43f33dc"><span class="section-number-3">2.9</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-9">
<div class="important">
<p>
@@ -986,7 +970,7 @@ In that case, maximizing the stiffness of the payload is a good idea.
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-26 jeu. 17:25</p>
<p class="date">Created: 2020-04-01 mer. 16:14</p>
</div>
</body>
</html>