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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
parent 2deb41939d
commit f7714a1449
33 changed files with 4627 additions and 2915 deletions
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docs/uncertainty_payload.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 payload&rsquo;s dynamics on the isolation platform dynamics</title>
<meta name="generator" content="Org mode" />
@@ -37,17 +37,17 @@
<ul>
<li><a href="#orgcc5f0ec">1. Simple Introductory Example</a>
<ul>
<li><a href="#orgf75e223">1.1. Equations of motion</a></li>
<li><a href="#org5ed1517">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="#orgd1e600e">1.5. Conclusion</a></li>
<li><a href="#org3f697cc">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="#org5ed1517">2.2. Equations of motion</a></li>
<li><a href="#org3367211">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>
@@ -60,7 +60,7 @@
<li><a href="#org9086831">2.8.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul>
</li>
<li><a href="#org3f697cc">2.9. Conclusion</a></li>
<li><a href="#org272e76a">2.9. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -114,8 +114,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the payloa
</div>
</div>
<div id="outline-container-orgf75e223" class="outline-3">
<h3 id="orgf75e223"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-org5ed1517" class="outline-3">
<h3 id="org5ed1517"><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:
@@ -152,8 +152,8 @@ Let the payload have:
kpi = 5e6;
cpi = 3e3;
kpi = (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50)<span class="org-type">^</span>2<span class="org-type">*</span>mpi;
cpi = 0.2<span class="org-type">*</span>sqrt(kpi<span class="org-type">*</span>mpi);
kpi = (2*pi*50)^2*mpi;
cpi = 0.2*sqrt(kpi*mpi);
</pre>
</div>
@@ -161,9 +161,9 @@ cpi = 0.2<span class="org-type">*</span>sqrt(kpi<span class="org-type">*</span>m
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">'Range'</span>, [1, 100]);
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, 'Range', [1, 100]);
cp = ureal('c', cpi, 'Percentage', 30);
kp = ureal('k', kpi, 'Percentage', 30);
</pre>
</div>
@@ -222,8 +222,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-orgd1e600e" class="outline-3">
<h3 id="orgd1e600e"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org3f697cc" class="outline-3">
<h3 id="org3f697cc"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@@ -275,8 +275,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org5ed1517" class="outline-3">
<h3 id="org5ed1517"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org3367211" class="outline-3">
<h3 id="org3367211"><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:
@@ -343,7 +343,7 @@ And we obtain
\end{align*}
<p>
Which is the same transfer function that was obtained in section <a href="#org971d11c">1</a> (Eq. \eqref{orge5d69a3}).
Which is the same transfer function that was obtained in section <a href="#org971d11c">1</a> (Eq. \eqref{eq:plant_simple_system}).
</p>
<p>
@@ -455,7 +455,7 @@ The main resonance of the payload is then \(\omega^\prime = \sqrt{\frac{m^\prime
c0 = 3e2;
k0 = 5e5;
Gp0 = (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))<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 = (m0*s^2 * (c0*s + k0))/(m0*s^2 + c0*s + k0);
</pre>
</div>
@@ -486,10 +486,10 @@ 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>(50<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>);
tau = 1/(50*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>
@@ -497,7 +497,7 @@ 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>
@@ -505,7 +505,7 @@ 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>
@@ -583,12 +583,12 @@ 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> 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;
G_soft = 1/(m*s^2 + c_soft*s + k_soft + Gp);
G0_soft = 1/(m*s^2 + c_soft*s + k_soft + Gp0);
wiI_soft = Gp0*G0_soft*wI;
</pre>
</div>
@@ -596,12 +596,12 @@ wiI_soft = Gp0<span class="org-type">*</span>G0_soft<span class="org-type">*</sp
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> 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;
G_mid = 1/(m*s^2 + c_mid*s + k_mid + Gp);
G0_mid = 1/(m*s^2 + c_mid*s + k_mid + Gp0);
wiI_mid = Gp0*G0_mid*wI;
</pre>
</div>
@@ -609,12 +609,12 @@ wiI_mid = Gp0<span class="org-type">*</span>G0_mid<span class="org-type">*</span
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> 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;
G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp);
G0_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp0);
wiI_stiff = Gp0*G0_stiff*wI;
</pre>
</div>
@@ -732,8 +732,8 @@ 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-org3f697cc" class="outline-3">
<h3 id="org3f697cc"><span class="section-number-3">2.9</span> Conclusion</h3>
<div id="outline-container-org272e76a" class="outline-3">
<h3 id="org272e76a"><span class="section-number-3">2.9</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-9">
<div class="important">
<p>
@@ -757,7 +757,7 @@ 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: 2020-04-17 ven. 09:35</p>
<p class="date">Created: 2020-05-05 mar. 10:33</p>
</div>
</body>
</html>