Use online CSS and JS
This commit is contained in:
@@ -1,79 +1,83 @@
|
||||
<?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>
|
||||
<!-- 2020-04-17 ven. 09:35 -->
|
||||
<!-- 2021-02-20 sam. 23:08 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<title>Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</title>
|
||||
<meta name="generator" content="Org mode" />
|
||||
<meta name="author" content="Dehaeze Thomas" />
|
||||
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
||||
<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
||||
<script src="./js/jquery.min.js"></script>
|
||||
<script src="./js/bootstrap.min.js"></script>
|
||||
<script src="./js/jquery.stickytableheaders.min.js"></script>
|
||||
<script src="./js/readtheorg.js"></script>
|
||||
<script>MathJax = {
|
||||
tex: {
|
||||
tags: 'ams',
|
||||
macros: {bm: ["\\boldsymbol{#1}",1],}
|
||||
}
|
||||
};
|
||||
</script>
|
||||
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
||||
<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
|
||||
<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
|
||||
<script>
|
||||
MathJax = {
|
||||
svg: {
|
||||
scale: 1,
|
||||
fontCache: "global"
|
||||
},
|
||||
tex: {
|
||||
tags: "ams",
|
||||
multlineWidth: "%MULTLINEWIDTH",
|
||||
tagSide: "right",
|
||||
macros: {bm: ["\\boldsymbol{#1}",1],},
|
||||
tagIndent: ".8em"
|
||||
}
|
||||
};
|
||||
</script>
|
||||
<script id="MathJax-script" async
|
||||
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
<div id="org-div-home-and-up">
|
||||
<a accesskey="h" href="./index.html"> UP </a>
|
||||
|
|
||||
<a accesskey="H" href="./index.html"> HOME </a>
|
||||
<a accesskey="H" href="../../index.html"> HOME </a>
|
||||
</div><div id="content">
|
||||
<h1 class="title">Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org9e9f810">1. Disturbances</a></li>
|
||||
<li><a href="#orgc44cf7e">2. Effect of disturbances on the position error</a>
|
||||
<li><a href="#org1d00c3b">1. Disturbances</a></li>
|
||||
<li><a href="#orgd8c957a">2. Effect of disturbances on the position error</a>
|
||||
<ul>
|
||||
<li><a href="#org524df41">2.1. Initialization</a></li>
|
||||
<li><a href="#orgaf88c9f">2.2. Identification</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="#org0448746">2.5. Direct Forces (Compliance).</a></li>
|
||||
<li><a href="#org626fe57">2.6. Conclusion</a></li>
|
||||
<li><a href="#org35d2f91">2.1. Initialization</a></li>
|
||||
<li><a href="#org094b5a7">2.2. Identification</a></li>
|
||||
<li><a href="#org7e3e79d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
|
||||
<li><a href="#org19c4b17">2.4. Effect of Ground motion (Transmissibility).</a></li>
|
||||
<li><a href="#org0890188">2.5. Direct Forces (Compliance).</a></li>
|
||||
<li><a href="#orgea2d185">2.6. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org6527e58">3. Effect of granite stiffness</a>
|
||||
<li><a href="#orgf6c1d0f">3. Effect of granite stiffness</a>
|
||||
<ul>
|
||||
<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a>
|
||||
<li><a href="#org9b72d9e">3.1. Analytical Analysis</a>
|
||||
<ul>
|
||||
<li><a href="#orgbc34a65">3.1.1. Simple mass-spring-damper model</a></li>
|
||||
<li><a href="#org4ddec32">3.1.2. General Case</a></li>
|
||||
<li><a href="#orgcb7fd26">3.1.1. Simple mass-spring-damper model</a></li>
|
||||
<li><a href="#orgb9ac8f4">3.1.2. General Case</a></li>
|
||||
</ul>
|
||||
</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="#orgf388e39">3.4. Conclusion</a></li>
|
||||
<li><a href="#orge9846e9">3.2. Soft Granite</a></li>
|
||||
<li><a href="#org699ef0a">3.3. Effect of the Granite transfer function</a></li>
|
||||
<li><a href="#org26b072f">3.4. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
|
||||
<li><a href="#org92ed7b0">4. Open Loop Budget Error</a>
|
||||
<ul>
|
||||
<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="#orgef96b89">4.3. Cumulative Amplitude Spectrum</a></li>
|
||||
<li><a href="#org0be3c47">4.4. Conclusion</a></li>
|
||||
<li><a href="#orgf8bb217">4.1. Noise Budgeting - Theory</a></li>
|
||||
<li><a href="#org0a97ffb">4.2. Power Spectral Densities</a></li>
|
||||
<li><a href="#org44eafed">4.3. Cumulative Amplitude Spectrum</a></li>
|
||||
<li><a href="#org091b440">4.4. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org34c0f38">5. Closed Loop Budget Error</a>
|
||||
<li><a href="#org5bb46c4">5. Closed Loop Budget Error</a>
|
||||
<ul>
|
||||
<li><a href="#orgdfef0eb">5.1. Approximation of the effect of feedback on the motion error</a></li>
|
||||
<li><a href="#orgf2d36a1">5.2. Reduction thanks to feedback - Required bandwidth</a></li>
|
||||
<li><a href="#org2cb5e9a">5.1. Approximation of the effect of feedback on the motion error</a></li>
|
||||
<li><a href="#org15ee341">5.2. Reduction thanks to feedback - Required bandwidth</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org08f4548">6. Conclusion</a></li>
|
||||
<li><a href="#orgf3b0566">6. Conclusion</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
@@ -86,18 +90,18 @@ In this document is studied how the stiffness of the nano-hexapod will impact th
|
||||
It is divided in the following sections:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Section <a href="#org17d3d6a">1</a>: the disturbances are listed and their Power Spectral Densities (PSD) are shown</li>
|
||||
<li>Section <a href="#orgf9e4300">2</a>: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses</li>
|
||||
<li>Section <a href="#orgd4105b6">3</a>:</li>
|
||||
<li>Section <a href="#org5d05990">4</a>: from both the PSD of the disturbances and the transfer function from disturbances to sample’s position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)</li>
|
||||
<li>Section <a href="#orgd3503fb">5</a>: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values</li>
|
||||
<li>Section <a href="#orge637f3d">1</a>: the disturbances are listed and their Power Spectral Densities (PSD) are shown</li>
|
||||
<li>Section <a href="#org8db9681">2</a>: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses</li>
|
||||
<li>Section <a href="#orge639dbd">3</a>:</li>
|
||||
<li>Section <a href="#org5714572">4</a>: from both the PSD of the disturbances and the transfer function from disturbances to sample’s position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)</li>
|
||||
<li>Section <a href="#org6ed8184">5</a>: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values</li>
|
||||
</ul>
|
||||
|
||||
<div id="outline-container-org9e9f810" class="outline-2">
|
||||
<h2 id="org9e9f810"><span class="section-number-2">1</span> Disturbances</h2>
|
||||
<div id="outline-container-org1d00c3b" class="outline-2">
|
||||
<h2 id="org1d00c3b"><span class="section-number-2">1</span> Disturbances</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
<a id="org17d3d6a"></a>
|
||||
<a id="orge637f3d"></a>
|
||||
</p>
|
||||
<p>
|
||||
The main disturbances considered here are:
|
||||
@@ -113,7 +117,7 @@ The main disturbances considered here are:
|
||||
The level of these disturbances has been identified form experiments which are detailed in <a href="disturbances.html">this</a> document.
|
||||
</p>
|
||||
<p>
|
||||
The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures <a href="#org6b4e47c">1</a> and <a href="#orgb7b8e77">2</a>.
|
||||
The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures <a href="#org3baaaea">1</a> and <a href="#orga5ce540">2</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -121,14 +125,14 @@ In this study, the expected frequency content of the direct forces applied to th
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6b4e47c" class="figure">
|
||||
<div id="org3baaaea" class="figure">
|
||||
<p><img src="figs/opt_stiff_dist_gm.png" alt="opt_stiff_dist_gm.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Amplitude Spectral Density of the Ground Displacement (<a href="./figs/opt_stiff_dist_gm.png">png</a>, <a href="./figs/opt_stiff_dist_gm.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb7b8e77" class="figure">
|
||||
<div id="orga5ce540" class="figure">
|
||||
<p><img src="figs/opt_stiff_dist_fty_frz.png" alt="opt_stiff_dist_fty_frz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Amplitude Spectral Density of the “parasitic” forces comming from the Translation stage and the spindle (<a href="./figs/opt_stiff_dist_fty_frz.png">png</a>, <a href="./figs/opt_stiff_dist_fty_frz.pdf">pdf</a>)</p>
|
||||
@@ -136,32 +140,32 @@ In this study, the expected frequency content of the direct forces applied to th
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc44cf7e" class="outline-2">
|
||||
<h2 id="orgc44cf7e"><span class="section-number-2">2</span> Effect of disturbances on the position error</h2>
|
||||
<div id="outline-container-orgd8c957a" class="outline-2">
|
||||
<h2 id="orgd8c957a"><span class="section-number-2">2</span> Effect of disturbances on the position error</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="orgf9e4300"></a>
|
||||
<a id="org8db9681"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section, we use the Simscape model to identify the transfer function from disturbances to the position error of the sample.
|
||||
We do that for a wide range of nano-hexapod stiffnesses and we compare the obtained results.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org524df41" class="outline-3">
|
||||
<h3 id="org524df41"><span class="section-number-3">2.1</span> Initialization</h3>
|
||||
<div id="outline-container-org35d2f91" class="outline-3">
|
||||
<h3 id="org35d2f91"><span class="section-number-3">2.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
<pre class="src src-matlab"> initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -169,7 +173,7 @@ initializeMirror();
|
||||
We use a sample mass of 10kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 10);
|
||||
<pre class="src src-matlab"> initializeSample(<span class="org-string">'mass'</span>, 10);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -177,17 +181,17 @@ We use a sample mass of 10kg.
|
||||
We include gravity, and we use no controller.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeController();
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeController();
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgaf88c9f" class="outline-3">
|
||||
<h3 id="orgaf88c9f"><span class="section-number-3">2.2</span> Identification</h3>
|
||||
<div id="outline-container-org094b5a7" class="outline-3">
|
||||
<h3 id="org094b5a7"><span class="section-number-3">2.2</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
The considered inputs are:
|
||||
@@ -210,40 +214,40 @@ The outputs are <code>Ex</code>, <code>Ey</code>, <code>Ez</code>, <code>Erx</co
|
||||
We initialize the set of the nano-hexapod stiffnesses, and for each of them, we identify the dynamics from defined inputs to defined outputs.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
<pre class="src src-matlab"> Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org78dd34d" class="outline-3">
|
||||
<h3 id="org78dd34d"><span class="section-number-3">2.3</span> Sensitivity to Stages vibration (Filtering)</h3>
|
||||
<div id="outline-container-org7e3e79d" class="outline-3">
|
||||
<h3 id="org7e3e79d"><span class="section-number-3">2.3</span> Sensitivity to Stages vibration (Filtering)</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
The sensitivity the stage vibrations are displayed:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#orgf55ba1b">3</a>: sensitivity to vertical spindle vibrations</li>
|
||||
<li>Figure <a href="#orgce1ac2c">4</a>: sensitivity to vertical translation stage vibrations</li>
|
||||
<li>Figure <a href="#org1a24ee2">5</a>: sensitivity to horizontal (x) translation stage vibrations</li>
|
||||
<li>Figure <a href="#org0073408">3</a>: sensitivity to vertical spindle vibrations</li>
|
||||
<li>Figure <a href="#org0999f05">4</a>: sensitivity to vertical translation stage vibrations</li>
|
||||
<li>Figure <a href="#org6337962">5</a>: sensitivity to horizontal (x) translation stage vibrations</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgf55ba1b" class="figure">
|
||||
<div id="org0073408" class="figure">
|
||||
<p><img src="figs/opt_stiff_sensitivity_Frz.png" alt="opt_stiff_sensitivity_Frz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Sensitivity to Spindle vertical motion error (\(F_{rz}\)) to the vertical error position of the sample (\(E_z\)) (<a href="./figs/opt_stiff_sensitivity_Frz.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Frz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgce1ac2c" class="figure">
|
||||
<div id="org0999f05" class="figure">
|
||||
<p><img src="figs/opt_stiff_sensitivity_Fty_z.png" alt="opt_stiff_sensitivity_Fty_z.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Sensitivity to Translation stage vertical motion error (\(F_{ty,z}\)) to the vertical error position of the sample (\(E_z\)) (<a href="./figs/opt_stiff_sensitivity_Fty_z.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Fty_z.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org1a24ee2" class="figure">
|
||||
<div id="org6337962" class="figure">
|
||||
<p><img src="figs/opt_stiff_sensitivity_Fty_x.png" alt="opt_stiff_sensitivity_Fty_x.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Sensitivity to Translation stage \(x\) motion error (\(F_{ty,x}\)) to the error position of the sample in the \(x\) direction (\(E_x\)) (<a href="./figs/opt_stiff_sensitivity_Fty_x.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Fty_x.pdf">pdf</a>)</p>
|
||||
@@ -251,15 +255,15 @@ The sensitivity the stage vibrations are displayed:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd4ea2f4" class="outline-3">
|
||||
<h3 id="orgd4ea2f4"><span class="section-number-3">2.4</span> Effect of Ground motion (Transmissibility).</h3>
|
||||
<div id="outline-container-org19c4b17" class="outline-3">
|
||||
<h3 id="org19c4b17"><span class="section-number-3">2.4</span> Effect of Ground motion (Transmissibility).</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
The effect of Ground motion on the position error of the sample is shown in Figure <a href="#org212587b">6</a>.
|
||||
The effect of Ground motion on the position error of the sample is shown in Figure <a href="#org671bbd6">6</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org212587b" class="figure">
|
||||
<div id="org671bbd6" class="figure">
|
||||
<p><img src="figs/opt_stiff_sensitivity_Dw.png" alt="opt_stiff_sensitivity_Dw.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Sensitivity to Ground motion (\(D_{w}\)) to the position error of the sample (\(E_y\) and \(E_z\)) (<a href="./figs/opt_stiff_sensitivity_Dw.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Dw.pdf">pdf</a>)</p>
|
||||
@@ -267,15 +271,15 @@ The effect of Ground motion on the position error of the sample is shown in Figu
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0448746" class="outline-3">
|
||||
<h3 id="org0448746"><span class="section-number-3">2.5</span> Direct Forces (Compliance).</h3>
|
||||
<div id="outline-container-org0890188" class="outline-3">
|
||||
<h3 id="org0890188"><span class="section-number-3">2.5</span> Direct Forces (Compliance).</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<p>
|
||||
The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure <a href="#orga33395f">7</a>.
|
||||
The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure <a href="#org022a37b">7</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga33395f" class="figure">
|
||||
<div id="org022a37b" class="figure">
|
||||
<p><img src="figs/opt_stiff_sensitivity_Fd.png" alt="opt_stiff_sensitivity_Fd.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Sensitivity to Direct forces and torques applied to the sample (\(F_d\), \(M_d\)) to the position error of the sample (<a href="./figs/opt_stiff_sensitivity_Fd.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Fd.pdf">pdf</a>)</p>
|
||||
@@ -283,16 +287,16 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org626fe57" class="outline-3">
|
||||
<h3 id="org626fe57"><span class="section-number-3">2.6</span> Conclusion</h3>
|
||||
<div id="outline-container-orgea2d185" class="outline-3">
|
||||
<h3 id="orgea2d185"><span class="section-number-3">2.6</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<div class="important">
|
||||
<div class="important" id="orgb58fd33">
|
||||
<p>
|
||||
Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section <a href="#org5d05990">4</a>.
|
||||
In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section <a href="#org5714572">4</a>.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
@@ -300,38 +304,38 @@ In order to conclude on the optimal stiffness that will yield the smallest sampl
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6527e58" class="outline-2">
|
||||
<h2 id="org6527e58"><span class="section-number-2">3</span> Effect of granite stiffness</h2>
|
||||
<div id="outline-container-orgf6c1d0f" class="outline-2">
|
||||
<h2 id="orgf6c1d0f"><span class="section-number-2">3</span> Effect of granite stiffness</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="orgd4105b6"></a>
|
||||
<a id="orge639dbd"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgd3e5fe1" class="outline-3">
|
||||
<h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
|
||||
<div id="outline-container-org9b72d9e" class="outline-3">
|
||||
<h3 id="org9b72d9e"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
</div>
|
||||
<div id="outline-container-orgbc34a65" class="outline-4">
|
||||
<h4 id="orgbc34a65"><span class="section-number-4">3.1.1</span> Simple mass-spring-damper model</h4>
|
||||
<div id="outline-container-orgcb7fd26" class="outline-4">
|
||||
<h4 id="orgcb7fd26"><span class="section-number-4">3.1.1</span> Simple mass-spring-damper model</h4>
|
||||
<div class="outline-text-4" id="text-3-1-1">
|
||||
<p>
|
||||
Let’s consider the system shown in Figure <a href="#org8fb9606">8</a> consisting of two stacked mass-spring-damper systems.
|
||||
Let’s consider the system shown in Figure <a href="#org210f3bb">8</a> consisting of two stacked mass-spring-damper systems.
|
||||
The bottom one represents the granite, and the top one all the positioning stages.
|
||||
We want the smallest stage “deformation” \(d = x^\prime - x\) due to ground motion \(w\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org8fb9606" class="figure">
|
||||
<div id="org210f3bb" class="figure">
|
||||
<p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Mass Spring Damper system consisting of a granite and a positioning stage</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
If we write the equation of motion of the system in Figure <a href="#org8fb9606">8</a>, we obtain:
|
||||
If we write the equation of motion of the system in Figure <a href="#org210f3bb">8</a>, we obtain:
|
||||
</p>
|
||||
\begin{align}
|
||||
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
|
||||
@@ -347,8 +351,8 @@ If we note \(d = x^\prime - x\), we obtain:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4ddec32" class="outline-4">
|
||||
<h4 id="org4ddec32"><span class="section-number-4">3.1.2</span> General Case</h4>
|
||||
<div id="outline-container-orgb9ac8f4" class="outline-4">
|
||||
<h4 id="orgb9ac8f4"><span class="section-number-4">3.1.2</span> General Case</h4>
|
||||
<div class="outline-text-4" id="text-3-1-2">
|
||||
<p>
|
||||
Let’s now considering a general positioning stage defined by:
|
||||
@@ -359,7 +363,7 @@ Let’s now considering a general positioning stage defined by:
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org9702e0f" class="figure">
|
||||
<div id="org82d8583" class="figure">
|
||||
<p><img src="figs/general_system_granite_stiffness.png" alt="general_system_granite_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Mass Spring Damper representing the granite and a general representation of positioning stages</p>
|
||||
@@ -380,7 +384,7 @@ where:
|
||||
<li>\(F\) is the force applied by the position stages on the granite mass</li>
|
||||
</ul>
|
||||
|
||||
<div class="important">
|
||||
<div class="important" id="org3106f21">
|
||||
<p>
|
||||
We can express \(d\) as a function of \(w\):
|
||||
</p>
|
||||
@@ -416,47 +420,47 @@ which is the same as the previously derived equation.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9215f81" class="outline-3">
|
||||
<h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3>
|
||||
<div id="outline-container-orge9846e9" class="outline-3">
|
||||
<h3 id="orge9846e9"><span class="section-number-3">3.2</span> Soft Granite</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
Let’s initialize a soft granite and see how the sensitivity to disturbances will change.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGranite(<span class="org-string">'K'</span>, 5e5<span class="org-type">*</span>ones(3,1), <span class="org-string">'C'</span>, 5e3<span class="org-type">*</span>ones(3,1));
|
||||
<pre class="src src-matlab"> initializeGranite(<span class="org-string">'K'</span>, 5e5<span class="org-type">*</span>ones(3,1), <span class="org-string">'C'</span>, 5e3<span class="org-type">*</span>ones(3,1));
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8878556" class="outline-3">
|
||||
<h3 id="org8878556"><span class="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
|
||||
<div id="outline-container-org699ef0a" class="outline-3">
|
||||
<h3 id="org699ef0a"><span class="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
From Figure <a href="#org38146da">10</a>, we can see that having a “soft” granite suspension greatly lowers the sensitivity to ground motion.
|
||||
From Figure <a href="#org5ecd9f8">10</a>, we can see that having a “soft” granite suspension greatly lowers the sensitivity to ground motion.
|
||||
The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From Figures <a href="#orgc4c14fb">11</a> and <a href="#org533cc4b">12</a>, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
|
||||
From Figures <a href="#org71cfcc7">11</a> and <a href="#org0bb4ee0">12</a>, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org38146da" class="figure">
|
||||
<div id="org5ecd9f8" class="figure">
|
||||
<p><img src="figs/opt_stiff_soft_granite_Dw.png" alt="opt_stiff_soft_granite_Dw.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Dw.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Dw.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc4c14fb" class="figure">
|
||||
<div id="org71cfcc7" class="figure">
|
||||
<p><img src="figs/opt_stiff_soft_granite_Frz.png" alt="opt_stiff_soft_granite_Frz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Frz.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Frz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org533cc4b" class="figure">
|
||||
<div id="org0bb4ee0" class="figure">
|
||||
<p><img src="figs/opt_stiff_soft_granite_Fd.png" alt="opt_stiff_soft_granite_Fd.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Fd.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Fd.pdf">pdf</a>)</p>
|
||||
@@ -464,10 +468,10 @@ From Figures <a href="#orgc4c14fb">11</a> and <a href="#org533cc4b">12</a>, we s
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf388e39" class="outline-3">
|
||||
<h3 id="orgf388e39"><span class="section-number-3">3.4</span> Conclusion</h3>
|
||||
<div id="outline-container-org26b072f" class="outline-3">
|
||||
<h3 id="org26b072f"><span class="section-number-3">3.4</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<div class="important">
|
||||
<div class="important" id="orge437d65">
|
||||
<p>
|
||||
Having a soft granite suspension greatly decreases the sensitivity the ground motion.
|
||||
Also, it does not affect much the sensitivity to stage vibration and direct forces.
|
||||
@@ -479,29 +483,29 @@ Thus the level of sample vibration can be reduced by using a soft granite suspen
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8a88fb0" class="outline-2">
|
||||
<h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
|
||||
<div id="outline-container-org92ed7b0" class="outline-2">
|
||||
<h2 id="org92ed7b0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org5d05990"></a>
|
||||
<a id="org5714572"></a>
|
||||
</p>
|
||||
<p>
|
||||
Now that the frequency content of disturbances have been estimated (Section <a href="#org17d3d6a">1</a>) and the transfer functions from disturbances to the position error of the sample have been identified (Section <a href="#orgf9e4300">2</a>), we can compute the level of sample vibration due to the disturbances.
|
||||
Now that the frequency content of disturbances have been estimated (Section <a href="#orge637f3d">1</a>) and the transfer functions from disturbances to the position error of the sample have been identified (Section <a href="#org8db9681">2</a>), we can compute the level of sample vibration due to the disturbances.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We then can conclude and the nano-hexapod stiffness that will lower the sample position error.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org6bd588f" class="outline-3">
|
||||
<h3 id="org6bd588f"><span class="section-number-3">4.1</span> Noise Budgeting - Theory</h3>
|
||||
<div id="outline-container-orgf8bb217" class="outline-3">
|
||||
<h3 id="orgf8bb217"><span class="section-number-3">4.1</span> Noise Budgeting - Theory</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
Let’s consider Figure <a href="#org7ff50a0">13</a> there \(G_d(s)\) is the transfer function from a signal \(d\) (the perturbation) to a signal \(y\) (the sample’s position error).
|
||||
Let’s consider Figure <a href="#orgf48ae3a">13</a> there \(G_d(s)\) is the transfer function from a signal \(d\) (the perturbation) to a signal \(y\) (the sample’s position error).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org7ff50a0" class="figure">
|
||||
<div id="orgf48ae3a" class="figure">
|
||||
<p><img src="figs/psd_change_tf.png" alt="psd_change_tf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Signal \(d\) going through and LTI transfer function \(G_d(s)\) to give a signal \(y\)</p>
|
||||
@@ -515,7 +519,7 @@ We can compute the Power Spectral Density (PSD) of signal \(y\) from the PSD of
|
||||
\end{equation}
|
||||
|
||||
<p>
|
||||
If we now consider multiple disturbances \(d_1, \dots, d_n\) as shown in Figure <a href="#orgc24bdf6">14</a>, we have that:
|
||||
If we now consider multiple disturbances \(d_1, \dots, d_n\) as shown in Figure <a href="#orga4616f6">14</a>, we have that:
|
||||
</p>
|
||||
\begin{equation}
|
||||
S_{y}(\omega) = \left|G_{d_1}(j\omega)\right|^2 S_{d_1}(\omega) + \dots + \left|G_{d_n}(j\omega)\right|^2 S_{d_n}(\omega) \label{eq:sum_psd}
|
||||
@@ -527,7 +531,7 @@ Sometimes, we prefer to compute the <b>Amplitude</b> Spectral Density (ASD) whic
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgc24bdf6" class="figure">
|
||||
<div id="orga4616f6" class="figure">
|
||||
<p><img src="figs/psd_change_tf_multiple_pert.png" alt="psd_change_tf_multiple_pert.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Block diagram showing and output \(y\) resulting from the addition of multiple perturbations \(d_i\)</p>
|
||||
@@ -554,8 +558,8 @@ The CAS evaluation for all frequency corresponds to the rms value of the conside
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcc86f59" class="outline-3">
|
||||
<h3 id="orgcc86f59"><span class="section-number-3">4.2</span> Power Spectral Densities</h3>
|
||||
<div id="outline-container-org0a97ffb" class="outline-3">
|
||||
<h3 id="org0a97ffb"><span class="section-number-3">4.2</span> Power Spectral Densities</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
We compute the effect of perturbations on the motion error thanks to Eq. \eqref{eq:psd_transfer_function}.
|
||||
@@ -565,29 +569,29 @@ We compute the effect of perturbations on the motion error thanks to Eq. \eqref{
|
||||
The result is shown in:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#orgd3d7b28">15</a>: PSD of the vertical sample’s motion error due to vertical ground motion</li>
|
||||
<li>Figure <a href="#orgd8e87cd">16</a>: PSD of the vertical sample’s motion error due to vertical vibrations of the Spindle</li>
|
||||
<li>Figure <a href="#orgd8bc100">15</a>: PSD of the vertical sample’s motion error due to vertical ground motion</li>
|
||||
<li>Figure <a href="#org631c8cb">16</a>: PSD of the vertical sample’s motion error due to vertical vibrations of the Spindle</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgd3d7b28" class="figure">
|
||||
<div id="orgd8bc100" class="figure">
|
||||
<p><img src="figs/opt_stiff_psd_dz_gm.png" alt="opt_stiff_psd_dz_gm.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Amplitude Spectral Density of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_psd_dz_gm.png">png</a>, <a href="./figs/opt_stiff_psd_dz_gm.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd8e87cd" class="figure">
|
||||
<div id="org631c8cb" class="figure">
|
||||
<p><img src="figs/opt_stiff_psd_dz_rz.png" alt="opt_stiff_psd_dz_rz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_psd_dz_rz.png">png</a>, <a href="./figs/opt_stiff_psd_dz_rz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We compute the effect of all perturbations on the vertical position error using Eq. \eqref{eq:sum_psd} and the resulting PSD is shown in Figure <a href="#orgdbfb5e0">17</a>.
|
||||
We compute the effect of all perturbations on the vertical position error using Eq. \eqref{eq:sum_psd} and the resulting PSD is shown in Figure <a href="#org5ce09d9">17</a>.
|
||||
</p>
|
||||
|
||||
<div id="orgdbfb5e0" class="figure">
|
||||
<div id="org5ce09d9" class="figure">
|
||||
<p><img src="figs/opt_stiff_psd_dz_tot.png" alt="opt_stiff_psd_dz_tot.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Amplitude Spectral Density of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_psd_dz_tot.png">png</a>, <a href="./figs/opt_stiff_psd_dz_tot.pdf">pdf</a>)</p>
|
||||
@@ -595,16 +599,16 @@ We compute the effect of all perturbations on the vertical position error using
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgef96b89" class="outline-3">
|
||||
<h3 id="orgef96b89"><span class="section-number-3">4.3</span> Cumulative Amplitude Spectrum</h3>
|
||||
<div id="outline-container-org44eafed" class="outline-3">
|
||||
<h3 id="org44eafed"><span class="section-number-3">4.3</span> Cumulative Amplitude Spectrum</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<p>
|
||||
Similarly, the Cumulative Amplitude Spectrum of the sample vibrations are shown:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org488d65f">18</a>: due to vertical ground motion</li>
|
||||
<li>Figure <a href="#orge5458c6">19</a>: due to vertical vibrations of the Spindle</li>
|
||||
<li>Figure <a href="#orgf6888f0">20</a>: due to all considered perturbations</li>
|
||||
<li>Figure <a href="#orgccc4019">18</a>: due to vertical ground motion</li>
|
||||
<li>Figure <a href="#orgbc68410">19</a>: due to vertical vibrations of the Spindle</li>
|
||||
<li>Figure <a href="#org0bcb1fa">20</a>: due to all considered perturbations</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
@@ -612,21 +616,21 @@ The black dashed line corresponds to the performance objective of a sample vibra
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org488d65f" class="figure">
|
||||
<div id="orgccc4019" class="figure">
|
||||
<p><img src="figs/opt_stiff_cas_dz_gm.png" alt="opt_stiff_cas_dz_gm.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_dz_gm.png">png</a>, <a href="./figs/opt_stiff_cas_dz_gm.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orge5458c6" class="figure">
|
||||
<div id="orgbc68410" class="figure">
|
||||
<p><img src="figs/opt_stiff_cas_dz_rz.png" alt="opt_stiff_cas_dz_rz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_dz_rz.png">png</a>, <a href="./figs/opt_stiff_cas_dz_rz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgf6888f0" class="figure">
|
||||
<div id="org0bcb1fa" class="figure">
|
||||
<p><img src="figs/opt_stiff_cas_dz_tot.png" alt="opt_stiff_cas_dz_tot.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_dz_tot.png">png</a>, <a href="./figs/opt_stiff_cas_dz_tot.pdf">pdf</a>)</p>
|
||||
@@ -634,12 +638,12 @@ The black dashed line corresponds to the performance objective of a sample vibra
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0be3c47" class="outline-3">
|
||||
<h3 id="org0be3c47"><span class="section-number-3">4.4</span> Conclusion</h3>
|
||||
<div id="outline-container-org091b440" class="outline-3">
|
||||
<h3 id="org091b440"><span class="section-number-3">4.4</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="important">
|
||||
<div class="important" id="org86258fc">
|
||||
<p>
|
||||
From Figure <a href="#orgf6888f0">20</a>, we can see that a soft nano-hexapod \(k<10^6\ [N/m]\) significantly reduces the effect of perturbations from 20Hz to 300Hz.
|
||||
From Figure <a href="#org0bcb1fa">20</a>, we can see that a soft nano-hexapod \(k<10^6\ [N/m]\) significantly reduces the effect of perturbations from 20Hz to 300Hz.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
@@ -647,25 +651,25 @@ From Figure <a href="#orgf6888f0">20</a>, we can see that a soft nano-hexapod \(
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org34c0f38" class="outline-2">
|
||||
<h2 id="org34c0f38"><span class="section-number-2">5</span> Closed Loop Budget Error</h2>
|
||||
<div id="outline-container-org5bb46c4" class="outline-2">
|
||||
<h2 id="org5bb46c4"><span class="section-number-2">5</span> Closed Loop Budget Error</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="orgd3503fb"></a>
|
||||
<a id="org6ed8184"></a>
|
||||
</p>
|
||||
<p>
|
||||
From the total open-loop power spectral density of the sample’s motion error, we can estimate what is the required control bandwidth for the sample’s motion error to be reduced down to \(10nm\).
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgdfef0eb" class="outline-3">
|
||||
<h3 id="orgdfef0eb"><span class="section-number-3">5.1</span> Approximation of the effect of feedback on the motion error</h3>
|
||||
<div id="outline-container-org2cb5e9a" class="outline-3">
|
||||
<h3 id="org2cb5e9a"><span class="section-number-3">5.1</span> Approximation of the effect of feedback on the motion error</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<p>
|
||||
Let’s consider Figure <a href="#org6308d80">21</a> where a controller \(K\) is used to reduce the effect of the disturbance \(d\) on the position error \(y\).
|
||||
Let’s consider Figure <a href="#orgd3c6de4">21</a> where a controller \(K\) is used to reduce the effect of the disturbance \(d\) on the position error \(y\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6308d80" class="figure">
|
||||
<div id="orgd3c6de4" class="figure">
|
||||
<p><img src="figs/effect_feedback_disturbance_diagram.png" alt="effect_feedback_disturbance_diagram.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 21: </span>Feedback System</p>
|
||||
@@ -678,7 +682,7 @@ The reduction of the impact of \(d\) on \(y\) thanks to feedback is described by
|
||||
\frac{y}{d} = \frac{G_d}{1 + KG}
|
||||
\end{equation}
|
||||
<p>
|
||||
The transfer functions corresponding to \(G_d\) are those identified in Section <a href="#orgf9e4300">2</a>.
|
||||
The transfer functions corresponding to \(G_d\) are those identified in Section <a href="#org8db9681">2</a>.
|
||||
</p>
|
||||
|
||||
|
||||
@@ -702,43 +706,43 @@ This will help to determine what is the approximate control bandwidth required s
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf2d36a1" class="outline-3">
|
||||
<h3 id="orgf2d36a1"><span class="section-number-3">5.2</span> Reduction thanks to feedback - Required bandwidth</h3>
|
||||
<div id="outline-container-org15ee341" class="outline-3">
|
||||
<h3 id="org15ee341"><span class="section-number-3">5.2</span> Reduction thanks to feedback - Required bandwidth</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
Let’s first see how does the Cumulative Amplitude Spectrum of the sample’s motion error is modified by the control.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In Figure <a href="#orgcbef465">22</a> is shown the Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses.
|
||||
In Figure <a href="#org67a0258">22</a> is shown the Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses.
|
||||
The controller used in this simulation is \(K_1\). The loop gain is then a pure integrator.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In Figure <a href="#orgd677910">23</a> is shown the expected RMS value of the sample’s position error as a function of the control bandwidth, both for \(K_1\) (left plot) and \(K_2\) (right plot).
|
||||
In Figure <a href="#orgdbc233d">23</a> is shown the expected RMS value of the sample’s position error as a function of the control bandwidth, both for \(K_1\) (left plot) and \(K_2\) (right plot).
|
||||
As expected, it is shown that \(K_2\) performs better than \(K_1\).
|
||||
This Figure tells us how much control bandwidth is required to attain a certain level of performance, and that for all the considered nano-hexapod stiffnesses.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample’s motion error less than 10nm rms are gathered in Table <a href="#org5ab4860">1</a>.
|
||||
The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample’s motion error less than 10nm rms are gathered in Table <a href="#org16b406d">1</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgcbef465" class="figure">
|
||||
<div id="org67a0258" class="figure">
|
||||
<p><img src="figs/opt_stiff_cas_closed_loop.png" alt="opt_stiff_cas_closed_loop.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 22: </span>Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth and 4 different nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_closed_loop.png">png</a>, <a href="./figs/opt_stiff_cas_closed_loop.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd677910" class="figure">
|
||||
<div id="orgdbc233d" class="figure">
|
||||
<p><img src="figs/opt_stiff_req_bandwidth_K1_K2.png" alt="opt_stiff_req_bandwidth_K1_K2.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 23: </span>Expected RMS value of the sample’s motion error \(E_z\) as a function of the control bandwidth when using \(K_1\) and \(K_2\) (<a href="./figs/opt_stiff_req_bandwidth_K1_K2.png">png</a>, <a href="./figs/opt_stiff_req_bandwidth_K1_K2.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
<table id="org5ab4860" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<table id="org16b406d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 1:</span> Approximate required control bandwidth such that the motion error is below \(10nm\)</caption>
|
||||
|
||||
<colgroup>
|
||||
@@ -798,12 +802,12 @@ The obtained required bandwidth (approximate upper and lower bounds) to obtained
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org08f4548" class="outline-2">
|
||||
<h2 id="org08f4548"><span class="section-number-2">6</span> Conclusion</h2>
|
||||
<div id="outline-container-orgf3b0566" class="outline-2">
|
||||
<h2 id="orgf3b0566"><span class="section-number-2">6</span> Conclusion</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<div class="important">
|
||||
<div class="important" id="org692b84e">
|
||||
<p>
|
||||
From Figure <a href="#orgd677910">23</a> and Table <a href="#org5ab4860">1</a>, we can clearly see three different results depending on the nano-hexapod stiffness:
|
||||
From Figure <a href="#orgdbc233d">23</a> and Table <a href="#org16b406d">1</a>, we can clearly see three different results depending on the nano-hexapod stiffness:
|
||||
</p>
|
||||
<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>
|
||||
@@ -817,7 +821,7 @@ From Figure <a href="#orgd677910">23</a> and Table <a href="#org5ab4860">1</a>,
|
||||
</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: 2021-02-20 sam. 23:08</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user