nass-simscape/docs/uniaxial.html

1903 lines
90 KiB
HTML
Raw Normal View History

<?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>
2020-02-25 18:21:17 +01:00
<!-- 2020-02-25 mar. 18:20 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Simscape Uniaxial Model</title>
<meta name="generator" content="Org mode" />
<meta name="author" content="Dehaeze Thomas" />
<style type="text/css">
<!--/*--><![CDATA[/*><!--*/
.title { text-align: center;
margin-bottom: .2em; }
.subtitle { text-align: center;
font-size: medium;
font-weight: bold;
margin-top:0; }
.todo { font-family: monospace; color: red; }
.done { font-family: monospace; color: green; }
.priority { font-family: monospace; color: orange; }
.tag { background-color: #eee; font-family: monospace;
padding: 2px; font-size: 80%; font-weight: normal; }
.timestamp { color: #bebebe; }
.timestamp-kwd { color: #5f9ea0; }
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
.underline { text-decoration: underline; }
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
p.verse { margin-left: 3%; }
pre {
border: 1px solid #ccc;
box-shadow: 3px 3px 3px #eee;
padding: 8pt;
font-family: monospace;
overflow: auto;
margin: 1.2em;
}
pre.src {
position: relative;
overflow: visible;
padding-top: 1.2em;
}
pre.src:before {
display: none;
position: absolute;
background-color: white;
top: -10px;
right: 10px;
padding: 3px;
border: 1px solid black;
}
pre.src:hover:before { display: inline;}
/* Languages per Org manual */
pre.src-asymptote:before { content: 'Asymptote'; }
pre.src-awk:before { content: 'Awk'; }
pre.src-C:before { content: 'C'; }
/* pre.src-C++ doesn't work in CSS */
pre.src-clojure:before { content: 'Clojure'; }
pre.src-css:before { content: 'CSS'; }
pre.src-D:before { content: 'D'; }
pre.src-ditaa:before { content: 'ditaa'; }
pre.src-dot:before { content: 'Graphviz'; }
pre.src-calc:before { content: 'Emacs Calc'; }
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
pre.src-fortran:before { content: 'Fortran'; }
pre.src-gnuplot:before { content: 'gnuplot'; }
pre.src-haskell:before { content: 'Haskell'; }
pre.src-hledger:before { content: 'hledger'; }
pre.src-java:before { content: 'Java'; }
pre.src-js:before { content: 'Javascript'; }
pre.src-latex:before { content: 'LaTeX'; }
pre.src-ledger:before { content: 'Ledger'; }
pre.src-lisp:before { content: 'Lisp'; }
pre.src-lilypond:before { content: 'Lilypond'; }
pre.src-lua:before { content: 'Lua'; }
pre.src-matlab:before { content: 'MATLAB'; }
pre.src-mscgen:before { content: 'Mscgen'; }
pre.src-ocaml:before { content: 'Objective Caml'; }
pre.src-octave:before { content: 'Octave'; }
pre.src-org:before { content: 'Org mode'; }
pre.src-oz:before { content: 'OZ'; }
pre.src-plantuml:before { content: 'Plantuml'; }
pre.src-processing:before { content: 'Processing.js'; }
pre.src-python:before { content: 'Python'; }
pre.src-R:before { content: 'R'; }
pre.src-ruby:before { content: 'Ruby'; }
pre.src-sass:before { content: 'Sass'; }
pre.src-scheme:before { content: 'Scheme'; }
pre.src-screen:before { content: 'Gnu Screen'; }
pre.src-sed:before { content: 'Sed'; }
pre.src-sh:before { content: 'shell'; }
pre.src-sql:before { content: 'SQL'; }
pre.src-sqlite:before { content: 'SQLite'; }
/* additional languages in org.el's org-babel-load-languages alist */
pre.src-forth:before { content: 'Forth'; }
pre.src-io:before { content: 'IO'; }
pre.src-J:before { content: 'J'; }
pre.src-makefile:before { content: 'Makefile'; }
pre.src-maxima:before { content: 'Maxima'; }
pre.src-perl:before { content: 'Perl'; }
pre.src-picolisp:before { content: 'Pico Lisp'; }
pre.src-scala:before { content: 'Scala'; }
pre.src-shell:before { content: 'Shell Script'; }
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
/* additional language identifiers per "defun org-babel-execute"
in ob-*.el */
pre.src-cpp:before { content: 'C++'; }
pre.src-abc:before { content: 'ABC'; }
pre.src-coq:before { content: 'Coq'; }
pre.src-groovy:before { content: 'Groovy'; }
/* additional language identifiers from org-babel-shell-names in
ob-shell.el: ob-shell is the only babel language using a lambda to put
the execution function name together. */
pre.src-bash:before { content: 'bash'; }
pre.src-csh:before { content: 'csh'; }
pre.src-ash:before { content: 'ash'; }
pre.src-dash:before { content: 'dash'; }
pre.src-ksh:before { content: 'ksh'; }
pre.src-mksh:before { content: 'mksh'; }
pre.src-posh:before { content: 'posh'; }
/* Additional Emacs modes also supported by the LaTeX listings package */
pre.src-ada:before { content: 'Ada'; }
pre.src-asm:before { content: 'Assembler'; }
pre.src-caml:before { content: 'Caml'; }
pre.src-delphi:before { content: 'Delphi'; }
pre.src-html:before { content: 'HTML'; }
pre.src-idl:before { content: 'IDL'; }
pre.src-mercury:before { content: 'Mercury'; }
pre.src-metapost:before { content: 'MetaPost'; }
pre.src-modula-2:before { content: 'Modula-2'; }
pre.src-pascal:before { content: 'Pascal'; }
pre.src-ps:before { content: 'PostScript'; }
pre.src-prolog:before { content: 'Prolog'; }
pre.src-simula:before { content: 'Simula'; }
pre.src-tcl:before { content: 'tcl'; }
pre.src-tex:before { content: 'TeX'; }
pre.src-plain-tex:before { content: 'Plain TeX'; }
pre.src-verilog:before { content: 'Verilog'; }
pre.src-vhdl:before { content: 'VHDL'; }
pre.src-xml:before { content: 'XML'; }
pre.src-nxml:before { content: 'XML'; }
/* add a generic configuration mode; LaTeX export needs an additional
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
pre.src-conf:before { content: 'Configuration File'; }
table { border-collapse:collapse; }
caption.t-above { caption-side: top; }
caption.t-bottom { caption-side: bottom; }
td, th { vertical-align:top; }
th.org-right { text-align: center; }
th.org-left { text-align: center; }
th.org-center { text-align: center; }
td.org-right { text-align: right; }
td.org-left { text-align: left; }
td.org-center { text-align: center; }
dt { font-weight: bold; }
.footpara { display: inline; }
.footdef { margin-bottom: 1em; }
.figure { padding: 1em; }
.figure p { text-align: center; }
.equation-container {
display: table;
text-align: center;
width: 100%;
}
.equation {
vertical-align: middle;
}
.equation-label {
display: table-cell;
text-align: right;
vertical-align: middle;
}
.inlinetask {
padding: 10px;
border: 2px solid gray;
margin: 10px;
background: #ffffcc;
}
#org-div-home-and-up
{ text-align: right; font-size: 70%; white-space: nowrap; }
textarea { overflow-x: auto; }
.linenr { font-size: smaller }
.code-highlighted { background-color: #ffff00; }
.org-info-js_info-navigation { border-style: none; }
#org-info-js_console-label
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
.org-info-js_search-highlight
{ background-color: #ffff00; color: #000000; font-weight: bold; }
.org-svg { width: 90%; }
/*]]>*/-->
</style>
2020-02-25 18:21:17 +01:00
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
<link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
<script type="text/javascript" src="./js/jquery.min.js"></script>
<script type="text/javascript" src="./js/bootstrap.min.js"></script>
<script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
<script type="text/javascript" src="./js/readtheorg.js"></script>
<script type="text/javascript">
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2020 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!--/*--><![CDATA[/*><!--*/
function CodeHighlightOn(elem, id)
{
var target = document.getElementById(id);
if(null != target) {
elem.cacheClassElem = elem.className;
elem.cacheClassTarget = target.className;
target.className = "code-highlighted";
elem.className = "code-highlighted";
}
}
function CodeHighlightOff(elem, id)
{
var target = document.getElementById(id);
if(elem.cacheClassElem)
elem.className = elem.cacheClassElem;
if(elem.cacheClassTarget)
target.className = elem.cacheClassTarget;
}
/*]]>*///-->
</script>
<script>
MathJax = {
tex: { macros: {
bm: ["\\boldsymbol{#1}",1],
}
}
};
</script>
<script type="text/javascript"
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>
<body>
<div id="org-div-home-and-up">
2020-02-25 18:21:17 +01:00
<a accesskey="h" href="./index.html"> UP </a>
|
2020-02-25 18:21:17 +01:00
<a accesskey="H" href="./index.html"> HOME </a>
</div><div id="content">
<h1 class="title">Simscape Uniaxial Model</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org227ba84">1. Simscape Model</a></li>
<li><a href="#org6bfb35d">2. Undamped System</a>
<ul>
2020-02-25 18:21:17 +01:00
<li><a href="#org5265d72">2.1. Init</a></li>
<li><a href="#org3b15675">2.2. Identification</a></li>
<li><a href="#org3492011">2.3. Sensitivity to Disturbances</a></li>
<li><a href="#orgd860077">2.4. Noise Budget</a></li>
<li><a href="#orgdb3535a">2.5. Plant</a></li>
</ul>
</li>
2020-02-25 18:21:17 +01:00
<li><a href="#org66a1e03">3. Integral Force Feedback</a>
<ul>
2020-02-25 18:21:17 +01:00
<li><a href="#org5d899b2">3.1. Control Design</a></li>
<li><a href="#orge386fb1">3.2. Identification</a></li>
<li><a href="#orgbd405f9">3.3. Sensitivity to Disturbance</a></li>
<li><a href="#orgbe8aa64">3.4. Damped Plant</a></li>
<li><a href="#orgfa342b9">3.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org07ff58f">4. Relative Motion Control</a>
<ul>
2020-02-25 18:21:17 +01:00
<li><a href="#org971bdb9">4.1. Control Design</a></li>
<li><a href="#orgfce3b27">4.2. Identification</a></li>
<li><a href="#org2123e6c">4.3. Sensitivity to Disturbance</a></li>
<li><a href="#org7e3e638">4.4. Damped Plant</a></li>
<li><a href="#org0ec985e">4.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgc9b3622">5. Direct Velocity Feedback</a>
<ul>
2020-02-25 18:21:17 +01:00
<li><a href="#orgd5df5f2">5.1. Control Design</a></li>
<li><a href="#orgca551cb">5.2. Identification</a></li>
<li><a href="#orga45e582">5.3. Sensitivity to Disturbance</a></li>
<li><a href="#orgdbf551f">5.4. Damped Plant</a></li>
<li><a href="#orge883751">5.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org5ac7dda">6. With Cedrat Piezo-electric Actuators</a>
<ul>
2020-02-25 18:21:17 +01:00
<li><a href="#orgddd7da8">6.1. Identification</a></li>
<li><a href="#orgb6ef594">6.2. Control Design</a></li>
<li><a href="#org5314588">6.3. Identification</a></li>
<li><a href="#orgd226771">6.4. Sensitivity to Disturbance</a></li>
<li><a href="#org294b781">6.5. Damped Plant</a></li>
<li><a href="#orga768e31">6.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#org77a79e6">7. Comparison of Active Damping Techniques</a>
<ul>
<li><a href="#orgb0afe4f">7.1. Load the plants</a></li>
2020-02-25 18:21:17 +01:00
<li><a href="#org948b91d">7.2. Sensitivity to Disturbance</a></li>
<li><a href="#org234479d">7.3. Noise Budget</a></li>
<li><a href="#orgf591690">7.4. Damped Plant</a></li>
<li><a href="#orgf332379">7.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org15965e0">8. Voice Coil</a>
<ul>
2020-02-25 18:21:17 +01:00
<li><a href="#orgefc326e">8.1. Init</a></li>
<li><a href="#orgae49d6e">8.2. Identification</a></li>
<li><a href="#org72047d5">8.3. Sensitivity to Disturbances</a></li>
<li><a href="#orgbcac95d">8.4. Noise Budget</a></li>
<li><a href="#org95b95ac">8.5. Integral Force Feedback</a></li>
<li><a href="#org2a655f7">8.6. Identification of the Damped Plant</a></li>
2020-02-25 18:21:17 +01:00
<li><a href="#orgc8d2daf">8.7. Noise Budget</a></li>
<li><a href="#orgb885664">8.8. Conclusion</a></li>
</ul>
</li>
</ul>
</div>
</div>
<p>
The idea is to use the same model as the full Simscape Model but to restrict the motion only in the vertical direction.
</p>
<p>
This is done in order to more easily study the system and evaluate control techniques.
</p>
<div id="outline-container-org227ba84" class="outline-2">
<h2 id="org227ba84"><span class="section-number-2">1</span> Simscape Model</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orge4f893b"></a>
</p>
<p>
A schematic of the uniaxial model used for simulations is represented in figure <a href="#org4f6d475">1</a>.
</p>
<p>
The perturbations \(w\) are:
</p>
<ul class="org-ul">
<li>\(F_s\): direct forces applied to the sample such as inertia forces and cable forces</li>
<li>\(F_{rz}\): parasitic forces due to the rotation of the spindle</li>
<li>\(F_{ty}\): parasitic forces due to scans with the translation stage</li>
<li>\(D_w\): ground motion</li>
</ul>
<p>
The quantity to \(z\) to control is:
</p>
<ul class="org-ul">
<li>\(D\): the position of the sample with respect to the granite</li>
</ul>
<p>
The measured quantities \(v\) are:
</p>
<ul class="org-ul">
<li>\(D\): the position of the sample with respect to the granite</li>
</ul>
<p>
We study the use of an additional sensor:
</p>
<ul class="org-ul">
<li>\(F_n\): a force sensor located in the nano-hexapod</li>
<li>\(v_n\): an absolute velocity sensor located on the top platform of the nano-hexapod</li>
<li>\(d_r\): a relative motion sensor located in the nano-hexapod</li>
</ul>
<p>
The control signal \(u\) is:
</p>
<ul class="org-ul">
<li>\(F\) the force applied by the nano-hexapod actuator</li>
</ul>
<div id="org4f6d475" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible.png" alt="uniaxial-model-nass-flexible.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the uniaxial model used</p>
</div>
<p>
Few active damping techniques will be compared in order to decide which sensor is to be included in the system.
Schematics of the active damping techniques are displayed in figure <a href="#orgc460f0d">2</a>.
</p>
<div id="orgc460f0d" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-active-damping.png" alt="uniaxial-model-nass-flexible-active-damping.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Comparison of used active damping techniques</p>
</div>
</div>
</div>
<div id="outline-container-org6bfb35d" class="outline-2">
<h2 id="org6bfb35d"><span class="section-number-2">2</span> Undamped System</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org83e5eca"></a>
</p>
<p>
Let&rsquo;s start by study the undamped system.
</p>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org5265d72" class="outline-3">
<h3 id="org5265d72"><span class="section-number-3">2.1</span> Init</h3>
<div class="outline-text-3" id="text-2-1">
<p>
We initialize all the stages with the default parameters.
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
</p>
<p>
All the controllers are set to 0 (Open Loop).
</p>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org3b15675" class="outline-3">
<h3 id="org3b15675"><span class="section-number-3">2.2</span> Identification</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We identify the dynamics of the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<p>
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<p>
Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, options);
G.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
<p>
Finally, we save the identified system dynamics for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>);
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org3492011" class="outline-3">
<h3 id="org3492011"><span class="section-number-3">2.3</span> Sensitivity to Disturbances</h3>
<div class="outline-text-3" id="text-2-3">
<p>
We show several plots representing the sensitivity to disturbances:
</p>
<ul class="org-ul">
<li>in figure <a href="#orgbfaf2d3">3</a> the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown</li>
<li>in figure <a href="#orge6ba7ee">4</a>, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown</li>
</ul>
<div id="orgbfaf2d3" class="figure">
<p><img src="figs/uniaxial-sensitivity-disturbances.png" alt="uniaxial-sensitivity-disturbances.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-disturbances.png">png</a>, <a href="./figs/uniaxial-sensitivity-disturbances.pdf">pdf</a>)</p>
</div>
<div id="orge6ba7ee" class="figure">
<p><img src="figs/uniaxial-sensitivity-force-dist.png" alt="uniaxial-sensitivity-force-dist.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-force-dist.png">png</a>, <a href="./figs/uniaxial-sensitivity-force-dist.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgd860077" class="outline-3">
<h3 id="orgd860077"><span class="section-number-3">2.4</span> Noise Budget</h3>
<div class="outline-text-3" id="text-2-4">
<p>
We first load the measured PSD of the disturbance.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>);
</pre>
</div>
<p>
The effect of these disturbances on the distance \(D\) is computed below.
The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure <a href="#org1c1d9b8">5</a> and the Cumulative Amplitude Spectrum is shown in figure <a href="#org2f678d5">6</a>.
</p>
<p>
The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure <a href="#org2f678d5">6</a>.
</p>
<pre class="example">
3.3793e-06
</pre>
<div id="org1c1d9b8" class="figure">
<p><img src="figs/uniaxial-psd-dist.png" alt="uniaxial-psd-dist.png" />
</p>
<p><span class="figure-number">Figure 5: </span>PSD of the effect of disturbances on \(D\) (<a href="./figs/uniaxial-psd-dist.png">png</a>, <a href="./figs/uniaxial-psd-dist.pdf">pdf</a>)</p>
</div>
<div id="org2f678d5" class="figure">
<p><img src="figs/uniaxial-cas-dist.png" alt="uniaxial-cas-dist.png" />
</p>
<p><span class="figure-number">Figure 6: </span>CAS of the effect of disturbances on \(D\) (<a href="./figs/uniaxial-cas-dist.png">png</a>, <a href="./figs/uniaxial-cas-dist.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgdb3535a" class="outline-3">
<h3 id="orgdb3535a"><span class="section-number-3">2.5</span> Plant</h3>
<div class="outline-text-3" id="text-2-5">
<p>
The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure <a href="#orgaaf3fcb">7</a>.
It corresponds to the plant to control.
</p>
<div id="orgaaf3fcb" class="figure">
<p><img src="figs/uniaxial-plant.png" alt="uniaxial-plant.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Bode plot of the Plant (<a href="./figs/uniaxial-plant.png">png</a>, <a href="./figs/uniaxial-plant.pdf">pdf</a>)</p>
</div>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org66a1e03" class="outline-2">
<h2 id="org66a1e03"><span class="section-number-2">3</span> Integral Force Feedback</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org37f1b7d"></a>
</p>
<div id="org884322f" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-iff.png" alt="uniaxial-model-nass-flexible-iff.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Uniaxial IFF Control Schematic</p>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org5d899b2" class="outline-3">
<h3 id="org5d899b2"><span class="section-number-3">3.1</span> Control Design</h3>
<div class="outline-text-3" id="text-3-1">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>);
</pre>
</div>
<p>
Let&rsquo;s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor.
</p>
<div id="orgde62a50" class="figure">
<p><img src="figs/uniaxial_iff_plant.png" alt="uniaxial_iff_plant.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_iff_plant.png">png</a>, <a href="./figs/uniaxial_iff_plant.pdf">pdf</a>)</p>
</div>
<p>
The controller for each pair of actuator/sensor is:
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = <span class="org-type">-</span>1000<span class="org-type">/</span>s;
</pre>
</div>
<div id="orgf0a3805" class="figure">
<p><img src="figs/uniaxial_iff_open_loop.png" alt="uniaxial_iff_open_loop.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_iff_open_loop.png">png</a>, <a href="./figs/uniaxial_iff_open_loop.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orge386fb1" class="outline-3">
<h3 id="orge386fb1"><span class="section-number-3">3.2</span> Identification</h3>
<div class="outline-text-3" id="text-3-2">
<p>
Let&rsquo;s initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeSample(<span class="org-string">'mass'</span>, 50);
</pre>
</div>
<p>
All the controllers are set to 0.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_iff = <span class="org-type">-</span>K_iff;
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_rmc = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
K_dvf = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_iff = linearize(mdl, io, options);
G_iff.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_iff.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgbd405f9" class="outline-3">
<h3 id="orgbd405f9"><span class="section-number-3">3.3</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-3-3">
<div id="orge528565" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_iff.png" alt="uniaxial_sensitivity_dist_iff.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Sensitivity to disturbance once the IFF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_iff.pdf">pdf</a>)</p>
</div>
<div id="org01074d3" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_iff.png" alt="uniaxial_sensitivity_dist_stages_iff.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Sensitivity to force disturbances in various stages when IFF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_iff.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgbe8aa64" class="outline-3">
<h3 id="orgbe8aa64"><span class="section-number-3">3.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-3-4">
<div id="org57280cb" class="figure">
<p><img src="figs/uniaxial_plant_iff_damped.png" alt="uniaxial_plant_iff_damped.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Damped Plant after IFF is applied (<a href="./figs/uniaxial_plant_iff_damped.png">png</a>, <a href="./figs/uniaxial_plant_iff_damped.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgfa342b9" class="outline-3">
<h3 id="orgfa342b9"><span class="section-number-3">3.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-5">
<div class="important">
<p>
Integral Force Feedback:
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org07ff58f" class="outline-2">
<h2 id="org07ff58f"><span class="section-number-2">4</span> Relative Motion Control</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org7d87e9d"></a>
</p>
<p>
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
</p>
<div id="orgd2f9465" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-rmc.png" alt="uniaxial-model-nass-flexible-rmc.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Uniaxial RMC Control Schematic</p>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org971bdb9" class="outline-3">
<h3 id="org971bdb9"><span class="section-number-3">4.1</span> Control Design</h3>
<div class="outline-text-3" id="text-4-1">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>);
</pre>
</div>
<p>
Let&rsquo;s look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor.
</p>
<div id="org8260ad7" class="figure">
<p><img src="figs/uniaxial_rmc_plant.png" alt="uniaxial_rmc_plant.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/uniaxial_rmc_plant.png">png</a>, <a href="./figs/uniaxial_rmc_plant.pdf">pdf</a>)</p>
</div>
<p>
The Relative Motion Controller is defined below.
A Low pass Filter is added to make the controller transfer function proper.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_rmc = s<span class="org-type">*</span>50000<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10000);
</pre>
</div>
<div id="orga5e2583" class="figure">
<p><img src="figs/uniaxial_rmc_open_loop.png" alt="uniaxial_rmc_open_loop.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_rmc_open_loop.png">png</a>, <a href="./figs/uniaxial_rmc_open_loop.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgfce3b27" class="outline-3">
<h3 id="orgfce3b27"><span class="section-number-3">4.2</span> Identification</h3>
<div class="outline-text-3" id="text-4-2">
<p>
Let&rsquo;s initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeSample(<span class="org-string">'mass'</span>, 50);
</pre>
</div>
<p>
And initialize the controllers.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_iff = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_rmc = <span class="org-type">-</span>K_rmc;
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
K_dvf = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_rmc = linearize(mdl, io, options);
G_rmc.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_rmc.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org2123e6c" class="outline-3">
<h3 id="org2123e6c"><span class="section-number-3">4.3</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-4-3">
<div id="org9e0dde4" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_rmc.png" alt="uniaxial_sensitivity_dist_rmc.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Sensitivity to disturbance once the RMC controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_rmc.pdf">pdf</a>)</p>
</div>
<div id="org38d5224" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_rmc.png" alt="uniaxial_sensitivity_dist_stages_rmc.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Sensitivity to force disturbances in various stages when RMC is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_rmc.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org7e3e638" class="outline-3">
<h3 id="org7e3e638"><span class="section-number-3">4.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-4-4">
<div id="org18786e8" class="figure">
<p><img src="figs/uniaxial_plant_rmc_damped.png" alt="uniaxial_plant_rmc_damped.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Damped Plant after RMC is applied (<a href="./figs/uniaxial_plant_rmc_damped.png">png</a>, <a href="./figs/uniaxial_plant_rmc_damped.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org0ec985e" class="outline-3">
<h3 id="org0ec985e"><span class="section-number-3">4.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-5">
<div class="important">
<p>
Relative Motion Control:
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgc9b3622" class="outline-2">
<h2 id="orgc9b3622"><span class="section-number-2">5</span> Direct Velocity Feedback</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="orgdffe2c2"></a>
</p>
<p>
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
</p>
<div id="orgba6ec08" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-dvf.png" alt="uniaxial-model-nass-flexible-dvf.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Uniaxial DVF Control Schematic</p>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgd5df5f2" class="outline-3">
<h3 id="orgd5df5f2"><span class="section-number-3">5.1</span> Control Design</h3>
<div class="outline-text-3" id="text-5-1">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>);
</pre>
</div>
<div id="org7b83a4a" class="figure">
<p><img src="figs/uniaxial_dvf_plant.png" alt="uniaxial_dvf_plant.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_plant.png">png</a>, <a href="./figs/uniaxial_dvf_plant.pdf">pdf</a>)</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">K_dvf = tf(5e4);
</pre>
</div>
<div id="org25567d5" class="figure">
<p><img src="figs/uniaxial_dvf_loop_gain.png" alt="uniaxial_dvf_loop_gain.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_loop_gain.png">png</a>, <a href="./figs/uniaxial_dvf_loop_gain.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgca551cb" class="outline-3">
<h3 id="orgca551cb"><span class="section-number-3">5.2</span> Identification</h3>
<div class="outline-text-3" id="text-5-2">
<p>
Let&rsquo;s initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeSample(<span class="org-string">'mass'</span>, 50);
</pre>
</div>
<p>
And initialize the controllers.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_iff = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_rmc = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
K_dvf = <span class="org-type">-</span>K_dvf;
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_dvf = linearize(mdl, io, options);
G_dvf.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_dvf.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orga45e582" class="outline-3">
<h3 id="orga45e582"><span class="section-number-3">5.3</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-5-3">
<div id="orgf9d4052" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_dvf.png" alt="uniaxial_sensitivity_dist_dvf.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Sensitivity to disturbance once the DVF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_dvf.pdf">pdf</a>)</p>
</div>
<div id="org98ddadb" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_dvf.png" alt="uniaxial_sensitivity_dist_stages_dvf.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Sensitivity to force disturbances in various stages when DVF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_dvf.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgdbf551f" class="outline-3">
<h3 id="orgdbf551f"><span class="section-number-3">5.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-5-4">
<div id="orgbc9c953" class="figure">
<p><img src="figs/uniaxial_plant_dvf_damped.png" alt="uniaxial_plant_dvf_damped.png" />
</p>
<p><span class="figure-number">Figure 25: </span>Damped Plant after DVF is applied (<a href="./figs/uniaxial_plant_dvf_damped.png">png</a>, <a href="./figs/uniaxial_plant_dvf_damped.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orge883751" class="outline-3">
<h3 id="orge883751"><span class="section-number-3">5.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-5">
<div class="important">
<p>
Direct Velocity Feedback:
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org5ac7dda" class="outline-2">
<h2 id="org5ac7dda"><span class="section-number-2">6</span> With Cedrat Piezo-electric Actuators</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="org604af95"></a>
</p>
<p>
The model used for the Cedrat actuator is shown in figure <a href="#org83591fa">26</a>.
</p>
<div id="org83591fa" class="figure">
<p><img src="figs/cedrat-uniaxial-actuator.png" alt="cedrat-uniaxial-actuator.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Schematic of the model used for the Cedrat Actuator</p>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgddd7da8" class="outline-3">
<h3 id="orgddd7da8"><span class="section-number-3">6.1</span> Identification</h3>
<div class="outline-text-3" id="text-6-1">
<p>
Let&rsquo;s initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeCedratPiezo();
initializeSample(<span class="org-string">'mass'</span>, 50);
</pre>
</div>
<p>
And initialize the controllers.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_iff = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_rmc = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
K_dvf = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<p>
We identify the dynamics of the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial_cedrat_bis'</span>;
</pre>
</div>
<p>
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<p>
Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, options);
G.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgb6ef594" class="outline-3">
<h3 id="orgb6ef594"><span class="section-number-3">6.2</span> Control Design</h3>
<div class="outline-text-3" id="text-6-2">
<p>
Let&rsquo;s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor.
</p>
<div id="org2a02461" class="figure">
<p><img src="figs/uniaxial_cedrat_plant.png" alt="uniaxial_cedrat_plant.png" />
</p>
<p><span class="figure-number">Figure 27: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_cedrat_plant.png">png</a>, <a href="./figs/uniaxial_cedrat_plant.pdf">pdf</a>)</p>
</div>
<p>
The controller for each pair of actuator/sensor is:
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_cedrat = <span class="org-type">-</span>5000<span class="org-type">/</span>s;
</pre>
</div>
<div id="org7da2401" class="figure">
<p><img src="figs/uniaxial_cedrat_open_loop.png" alt="uniaxial_cedrat_open_loop.png" />
</p>
<p><span class="figure-number">Figure 28: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_cedrat_open_loop.png">png</a>, <a href="./figs/uniaxial_cedrat_open_loop.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org5314588" class="outline-3">
<h3 id="org5314588"><span class="section-number-3">6.3</span> Identification</h3>
<div class="outline-text-3" id="text-6-3">
<p>
Let&rsquo;s initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeCedratPiezo();
initializeSample(<span class="org-string">'mass'</span>, 50);
</pre>
</div>
<p>
All the controllers are set to 0.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_iff = <span class="org-type">-</span>K_cedrat;
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_rmc = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
K_dvf = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial_cedrat_bis'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_cedrat = linearize(mdl, io, options);
G_cedrat.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_cedrat.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% save('./uniaxial/mat/plants.mat', 'G_cedrat', '-append');</span>
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgd226771" class="outline-3">
<h3 id="orgd226771"><span class="section-number-3">6.4</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-6-4">
<div id="org90f5c2b" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_cedrat.png" alt="uniaxial_sensitivity_dist_cedrat.png" />
</p>
<p><span class="figure-number">Figure 29: </span>Sensitivity to disturbance once the CEDRAT controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_cedrat.pdf">pdf</a>)</p>
</div>
<div id="org5acd68e" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_cedrat.png" alt="uniaxial_sensitivity_dist_stages_cedrat.png" />
</p>
<p><span class="figure-number">Figure 30: </span>Sensitivity to force disturbances in various stages when CEDRAT is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org294b781" class="outline-3">
<h3 id="org294b781"><span class="section-number-3">6.5</span> Damped Plant</h3>
<div class="outline-text-3" id="text-6-5">
<div id="org0c4e61a" class="figure">
<p><img src="figs/uniaxial_plant_cedrat_damped.png" alt="uniaxial_plant_cedrat_damped.png" />
</p>
<p><span class="figure-number">Figure 31: </span>Damped Plant after CEDRAT is applied (<a href="./figs/uniaxial_plant_cedrat_damped.png">png</a>, <a href="./figs/uniaxial_plant_cedrat_damped.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orga768e31" class="outline-3">
<h3 id="orga768e31"><span class="section-number-3">6.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-6-6">
<div class="important">
<p>
This gives similar results than with a classical force sensor.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org77a79e6" class="outline-2">
<h2 id="org77a79e6"><span class="section-number-2">7</span> Comparison of Active Damping Techniques</h2>
<div class="outline-text-2" id="text-7">
<p>
<a id="orgd5640a5"></a>
</p>
</div>
<div id="outline-container-orgb0afe4f" class="outline-3">
<h3 id="orgb0afe4f"><span class="section-number-3">7.1</span> Load the plants</h3>
<div class="outline-text-3" id="text-7-1">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'G_dvf'</span>);
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org948b91d" class="outline-3">
<h3 id="org948b91d"><span class="section-number-3">7.2</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-7-2">
<div id="org8c59440" class="figure">
<p><img src="figs/uniaxial_sensitivity_ground_motion.png" alt="uniaxial_sensitivity_ground_motion.png" />
</p>
<p><span class="figure-number">Figure 32: </span>Sensitivity to Ground Motion - Comparison (<a href="./figs/uniaxial_sensitivity_ground_motion.png">png</a>, <a href="./figs/uniaxial_sensitivity_ground_motion.pdf">pdf</a>)</p>
</div>
<div id="orgf1c26c0" class="figure">
<p><img src="figs/uniaxial_sensitivity_direct_force.png" alt="uniaxial_sensitivity_direct_force.png" />
</p>
<p><span class="figure-number">Figure 33: </span>Sensitivity to disturbance - Comparison (<a href="./figs/uniaxial_sensitivity_direct_force.png">png</a>, <a href="./figs/uniaxial_sensitivity_direct_force.pdf">pdf</a>)</p>
</div>
<div id="orga77aca7" class="figure">
<p><img src="figs/uniaxial_sensitivity_fty.png" alt="uniaxial_sensitivity_fty.png" />
</p>
<p><span class="figure-number">Figure 34: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_fty.png">png</a>, <a href="./figs/uniaxial_sensitivity_fty.pdf">pdf</a>)</p>
</div>
<div id="orga6c1630" class="figure">
<p><img src="figs/uniaxial_sensitivity_frz.png" alt="uniaxial_sensitivity_frz.png" />
</p>
<p><span class="figure-number">Figure 35: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_frz.png">png</a>, <a href="./figs/uniaxial_sensitivity_frz.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org234479d" class="outline-3">
<h3 id="org234479d"><span class="section-number-3">7.3</span> Noise Budget</h3>
<div class="outline-text-3" id="text-7-3">
<p>
We first load the measured PSD of the disturbance.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>);
</pre>
</div>
<p>
The effect of these disturbances on the distance \(D\) is computed for all active damping techniques.
We then compute the Cumulative Amplitude Spectrum (figure <a href="#org68d1f1c">36</a>).
</p>
<div id="org68d1f1c" class="figure">
<p><img src="figs/uniaxial-comp-cas-dist.png" alt="uniaxial-comp-cas-dist.png" />
</p>
<p><span class="figure-number">Figure 36: </span>Comparison of the Cumulative Amplitude Spectrum of \(D\) for different active damping techniques (<a href="./figs/uniaxial-comp-cas-dist.png">png</a>, <a href="./figs/uniaxial-comp-cas-dist.pdf">pdf</a>)</p>
</div>
<p>
The obtained Root Mean Square Value for each active damping technique is shown below.
</p>
<table id="org63699f5" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Obtain Root Mean Square value of \(D\) for each Active Damping Technique applied</caption>
<colgroup>
<col class="org-left" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">D [m rms]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">OL</td>
<td class="org-right">3.38e-06</td>
</tr>
<tr>
<td class="org-left">IFF</td>
<td class="org-right">3.40e-06</td>
</tr>
<tr>
<td class="org-left">RMC</td>
<td class="org-right">3.37e-06</td>
</tr>
<tr>
<td class="org-left">DVF</td>
<td class="org-right">3.38e-06</td>
</tr>
</tbody>
</table>
<p>
It is important to note that the effect of direct forces applied to the sample are not taken into account here.
</p>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgf591690" class="outline-3">
<h3 id="orgf591690"><span class="section-number-3">7.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-7-4">
<div id="orga0c1298" class="figure">
<p><img src="figs/uniaxial_plant_damped_comp.png" alt="uniaxial_plant_damped_comp.png" />
</p>
<p><span class="figure-number">Figure 37: </span>Damped Plant - Comparison (<a href="./figs/uniaxial_plant_damped_comp.png">png</a>, <a href="./figs/uniaxial_plant_damped_comp.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgf332379" class="outline-3">
<h3 id="orgf332379"><span class="section-number-3">7.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-7-5">
<table id="org46e95c2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Comparison of proposed active damping techniques</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">IFF</th>
<th scope="col" class="org-left">RMC</th>
<th scope="col" class="org-left">DVF</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Sensor Type</td>
<td class="org-left">Force sensor</td>
<td class="org-left">Relative Motion</td>
<td class="org-left">Inertial</td>
</tr>
<tr>
<td class="org-left">Guaranteed Stability</td>
<td class="org-left">+</td>
<td class="org-left">+</td>
<td class="org-left">-</td>
</tr>
<tr>
<td class="org-left">Sensitivity (\(D_w\))</td>
<td class="org-left">-</td>
<td class="org-left">+</td>
<td class="org-left">-</td>
</tr>
<tr>
<td class="org-left">Sensitivity (\(F_s\))</td>
<td class="org-left">- (at low freq)</td>
<td class="org-left">+</td>
<td class="org-left">+</td>
</tr>
<tr>
<td class="org-left">Sensitivity (\(F_{ty,rz}\))</td>
<td class="org-left">+</td>
<td class="org-left">-</td>
<td class="org-left">+</td>
</tr>
<tr>
<td class="org-left">Overall RMS of \(D\)</td>
<td class="org-left">=</td>
<td class="org-left">=</td>
<td class="org-left">=</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-org15965e0" class="outline-2">
<h2 id="org15965e0"><span class="section-number-2">8</span> Voice Coil</h2>
<div class="outline-text-2" id="text-8">
<p>
<a id="orgf9e9d51"></a>
</p>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgefc326e" class="outline-3">
<h3 id="orgefc326e"><span class="section-number-3">8.1</span> Init</h3>
<div class="outline-text-3" id="text-8-1">
<p>
We initialize all the stages with the default parameters.
The nano-hexapod is an hexapod with voice coils and the sample has a mass of 50kg.
</p>
<p>
All the controllers are set to 0 (Open Loop).
</p>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgae49d6e" class="outline-3">
<h3 id="orgae49d6e"><span class="section-number-3">8.2</span> Identification</h3>
<div class="outline-text-3" id="text-8-2">
<p>
We identify the dynamics of the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<p>
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<p>
Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_vc = linearize(mdl, io, options);
G_vc.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_vc.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
<p>
Finally, we save the identified system dynamics for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_vc'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org72047d5" class="outline-3">
<h3 id="org72047d5"><span class="section-number-3">8.3</span> Sensitivity to Disturbances</h3>
<div class="outline-text-3" id="text-8-3">
<p>
We load the dynamics when using a piezo-electric nano hexapod to compare the results.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>);
</pre>
</div>
<p>
We show several plots representing the sensitivity to disturbances:
</p>
<ul class="org-ul">
<li>in figure <a href="#orgc80c94c">38</a> the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown</li>
<li>in figure <a href="#org96743f2">39</a>, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown</li>
</ul>
<div id="orgc80c94c" class="figure">
<p><img src="figs/uniaxial-sensitivity-vc-disturbances.png" alt="uniaxial-sensitivity-vc-disturbances.png" />
</p>
<p><span class="figure-number">Figure 38: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-vc-disturbances.png">png</a>, <a href="./figs/uniaxial-sensitivity-vc-disturbances.pdf">pdf</a>)</p>
</div>
<div id="org96743f2" class="figure">
<p><img src="figs/uniaxial-sensitivity-vc-force-dist.png" alt="uniaxial-sensitivity-vc-force-dist.png" />
</p>
<p><span class="figure-number">Figure 39: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-vc-force-dist.png">png</a>, <a href="./figs/uniaxial-sensitivity-vc-force-dist.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgbcac95d" class="outline-3">
<h3 id="orgbcac95d"><span class="section-number-3">8.4</span> Noise Budget</h3>
<div class="outline-text-3" id="text-8-4">
<p>
We first load the measured PSD of the disturbance.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>);
</pre>
</div>
<p>
The effect of these disturbances on the distance \(D\) is computed below.
The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure <a href="#orgd3fd39b">40</a> and the Cumulative Amplitude Spectrum is shown in figure <a href="#org2ccc8a7">41</a>.
</p>
<p>
The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure <a href="#org2ccc8a7">41</a>.
</p>
<pre class="example">
4.8793e-06
</pre>
<div id="orgd3fd39b" class="figure">
<p><img src="figs/uniaxial-vc-psd-dist.png" alt="uniaxial-vc-psd-dist.png" />
</p>
<p><span class="figure-number">Figure 40: </span>PSD of the displacement \(D\) due to disturbances (<a href="./figs/uniaxial-vc-psd-dist.png">png</a>, <a href="./figs/uniaxial-vc-psd-dist.pdf">pdf</a>)</p>
</div>
<div id="org2ccc8a7" class="figure">
<p><img src="figs/uniaxial-vc-cas-dist.png" alt="uniaxial-vc-cas-dist.png" />
</p>
<p><span class="figure-number">Figure 41: </span>CAS of the displacement \(D\) due the disturbances (<a href="./figs/uniaxial-vc-cas-dist.png">png</a>, <a href="./figs/uniaxial-vc-cas-dist.pdf">pdf</a>)</p>
</div>
<div class="important">
<p>
Even though the RMS value of the displacement \(D\) is lower when using a piezo-electric actuator, the motion is mainly due to high frequency disturbances which are more difficult to control (an higher control bandwidth is required).
</p>
<p>
Thus, it may be desirable to use voice coil actuators.
</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-org95b95ac" class="outline-3">
<h3 id="org95b95ac"><span class="section-number-3">8.5</span> Integral Force Feedback</h3>
<div class="outline-text-3" id="text-8-5">
<div class="org-src-container">
<pre class="src src-matlab">K_iff = <span class="org-type">-</span>20<span class="org-type">/</span>s;
</pre>
</div>
<div id="orgb6bf4fa" class="figure">
<p><img src="figs/uniaxial_iff_vc_open_loop.png" alt="uniaxial_iff_vc_open_loop.png" />
</p>
<p><span class="figure-number">Figure 42: </span>Open Loop Transfer Function for IFF control when using a voice coil actuator (<a href="./figs/uniaxial_iff_vc_open_loop.png">png</a>, <a href="./figs/uniaxial_iff_vc_open_loop.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org2a655f7" class="outline-3">
<h3 id="org2a655f7"><span class="section-number-3">8.6</span> Identification of the Damped Plant</h3>
<div class="outline-text-3" id="text-8-6">
<p>
Let&rsquo;s initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'lorentz'</span>);
initializeSample(<span class="org-string">'mass'</span>, 50);
</pre>
</div>
<p>
All the controllers are set to 0.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_iff = <span class="org-type">-</span>K_iff;
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_rmc = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
K_dvf = tf(0);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span>
io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span>
io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span>
io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_vc_iff = linearize(mdl, io, options);
G_vc_iff.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_vc_iff.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgc8d2daf" class="outline-3">
<h3 id="orgc8d2daf"><span class="section-number-3">8.7</span> Noise Budget</h3>
<div class="outline-text-3" id="text-8-7">
<p>
We compute the obtain PSD of the displacement \(D\) when using IFF.
</p>
<div id="org60accee" class="figure">
<p><img src="figs/uniaxial-cas-iff-vc.png" alt="uniaxial-cas-iff-vc.png" />
</p>
<p><span class="figure-number">Figure 43: </span>CAS of the displacement \(D\) (<a href="./figs/uniaxial-cas-iff-vc.png">png</a>, <a href="./figs/uniaxial-cas-iff-vc.pdf">pdf</a>)</p>
</div>
</div>
</div>
2020-02-25 18:21:17 +01:00
<div id="outline-container-orgb885664" class="outline-3">
<h3 id="orgb885664"><span class="section-number-3">8.8</span> Conclusion</h3>
<div class="outline-text-3" id="text-8-8">
<div class="important">
<p>
The use of voice coil actuators would allow a better disturbance rejection for a fixed bandwidth compared with a piezo-electric hexapod.
</p>
<p>
Similarly, it would require much lower bandwidth to attain the same level of disturbance rejection for \(D\).
</p>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
2020-02-25 18:21:17 +01:00
<p class="date">Created: 2020-02-25 mar. 18:20</p>
</div>
</body>
</html>