1546 lines
73 KiB
HTML
1546 lines
73 KiB
HTML
<?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>
|
|
<!-- 2019-10-25 ven. 16:00 -->
|
|
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
|
<title>Active Damping</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>
|
|
<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) 2012-2019 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 type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
displayAlign: "center",
|
|
displayIndent: "0em",
|
|
|
|
"HTML-CSS": { scale: 100,
|
|
linebreaks: { automatic: "false" },
|
|
webFont: "TeX"
|
|
},
|
|
SVG: {scale: 100,
|
|
linebreaks: { automatic: "false" },
|
|
font: "TeX"},
|
|
NativeMML: {scale: 100},
|
|
TeX: { equationNumbers: {autoNumber: "AMS"},
|
|
MultLineWidth: "85%",
|
|
TagSide: "right",
|
|
TagIndent: ".8em"
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript"
|
|
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></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>
|
|
</div><div id="content">
|
|
<h1 class="title">Active Damping</h1>
|
|
<div id="table-of-contents">
|
|
<h2>Table of Contents</h2>
|
|
<div id="text-table-of-contents">
|
|
<ul>
|
|
<li><a href="#orgbcf03a9">1. Undamped System</a>
|
|
<ul>
|
|
<li><a href="#org46a3e46">1.1. Init</a></li>
|
|
<li><a href="#org182ef44">1.2. Identification</a></li>
|
|
<li><a href="#org78e6984">1.3. Sensitivity to disturbances</a></li>
|
|
<li><a href="#orgac6dd79">1.4. Undamped Plant</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#orgeea54a1">2. Integral Force Feedback</a>
|
|
<ul>
|
|
<li><a href="#org2c83f94">2.1. One degree-of-freedom example</a>
|
|
<ul>
|
|
<li><a href="#org4efeef1">2.1.1. Equations</a></li>
|
|
<li><a href="#orgb3880fc">2.1.2. Matlab Example</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org731eb41">2.2. Control Design</a></li>
|
|
<li><a href="#orgcaee5e8">2.3. Identification of the damped plant</a></li>
|
|
<li><a href="#org92c8ec7">2.4. Sensitivity to disturbances</a></li>
|
|
<li><a href="#orgacc1bc1">2.5. Damped Plant</a></li>
|
|
<li><a href="#orged93d15">2.6. Conclusion</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org087ecf6">3. Relative Motion Control</a>
|
|
<ul>
|
|
<li><a href="#orga33a4fa">3.1. One degree-of-freedom example</a>
|
|
<ul>
|
|
<li><a href="#orge6e79c5">3.1.1. Equations</a></li>
|
|
<li><a href="#orgfb8caad">3.1.2. Matlab Example</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#orgcb24491">3.2. Control Design</a></li>
|
|
<li><a href="#orgaeb6872">3.3. Identification of the damped plant</a></li>
|
|
<li><a href="#org2fc9fe6">3.4. Sensitivity to disturbances</a></li>
|
|
<li><a href="#org846d098">3.5. Damped Plant</a></li>
|
|
<li><a href="#org12f4764">3.6. Conclusion</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#orgd92711d">4. Direct Velocity Feedback</a>
|
|
<ul>
|
|
<li><a href="#org189fb3b">4.1. One degree-of-freedom example</a>
|
|
<ul>
|
|
<li><a href="#org5318c89">4.1.1. Equations</a></li>
|
|
<li><a href="#orgfb2a947">4.1.2. Matlab Example</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org3c87599">4.2. Control Design</a></li>
|
|
<li><a href="#org07e7826">4.3. Identification of the damped plant</a></li>
|
|
<li><a href="#org901fc2b">4.4. Sensitivity to disturbances</a></li>
|
|
<li><a href="#org22a4374">4.5. Damped Plant</a></li>
|
|
<li><a href="#org1c0a92c">4.6. Conclusion</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org099228a">5. Comparison</a>
|
|
<ul>
|
|
<li><a href="#org10b67a5">5.1. Load the plants</a></li>
|
|
<li><a href="#org0dcb40a">5.2. Sensitivity to Disturbance</a></li>
|
|
<li><a href="#orgcc0be93">5.3. Damped Plant</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org0476571">6. Conclusion</a></li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<p>
|
|
First, in section <a href="#org86b2b5e">1</a>, we will looked at the undamped system.
|
|
</p>
|
|
|
|
<p>
|
|
Then, we will compare three active damping techniques:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>In section <a href="#org1e5669e">2</a>: the integral force feedback is used</li>
|
|
<li>In section <a href="#org42ae53a">3</a>: the relative motion control is used</li>
|
|
<li>In section <a href="#org556890f">4</a>: the direct velocity feedback is used</li>
|
|
</ul>
|
|
|
|
<p>
|
|
For each of the active damping technique, we will:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Compare the sensitivity from disturbances</li>
|
|
<li>Look at the damped plant</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The disturbances are:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Ground motion</li>
|
|
<li>Direct forces</li>
|
|
<li>Motion errors of all the stages</li>
|
|
</ul>
|
|
|
|
<div id="outline-container-orgbcf03a9" class="outline-2">
|
|
<h2 id="orgbcf03a9"><span class="section-number-2">1</span> Undamped System</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
<a id="org86b2b5e"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/undamped_system.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<p>
|
|
We first look at the undamped system.
|
|
The performance of this undamped system will be compared with the damped system using various techniques.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org46a3e46" class="outline-3">
|
|
<h3 id="org46a3e46"><span class="section-number-3">1.1</span> Init</h3>
|
|
<div class="outline-text-3" id="text-1-1">
|
|
<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>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeInputs<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
All the controllers are set to 0.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org182ef44" class="outline-3">
|
|
<h3 id="org182ef44"><span class="section-number-3">1.2</span> Identification</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
<p>
|
|
We identify the various transfer functions of the system
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G = identifyPlant<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we save it for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org78e6984" class="outline-3">
|
|
<h3 id="org78e6984"><span class="section-number-3">1.3</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<p>
|
|
The sensitivity to disturbances are shown on figure <a href="#orgf3b9fba">1</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgf3b9fba" class="figure">
|
|
<p><img src="figs/sensitivity_dist_undamped.png" alt="sensitivity_dist_undamped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Undamped sensitivity to disturbances (<a href="./figs/sensitivity_dist_undamped.png">png</a>, <a href="./figs/sensitivity_dist_undamped.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgb418189" class="figure">
|
|
<p><img src="figs/sensitivity_dist_stages.png" alt="sensitivity_dist_stages.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Sensitivity to force disturbances in various stages (<a href="./figs/sensitivity_dist_stages.png">png</a>, <a href="./figs/sensitivity_dist_stages.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgac6dd79" class="outline-3">
|
|
<h3 id="orgac6dd79"><span class="section-number-3">1.4</span> Undamped Plant</h3>
|
|
<div class="outline-text-3" id="text-1-4">
|
|
<p>
|
|
The "plant" (transfer function from forces applied by the nano-hexapod to the measured displacement of the sample with respect to the granite) bode plot is shown on figure <a href="#orgf3b9fba">1</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgcc5c0db" class="figure">
|
|
<p><img src="figs/plant_undamped.png" alt="plant_undamped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Transfer Function from cartesian forces to displacement for the undamped plant (<a href="./figs/plant_undamped.png">png</a>, <a href="./figs/plant_undamped.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgeea54a1" class="outline-2">
|
|
<h2 id="orgeea54a1"><span class="section-number-2">2</span> Integral Force Feedback</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="org1e5669e"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/iff.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<p>
|
|
Integral Force Feedback is applied.
|
|
In section <a href="#org9620ec3">2.1</a>, IFF is applied on a uni-axial system to understand its behavior.
|
|
Then, it is applied on the simscape model.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org2c83f94" class="outline-3">
|
|
<h3 id="org2c83f94"><span class="section-number-3">2.1</span> One degree-of-freedom example</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
<a id="org9620ec3"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org4efeef1" class="outline-4">
|
|
<h4 id="org4efeef1"><span class="section-number-4">2.1.1</span> Equations</h4>
|
|
<div class="outline-text-4" id="text-2-1-1">
|
|
|
|
<div id="orge1d8b91" class="figure">
|
|
<p><img src="figs/iff_1dof.png" alt="iff_1dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Integral Force Feedback applied to a 1dof system</p>
|
|
</div>
|
|
|
|
<p>
|
|
The dynamic of the system is described by the following equation:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
<p>
|
|
The measured force \(F_m\) is:
|
|
</p>
|
|
\begin{align}
|
|
F_m &= F - kx - csx + kw + csw \\
|
|
&= ms^2 x - F_d
|
|
\end{align}
|
|
<p>
|
|
The Integral Force Feedback controller is \(K = -\frac{g}{s}\), and thus the applied force by this controller is:
|
|
</p>
|
|
\begin{equation}
|
|
F_{\text{IFF}} = -\frac{g}{s} F_m = -\frac{g}{s} (ms^2 x - F_d)
|
|
\end{equation}
|
|
<p>
|
|
Once the IFF is applied, the new dynamics of the system is:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d + F - kx - csx + kw + csw - \frac{g}{s} (ms^2x - F_d)
|
|
\end{equation}
|
|
|
|
<p>
|
|
And finally:
|
|
</p>
|
|
\begin{equation}
|
|
x = F_d \frac{1 + \frac{g}{s}}{ms^2 + (mg + c)s + k} + F \frac{1}{ms^2 + (mg + c)s + k} + w \frac{k + cs}{ms^2 + (mg + c)s + k}
|
|
\end{equation}
|
|
|
|
<p>
|
|
We can see that this:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>adds damping to the system by a value \(mg\)</li>
|
|
<li>lower the compliance as low frequency by a factor: \(1 + g/s\)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
If we want critical damping:
|
|
</p>
|
|
\begin{equation}
|
|
\xi = \frac{1}{2} \frac{c + gm}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
|
|
<p>
|
|
This is attainable if we have:
|
|
</p>
|
|
\begin{equation}
|
|
g = \frac{\sqrt{km} - c}{m}
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb3880fc" class="outline-4">
|
|
<h4 id="orgb3880fc"><span class="section-number-4">2.1.2</span> Matlab Example</h4>
|
|
<div class="outline-text-4" id="text-2-1-2">
|
|
<p>
|
|
Let define the system parameters.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = <span class="org-highlight-numbers-number">50</span>; <span class="org-comment">% [kg]</span>
|
|
k = <span class="org-highlight-numbers-number">1e6</span>; <span class="org-comment">% [N/m]</span>
|
|
c = <span class="org-highlight-numbers-number">1e3</span>; <span class="org-comment">% [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The state space model of the system is defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">A = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
B = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
C = <span class="org-rainbow-delimiters-depth-1">[</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-type">-</span>c <span class="org-type">-</span>k<span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
D = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
sys = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
sys.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'d'</span>, <span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.StateName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The controller \(K_\text{IFF}\) is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kiff = <span class="org-type">-</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>k<span class="org-type">*</span>m<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>c<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span>m<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span>s;
|
|
Kiff.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
Kiff.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the closed loop system is computed below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys_iff = feedback<span class="org-rainbow-delimiters-depth-1">(</span>sys, Kiff, <span class="org-string">'name'</span>, <span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgb9563d1" class="figure">
|
|
<p><img src="figs/iff_1dof_sensitivitiy.png" alt="iff_1dof_sensitivitiy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Sensitivity to disturbance when IFF is applied on the 1dof system (<a href="./figs/iff_1dof_sensitivitiy.png">png</a>, <a href="./figs/iff_1dof_sensitivitiy.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org731eb41" class="outline-3">
|
|
<h3 id="org731eb41"><span class="section-number-3">2.2</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
Let's load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let'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 (figure <a href="#orgeb9eff4">6</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgeb9eff4" class="figure">
|
|
<p><img src="figs/iff_plant.png" alt="iff_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/iff_plant.png">png</a>, <a href="./figs/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><span class="org-highlight-numbers-number">1000</span><span class="org-type">/</span>s;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The corresponding loop gains are shown in figure <a href="#org4e84d34">7</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org4e84d34" class="figure">
|
|
<p><img src="figs/iff_open_loop.png" alt="iff_open_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/iff_open_loop.png">png</a>, <a href="./figs/iff_open_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcaee5e8" class="outline-3">
|
|
<h3 id="orgcaee5e8"><span class="section-number-3">2.3</span> Identification of the damped plant</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
Let's initialize the system prior to identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeInputs<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
All the controllers are set to 0.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_iff = <span class="org-type">-</span>K_iff<span class="org-type">*</span>eye<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We identify the system dynamics now that the IFF controller is ON.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_iff = identifyPlant<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we save the damped plant for further analysis
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org92c8ec7" class="outline-3">
|
|
<h3 id="org92c8ec7"><span class="section-number-3">2.4</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
As shown on figure <a href="#orgb6249ff">8</a>:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>The top platform of the nano-hexapod how behaves as a "free-mass".</li>
|
|
<li>The transfer function from direct forces \(F_s\) to the relative displacement \(D\) is equivalent to the one of an isolated mass.</li>
|
|
<li>The transfer function from ground motion \(D_g\) to the relative displacement \(D\) tends to the transfer function from \(D_g\) to the displacement of the granite (the sample is being isolated thanks to IFF).
|
|
However, as the goal is to make the relative displacement \(D\) as small as possible (e.g. to make the sample motion follows the granite motion), this is not a good thing.</li>
|
|
</ul>
|
|
|
|
|
|
<div id="orgb6249ff" class="figure">
|
|
<p><img src="figs/sensitivity_dist_iff.png" alt="sensitivity_dist_iff.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Sensitivity to disturbance once the IFF controller is applied to the system (<a href="./figs/sensitivity_dist_iff.png">png</a>, <a href="./figs/sensitivity_dist_iff.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<div class="warning">
|
|
<p>
|
|
The order of the models are very high and thus the plots may be wrong.
|
|
For instance, the plots are not the same when using <code>minreal</code>.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
|
|
<div id="org89f676a" class="figure">
|
|
<p><img src="figs/sensitivity_dist_stages_iff.png" alt="sensitivity_dist_stages_iff.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Sensitivity to force disturbances in various stages when IFF is applied (<a href="./figs/sensitivity_dist_stages_iff.png">png</a>, <a href="./figs/sensitivity_dist_stages_iff.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgacc1bc1" class="outline-3">
|
|
<h3 id="orgacc1bc1"><span class="section-number-3">2.5</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-2-5">
|
|
<p>
|
|
Now, look at the new damped plant to control.
|
|
</p>
|
|
|
|
<p>
|
|
It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure <a href="#orgef38522">10</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgef38522" class="figure">
|
|
<p><img src="figs/plant_iff_damped.png" alt="plant_iff_damped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Damped Plant after IFF is applied (<a href="./figs/plant_iff_damped.png">png</a>, <a href="./figs/plant_iff_damped.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
However, it increases coupling at low frequency (figure <a href="#org4f7739c">11</a>).
|
|
</p>
|
|
|
|
<div id="org4f7739c" class="figure">
|
|
<p><img src="figs/plant_iff_coupling.png" alt="plant_iff_coupling.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Coupling induced by IFF (<a href="./figs/plant_iff_coupling.png">png</a>, <a href="./figs/plant_iff_coupling.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orged93d15" class="outline-3">
|
|
<h3 id="orged93d15"><span class="section-number-3">2.6</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-6">
|
|
<div class="important">
|
|
<p>
|
|
Integral Force Feedback:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Robust (guaranteed stability)</li>
|
|
<li>Acceptable Damping</li>
|
|
<li>Increase the sensitivity to disturbances at low frequencies</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org087ecf6" class="outline-2">
|
|
<h2 id="org087ecf6"><span class="section-number-2">3</span> Relative Motion Control</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org42ae53a"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/rmc.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<p>
|
|
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orga33a4fa" class="outline-3">
|
|
<h3 id="orga33a4fa"><span class="section-number-3">3.1</span> One degree-of-freedom example</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
<a id="org80ae5dd"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orge6e79c5" class="outline-4">
|
|
<h4 id="orge6e79c5"><span class="section-number-4">3.1.1</span> Equations</h4>
|
|
<div class="outline-text-4" id="text-3-1-1">
|
|
|
|
<div id="orgdd8eb1b" class="figure">
|
|
<p><img src="figs/rmc_1dof.png" alt="rmc_1dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Relative Motion Control applied to a 1dof system</p>
|
|
</div>
|
|
|
|
<p>
|
|
The dynamic of the system is:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
<p>
|
|
In terms of the stage deformation \(d = x - w\):
|
|
</p>
|
|
\begin{equation}
|
|
(ms^2 + cs + k) d = -ms^2 w + F_d + F
|
|
\end{equation}
|
|
<p>
|
|
The relative motion control law is:
|
|
</p>
|
|
\begin{equation}
|
|
K = -g s
|
|
\end{equation}
|
|
<p>
|
|
Thus, the applied force is:
|
|
</p>
|
|
\begin{equation}
|
|
F = -g s d
|
|
\end{equation}
|
|
<p>
|
|
And the new dynamics will be:
|
|
</p>
|
|
\begin{equation}
|
|
d = w \frac{-ms^2}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
|
|
\end{equation}
|
|
|
|
<p>
|
|
And thus damping is added.
|
|
</p>
|
|
|
|
<p>
|
|
If critical damping is wanted:
|
|
</p>
|
|
\begin{equation}
|
|
\xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
<p>
|
|
This corresponds to a gain:
|
|
</p>
|
|
\begin{equation}
|
|
g = \sqrt{km} - c
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfb8caad" class="outline-4">
|
|
<h4 id="orgfb8caad"><span class="section-number-4">3.1.2</span> Matlab Example</h4>
|
|
<div class="outline-text-4" id="text-3-1-2">
|
|
<p>
|
|
Let define the system parameters.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = <span class="org-highlight-numbers-number">50</span>; <span class="org-comment">% [kg]</span>
|
|
k = <span class="org-highlight-numbers-number">1e6</span>; <span class="org-comment">% [N/m]</span>
|
|
c = <span class="org-highlight-numbers-number">1e3</span>; <span class="org-comment">% [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The state space model of the system is defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">A = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
B = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
C = <span class="org-rainbow-delimiters-depth-1">[</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-type">-</span>c <span class="org-type">-</span>k<span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
D = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
sys = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
sys.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'d'</span>, <span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.StateName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The controller \(K_\text{RMC}\) is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Krmc = <span class="org-type">-</span><span class="org-rainbow-delimiters-depth-1">(</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>k<span class="org-type">*</span>m<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">-</span>c<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span>s;
|
|
Krmc.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
Krmc.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the closed loop system is computed below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys_rmc = feedback<span class="org-rainbow-delimiters-depth-1">(</span>sys, Krmc, <span class="org-string">'name'</span>, <span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org32a5d85" class="figure">
|
|
<p><img src="figs/rmc_1dof_sensitivitiy.png" alt="rmc_1dof_sensitivitiy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Sensitivity to disturbance when RMC is applied on the 1dof system (<a href="./figs/rmc_1dof_sensitivitiy.png">png</a>, <a href="./figs/rmc_1dof_sensitivitiy.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcb24491" class="outline-3">
|
|
<h3 id="orgcb24491"><span class="section-number-3">3.2</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
Let's load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let'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 (figure <a href="#org121c8a8">14</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org121c8a8" class="figure">
|
|
<p><img src="figs/rmc_plant.png" alt="rmc_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/rmc_plant.png">png</a>, <a href="./figs/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><span class="org-highlight-numbers-number">50000</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">10000</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained loop gains are shown in figure <a href="#orgcbece09">15</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgcbece09" class="figure">
|
|
<p><img src="figs/rmc_open_loop.png" alt="rmc_open_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/rmc_open_loop.png">png</a>, <a href="./figs/rmc_open_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaeb6872" class="outline-3">
|
|
<h3 id="orgaeb6872"><span class="section-number-3">3.3</span> Identification of the damped plant</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
Let's initialize the system prior to identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeInputs<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And initialize the controllers.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_rmc = <span class="org-type">-</span>K_rmc<span class="org-type">*</span>eye<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We identify the system dynamics now that the RMC controller is ON.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_rmc = identifyPlant<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we save the damped plant for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2fc9fe6" class="outline-3">
|
|
<h3 id="org2fc9fe6"><span class="section-number-3">3.4</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<p>
|
|
As shown in figure <a href="#org826fff4">16</a>, RMC control succeed in lowering the sensitivity to disturbances near resonance of the system.
|
|
</p>
|
|
|
|
|
|
<div id="org826fff4" class="figure">
|
|
<p><img src="figs/sensitivity_dist_rmc.png" alt="sensitivity_dist_rmc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Sensitivity to disturbance once the RMC controller is applied to the system (<a href="./figs/sensitivity_dist_rmc.png">png</a>, <a href="./figs/sensitivity_dist_rmc.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org9de1750" class="figure">
|
|
<p><img src="figs/sensitivity_dist_stages_rmc.png" alt="sensitivity_dist_stages_rmc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Sensitivity to force disturbances in various stages when RMC is applied (<a href="./figs/sensitivity_dist_stages_rmc.png">png</a>, <a href="./figs/sensitivity_dist_stages_rmc.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org846d098" class="outline-3">
|
|
<h3 id="org846d098"><span class="section-number-3">3.5</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-3-5">
|
|
|
|
<div id="orgd532fac" class="figure">
|
|
<p><img src="figs/plant_rmc_damped.png" alt="plant_rmc_damped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Damped Plant after RMC is applied (<a href="./figs/plant_rmc_damped.png">png</a>, <a href="./figs/plant_rmc_damped.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org12f4764" class="outline-3">
|
|
<h3 id="org12f4764"><span class="section-number-3">3.6</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-3-6">
|
|
<div class="important">
|
|
<p>
|
|
Relative Motion Control:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li></li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd92711d" class="outline-2">
|
|
<h2 id="orgd92711d"><span class="section-number-2">4</span> Direct Velocity Feedback</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org556890f"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/dvf.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<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>
|
|
|
|
<div id="outline-container-org189fb3b" class="outline-3">
|
|
<h3 id="org189fb3b"><span class="section-number-3">4.1</span> One degree-of-freedom example</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
<a id="org61e898c"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org5318c89" class="outline-4">
|
|
<h4 id="org5318c89"><span class="section-number-4">4.1.1</span> Equations</h4>
|
|
<div class="outline-text-4" id="text-4-1-1">
|
|
|
|
<div id="org558eae6" class="figure">
|
|
<p><img src="figs/dvf_1dof.png" alt="dvf_1dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Direct Velocity Feedback applied to a 1dof system</p>
|
|
</div>
|
|
|
|
<p>
|
|
The dynamic of the system is:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
<p>
|
|
In terms of the stage deformation \(d = x - w\):
|
|
</p>
|
|
\begin{equation}
|
|
(ms^2 + cs + k) d = -ms^2 w + F_d + F
|
|
\end{equation}
|
|
<p>
|
|
The direct velocity feedback law shown in figure <a href="#org558eae6">19</a> is:
|
|
</p>
|
|
\begin{equation}
|
|
K = -g
|
|
\end{equation}
|
|
<p>
|
|
Thus, the applied force is:
|
|
</p>
|
|
\begin{equation}
|
|
F = -g \dot{x}
|
|
\end{equation}
|
|
<p>
|
|
And the new dynamics will be:
|
|
</p>
|
|
\begin{equation}
|
|
d = w \frac{-ms^2 - gs}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
|
|
\end{equation}
|
|
|
|
<p>
|
|
And thus damping is added.
|
|
</p>
|
|
|
|
<p>
|
|
If critical damping is wanted:
|
|
</p>
|
|
\begin{equation}
|
|
\xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
<p>
|
|
This corresponds to a gain:
|
|
</p>
|
|
\begin{equation}
|
|
g = \sqrt{km} - c
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfb2a947" class="outline-4">
|
|
<h4 id="orgfb2a947"><span class="section-number-4">4.1.2</span> Matlab Example</h4>
|
|
<div class="outline-text-4" id="text-4-1-2">
|
|
<p>
|
|
Let define the system parameters.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = <span class="org-highlight-numbers-number">50</span>; <span class="org-comment">% [kg]</span>
|
|
k = <span class="org-highlight-numbers-number">1e6</span>; <span class="org-comment">% [N/m]</span>
|
|
c = <span class="org-highlight-numbers-number">1e3</span>; <span class="org-comment">% [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The state space model of the system is defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">A = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
B = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
C = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
D = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
sys = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
sys.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.StateName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Because we need \(\dot{x}\) for feedback, we compute it from the outputs
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_xdot = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>s;
|
|
<span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
G_xdot.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
G_xdot.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'xdot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, the system is described by <code>sys</code> as defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys = series<span class="org-rainbow-delimiters-depth-1">(</span>sys, G_xdot, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">2</span> <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">2</span> <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The controller \(K_\text{DVF}\) is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kdvf = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>k<span class="org-type">*</span>m<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>c<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
Kdvf.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'xdot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
Kdvf.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the closed loop system is computed below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys_dvf = feedback<span class="org-rainbow-delimiters-depth-1">(</span>sys, Kdvf, <span class="org-string">'name'</span>, <span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained sensitivity to disturbances is shown in figure <a href="#org8ff062e">20</a>.
|
|
</p>
|
|
|
|
<div id="org8ff062e" class="figure">
|
|
<p><img src="figs/dvf_1dof_sensitivitiy.png" alt="dvf_1dof_sensitivitiy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Sensitivity to disturbance when DVF is applied on the 1dof system (<a href="./figs/dvf_1dof_sensitivitiy.png">png</a>, <a href="./figs/dvf_1dof_sensitivitiy.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3c87599" class="outline-3">
|
|
<h3 id="org3c87599"><span class="section-number-3">4.2</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
Let's load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure <a href="#orgffe220b">21</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgffe220b" class="figure">
|
|
<p><img src="figs/dvf_plant.png" alt="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/dvf_plant.png">png</a>, <a href="./figs/dvf_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
The controller is defined below and the obtained loop gain is shown in figure <a href="#org9506f40">22</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3e4</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org9506f40" class="figure">
|
|
<p><img src="figs/dvf_open_loop_gain.png" alt="dvf_open_loop_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>Loop Gain for DVF (<a href="./figs/dvf_open_loop_gain.png">png</a>, <a href="./figs/dvf_open_loop_gain.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org07e7826" class="outline-3">
|
|
<h3 id="org07e7826"><span class="section-number-3">4.3</span> Identification of the damped plant</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<p>
|
|
Let's initialize the system prior to identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeInputs<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And initialize the controllers.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_dvf = <span class="org-type">-</span>K_dvf<span class="org-type">*</span>eye<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We identify the system dynamics now that the RMC controller is ON.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_dvf = identifyPlant<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we save the damped plant for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org901fc2b" class="outline-3">
|
|
<h3 id="org901fc2b"><span class="section-number-3">4.4</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
|
|
<div id="orgc1ce7ef" class="figure">
|
|
<p><img src="figs/sensitivity_dist_dvf.png" alt="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/sensitivity_dist_dvf.png">png</a>, <a href="./figs/sensitivity_dist_dvf.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org19e919e" class="figure">
|
|
<p><img src="figs/sensitivity_dist_stages_dvf.png" alt="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/sensitivity_dist_stages_dvf.png">png</a>, <a href="./figs/sensitivity_dist_stages_dvf.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org22a4374" class="outline-3">
|
|
<h3 id="org22a4374"><span class="section-number-3">4.5</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
|
|
<div id="org192575d" class="figure">
|
|
<p><img src="figs/plant_dvf_damped.png" alt="plant_dvf_damped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 25: </span>Damped Plant after DVF is applied (<a href="./figs/plant_dvf_damped.png">png</a>, <a href="./figs/plant_dvf_damped.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1c0a92c" class="outline-3">
|
|
<h3 id="org1c0a92c"><span class="section-number-3">4.6</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-6">
|
|
<div class="important">
|
|
<p>
|
|
Direct Velocity Feedback:
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org099228a" class="outline-2">
|
|
<h2 id="org099228a"><span class="section-number-2">5</span> Comparison</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="orgcff0bce"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org10b67a5" class="outline-3">
|
|
<h3 id="org10b67a5"><span class="section-number-3">5.1</span> Load the plants</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/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><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0dcb40a" class="outline-3">
|
|
<h3 id="org0dcb40a"><span class="section-number-3">5.2</span> Sensitivity to Disturbance</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
|
|
<div id="org919b8db" class="figure">
|
|
<p><img src="figs/sensitivity_comp_ground_motion_z.png" alt="sensitivity_comp_ground_motion_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 26: </span>caption (<a href="./figs/sensitivity_comp_ground_motion_z.png">png</a>, <a href="./figs/sensitivity_comp_ground_motion_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="orge4a04c8" class="figure">
|
|
<p><img src="figs/sensitivity_comp_direct_forces_z.png" alt="sensitivity_comp_direct_forces_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 27: </span>caption (<a href="./figs/sensitivity_comp_direct_forces_z.png">png</a>, <a href="./figs/sensitivity_comp_direct_forces_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgd15d59b" class="figure">
|
|
<p><img src="figs/sensitivity_comp_spindle_z.png" alt="sensitivity_comp_spindle_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 28: </span>caption (<a href="./figs/sensitivity_comp_spindle_z.png">png</a>, <a href="./figs/sensitivity_comp_spindle_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgcb6a783" class="figure">
|
|
<p><img src="figs/sensitivity_comp_ty_z.png" alt="sensitivity_comp_ty_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 29: </span>caption (<a href="./figs/sensitivity_comp_ty_z.png">png</a>, <a href="./figs/sensitivity_comp_ty_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org154b81b" class="figure">
|
|
<p><img src="figs/sensitivity_comp_ty_x.png" alt="sensitivity_comp_ty_x.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 30: </span>caption (<a href="./figs/sensitivity_comp_ty_x.png">png</a>, <a href="./figs/sensitivity_comp_ty_x.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcc0be93" class="outline-3">
|
|
<h3 id="orgcc0be93"><span class="section-number-3">5.3</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
|
|
<div id="orgf41f2bc" class="figure">
|
|
<p><img src="figs/plant_comp_damping_z.png" alt="plant_comp_damping_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 31: </span>Plant for the \(z\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_z.png">png</a>, <a href="./figs/plant_comp_damping_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org5183422" class="figure">
|
|
<p><img src="figs/plant_comp_damping_x.png" alt="plant_comp_damping_x.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 32: </span>Plant for the \(x\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_x.png">png</a>, <a href="./figs/plant_comp_damping_x.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org74505ba" class="figure">
|
|
<p><img src="figs/plant_comp_damping_coupling.png" alt="plant_comp_damping_coupling.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 33: </span>Comparison of one off-diagonal plant for different damping technique applied (<a href="./figs/plant_comp_damping_coupling.png">png</a>, <a href="./figs/plant_comp_damping_coupling.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0476571" class="outline-2">
|
|
<h2 id="org0476571"><span class="section-number-2">6</span> Conclusion</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="org158aa9e"></a>
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2019-10-25 ven. 16:00</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|