nass-simscape/docs/hac_lac.html

535 lines
22 KiB
HTML

<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-03-13 ven. 17:39 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>HAC-LAC applied on the Simscape 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>
<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">
// @license magnet:?xt=urn:btih:1f739d935676111cfff4b4693e3816e664797050&dn=gpl-3.0.txt GPL-v3-or-Later
<!--/*--><![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;
}
/*]]>*///-->
// @license-end
</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">
<a accesskey="h" href="./index.html"> UP </a>
|
<a accesskey="H" href="./index.html"> HOME </a>
</div><div id="content">
<h1 class="title">HAC-LAC applied on the Simscape Model</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org9a782be">1. Initialization</a></li>
<li><a href="#org5f71356">2. Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)</a>
<ul>
<li><a href="#orgd91db57">2.1. Identification</a></li>
<li><a href="#org09c8990">2.2. Plant</a></li>
<li><a href="#orgc21ad83">2.3. Root Locus</a></li>
<li><a href="#orgd18d476">2.4. Controller and Loop Gain</a></li>
</ul>
</li>
<li><a href="#orgbba86f0">3. High Authority Control - \(\bm{K}_\mathcal{X}\)</a>
<ul>
<li><a href="#orgdd75cdd">3.1. Identification of the damped plant</a></li>
<li><a href="#orgf70f020">3.2. Controller Design</a></li>
</ul>
</li>
<li><a href="#org5a1507e">4. Simulation</a></li>
<li><a href="#org9498b7b">5. Results</a></li>
</ul>
</div>
</div>
<p>
The position \(\bm{\mathcal{X}}\) of the Sample with respect to the granite is measured.
</p>
<p>
It is then compare to the wanted position of the Sample \(\bm{r}_\mathcal{X}\) in order to obtain the position error \(\bm{\epsilon}_\mathcal{X}\) of the Sample with respect to a frame attached to the Stewart top platform.
</p>
<div class="figure">
<p><img src="figs/hac_lac_control_schematic.png" alt="hac_lac_control_schematic.png" />
</p>
</div>
<div id="outline-container-org9a782be" class="outline-2">
<h2 id="org9a782be"><span class="section-number-2">1</span> Initialization</h2>
<div class="outline-text-2" id="text-1">
<p>
We initialize all the stages with the default parameters.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
</pre>
</div>
<p>
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">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeSample(<span class="org-string">'mass'</span>, 1);
</pre>
</div>
<p>
We set the references that corresponds to a tomography experiment.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, 1);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeDisturbances();
</pre>
</div>
<p>
Open Loop.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
<p>
And we put some gravity.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
</pre>
</div>
<p>
We log the signals.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org5f71356" class="outline-2">
<h2 id="org5f71356"><span class="section-number-2">2</span> Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)</h2>
<div class="outline-text-2" id="text-2">
<p>
The first loop closed corresponds to a direct velocity feedback loop.
</p>
<p>
The design of the associated decentralized controller is explained in <a href="active_damping.html">this</a> file.
</p>
</div>
<div id="outline-container-orgd91db57" class="outline-3">
<h3 id="orgd91db57"><span class="section-number-3">2.1</span> Identification</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'nass_model'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Dnlm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Motion Outputs</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_dvf = linearize(mdl, io, 0);
G_dvf.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
G_dvf.OutputName = {<span class="org-string">'Dnlm1'</span>, <span class="org-string">'Dnlm2'</span>, <span class="org-string">'Dnlm3'</span>, <span class="org-string">'Dnlm4'</span>, <span class="org-string">'Dnlm5'</span>, <span class="org-string">'Dnlm6'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-org09c8990" class="outline-3">
<h3 id="org09c8990"><span class="section-number-3">2.2</span> Plant</h3>
</div>
<div id="outline-container-orgc21ad83" class="outline-3">
<h3 id="orgc21ad83"><span class="section-number-3">2.3</span> Root Locus</h3>
</div>
<div id="outline-container-orgd18d476" class="outline-3">
<h3 id="orgd18d476"><span class="section-number-3">2.4</span> Controller and Loop Gain</h3>
<div class="outline-text-3" id="text-2-4">
<div class="org-src-container">
<pre class="src src-matlab">K_dvf = s<span class="org-type">*</span>15000<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 class="org-src-container">
<pre class="src src-matlab">K_dvf = <span class="org-type">-</span>K_dvf<span class="org-type">*</span>eye(6);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgbba86f0" class="outline-2">
<h2 id="orgbba86f0"><span class="section-number-2">3</span> High Authority Control - \(\bm{K}_\mathcal{X}\)</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orgdd75cdd" class="outline-3">
<h3 id="orgdd75cdd"><span class="section-number-3">3.1</span> Identification of the damped plant</h3>
<div class="outline-text-3" id="text-3-1">
<div class="org-src-container">
<pre class="src src-matlab">Kx = tf(zeros(6));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'nass_model'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'output'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Errror</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, 0);
G.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
</pre>
</div>
<p>
The minus sine is put here because there is already a minus sign included due to the computation of the position error.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
Gx = <span class="org-type">-</span>G<span class="org-type">*</span>inv(nano_hexapod.J<span class="org-type">'</span>);
Gx.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-orgf70f020" class="outline-3">
<h3 id="orgf70f020"><span class="section-number-3">3.2</span> Controller Design</h3>
<div class="outline-text-3" id="text-3-2">
<p>
The controller consists of:
</p>
<ul class="org-ul">
<li>A pure integrator</li>
<li>A Second integrator up to half the wanted bandwidth</li>
<li>A Lead around the cross-over frequency</li>
<li>A low pass filter with a cut-off equal to two times the wanted bandwidth</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>15; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
h = 1.5; <span class="org-comment">% Lead parameter</span>
Kx = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> ((s<span class="org-type">/</span>wc<span class="org-type">*</span>2 <span class="org-type">+</span> 1)<span class="org-type">/</span>(s<span class="org-type">/</span>wc<span class="org-type">*</span>2)) <span class="org-type">*</span> (1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>2));
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
Kx = Kx<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kx, wc))));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">isstable(feedback(Gx<span class="org-type">*</span>Kx, eye(6), <span class="org-type">-</span>1))
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Kx = inv(nano_hexapod.J<span class="org-type">'</span>)<span class="org-type">*</span>Kx;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">isstable(feedback(G<span class="org-type">*</span>Kx, eye(6), 1))
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org5a1507e" class="outline-2">
<h2 id="org5a1507e"><span class="section-number-2">4</span> Simulation</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'1.5'</span>);
</pre>
</div>
<p>
And we simulate the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'simout'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org9498b7b" class="outline-2">
<h2 id="org9498b7b"><span class="section-number-2">5</span> Results</h2>
<div class="outline-text-2" id="text-5">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/tomo_exp_hac_lac.mat'</span>, <span class="org-string">'simout'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-13 ven. 17:39</p>
</div>
</body>
</html>