2020-03-23 10:05:32 +01:00
<?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 >
2020-03-26 17:25:43 +01:00
<!-- 2020 - 03 - 26 jeu. 17:25 -->
2020-03-23 10:05:32 +01:00
< meta http-equiv = "Content-Type" content = "text/html;charset=utf-8" / >
< meta name = "viewport" content = "width=device-width, initial-scale=1" / >
< title > Control in the Frame of the Legs 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" >
2020-03-26 17:25:43 +01:00
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2020 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
2020-03-23 10:05:32 +01:00
<!-- /* --> <![CDATA[/*> <!-- */
2020-03-26 17:25:43 +01:00
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;
}
/*]]>*///-->
2020-03-23 10:05:32 +01:00
< / 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" > Control in the Frame of the Legs 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 = "#org1f22cdb" > 1. Decentralized Control< / a >
< ul >
2020-03-26 17:25:43 +01:00
< li > < a href = "#orgcd58f5e" > 1.1. Control Schematic< / a > < / li >
< li > < a href = "#org4318f13" > 1.2. Initialize the Simscape Model< / a > < / li >
2020-03-23 10:05:32 +01:00
< li > < a href = "#org42afc36" > 1.3. Identification of the plant< / a > < / li >
< li > < a href = "#org871560e" > 1.4. Plant Analysis< / a > < / li >
2020-03-26 17:25:43 +01:00
< li > < a href = "#orgdf3c9aa" > 1.5. Controller Design< / a > < / li >
< li > < a href = "#org753f2c4" > 1.6. Simulation< / a > < / li >
< li > < a href = "#orgd8c14ee" > 1.7. Results< / a > < / li >
2020-03-23 10:05:32 +01:00
< / ul >
< / li >
< li > < a href = "#orgfa4be92" > 2. HAC-LAC (IFF) Decentralized Control< / a >
< ul >
2020-03-26 17:25:43 +01:00
< li > < a href = "#org744f1a3" > 2.1. Control Schematic< / a > < / li >
< li > < a href = "#org0e6472e" > 2.2. Initialize the Simscape Model< / a > < / li >
2020-03-23 10:05:32 +01:00
< li > < a href = "#org0903106" > 2.3. Initialization< / a > < / li >
< li > < a href = "#orge739f61" > 2.4. Identification for IFF< / a > < / li >
< li > < a href = "#org08b6a99" > 2.5. Integral Force Feedback Controller< / a > < / li >
< li > < a href = "#org06b5c75" > 2.6. Identification of the damped plant< / a > < / li >
2020-03-26 17:25:43 +01:00
< li > < a href = "#orgf258d8b" > 2.7. Controller Design< / a > < / li >
< li > < a href = "#org5f74b9e" > 2.8. Simulation< / a > < / li >
< li > < a href = "#org42ecd75" > 2.9. Results< / a > < / li >
2020-03-23 10:05:32 +01:00
< / ul >
< / li >
< li > < a href = "#org21a9294" > 3. Conclusion< / a > < / li >
< / ul >
< / div >
< / div >
< p >
In this document, we apply some decentralized control to the NASS and see what level of performance can be obtained.
< / p >
< div id = "outline-container-org1f22cdb" class = "outline-2" >
< h2 id = "org1f22cdb" > < span class = "section-number-2" > 1< / span > Decentralized Control< / h2 >
< div class = "outline-text-2" id = "text-1" >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-orgcd58f5e" class = "outline-3" >
< h3 id = "orgcd58f5e" > < span class = "section-number-3" > 1.1< / span > Control Schematic< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-1-1" >
< p >
2020-03-26 17:25:43 +01:00
The control architecture is shown in Figure < a href = "#org5b991df" > 1< / a > .
2020-03-23 10:05:32 +01:00
< / p >
< p >
The signals are:
< / p >
< ul class = "org-ul" >
< li > \(\bm{r}_\mathcal{X}\): wanted position of the sample with respect to the granite< / li >
< li > \(\bm{r}_{\mathcal{X}_n}\): wanted position of the sample with respect to the nano-hexapod< / li >
< li > \(\bm{r}_\mathcal{L}\): wanted length of each of the nano-hexapod’ s legs< / li >
< li > \(\bm{\tau}\): forces applied in each actuator< / li >
< li > \(\bm{\mathcal{L}}\): measured displacement of each leg< / li >
< li > \(\bm{\mathcal{X}}\): measured position of the sample with respect to the granite< / li >
< / ul >
2020-03-26 17:25:43 +01:00
< div id = "org5b991df" class = "figure" >
2020-03-23 10:05:32 +01:00
< p > < img src = "figs/decentralized_reference_tracking_L.png" alt = "decentralized_reference_tracking_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Decentralized control for reference tracking< / p >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-org4318f13" class = "outline-3" >
< h3 id = "org4318f13" > < span class = "section-number-3" > 1.2< / span > Initialize the Simscape Model< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-1-2" >
< 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" > 'ref-track-L'< / span > );
Kl = tf(zeros(6));
< / 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-org42afc36" class = "outline-3" >
< h3 id = "org42afc36" > < span class = "section-number-3" > 1.3< / span > Identification of the plant< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
< p >
Let’ s identify the transfer function from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\).
< / p >
< 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" > '/Controller/Reference-Tracking-L/Sum'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Leg length error< / 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" > 'El1'< / span > , < span class = "org-string" > 'El2'< / span > , < span class = "org-string" > 'El3'< / span > , < span class = "org-string" > 'El4'< / span > , < span class = "org-string" > 'El5'< / span > , < span class = "org-string" > 'El6'< / span > };
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org871560e" class = "outline-3" >
< h3 id = "org871560e" > < span class = "section-number-3" > 1.4< / span > Plant Analysis< / h3 >
< div class = "outline-text-3" id = "text-1-4" >
< p >
The diagonal and off-diagonal terms of the plant are shown in Figure < a href = "#org68ea839" > 2< / a > .
< / p >
< p >
We can see that:
< / p >
< ul class = "org-ul" >
< li > the diagonal terms have similar dynamics< / li >
< li > the plant is decoupled at low frequency< / li >
< / ul >
< div id = "org68ea839" class = "figure" >
< p > < img src = "figs/decentralized_control_plant_L.png" alt = "decentralized_control_plant_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Transfer Functions from forces applied in each actuator \(\tau_i\) to the relative motion of each leg \(d\mathcal{L}_i\) (< a href = "./figs/decentralized_control_plant_L.png" > png< / a > , < a href = "./figs/decentralized_control_plant_L.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-orgdf3c9aa" class = "outline-3" >
< h3 id = "orgdf3c9aa" > < span class = "section-number-3" > 1.5< / span > Controller Design< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-1-5" >
< p >
The controller consists of:
< / p >
< ul class = "org-ul" >
< li > A pure integrator< / li >
< li > An integrator up to little before the crossover< / li >
< li > A lead around the crossover< / li >
< li > A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin< / li >
< / ul >
< p >
The obtained loop gains corresponding to the diagonal elements are shown in Figure < a href = "#org3bd29d4" > 3< / a > .
< / p >
< 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 > 20;
h = 1.5;
Kl = diag(1< span class = "org-type" > ./< / span > diag(abs(freqresp(G, wc)))) < span class = "org-type" > *< / span > ...
wc< span class = "org-type" > /< / span > s < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Pure Integrator< / 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 > ...< span class = "org-comment" > % Integrator up to wc/2< / span >
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 > ...< span class = "org-comment" > % Lead< / span >
1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 3< span class = "org-type" > /< / span > wc) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Low pass Filter< / span >
1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 3< span class = "org-type" > /< / span > wc);
< / pre >
< / div >
< div id = "org3bd29d4" class = "figure" >
< p > < img src = "figs/decentralized_control_L_loop_gain.png" alt = "decentralized_control_L_loop_gain.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Obtained Loop Gain (< a href = "./figs/decentralized_control_L_loop_gain.png" > png< / a > , < a href = "./figs/decentralized_control_L_loop_gain.pdf" > pdf< / a > )< / p >
< / div >
< p >
We add a minus sign to the controller as it is not included in the Simscape model.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kl = < span class = "org-type" > -< / span > Kl;
< / pre >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-org753f2c4" class = "outline-3" >
< h3 id = "org753f2c4" > < span class = "section-number-3" > 1.6< / span > Simulation< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-1-6" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'ref-track-L'< / span > );
< / pre >
< / div >
< 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" > '2'< / span > );
< / pre >
< / div >
< 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" > decentralized_L = simout;
save(< span class = "org-string" > './mat/tomo_exp_decentalized.mat'< / span > , < span class = "org-string" > 'decentralized_L'< / span > , < span class = "org-string" > '-append'< / span > );
< / pre >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-orgd8c14ee" class = "outline-3" >
< h3 id = "orgd8c14ee" > < span class = "section-number-3" > 1.7< / span > Results< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-1-7" >
< p >
The reference path and the position of the mobile platform are shown in Figure < a href = "#org247b3a0" > 4< / a > .
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/experiment_tomography.mat'< / span > , < span class = "org-string" > 'tomo_align_dist'< / span > );
load(< span class = "org-string" > './mat/tomo_exp_decentalized.mat'< / span > , < span class = "org-string" > 'decentralized_L'< / span > );
< / pre >
< / div >
< div id = "org247b3a0" class = "figure" >
< p > < img src = "figs/decentralized_L_position_errors.png" alt = "decentralized_L_position_errors.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Position Errors when using the Decentralized Control Architecture (< a href = "./figs/decentralized_L_position_errors.png" > png< / a > , < a href = "./figs/decentralized_L_position_errors.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgfa4be92" class = "outline-2" >
< h2 id = "orgfa4be92" > < span class = "section-number-2" > 2< / span > HAC-LAC (IFF) Decentralized Control< / h2 >
< div class = "outline-text-2" id = "text-2" >
< p >
We here add an Active Damping Loop (Integral Force Feedback) prior to using the Decentralized control architecture using \(\bm{\mathcal{L}}\).
< / p >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-org744f1a3" class = "outline-3" >
< h3 id = "org744f1a3" > < span class = "section-number-3" > 2.1< / span > Control Schematic< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-2-1" >
< p >
2020-03-26 17:25:43 +01:00
The control architecture is shown in Figure < a href = "#org5b991df" > 1< / a > .
2020-03-23 10:05:32 +01:00
< / p >
< p >
The signals are:
< / p >
< ul class = "org-ul" >
< li > \(\bm{r}_\mathcal{X}\): wanted position of the sample with respect to the granite< / li >
< li > \(\bm{r}_{\mathcal{X}_n}\): wanted position of the sample with respect to the nano-hexapod< / li >
< li > \(\bm{r}_\mathcal{L}\): wanted length of each of the nano-hexapod’ s legs< / li >
< li > \(\bm{\tau}\): forces applied in each actuator< / li >
< li > \(\bm{\mathcal{L}}\): measured displacement of each leg< / li >
< li > \(\bm{\mathcal{X}}\): measured position of the sample with respect to the granite< / li >
< / ul >
2020-03-26 17:25:43 +01:00
< div id = "orgda1b906" class = "figure" >
2020-03-23 10:05:32 +01:00
< p > < img src = "figs/decentralized_reference_tracking_L.png" alt = "decentralized_reference_tracking_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Decentralized control for reference tracking< / p >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-org0e6472e" class = "outline-3" >
< h3 id = "org0e6472e" > < span class = "section-number-3" > 2.2< / span > Initialize the Simscape Model< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-2-2" >
< 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" > 'ref-track-L'< / span > );
Kl = tf(zeros(6));
< / 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-org0903106" class = "outline-3" >
< h3 id = "org0903106" > < span class = "section-number-3" > 2.3< / span > Initialization< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'ref-track-iff-L'< / span > );
K_iff = tf(zeros(6));
Kl = tf(zeros(6));
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orge739f61" class = "outline-3" >
< h3 id = "orge739f61" > < span class = "section-number-3" > 2.4< / span > Identification for IFF< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
< 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" > 'Fnlm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Force Sensors< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_iff = linearize(mdl, io, 0);
G_iff.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_iff.OutputName = {< span class = "org-string" > 'Fnlm1'< / span > , < span class = "org-string" > 'Fnlm2'< / span > , < span class = "org-string" > 'Fnlm3'< / span > , < span class = "org-string" > 'Fnlm4'< / span > , < span class = "org-string" > 'Fnlm5'< / span > , < span class = "org-string" > 'Fnlm6'< / span > };
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org08b6a99" class = "outline-3" >
< h3 id = "org08b6a99" > < span class = "section-number-3" > 2.5< / span > Integral Force Feedback Controller< / h3 >
< div class = "outline-text-3" id = "text-2-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50;
K_iff = < span class = "org-type" > -< / span > 5000< span class = "org-type" > /< / span > s < span class = "org-type" > *< / span > (s< span class = "org-type" > /< / span > w0)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > w0) < span class = "org-type" > *< / span > eye(6);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K_iff = < span class = "org-type" > -< / span > K_iff;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org06b5c75" class = "outline-3" >
< h3 id = "org06b5c75" > < span class = "section-number-3" > 2.6< / span > Identification of the damped plant< / h3 >
< div class = "outline-text-3" id = "text-2-6" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink DehaezeFile< / 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" > '/Controller/Reference-Tracking-IFF-L/Sum'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Leg length error< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
Gd = linearize(mdl, io, 0);
Gd.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 > };
Gd.OutputName = {< span class = "org-string" > 'El1'< / span > , < span class = "org-string" > 'El2'< / span > , < span class = "org-string" > 'El3'< / span > , < span class = "org-string" > 'El4'< / span > , < span class = "org-string" > 'El5'< / span > , < span class = "org-string" > 'El6'< / span > };
< / pre >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-orgf258d8b" class = "outline-3" >
< h3 id = "orgf258d8b" > < span class = "section-number-3" > 2.7< / span > Controller Design< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-2-7" >
< 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 > 300;
h = 3;
Kl = diag(1< span class = "org-type" > ./< / span > diag(abs(freqresp(Gd, wc)))) < 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 > 20) < 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 > 20))) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Pure Integrator< / span >
((s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50) < span class = "org-type" > +< / span > 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 > 50))) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Integrator up to wc/2< / span >
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 > ...
1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > wc)) < span class = "org-type" > *< / span > ...
1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (3< span class = "org-type" > *< / span > wc));
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > isstable(feedback(Gd< span class = "org-type" > *< / span > Kl, eye(6), < span class = "org-type" > -< / span > 1))
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kl = < span class = "org-type" > -< / span > Kl;
< / pre >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-org5f74b9e" class = "outline-3" >
< h3 id = "org5f74b9e" > < span class = "section-number-3" > 2.8< / span > Simulation< / h3 >
2020-03-23 10:05:32 +01:00
< div class = "outline-text-3" id = "text-2-8" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'ref-track-iff-L'< / span > );
< / pre >
< / div >
< 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" > '2'< / span > );
< / pre >
< / div >
< 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" > decentralized_iff_L = simout;
save(< span class = "org-string" > './mat/tomo_exp_decentalized.mat'< / span > , < span class = "org-string" > 'decentralized_iff_L'< / span > , < span class = "org-string" > '-append'< / span > );
< / pre >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-org42ecd75" class = "outline-3" >
< h3 id = "org42ecd75" > < span class = "section-number-3" > 2.9< / span > Results< / h3 >
2020-03-23 10:05:32 +01:00
< / div >
< / div >
< div id = "outline-container-org21a9294" class = "outline-2" >
< h2 id = "org21a9294" > < span class = "section-number-2" > 3< / span > Conclusion< / h2 >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-03-26 17:25:43 +01:00
< p class = "date" > Created: 2020-03-26 jeu. 17:25< / p >
2020-03-23 10:05:32 +01:00
< / div >
< / body >
< / html >
