699 lines
30 KiB
HTML
699 lines
30 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>
|
|
<!-- 2020-02-28 ven. 17:34 -->
|
|
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
|
<title>Stewart Platform - Dynamics Study</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"/>
|
|
<script src="./js/jquery.min.js"></script>
|
|
<script src="./js/bootstrap.min.js"></script>
|
|
<script src="./js/jquery.stickytableheaders.min.js"></script>
|
|
<script src="./js/readtheorg.js"></script>
|
|
<script 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">Stewart Platform - Dynamics Study</h1>
|
|
<div id="table-of-contents">
|
|
<h2>Table of Contents</h2>
|
|
<div id="text-table-of-contents">
|
|
<ul>
|
|
<li><a href="#orgc59e712">1. Compare external forces and forces applied by the actuators</a>
|
|
<ul>
|
|
<li><a href="#org4509b7d">1.1. Comparison with fixed support</a></li>
|
|
<li><a href="#org8662186">1.2. Comparison with a flexible support</a></li>
|
|
<li><a href="#org920d3c4">1.3. Conclusion</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org81ab204">2. Comparison of the static transfer function and the Compliance matrix</a>
|
|
<ul>
|
|
<li><a href="#orge7e7242">2.1. Analysis</a></li>
|
|
<li><a href="#orgbb930ae">2.2. Conclusion</a></li>
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc59e712" class="outline-2">
|
|
<h2 id="orgc59e712"><span class="section-number-2">1</span> Compare external forces and forces applied by the actuators</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
In this section, we wish to compare the effect of forces/torques applied by the actuators with the effect of external forces/torques on the displacement of the mobile platform.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org4509b7d" class="outline-3">
|
|
<h3 id="org4509b7d"><span class="section-number-3">1.1</span> Comparison with fixed support</h3>
|
|
<div class="outline-text-3" id="text-1-1">
|
|
<p>
|
|
Let’s generate a Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We don’t put any flexibility below the Stewart platform such that <b>its base is fixed to an inertial frame</b>.
|
|
We also don’t put any payload on top of the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The transfer function from actuator forces \(\bm{\tau}\) to the relative displacement of the mobile platform \(\mathcal{\bm{X}}\) is extracted.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'stewart_platform_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 Force Inputs [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
|
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Using the Jacobian matrix, we compute the transfer function from force/torques applied by the actuators on the frame \(\{B\}\) fixed to the mobile platform:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
|
|
Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We also extract the transfer function from external forces \(\bm{\mathcal{F}}_{\text{ext}}\) on the frame \(\{B\}\) fixed to the mobile platform to the relative displacement \(\mathcal{\bm{X}}\) of \(\{B\}\) with respect to frame \(\{A\}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'F_ext'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces/torques applied on {B}</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
Gd = linearize(mdl, io, options);
|
|
Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string">'Fey'</span>, <span class="org-string">'Fez'</span>, <span class="org-string">'Mex'</span>, <span class="org-string">'Mey'</span>, <span class="org-string">'Mez'</span>};
|
|
Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The comparison of the two transfer functions is shown in Figure <a href="#orgbf9a54a">1</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgbf9a54a" class="figure">
|
|
<p><img src="figs/comparison_Fext_F_fixed_base.png" alt="comparison_Fext_F_fixed_base.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_fixed_base.png">png</a>, <a href="./figs/comparison_Fext_F_fixed_base.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
This can be understood from figure <a href="#org8bd3e63">2</a> where \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly the same effect on \(\mathcal{X}_{x}\).
|
|
</p>
|
|
|
|
|
|
<div id="org8bd3e63" class="figure">
|
|
<p><img src="figs/1dof_actuator_external_forces.png" alt="1dof_actuator_external_forces.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Schematic representation of the stewart platform on a rigid support</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8662186" class="outline-3">
|
|
<h3 id="org8662186"><span class="section-number-3">1.2</span> Comparison with a flexible support</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
<p>
|
|
We now add a flexible support under the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'flexible'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we perform again the identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
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 Force Inputs [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
|
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
|
|
Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
|
|
Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</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">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'F_ext'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces/torques applied on {B}</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
Gd = linearize(mdl, io, options);
|
|
Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string">'Fey'</span>, <span class="org-string">'Fez'</span>, <span class="org-string">'Mex'</span>, <span class="org-string">'Mey'</span>, <span class="org-string">'Mez'</span>};
|
|
Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The comparison between the obtained transfer functions is shown in Figure <a href="#orga2f2bd5">3</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orga2f2bd5" class="figure">
|
|
<p><img src="figs/comparison_Fext_F_flexible_base.png" alt="comparison_Fext_F_flexible_base.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_flexible_base.png">png</a>, <a href="./figs/comparison_Fext_F_flexible_base.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
The addition of a flexible support can be schematically represented in Figure <a href="#orgee3ecbe">4</a>.
|
|
We see that \(\mathcal{F}_{x}\) applies a force both on \(m\) and \(m^{\prime}\) whereas \(\mathcal{F}_{x,\text{ext}}\) only applies a force on \(m\).
|
|
And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly <b>not</b> the same effect on \(\mathcal{X}_{x}\).
|
|
</p>
|
|
|
|
|
|
<div id="orgee3ecbe" class="figure">
|
|
<p><img src="figs/2dof_actuator_external_forces.png" alt="2dof_actuator_external_forces.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Schematic representation of the stewart platform on top of a flexible support</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org920d3c4" class="outline-3">
|
|
<h3 id="org920d3c4"><span class="section-number-3">1.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<div class="important">
|
|
<p>
|
|
The transfer function from forces/torques applied by the actuators on the payload \(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\) to the pose of the mobile platform \(\bm{\mathcal{X}}\) is the same as the transfer function from external forces/torques to \(\bm{\mathcal{X}}\) as long as the Stewart platform’s base is fixed.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org81ab204" class="outline-2">
|
|
<h2 id="org81ab204"><span class="section-number-2">2</span> Comparison of the static transfer function and the Compliance matrix</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orge7e7242" class="outline-3">
|
|
<h3 id="orge7e7242"><span class="section-number-3">2.1</span> Analysis</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
Initialization of the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
No flexibility below the Stewart platform and no payload.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'stewart_platform_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 Force Inputs [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
|
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
|
|
Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s first look at the low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-7.2e-19</td>
|
|
<td class="org-right">5.0e-18</td>
|
|
<td class="org-right">-8.9e-18</td>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">9.9e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.7e-18</td>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-5.7e-18</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">-1.6e-17</td>
|
|
<td class="org-right">-1.7e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.3e-18</td>
|
|
<td class="org-right">-6.3e-18</td>
|
|
<td class="org-right">2.1e-08</td>
|
|
<td class="org-right">4.4e-17</td>
|
|
<td class="org-right">6.6e-18</td>
|
|
<td class="org-right">7.4e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-3.2e-17</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">6.2e-18</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-3.5e-16</td>
|
|
<td class="org-right">6.3e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">2.7e-17</td>
|
|
<td class="org-right">4.8e-17</td>
|
|
<td class="org-right">-4.5e-16</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-1.2e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.0e-17</td>
|
|
<td class="org-right">-9.5e-17</td>
|
|
<td class="org-right">8.4e-18</td>
|
|
<td class="org-right">4.3e-16</td>
|
|
<td class="org-right">5.8e-16</td>
|
|
<td class="org-right">1.7e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
And now at the Compliance matrix.
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-2.0e-24</td>
|
|
<td class="org-right">7.4e-25</td>
|
|
<td class="org-right">5.9e-23</td>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">5.9e-24</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-7.1e-25</td>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">2.9e-25</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">-5.4e-24</td>
|
|
<td class="org-right">-3.3e-23</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.9e-26</td>
|
|
<td class="org-right">-6.4e-25</td>
|
|
<td class="org-right">2.1e-08</td>
|
|
<td class="org-right">1.9e-23</td>
|
|
<td class="org-right">5.3e-25</td>
|
|
<td class="org-right">-6.5e-40</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.4e-23</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">1.3e-23</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">4.9e-22</td>
|
|
<td class="org-right">-3.8e-24</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">7.6e-24</td>
|
|
<td class="org-right">1.2e-23</td>
|
|
<td class="org-right">6.9e-22</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-2.6e-22</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.3e-24</td>
|
|
<td class="org-right">-3.2e-23</td>
|
|
<td class="org-right">-1.6e-39</td>
|
|
<td class="org-right">9.9e-23</td>
|
|
<td class="org-right">-3.3e-22</td>
|
|
<td class="org-right">1.7e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbb930ae" class="outline-3">
|
|
<h3 id="orgbb930ae"><span class="section-number-3">2.2</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<div class="important">
|
|
<p>
|
|
The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\) corresponds to the compliance matrix of the Stewart platform.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-02-28 ven. 17:34</p>
|
|
</div>
|
|
</body>
|
|
</html>
|