2020-01-22 16:31:44 +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 >
2020-02-13 15:37:06 +01:00
<!-- 2020 - 02 - 13 jeu. 15:36 -->
2020-01-22 16:31:44 +01:00
< 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" >
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2020 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!-- /* --> <![CDATA[/*> <!-- */
function CodeHighlightOn(elem, id)
{
var target = document.getElementById(id);
if(null != target) {
elem.cacheClassElem = elem.className;
elem.cacheClassTarget = target.className;
target.className = "code-highlighted";
elem.className = "code-highlighted";
}
}
function CodeHighlightOff(elem, id)
{
var target = document.getElementById(id);
if(elem.cacheClassElem)
elem.className = elem.cacheClassElem;
if(elem.cacheClassTarget)
target.className = elem.cacheClassTarget;
}
/*]]>*///-->
< / script >
2020-02-11 15:27:39 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / 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 >
2020-02-13 15:19:30 +01:00
< li > < a href = "#orgc59e712" > 1. Compare external forces and forces applied by the actuators< / a >
2020-01-22 16:31:44 +01:00
< ul >
2020-02-13 15:19:30 +01:00
< 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 >
2020-02-13 15:37:06 +01:00
< li > < a href = "#org230655f" > 1.3. Conclusion< / a > < / li >
2020-02-13 15:19:30 +01:00
< / 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 >
2020-02-13 15:37:06 +01:00
< li > < a href = "#org1cbdf9a" > 2.2. Conclusion< / a > < / li >
2020-01-22 16:31:44 +01:00
< / ul >
< / li >
< / ul >
< / div >
< / div >
2020-02-13 15:19:30 +01:00
< 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 >
2020-02-11 15:50:52 +01:00
< div class = "outline-text-2" id = "text-1" >
2020-02-13 15:37:06 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / div >
2020-02-13 15:37:06 +01:00
2020-02-13 15:19:30 +01:00
< div id = "outline-container-org4509b7d" class = "outline-3" >
< h3 id = "org4509b7d" > < span class = "section-number-3" > 1.1< / span > Comparison with fixed support< / h3 >
2020-02-11 15:50:52 +01:00
< div class = "outline-text-3" id = "text-1-1" >
2020-02-13 15:37:06 +01:00
< p >
Let’ s generate a Stewart platform.
< / p >
2020-01-22 16:31:44 +01:00
< div class = "org-src-container" >
2020-02-11 15:27:39 +01:00
< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
2020-02-13 15:19:30 +01:00
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);
2020-02-11 15:27:39 +01:00
stewart = generateGeneralConfiguration(stewart);
2020-01-22 16:31:44 +01:00
stewart = computeJointsPose(stewart);
2020-02-11 15:27:39 +01:00
stewart = initializeStrutDynamics(stewart);
2020-02-13 15:19:30 +01:00
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 > );
2020-02-11 15:27:39 +01:00
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
2020-01-22 16:31:44 +01:00
stewart = computeJacobian(stewart);
2020-02-11 15:27:39 +01:00
stewart = initializeStewartPose(stewart);
2020-02-13 15:19:30 +01:00
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
2020-01-22 16:31:44 +01:00
< / pre >
< / div >
2020-02-13 15:37:06 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< div class = "org-src-container" >
2020-02-13 15:19:30 +01:00
< 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 > );
2020-01-22 16:31:44 +01:00
< / pre >
< / div >
2020-02-13 15:19:30 +01:00
< p >
2020-02-13 15:37:06 +01:00
The transfer function from actuator forces \(\bm{\tau}\) to the relative displacement of the mobile platform \(\mathcal{\bm{X}}\) is extracted.
2020-02-13 15:19:30 +01:00
< / p >
2020-01-22 16:31:44 +01:00
< 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 >
2020-02-13 15:19:30 +01:00
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
2020-01-22 16:31:44 +01:00
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
2020-02-13 15:19:30 +01:00
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 >
2020-01-22 16:31:44 +01:00
< 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 >
2020-02-13 15:37:06 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< div class = "org-src-container" >
2020-02-13 15:19:30 +01:00
< 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 > };
2020-01-22 16:31:44 +01:00
< / pre >
< / div >
2020-02-13 15:19:30 +01:00
< p >
2020-02-13 15:37:06 +01:00
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 \(\{A\}\):
2020-02-13 15:19:30 +01:00
< / p >
2020-01-22 16:31:44 +01:00
< div class = "org-src-container" >
2020-02-13 15:19:30 +01:00
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
2020-01-22 16:31:44 +01:00
clear io; io_i = 1;
2020-02-13 15:19:30 +01:00
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 >
2020-01-22 16:31:44 +01:00
< 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 >
2020-02-13 15:37:06 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / div >
< / div >
2020-02-13 15:19:30 +01:00
< 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" >
2020-01-22 16:31:44 +01:00
< p >
2020-02-13 15:37:06 +01:00
We now add a flexible support under the Stewart platform.
2020-01-22 16:31:44 +01:00
< / p >
< div class = "org-src-container" >
2020-02-13 15:19:30 +01:00
< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'flexible'< / span > );
2020-01-22 16:31:44 +01:00
< / pre >
< / div >
< p >
2020-02-13 15:37:06 +01:00
And we perform again the identification.
2020-01-22 16:31:44 +01:00
< / p >
< div class = "org-src-container" >
2020-02-13 15:37:06 +01:00
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
2020-01-22 16:31:44 +01:00
clear io; io_i = 1;
2020-02-13 15:19:30 +01:00
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 >
2020-01-22 16:31:44 +01:00
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, options);
2020-02-13 15:19:30 +01:00
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 > };
2020-01-22 16:31:44 +01:00
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 > };
2020-02-13 15:37:06 +01:00
Gc = minreal(G< span class = "org-type" > *< / span > inv(stewart.kinematics.J< span class = "org-type" > '< / span > ));
2020-02-13 15:19:30 +01:00
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 > };
2020-02-13 15:37:06 +01:00
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
2020-01-22 16:31:44 +01:00
clear io; io_i = 1;
2020-02-13 15:19:30 +01:00
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 >
2020-01-22 16:31:44 +01:00
< 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 >
2020-02-13 15:37:06 +01:00
< p >
The comparison between the obtained transfer functions is shown in Figure < a href = "#orga2f2bd5" > 2< / 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 2: < / 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 >
2020-02-13 15:19:30 +01:00
< / div >
2020-01-22 16:31:44 +01:00
< / div >
2020-02-13 15:37:06 +01:00
< div id = "outline-container-org230655f" class = "outline-3" >
< h3 id = "org230655f" > < span class = "section-number-3" > 1.3< / span > Conclusion< / h3 >
2020-02-13 15:19:30 +01:00
< div class = "outline-text-3" id = "text-1-3" >
2020-01-22 16:31:44 +01:00
< div class = "important" >
< p >
2020-02-13 15:19:30 +01:00
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.
2020-01-22 16:31:44 +01:00
< / p >
< / div >
< / div >
< / div >
2020-02-13 15:19:30 +01:00
< / div >
2020-01-22 16:31:44 +01:00
2020-02-13 15:19:30 +01:00
< 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" >
2020-02-13 15:37:06 +01:00
< 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 >
2020-02-13 15:19:30 +01:00
< / div >
2020-02-13 15:37:06 +01:00
2020-02-13 15:19:30 +01:00
< 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" >
2020-01-22 16:31:44 +01:00
< p >
Initialization of the Stewart platform.
< / p >
< div class = "org-src-container" >
2020-02-11 15:27:39 +01:00
< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
2020-02-13 15:19:30 +01:00
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);
2020-02-11 15:27:39 +01:00
stewart = generateGeneralConfiguration(stewart);
2020-01-22 16:31:44 +01:00
stewart = computeJointsPose(stewart);
2020-02-11 15:27:39 +01:00
stewart = initializeStrutDynamics(stewart);
2020-02-13 15:19:30 +01:00
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 > );
2020-02-11 15:27:39 +01:00
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
2020-01-22 16:31:44 +01:00
stewart = computeJacobian(stewart);
2020-02-11 15:27:39 +01:00
stewart = initializeStewartPose(stewart);
2020-02-13 15:19:30 +01:00
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
< / pre >
< / div >
2020-02-13 15:37:06 +01:00
< p >
No flexibility below the Stewart platform and no payload.
< / p >
2020-02-13 15:19:30 +01:00
< 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 > );
2020-01-22 16:31:44 +01:00
< / 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 >
2020-02-13 15:19:30 +01:00
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
2020-01-22 16:31:44 +01:00
< 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" > '/F'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/X'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1;
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, options);
2020-02-13 15:19:30 +01:00
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 > };
2020-01-22 16:31:44 +01:00
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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< 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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / 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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< / tbody >
< / table >
< / div >
< / div >
2020-02-13 15:37:06 +01:00
< div id = "outline-container-org1cbdf9a" class = "outline-3" >
< h3 id = "org1cbdf9a" > < span class = "section-number-3" > 2.2< / span > Conclusion< / h3 >
2020-02-13 15:19:30 +01:00
< div class = "outline-text-3" id = "text-2-2" >
< div class = "important" >
2020-01-22 16:31:44 +01:00
< p >
2020-02-13 15:19:30 +01:00
The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\) corresponds to the compliance matrix of the Stewart platform.
2020-01-22 16:31:44 +01:00
< / p >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-02-13 15:37:06 +01:00
< p class = "date" > Created: 2020-02-13 jeu. 15:36< / p >
2020-01-22 16:31:44 +01:00
< / div >
< / body >
< / html >
