<?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>
<!-- 2020-02-11 mar. 15:26 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Identification of the Stewart Platform using Simscape</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>
<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">Identification of the Stewart Platform using Simscape</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgf65174f">Identification</a>
<ul>
<li><a href="#org5b89813">Simscape Model</a></li>
<li><a href="#org2bfdf1b">Initialize the Stewart Platform</a></li>
<li><a href="#org0d97b27">Identification</a></li>
</ul>
</li>
<li><a href="#orge464de2">States as the motion of the mobile platform</a>
<ul>
<li><a href="#org987daca">Initialize the Stewart Platform</a></li>
<li><a href="#orgc808316">Identification</a></li>
<li><a href="#orge68adea">Coordinate transformation</a></li>
<li><a href="#org4973ae1">Analysis</a></li>
<li><a href="#orge7b97c8">Visualizing the modes</a></li>
<li><a href="#org5d63457">Identification</a></li>
<li><a href="#orgf7a52cb">Change of states</a></li>
</ul>
</li>
<li><a href="#org23d7e7b">Simple Model without any sensor</a>
<ul>
<li><a href="#org69b8a98">Simscape Model</a></li>
<li><a href="#org4aef27a">Initialize the Stewart Platform</a></li>
<li><a href="#orgb9fd532">Identification</a></li>
</ul>
</li>
<li><a href="#org0502cd2">Cartesian Plot</a></li>
<li><a href="#org32e2eb3">From a force to force sensor</a></li>
<li><a href="#org8ddfd2c">From a force applied in the leg to the displacement of the leg</a></li>
<li><a href="#org5685537">Transmissibility</a></li>
<li><a href="#org3335d1e">Compliance</a></li>
<li><a href="#org5ca7af8">Inertial</a></li>
</ul>
</div>
</div>
<p>
We would like to extract a state space model of the Stewart Platform from the Simscape model.
</p>
<p>
The inputs are:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Symbol</th>
<th scope="col" class="org-left">Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(\bm{\mathcal{F}}_{d}\)</td>
<td class="org-left">External forces applied in {B}</td>
</tr>
<tr>
<td class="org-left">\(\bm{\tau}\)</td>
<td class="org-left">Joint forces</td>
</tr>
<tr>
<td class="org-left">\(\bm{\mathcal{F}}\)</td>
<td class="org-left">Cartesian forces applied by the Joints</td>
</tr>
<tr>
<td class="org-left">\(\bm{D}_{w}\)</td>
<td class="org-left">Fixed Based translation and rotations around {A}</td>
</tr>
</tbody>
</table>
<p>
The outputs are:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Symbol</th>
<th scope="col" class="org-left">Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(\bm{\mathcal{X}}\)</td>
<td class="org-left">Relative Motion of {B} with respect to {A}</td>
</tr>
<tr>
<td class="org-left">\(\bm{\mathcal{L}}\)</td>
<td class="org-left">Joint Displacement</td>
</tr>
<tr>
<td class="org-left">\(\bm{F}_{m}\)</td>
<td class="org-left">Force Sensors in each strut</td>
</tr>
<tr>
<td class="org-left">\(\bm{v}_{m}\)</td>
<td class="org-left">Inertial Sensors located at \(b_i\) measuring in the direction of the strut</td>
</tr>
</tbody>
</table>
<blockquote>
<p>
An important difference from basic Simulink models is that the states in a physical network are not independent in general, because some states have dependencies on other states through constraints.
</p>
</blockquote>
<div id="outline-container-orgf65174f" class="outline-2">
<h2 id="orgf65174f">Identification</h2>
<div class="outline-text-2" id="text-orgf65174f">
</div>
<div id="outline-container-org5b89813" class="outline-3">
<h3 id="org5b89813">Simscape Model</h3>
</div>
<div id="outline-container-org2bfdf1b" class="outline-3">
<h3 id="org2bfdf1b">Initialize the Stewart Platform</h3>
<div class="outline-text-3" id="text-org2bfdf1b">
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
</pre>
</div>
</div>
</div>
<div id="outline-container-org0d97b27" class="outline-3">
<h3 id="org0d97b27">Identification</h3>
<div class="outline-text-3" id="text-org0d97b27">
<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_identification'</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">'/tau'</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">'/Fext'</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;
io(io_i) = linio([mdl, <span class="org-string">'/Vm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Taum'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Lm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
<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">'tau1'</span>, <span class="org-string">'tau2'</span>, <span class="org-string">'tau3'</span>, <span class="org-string">'tau4'</span>, <span class="org-string">'tau5'</span>, <span class="org-string">'tau6'</span>, ...
<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
G.OutputName = {<span class="org-string">'Xdx'</span>, <span class="org-string">'Xdy'</span>, <span class="org-string">'Xdz'</span>, <span class="org-string">'Xrx'</span>, <span class="org-string">'Xry'</span>, <span class="org-string">'Xrz'</span>, ...
<span class="org-string">'Vm1'</span>, <span class="org-string">'Vm2'</span>, <span class="org-string">'Vm3'</span>, <span class="org-string">'Vm4'</span>, <span class="org-string">'Vm5'</span>, <span class="org-string">'Vm6'</span>, ...
<span class="org-string">'taum1'</span>, <span class="org-string">'taum2'</span>, <span class="org-string">'taum3'</span>, <span class="org-string">'taum4'</span>, <span class="org-string">'taum5'</span>, <span class="org-string">'taum6'</span>, ...
<span class="org-string">'Lm1'</span>, <span class="org-string">'Lm2'</span>, <span class="org-string">'Lm3'</span>, <span class="org-string">'Lm4'</span>, <span class="org-string">'Lm5'</span>, <span class="org-string">'Lm6'</span>};
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orge464de2" class="outline-2">
<h2 id="orge464de2">States as the motion of the mobile platform</h2>
<div class="outline-text-2" id="text-orge464de2">
</div>
<div id="outline-container-org987daca" class="outline-3">
<h3 id="org987daca">Initialize the Stewart Platform</h3>
<div class="outline-text-3" id="text-org987daca">
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgc808316" class="outline-3">
<h3 id="orgc808316">Identification</h3>
<div class="outline-text-3" id="text-orgc808316">
<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_identification_simple'</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">'/tau'</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;
io(io_i) = linio([mdl, <span class="org-string">'/Xdot'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io);
<span class="org-comment">% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};</span>
<span class="org-comment">% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};</span>
</pre>
</div>
<p>
Let’s check the size of <code>G</code>:
</p>
<div class="org-src-container">
<pre class="src src-matlab">size(G)
</pre>
</div>
<pre class="example">
size(G)
State-space model with 12 outputs, 6 inputs, and 18 states.
'org_babel_eoe'
ans =
'org_babel_eoe'
</pre>
<p>
We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gm = minreal(G);
</pre>
</div>
<pre class="example">
Gm = minreal(G);
6 states removed.
</pre>
<p>
And indeed, we obtain 12 states.
</p>
</div>
</div>
<div id="outline-container-orge68adea" class="outline-3">
<h3 id="orge68adea">Coordinate transformation</h3>
<div class="outline-text-3" id="text-orge68adea">
<p>
We can perform the following transformation using the <code>ss2ss</code> command.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gt = ss2ss(Gm, Gm.C);
</pre>
</div>
<p>
Then, the <code>C</code> matrix of <code>Gt</code> is the unity matrix which means that the states of the state space model are equal to the measurements \(\bm{Y}\).
</p>
<p>
The measurements are the 6 displacement and 6 velocities of mobile platform with respect to \(\{B\}\).
</p>
<p>
We could perform the transformation by hand:
</p>
<div class="org-src-container">
<pre class="src src-matlab">At = Gm.C<span class="org-type">*</span>Gm.A<span class="org-type">*</span>pinv(Gm.C);
Bt = Gm.C<span class="org-type">*</span>Gm.B;
Ct = eye(12);
Dt = zeros(12, 6);
Gt = ss(At, Bt, Ct, Dt);
</pre>
</div>
</div>
</div>
<div id="outline-container-org4973ae1" class="outline-3">
<h3 id="org4973ae1">Analysis</h3>
<div class="outline-text-3" id="text-org4973ae1">
<div class="org-src-container">
<pre class="src src-matlab">[V,D] = eig(Gt.A);
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Mode Number</th>
<th scope="col" class="org-right">Resonance Frequency [Hz]</th>
<th scope="col" class="org-right">Damping Ratio [%]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">174.5</td>
<td class="org-right">0.9</td>
</tr>
<tr>
<td class="org-right">2.0</td>
<td class="org-right">174.5</td>
<td class="org-right">0.7</td>
</tr>
<tr>
<td class="org-right">3.0</td>
<td class="org-right">202.1</td>
<td class="org-right">0.7</td>
</tr>
<tr>
<td class="org-right">4.0</td>
<td class="org-right">237.3</td>
<td class="org-right">0.6</td>
</tr>
<tr>
<td class="org-right">5.0</td>
<td class="org-right">237.3</td>
<td class="org-right">0.5</td>
</tr>
<tr>
<td class="org-right">6.0</td>
<td class="org-right">283.8</td>
<td class="org-right">0.5</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orge7b97c8" class="outline-3">
<h3 id="orge7b97c8">Visualizing the modes</h3>
<div class="outline-text-3" id="text-orge7b97c8">
<p>
To visualize the i’th mode, we may excite the system using the inputs \(U_i\) such that \(B U_i\) is co-linear to \(\xi_i\) (the mode we want to excite).
</p>
<p>
\[ U(t) = e^{\alpha t} ( ) \]
</p>
<p>
Let’s first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ws = imag(diag(D));
[ws,I] = sort(ws)
ws = ws(7<span class="org-type">:</span>end); I = I(7<span class="org-type">:</span>end);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(ws)</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">i_mode = I(<span class="org-constant">i</span>); <span class="org-comment">% the argument is the i'th mode</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">lambda_i = D(i_mode, i_mode);
xi_i = V(<span class="org-type">:</span>,i_mode);
a_i = real(lambda_i);
b_i = imag(lambda_i);
</pre>
</div>
<p>
Let do 10 periods of the mode.
</p>
<div class="org-src-container">
<pre class="src src-matlab">t = linspace(0, 10<span class="org-type">/</span>(imag(lambda_i)<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span>), 1000);
U_i = pinv(Gt.B) <span class="org-type">*</span> real(xi_i <span class="org-type">*</span> lambda_i <span class="org-type">*</span> (cos(b_i <span class="org-type">*</span> t) <span class="org-type">+</span> 1<span class="org-constant">i</span><span class="org-type">*</span>sin(b_i <span class="org-type">*</span> t)));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">U = timeseries(U_i, t);
</pre>
</div>
<p>
Simulation:
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, num2str(t(<span class="org-variable-name">end</span>)));
<span class="org-matlab-simulink-keyword">sim</span>(mdl);
</pre>
</div>
<p>
Save the movie of the mode shape.
</p>
<div class="org-src-container">
<pre class="src src-matlab">smwritevideo(mdl, sprintf(<span class="org-string">'figs/mode%i'</span>, <span class="org-constant">i</span>), ...
<span class="org-string">'PlaybackSpeedRatio'</span>, 1<span class="org-type">/</span>(b_i<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span>), ...
<span class="org-string">'FrameRate'</span>, 30, ...
<span class="org-string">'FrameSize'</span>, [800, 400]);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">end</span>
</pre>
</div>
<div id="orgb15855a" class="figure">
<p><img src="figs/mode1.gif" alt="mode1.gif" />
</p>
<p><span class="figure-number">Figure 1: </span>Identified mode - 1</p>
</div>
<div id="org1816e59" class="figure">
<p><img src="figs/mode3.gif" alt="mode3.gif" />
</p>
<p><span class="figure-number">Figure 2: </span>Identified mode - 3</p>
</div>
<div id="org01c8dca" class="figure">
<p><img src="figs/mode5.gif" alt="mode5.gif" />
</p>
<p><span class="figure-number">Figure 3: </span>Identified mode - 5</p>
</div>
</div>
</div>
<div id="outline-container-org5d63457" class="outline-3">
<h3 id="org5d63457">Identification</h3>
<div class="outline-text-3" id="text-org5d63457">
<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_identification'</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">'/tau'</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">'/Lm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, options);
<span class="org-comment">% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};</span>
<span class="org-comment">% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">size(G)
</pre>
</div>
</div>
</div>
<div id="outline-container-orgf7a52cb" class="outline-3">
<h3 id="orgf7a52cb">Change of states</h3>
<div class="outline-text-3" id="text-orgf7a52cb">
<div class="org-src-container">
<pre class="src src-matlab">At = G.C<span class="org-type">*</span>G.A<span class="org-type">*</span>pinv(G.C);
Bt = G.C<span class="org-type">*</span>G.B;
Ct = eye(12);
Dt = zeros(12, 6);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Gt = ss(At, Bt, Ct, Dt);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">size(Gt)
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org23d7e7b" class="outline-2">
<h2 id="org23d7e7b">Simple Model without any sensor</h2>
<div class="outline-text-2" id="text-org23d7e7b">
</div>
<div id="outline-container-org69b8a98" class="outline-3">
<h3 id="org69b8a98">Simscape Model</h3>
<div class="outline-text-3" id="text-org69b8a98">
<div class="org-src-container">
<pre class="src src-matlab">open <span class="org-string">'stewart_identification_simple.slx'</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org4aef27a" class="outline-3">
<h3 id="org4aef27a">Initialize the Stewart Platform</h3>
<div class="outline-text-3" id="text-org4aef27a">
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb9fd532" class="outline-3">
<h3 id="orgb9fd532">Identification</h3>
<div class="outline-text-3" id="text-orgb9fd532">
<div class="org-src-container">
<pre class="src src-matlab">stateorder = {...
<span class="org-string">'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_1_1_1'</span>,...
<span class="org-string">'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_2_1_1'</span>,...
<span class="org-string">'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_3_1_1'</span>,...
<span class="org-string">'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_4_1_1'</span>,...
<span class="org-string">'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_5_1_1'</span>,...
<span class="org-string">'stewart_platform_identification_simple/Solver Configuration/EVAL_KEY/INPUT_6_1_1'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Rz.q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Rz.q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Rz.q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Rz.q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Rz.q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Rz.q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Pz.p'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Pz.p'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Pz.p'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Pz.p'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Pz.p'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Pz.p'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Rz.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Rz.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Rz.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Rz.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Rz.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Rz.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.cylindrical_joint.Pz.v'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.cylindrical_joint.Pz.v'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.cylindrical_joint.Pz.v'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.cylindrical_joint.Pz.v'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.cylindrical_joint.Pz.v'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.cylindrical_joint.Pz.v'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_F.S.Q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.spherical_joint_F.S.Q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.spherical_joint_F.S.Q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.spherical_joint_F.S.Q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.spherical_joint_F.S.Q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.spherical_joint_F.S.Q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_2.Subsystem.spherical_joint_F.S.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_3.Subsystem.spherical_joint_F.S.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_4.Subsystem.spherical_joint_F.S.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_5.Subsystem.spherical_joint_F.S.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_6.Subsystem.spherical_joint_F.S.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_F.S.w'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_M.S.Q'</span>,...
<span class="org-string">'stewart_platform_identification_simple.Stewart_Platform.Strut_1.Subsystem.spherical_joint_M.S.w'</span>};
</pre>
</div>
<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_identification_simple'</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">'/tau'</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;
io(io_i) = linio([mdl, <span class="org-string">'/Xdot'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
<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">'tau1'</span>, <span class="org-string">'tau2'</span>, <span class="org-string">'tau3'</span>, <span class="org-string">'tau4'</span>, <span class="org-string">'tau5'</span>, <span class="org-string">'tau6'</span>};
G.OutputName = {<span class="org-string">'Xdx'</span>, <span class="org-string">'Xdy'</span>, <span class="org-string">'Xdz'</span>, <span class="org-string">'Xrx'</span>, <span class="org-string">'Xry'</span>, <span class="org-string">'Xrz'</span>, <span class="org-string">'Vdx'</span>, <span class="org-string">'Vdy'</span>, <span class="org-string">'Vdz'</span>, <span class="org-string">'Vrx'</span>, <span class="org-string">'Vry'</span>, <span class="org-string">'Vrz'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">size(G)
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G.StateName
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org0502cd2" class="outline-2">
<h2 id="org0502cd2">Cartesian Plot</h2>
<div class="outline-text-2" id="text-org0502cd2">
<p>
From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_cart(1, 1), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_cart(2, 1), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_cart(3, 1), freqs, <span class="org-string">'Hz'</span>))));
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
bode(G.G_cart, freqs);
</pre>
</div>
</div>
</div>
<div id="outline-container-org32e2eb3" class="outline-2">
<h2 id="org32e2eb3">From a force to force sensor</h2>
<div class="outline-text-2" id="text-org32e2eb3">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$F_{m_i}/F_{i}$'</span>);
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [N/N]'</span>);
legend(<span class="org-string">'location'</span>, <span class="org-string">'southeast'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_forc(1, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$F_{m_i}/F_{i}$'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_forc(2, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$F_{m_j}/F_{i}$'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_forc(3, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_forc(4, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_forc(5, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_forc(6, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [N/N]'</span>);
legend(<span class="org-string">'location'</span>, <span class="org-string">'southeast'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org8ddfd2c" class="outline-2">
<h2 id="org8ddfd2c">From a force applied in the leg to the displacement of the leg</h2>
<div class="outline-text-2" id="text-org8ddfd2c">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$D_{i}/F_{i}$'</span>);
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/N]'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_legs(1, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$D_{i}/F_{i}$'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_legs(2, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$D_{j}/F_{i}$'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_legs(3, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_legs(4, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_legs(5, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
plot(freqs, abs(squeeze(freqresp(G.G_legs(6, 1), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/N]'</span>);
legend(<span class="org-string">'location'</span>, <span class="org-string">'northeast'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org5685537" class="outline-2">
<h2 id="org5685537">Transmissibility</h2>
<div class="outline-text-2" id="text-org5685537">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_tran(2, 2), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_tran(3, 3), freqs, <span class="org-string">'Hz'</span>))));
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/m]'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_tran(4, 4), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_tran(5, 5), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_tran(6, 6), freqs, <span class="org-string">'Hz'</span>))));
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [$\frac{rad/s}{rad/s}$]'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 1), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 2), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_tran(1, 3), freqs, <span class="org-string">'Hz'</span>))));
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/m]'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org3335d1e" class="outline-2">
<h2 id="org3335d1e">Compliance</h2>
<div class="outline-text-2" id="text-org3335d1e">
<p>
From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_comp(1, 1), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_comp(2, 2), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_comp(3, 3), freqs, <span class="org-string">'Hz'</span>))));
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/N]'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org5ca7af8" class="outline-2">
<h2 id="org5ca7af8">Inertial</h2>
<div class="outline-text-2" id="text-org5ca7af8">
<p>
From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot(freqs, abs(squeeze(freqresp(G.G_iner(1, 1), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_iner(2, 2), freqs, <span class="org-string">'Hz'</span>))));
plot(freqs, abs(squeeze(freqresp(G.G_iner(3, 3), freqs, <span class="org-string">'Hz'</span>))));
hold off;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'Amplitude [m/N]'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-11 mar. 15:26</p>
</div>
</body>
</html>