676 lines
27 KiB
HTML
676 lines
27 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>
|
|
<!-- 2020-02-11 mar. 15:50 -->
|
|
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
|
<title>Stewart Platform - Simscape Model</title>
|
|
<meta name="generator" content="Org mode" />
|
|
<meta name="author" content="Dehaeze Thomas" />
|
|
<style type="text/css">
|
|
<!--/*--><![CDATA[/*><!--*/
|
|
.title { text-align: center;
|
|
margin-bottom: .2em; }
|
|
.subtitle { text-align: center;
|
|
font-size: medium;
|
|
font-weight: bold;
|
|
margin-top:0; }
|
|
.todo { font-family: monospace; color: red; }
|
|
.done { font-family: monospace; color: green; }
|
|
.priority { font-family: monospace; color: orange; }
|
|
.tag { background-color: #eee; font-family: monospace;
|
|
padding: 2px; font-size: 80%; font-weight: normal; }
|
|
.timestamp { color: #bebebe; }
|
|
.timestamp-kwd { color: #5f9ea0; }
|
|
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
|
|
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
|
|
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
|
|
.underline { text-decoration: underline; }
|
|
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
|
|
p.verse { margin-left: 3%; }
|
|
pre {
|
|
border: 1px solid #ccc;
|
|
box-shadow: 3px 3px 3px #eee;
|
|
padding: 8pt;
|
|
font-family: monospace;
|
|
overflow: auto;
|
|
margin: 1.2em;
|
|
}
|
|
pre.src {
|
|
position: relative;
|
|
overflow: visible;
|
|
padding-top: 1.2em;
|
|
}
|
|
pre.src:before {
|
|
display: none;
|
|
position: absolute;
|
|
background-color: white;
|
|
top: -10px;
|
|
right: 10px;
|
|
padding: 3px;
|
|
border: 1px solid black;
|
|
}
|
|
pre.src:hover:before { display: inline;}
|
|
/* Languages per Org manual */
|
|
pre.src-asymptote:before { content: 'Asymptote'; }
|
|
pre.src-awk:before { content: 'Awk'; }
|
|
pre.src-C:before { content: 'C'; }
|
|
/* pre.src-C++ doesn't work in CSS */
|
|
pre.src-clojure:before { content: 'Clojure'; }
|
|
pre.src-css:before { content: 'CSS'; }
|
|
pre.src-D:before { content: 'D'; }
|
|
pre.src-ditaa:before { content: 'ditaa'; }
|
|
pre.src-dot:before { content: 'Graphviz'; }
|
|
pre.src-calc:before { content: 'Emacs Calc'; }
|
|
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
|
|
pre.src-fortran:before { content: 'Fortran'; }
|
|
pre.src-gnuplot:before { content: 'gnuplot'; }
|
|
pre.src-haskell:before { content: 'Haskell'; }
|
|
pre.src-hledger:before { content: 'hledger'; }
|
|
pre.src-java:before { content: 'Java'; }
|
|
pre.src-js:before { content: 'Javascript'; }
|
|
pre.src-latex:before { content: 'LaTeX'; }
|
|
pre.src-ledger:before { content: 'Ledger'; }
|
|
pre.src-lisp:before { content: 'Lisp'; }
|
|
pre.src-lilypond:before { content: 'Lilypond'; }
|
|
pre.src-lua:before { content: 'Lua'; }
|
|
pre.src-matlab:before { content: 'MATLAB'; }
|
|
pre.src-mscgen:before { content: 'Mscgen'; }
|
|
pre.src-ocaml:before { content: 'Objective Caml'; }
|
|
pre.src-octave:before { content: 'Octave'; }
|
|
pre.src-org:before { content: 'Org mode'; }
|
|
pre.src-oz:before { content: 'OZ'; }
|
|
pre.src-plantuml:before { content: 'Plantuml'; }
|
|
pre.src-processing:before { content: 'Processing.js'; }
|
|
pre.src-python:before { content: 'Python'; }
|
|
pre.src-R:before { content: 'R'; }
|
|
pre.src-ruby:before { content: 'Ruby'; }
|
|
pre.src-sass:before { content: 'Sass'; }
|
|
pre.src-scheme:before { content: 'Scheme'; }
|
|
pre.src-screen:before { content: 'Gnu Screen'; }
|
|
pre.src-sed:before { content: 'Sed'; }
|
|
pre.src-sh:before { content: 'shell'; }
|
|
pre.src-sql:before { content: 'SQL'; }
|
|
pre.src-sqlite:before { content: 'SQLite'; }
|
|
/* additional languages in org.el's org-babel-load-languages alist */
|
|
pre.src-forth:before { content: 'Forth'; }
|
|
pre.src-io:before { content: 'IO'; }
|
|
pre.src-J:before { content: 'J'; }
|
|
pre.src-makefile:before { content: 'Makefile'; }
|
|
pre.src-maxima:before { content: 'Maxima'; }
|
|
pre.src-perl:before { content: 'Perl'; }
|
|
pre.src-picolisp:before { content: 'Pico Lisp'; }
|
|
pre.src-scala:before { content: 'Scala'; }
|
|
pre.src-shell:before { content: 'Shell Script'; }
|
|
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
|
|
/* additional language identifiers per "defun org-babel-execute"
|
|
in ob-*.el */
|
|
pre.src-cpp:before { content: 'C++'; }
|
|
pre.src-abc:before { content: 'ABC'; }
|
|
pre.src-coq:before { content: 'Coq'; }
|
|
pre.src-groovy:before { content: 'Groovy'; }
|
|
/* additional language identifiers from org-babel-shell-names in
|
|
ob-shell.el: ob-shell is the only babel language using a lambda to put
|
|
the execution function name together. */
|
|
pre.src-bash:before { content: 'bash'; }
|
|
pre.src-csh:before { content: 'csh'; }
|
|
pre.src-ash:before { content: 'ash'; }
|
|
pre.src-dash:before { content: 'dash'; }
|
|
pre.src-ksh:before { content: 'ksh'; }
|
|
pre.src-mksh:before { content: 'mksh'; }
|
|
pre.src-posh:before { content: 'posh'; }
|
|
/* Additional Emacs modes also supported by the LaTeX listings package */
|
|
pre.src-ada:before { content: 'Ada'; }
|
|
pre.src-asm:before { content: 'Assembler'; }
|
|
pre.src-caml:before { content: 'Caml'; }
|
|
pre.src-delphi:before { content: 'Delphi'; }
|
|
pre.src-html:before { content: 'HTML'; }
|
|
pre.src-idl:before { content: 'IDL'; }
|
|
pre.src-mercury:before { content: 'Mercury'; }
|
|
pre.src-metapost:before { content: 'MetaPost'; }
|
|
pre.src-modula-2:before { content: 'Modula-2'; }
|
|
pre.src-pascal:before { content: 'Pascal'; }
|
|
pre.src-ps:before { content: 'PostScript'; }
|
|
pre.src-prolog:before { content: 'Prolog'; }
|
|
pre.src-simula:before { content: 'Simula'; }
|
|
pre.src-tcl:before { content: 'tcl'; }
|
|
pre.src-tex:before { content: 'TeX'; }
|
|
pre.src-plain-tex:before { content: 'Plain TeX'; }
|
|
pre.src-verilog:before { content: 'Verilog'; }
|
|
pre.src-vhdl:before { content: 'VHDL'; }
|
|
pre.src-xml:before { content: 'XML'; }
|
|
pre.src-nxml:before { content: 'XML'; }
|
|
/* add a generic configuration mode; LaTeX export needs an additional
|
|
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
|
|
pre.src-conf:before { content: 'Configuration File'; }
|
|
|
|
table { border-collapse:collapse; }
|
|
caption.t-above { caption-side: top; }
|
|
caption.t-bottom { caption-side: bottom; }
|
|
td, th { vertical-align:top; }
|
|
th.org-right { text-align: center; }
|
|
th.org-left { text-align: center; }
|
|
th.org-center { text-align: center; }
|
|
td.org-right { text-align: right; }
|
|
td.org-left { text-align: left; }
|
|
td.org-center { text-align: center; }
|
|
dt { font-weight: bold; }
|
|
.footpara { display: inline; }
|
|
.footdef { margin-bottom: 1em; }
|
|
.figure { padding: 1em; }
|
|
.figure p { text-align: center; }
|
|
.equation-container {
|
|
display: table;
|
|
text-align: center;
|
|
width: 100%;
|
|
}
|
|
.equation {
|
|
vertical-align: middle;
|
|
}
|
|
.equation-label {
|
|
display: table-cell;
|
|
text-align: right;
|
|
vertical-align: middle;
|
|
}
|
|
.inlinetask {
|
|
padding: 10px;
|
|
border: 2px solid gray;
|
|
margin: 10px;
|
|
background: #ffffcc;
|
|
}
|
|
#org-div-home-and-up
|
|
{ text-align: right; font-size: 70%; white-space: nowrap; }
|
|
textarea { overflow-x: auto; }
|
|
.linenr { font-size: smaller }
|
|
.code-highlighted { background-color: #ffff00; }
|
|
.org-info-js_info-navigation { border-style: none; }
|
|
#org-info-js_console-label
|
|
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
|
|
.org-info-js_search-highlight
|
|
{ background-color: #ffff00; color: #000000; font-weight: bold; }
|
|
.org-svg { width: 90%; }
|
|
/*]]>*/-->
|
|
</style>
|
|
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
|
<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
|
<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">Stewart Platform - Simscape Model</h1>
|
|
<div id="table-of-contents">
|
|
<h2>Table of Contents</h2>
|
|
<div id="text-table-of-contents">
|
|
<ul>
|
|
<li><a href="#orgc6e0b93">1. Parameters used for the Simscape Model</a></li>
|
|
<li><a href="#org66977e8">2. Simulation Configuration - Configuration reference</a></li>
|
|
<li><a href="#orgb2362eb">3. Subsystem Reference</a></li>
|
|
<li><a href="#orgdfad86d">4. Subsystem - Fixed base and Mobile Platform</a></li>
|
|
<li><a href="#org9d4af75">5. Subsystem - Struts</a>
|
|
<ul>
|
|
<li><a href="#org45d9234">5.1. Strut Configuration</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org7e2c432">6. Other Elements</a>
|
|
<ul>
|
|
<li><a href="#org4bdfc33">6.1. Z-Axis Geophone</a>
|
|
<ul>
|
|
<li><a href="#org01abf4c">6.1.1. Working Principle</a></li>
|
|
<li><a href="#org5da3f93">6.1.2. Initialization function</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org99786f1">6.2. Z-Axis Accelerometer</a>
|
|
<ul>
|
|
<li><a href="#org01c45ef">6.2.1. Working Principle</a></li>
|
|
<li><a href="#orga80b649">6.2.2. Initialization function</a></li>
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<p>
|
|
In this document is explained how the Simscape model of the Stewart Platform is implemented.
|
|
</p>
|
|
|
|
<p>
|
|
It is divided in the following sections:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>section <a href="#org8d965c3">1</a>: is explained how the parameters of the Stewart platform are set for the Simscape model</li>
|
|
<li>section <a href="#org354bfdb">2</a>: the Simulink configuration (solver, simulation time, …) is shared among all the Simulink files. It is explain how this is done.</li>
|
|
<li>section <a href="#org66bbae2">3</a>: All the elements (platforms, struts, sensors, …) are saved in separate files and imported in Simulink files using “subsystem referenced”.</li>
|
|
<li>section <a href="#orga4915c4">4</a>: The simscape model for the fixed base and mobile platform are described in this section.</li>
|
|
<li>section <a href="#orgdb5206f">5</a>: The simscape model for the Stewart platform struts is described in this section.</li>
|
|
</ul>
|
|
|
|
<div id="outline-container-orgc6e0b93" class="outline-2">
|
|
<h2 id="orgc6e0b93"><span class="section-number-2">1</span> Parameters used for the Simscape Model</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
<a id="org8d965c3"></a>
|
|
The Simscape Model of the Stewart Platform is working with the <code>stewart</code> structure generated using the functions described <a href="stewart-architecture.html">here</a>.
|
|
</p>
|
|
|
|
<p>
|
|
All the geometry and inertia of the mechanical elements are defined in the <code>stewart</code> structure.
|
|
</p>
|
|
|
|
<p>
|
|
By updating the <code>stewart</code> structure in the workspace, the Simscape model will be automatically updated.
|
|
</p>
|
|
|
|
<p>
|
|
Thus, nothing should be changed by hand inside the Simscape model.
|
|
</p>
|
|
|
|
<p>
|
|
The main advantage to have all the parameters defined in one structure (and not hard-coded in some simulink blocs) it that we can easily change the Stewart architecture/parameters in a Matlab script to perform some parametric study for instance.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org66977e8" class="outline-2">
|
|
<h2 id="org66977e8"><span class="section-number-2">2</span> Simulation Configuration - Configuration reference</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="org354bfdb"></a>
|
|
As multiple simulink files will be used for simulation and tests, it is very useful to determine good simulation configuration that will be <b>shared</b> among all the simulink files.
|
|
</p>
|
|
|
|
<p>
|
|
This is done using something called “<b>Configuration Reference</b>” (<a href="https://fr.mathworks.com/help/simulink/ug/more-about-configuration-references.html">documentation</a>).
|
|
</p>
|
|
|
|
<p>
|
|
Basically, the configuration is stored in a mat file <code>conf_simscape.mat</code> and then loaded in the workspace for it to be accessible to all the simulink models.
|
|
It is automatically loaded when the Simulink project is open. It can be loaded manually with the command:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
It is however possible to modify specific parameters just for one simulation using the <code>set_param</code> command:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, 1);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb2362eb" class="outline-2">
|
|
<h2 id="orgb2362eb"><span class="section-number-2">3</span> Subsystem Reference</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org66bbae2"></a>
|
|
Several Stewart platform models are used, for instance one is use to study the dynamics while the other is used to apply active damping techniques.
|
|
</p>
|
|
|
|
<p>
|
|
However, all the Simscape models share some subsystems using the <b>Subsystem Reference</b> Simulink block (<a href="https://fr.mathworks.com/help/simulink/ug/referenced-subsystem-1.html">documentation</a>).
|
|
</p>
|
|
|
|
<p>
|
|
These shared subsystems are:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><code>Fixed_Based.slx</code> - Fixed base of the Stewart Platform</li>
|
|
<li><code>Mobile_Platform.slx</code> - Mobile platform of the Stewart Platform</li>
|
|
<li><code>stewart_strut.slx</code> - One strut containing two spherical/universal joints, the actuator as well as the included sensors. A parameter <code>i</code> is initialized to determine what it the “number” of the strut.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
These subsystems are referenced from another subsystem called <code>Stewart_Platform.slx</code> shown in figure <a href="#orgf687c71">1</a>, that basically connect them correctly.
|
|
This subsystem is then referenced in other simulink models for various purposes (control, analysis, simulation, …).
|
|
</p>
|
|
|
|
|
|
<div id="orgf687c71" class="figure">
|
|
<p><img src="figs/simscape_stewart_platform.png" alt="simscape_stewart_platform.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Simscape Subsystem of the Stewart platform. Encapsulate the Subsystems corresponding to the fixed base, mobile platform and all the struts.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgdfad86d" class="outline-2">
|
|
<h2 id="orgdfad86d"><span class="section-number-2">4</span> Subsystem - Fixed base and Mobile Platform</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="orga4915c4"></a>
|
|
Both the fixed base and the mobile platform simscape models share many similarities.
|
|
</p>
|
|
|
|
<p>
|
|
Their are both composed of:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>a solid body representing the platform</li>
|
|
<li>6 rigid transform blocks to go from the frame \(\{F\}\) (resp. \(\{M\}\)) to the location of the joints.
|
|
These rigid transform are using \({}^F\bm{a}_i\) (resp. \({}^M\bm{b}_i\)) for the position of the joint and \({}^F\bm{R}_{a_i}\) (resp. \({}^M\bm{R}_{b_i}\)) for the orientation of the joint.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
As always, the parameters that define the geometry are taken from the <code>stewart</code> structure.
|
|
</p>
|
|
|
|
|
|
<div id="org858f0b4" class="figure">
|
|
<p><img src="figs/simscape_fixed_base.png" alt="simscape_fixed_base.png" width="1000px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Simscape Model of the Fixed base</p>
|
|
</div>
|
|
|
|
|
|
<div id="org4b31aa3" class="figure">
|
|
<p><img src="figs/simscape_mobile_platform.png" alt="simscape_mobile_platform.png" width="800px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Simscape Model of the Mobile platform</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9d4af75" class="outline-2">
|
|
<h2 id="org9d4af75"><span class="section-number-2">5</span> Subsystem - Struts</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="orgdb5206f"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org45d9234" class="outline-3">
|
|
<h3 id="org45d9234"><span class="section-number-3">5.1</span> Strut Configuration</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
For the Stewart platform, the 6 struts are identical.
|
|
Thus, all the struts used in the Stewart platform are referring to the same subsystem called <code>stewart_strut.slx</code> and shown in Figure <a href="#org1dc8fce">4</a>.
|
|
</p>
|
|
|
|
<p>
|
|
This strut as the following structure:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Universal Joint*</b> connected on the Fixed base</li>
|
|
<li><b>Prismatic Joint*</b> for the actuator</li>
|
|
<li><b>Spherical Joint*</b> connected on the Mobile platform</li>
|
|
</ul>
|
|
|
|
<p>
|
|
This configuration is called <b>UPS</b>.
|
|
</p>
|
|
|
|
<p>
|
|
The other common configuration <b>SPS</b> has the disadvantage of having additional passive degrees-of-freedom corresponding to the rotation of the strut around its main axis.
|
|
This is why the <b>UPS</b> configuration is used, but other configuration can be easily implemented.
|
|
</p>
|
|
|
|
|
|
<div id="org1dc8fce" class="figure">
|
|
<p><img src="figs/simscape_strut.png" alt="simscape_strut.png" width="800px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Simscape model of the Stewart platform’s strut</p>
|
|
</div>
|
|
|
|
<p>
|
|
Several sensors are included in the strut that may or may not be used for control:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Relative Displacement sensor: gives the relative displacement of the strut.</li>
|
|
<li>Force sensor: measure the total force applied by the force actuator, the stiffness and damping forces in the direction of the strut.</li>
|
|
<li>Inertial sensor: measure the absolute motion (velocity) of the top part of the strut in the direction of the strut.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
There is two main types of inertial sensor that can be used to measure the absolute motion of the top part of the strut in the direction of the strut:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>a geophone that measures the absolute velocity above some frequency</li>
|
|
<li>an accelerometer that measures the absolute acceleration below some frequency</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Both inertial sensors are described bellow.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7e2c432" class="outline-2">
|
|
<h2 id="org7e2c432"><span class="section-number-2">6</span> Other Elements</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
</div>
|
|
<div id="outline-container-org4bdfc33" class="outline-3">
|
|
<h3 id="org4bdfc33"><span class="section-number-3">6.1</span> Z-Axis Geophone</h3>
|
|
<div class="outline-text-3" id="text-6-1">
|
|
</div>
|
|
<div id="outline-container-org01abf4c" class="outline-4">
|
|
<h4 id="org01abf4c"><span class="section-number-4">6.1.1</span> Working Principle</h4>
|
|
<div class="outline-text-4" id="text-6-1-1">
|
|
<p>
|
|
From the schematic of the Z-axis geophone shown in Figure <a href="#org819fba8">5</a>, we can write the transfer function from the support velocity \(\dot{w}\) to the relative velocity of the inertial mass \(\dot{d}\):
|
|
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
|
with:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\omega_0 = \sqrt{\frac{k}{m}}\)</li>
|
|
<li>\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org819fba8" class="figure">
|
|
<p><img src="figs/inertial_sensor.png" alt="inertial_sensor.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Schematic of a Z-Axis geophone</p>
|
|
</div>
|
|
|
|
<p>
|
|
We see that at frequencies above \(\omega_0\):
|
|
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]
|
|
</p>
|
|
|
|
<p>
|
|
And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.
|
|
</p>
|
|
|
|
<p>
|
|
We generally want to have the smallest resonant frequency \(\omega_0\) to measure low frequency absolute velocity, however there is a trade-off between \(\omega_0\) and the mass of the inertial mass.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5da3f93" class="outline-4">
|
|
<h4 id="org5da3f93"><span class="section-number-4">6.1.2</span> Initialization function</h4>
|
|
<div class="outline-text-4" id="text-6-1-2">
|
|
<p>
|
|
<a id="orgd31bda9"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="../src/initializeZAxisGeophone.m">here</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[geophone]</span> = <span class="org-function-name">initializeZAxisGeophone</span>(<span class="org-variable-name">args</span>)
|
|
arguments
|
|
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e<span class="org-type">-</span>3 <span class="org-comment">% [kg]</span>
|
|
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 1 <span class="org-comment">% [Hz]</span>
|
|
<span class="org-keyword">end</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%%</span></span>
|
|
geophone.m = args.mass;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% The Stiffness is set to have the damping resonance frequency</span></span>
|
|
geophone.k = geophone.m <span class="org-type">*</span> (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>args.freq)<span class="org-type">^</span>2;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% We set the damping value to have critical damping</span></span>
|
|
geophone.c = 2<span class="org-type">*</span>sqrt(geophone.m <span class="org-type">*</span> geophone.k);
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Save</span></span>
|
|
save(<span class="org-string">'./mat/geophone_z_axis.mat'</span>, <span class="org-string">'geophone'</span>);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org99786f1" class="outline-3">
|
|
<h3 id="org99786f1"><span class="section-number-3">6.2</span> Z-Axis Accelerometer</h3>
|
|
<div class="outline-text-3" id="text-6-2">
|
|
</div>
|
|
<div id="outline-container-org01c45ef" class="outline-4">
|
|
<h4 id="org01c45ef"><span class="section-number-4">6.2.1</span> Working Principle</h4>
|
|
<div class="outline-text-4" id="text-6-2-1">
|
|
<p>
|
|
From the schematic of the Z-axis accelerometer shown in Figure <a href="#org1274602">6</a>, we can write the transfer function from the support acceleration \(\ddot{w}\) to the relative position of the inertial mass \(d\):
|
|
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
|
with:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\omega_0 = \sqrt{\frac{k}{m}}\)</li>
|
|
<li>\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org1274602" class="figure">
|
|
<p><img src="figs/inertial_sensor.png" alt="inertial_sensor.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Schematic of a Z-Axis geophone</p>
|
|
</div>
|
|
|
|
<p>
|
|
We see that at frequencies below \(\omega_0\):
|
|
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]
|
|
</p>
|
|
|
|
<p>
|
|
And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.
|
|
</p>
|
|
|
|
<p>
|
|
Note that there is trade-off between:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>the highest measurable acceleration \(\omega_0\)</li>
|
|
<li>the sensitivity of the accelerometer which is equal to \(-\frac{1}{{\omega_0}^2}\)</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga80b649" class="outline-4">
|
|
<h4 id="orga80b649"><span class="section-number-4">6.2.2</span> Initialization function</h4>
|
|
<div class="outline-text-4" id="text-6-2-2">
|
|
<p>
|
|
<a id="orge91f65f"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="../src/initializeZAxisAccelerometer.m">here</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[accelerometer]</span> = <span class="org-function-name">initializeZAxisAccelerometer</span>(<span class="org-variable-name">args</span>)
|
|
arguments
|
|
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e<span class="org-type">-</span>3 <span class="org-comment">% [kg]</span>
|
|
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 5e3 <span class="org-comment">% [Hz]</span>
|
|
<span class="org-keyword">end</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%%</span></span>
|
|
accelerometer.m = args.mass;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% The Stiffness is set to have the damping resonance frequency</span></span>
|
|
accelerometer.k = accelerometer.m <span class="org-type">*</span> (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>args.freq)<span class="org-type">^</span>2;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% We set the damping value to have critical damping</span></span>
|
|
accelerometer.c = 2<span class="org-type">*</span>sqrt(accelerometer.m <span class="org-type">*</span> accelerometer.k);
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Gain correction of the accelerometer to have a unity gain until the resonance</span></span>
|
|
accelerometer.gain = <span class="org-type">-</span>accelerometer.k<span class="org-type">/</span>accelerometer.m;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Save</span></span>
|
|
save(<span class="org-string">'./mat/accelerometer_z_axis.mat'</span>, <span class="org-string">'accelerometer'</span>);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-02-11 mar. 15:50</p>
|
|
</div>
|
|
</body>
|
|
</html>
|