1568 lines
70 KiB
HTML
1568 lines
70 KiB
HTML
<?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>
|
|
<!-- 2019-01-24 jeu. 15:17 -->
|
|
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
|
<title>Control in a rotating frame</title>
|
|
<meta name="generator" content="Org mode" />
|
|
<meta name="author" content="Thomas Dehaeze" />
|
|
<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 type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
|
|
<script type="text/javascript" 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) 2012-2019 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 type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
displayAlign: "center",
|
|
displayIndent: "0em",
|
|
|
|
"HTML-CSS": { scale: 100,
|
|
linebreaks: { automatic: "false" },
|
|
webFont: "TeX"
|
|
},
|
|
SVG: {scale: 100,
|
|
linebreaks: { automatic: "false" },
|
|
font: "TeX"},
|
|
NativeMML: {scale: 100},
|
|
TeX: { equationNumbers: {autoNumber: "AMS"},
|
|
MultLineWidth: "85%",
|
|
TagSide: "right",
|
|
TagIndent: ".8em"
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript"
|
|
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></script>
|
|
</head>
|
|
<body>
|
|
<div id="content">
|
|
<h1 class="title">Control in a rotating frame</h1>
|
|
<div id="table-of-contents">
|
|
<h2>Table of Contents</h2>
|
|
<div id="text-table-of-contents">
|
|
<ul>
|
|
<li><a href="#org35986a6">1. Introduction</a></li>
|
|
<li><a href="#org2cfc65e">2. System Description and Analysis</a>
|
|
<ul>
|
|
<li><a href="#org52d1b39">2.1. System description</a></li>
|
|
<li><a href="#org56f1c8e">2.2. Equations</a></li>
|
|
<li><a href="#org23e861a">2.3. Numerical Values for the NASS</a></li>
|
|
<li><a href="#org8834a4b">2.4. Euler and Coriolis forces - Numerical Result</a></li>
|
|
<li><a href="#org3fc75f8">2.5. Negative Spring Effect - Numerical Result</a></li>
|
|
<li><a href="#orgca44f56">2.6. Limitations due to coupling</a>
|
|
<ul>
|
|
<li><a href="#org972ba28">2.6.1. Numerical Analysis</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org24a2547">2.7. Limitations due to negative stiffness effect</a></li>
|
|
<li><a href="#org90bd4c5">2.8. Effect of rotation speed on the plant</a>
|
|
<ul>
|
|
<li><a href="#orgb2a8b4a">2.8.1. Voice coil actuator</a></li>
|
|
<li><a href="#org34e6778">2.8.2. Piezoelectric actuator</a></li>
|
|
<li><a href="#org36cd742">2.8.3. Analysis</a></li>
|
|
<li><a href="#org23ea4ed">2.8.4. Campbell diagram</a></li>
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org89b80ab">3. Control Strategies</a>
|
|
<ul>
|
|
<li><a href="#orgbdd9948">3.1. Measurement in the fixed reference frame</a></li>
|
|
<li><a href="#org724b218">3.2. Measurement in the rotating frame</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org30fbee8">4. Multi Body Model - Simscape</a>
|
|
<ul>
|
|
<li><a href="#orge1f000c">4.1. Initialize</a></li>
|
|
<li><a href="#org8b4df15">4.2. Parameter for the Simscape simulations</a></li>
|
|
<li><a href="#orga3ac610">4.3. Identification in the rotating referenced frame</a></li>
|
|
<li><a href="#orga381ded">4.4. Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</a></li>
|
|
<li><a href="#org6b388ff">4.5. Plant Control</a>
|
|
<ul>
|
|
<li><a href="#orgdb709bf">4.5.1. Low rotation speed and High rotation speed</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org5822ce2">4.6. Identification in the fixed frame</a></li>
|
|
<li><a href="#orgfa9ed99">4.7. Identification from actuator forces to displacement in the fixed frame</a></li>
|
|
<li><a href="#orgbc833bb">4.8. Effect of the rotating Speed</a>
|
|
<ul>
|
|
<li><a href="#orgaf21bf8">4.8.1. <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</a></li>
|
|
<li><a href="#orgdd964cc">4.8.2. <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#orgc30bae9">4.9. Effect of the X-Y stage stiffness</a>
|
|
<ul>
|
|
<li><a href="#org3a4478a">4.9.1. <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</a></li>
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org12e1d75">5. Control Implementation</a>
|
|
<ul>
|
|
<li><a href="#org70652b4">5.1. Measurement in the fixed reference frame</a></li>
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org35986a6" class="outline-2">
|
|
<h2 id="org35986a6"><span class="section-number-2">1</span> Introduction</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
The objective of this note it to highlight some control problems that arises when controlling the position of an object using actuators that are rotating with respect to a fixed reference frame.
|
|
</p>
|
|
|
|
<p>
|
|
In section <a href="#org0986a46">2</a>, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written.
|
|
The rotation induces some coupling between the actuators and their displacement, and modifies the dynamics of the system.
|
|
This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators.
|
|
</p>
|
|
|
|
<p>
|
|
Then, in section <a href="#org786bfb0">3</a>, two different control approach are compared where:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>the measurement is made in the fixed frame</li>
|
|
<li>the measurement is made in the rotating frame</li>
|
|
</ul>
|
|
|
|
<p>
|
|
In section <a href="#orgfce2ea4">4</a>, the analytical study will be validated using a multi body model of the studied system.
|
|
</p>
|
|
|
|
<p>
|
|
Finally, in section <a href="#org4a3b8a3">5</a>, the control strategies are implemented using Simulink and Simscape and compared.
|
|
</p>
|
|
|
|
<p>
|
|
Test citation: [<a href="#smith99_scien_engin_guide_digit_signal">1</a>].
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2cfc65e" class="outline-2">
|
|
<h2 id="org2cfc65e"><span class="section-number-2">2</span> System Description and Analysis</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="org0986a46"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org52d1b39" class="outline-3">
|
|
<h3 id="org52d1b39"><span class="section-number-3">2.1</span> System description</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#org455bae8">1</a>).
|
|
</p>
|
|
|
|
<p>
|
|
The control inputs are the forces applied by the actuators of the translation stage (\(F_u\) and \(F_v\)).
|
|
As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(u\) and \(v\).
|
|
</p>
|
|
|
|
<p>
|
|
The measurement is either the \(x-y\) displacement of the object located on top of the translation stage or the \(u-v\) displacement of the sample with respect to a fixed reference frame.
|
|
</p>
|
|
|
|
|
|
<div id="org455bae8" class="figure">
|
|
<p><img src="./Figures/rotating_frame_2dof.png" alt="rotating_frame_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Schematic of the mecanical system</p>
|
|
</div>
|
|
|
|
<p>
|
|
In the following block diagram:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(G\) is the transfer function from the forces applied in the actuators to the measurement</li>
|
|
<li>\(K\) is the controller to design</li>
|
|
<li>\(J\) is a Jacobian matrix usually used to change the reference frame</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Indices \(x\) and \(y\) corresponds to signals in the fixed reference frame (along \(\vec{i}_x\) and \(\vec{i}_y\)):
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(D_x\) is the measured position of the sample</li>
|
|
<li>\(r_x\) is the reference signal which corresponds to the wanted \(D_x\)</li>
|
|
<li>\(\epsilon_x\) is the position error</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (\(\vec{i}_u\) and \(\vec{i}_v\)):
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(F_u\) and \(F_v\) are forces applied by the actuators</li>
|
|
<li>\(\epsilon_u\) and \(\epsilon_v\) are position error of the sample along \(\vec{i}_u\) and \(\vec{i}_v\)</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org56f1c8e" class="outline-3">
|
|
<h3 id="org56f1c8e"><span class="section-number-3">2.2</span> Equations</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
<a id="org8074d39"></a>
|
|
Based on the figure <a href="#org455bae8">1</a>, we can write the equations of motion of the system.
|
|
</p>
|
|
|
|
<p>
|
|
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
|
|
</p>
|
|
\begin{align}
|
|
\label{org9b4a615}
|
|
T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
|
|
V & = \frac{1}{2} k \left( x^2 + y^2 \right)
|
|
\end{align}
|
|
|
|
<p>
|
|
The Lagrangian is the kinetic energy minus the potential energy.
|
|
</p>
|
|
\begin{equation}
|
|
\label{org81b342f}
|
|
L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
|
|
\end{equation}
|
|
|
|
|
|
<p>
|
|
The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:
|
|
</p>
|
|
\begin{align*}
|
|
\label{orgf5d2cb1}
|
|
\frac{\partial L}{\partial x} & = -kx \\
|
|
\frac{\partial L}{\partial y} & = -ky \\
|
|
\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
|
|
\frac{d}{dt}\frac{\partial L}{\partial \dot{y}} & = m\ddot{y}
|
|
\end{align*}
|
|
|
|
<p>
|
|
The external forces applied to the mass are:
|
|
</p>
|
|
\begin{align*}
|
|
F_{\text{ext}, x} &= F_u \cos{\theta} - F_v \sin{\theta}\\
|
|
F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta}
|
|
\end{align*}
|
|
|
|
<p>
|
|
By appling the Lagrangian equations, we obtain:
|
|
</p>
|
|
\begin{align}
|
|
m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
|
|
m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
|
|
\end{align}
|
|
|
|
<p>
|
|
We then change coordinates from \((x, y)\) to \((d_x, d_y, \theta)\).
|
|
</p>
|
|
\begin{align*}
|
|
x & = d_u \cos{\theta} - d_v \sin{\theta}\\
|
|
y & = d_u \sin{\theta} + d_v \cos{\theta}
|
|
\end{align*}
|
|
|
|
<p>
|
|
We obtain:
|
|
</p>
|
|
\begin{align*}
|
|
\ddot{x} & = \ddot{d_u} \cos{\theta} - 2\dot{d_u}\dot{\theta}\sin{\theta} - d_u\ddot{\theta}\sin{\theta} - d_u\dot{\theta}^2 \cos{\theta}
|
|
- \ddot{d_v} \sin{\theta} - 2\dot{d_v}\dot{\theta}\cos{\theta} - d_v\ddot{\theta}\cos{\theta} + d_v\dot{\theta}^2 \sin{\theta} \\
|
|
\ddot{y} & = \ddot{d_u} \sin{\theta} + 2\dot{d_u}\dot{\theta}\cos{\theta} + d_u\ddot{\theta}\cos{\theta} - d_u\dot{\theta}^2 \sin{\theta}
|
|
+ \ddot{d_v} \cos{\theta} - 2\dot{d_v}\dot{\theta}\sin{\theta} - d_v\ddot{\theta}\sin{\theta} - d_v\dot{\theta}^2 \cos{\theta} \\
|
|
\end{align*}
|
|
|
|
<p>
|
|
By injecting the previous result into the Lagrangian equation, we obtain:
|
|
</p>
|
|
\begin{align*}
|
|
m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta}
|
|
-m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta}
|
|
+ k d_u \cos{\theta} - k d_v \sin{\theta} = F_u \cos{\theta} - F_v \sin{\theta} \\
|
|
m \ddot{d_u} \sin{\theta} + 2m\dot{d_u}\dot{\theta}\cos{\theta} + m d_u\ddot{\theta}\cos{\theta} - m d_u\dot{\theta}^2 \sin{\theta}
|
|
+ m \ddot{d_v} \cos{\theta} - 2m\dot{d_v}\dot{\theta}\sin{\theta} - m d_v\ddot{\theta}\sin{\theta} - m d_v\dot{\theta}^2 \cos{\theta}
|
|
+ k d_u \sin{\theta} + k d_v \cos{\theta} = F_u \sin{\theta} + F_v \cos{\theta}
|
|
\end{align*}
|
|
|
|
<p>
|
|
Which is equivalent to:
|
|
</p>
|
|
\begin{align*}
|
|
m \ddot{d_u} - 2m\dot{d_u}\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\ddot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\dot{\theta}^2
|
|
-m \ddot{d_v} \frac{\sin{\theta}}{\cos{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} + m d_v\dot{\theta}^2 \frac{\sin{\theta}}{\cos{\theta}}
|
|
+ k d_u - k d_v \frac{\sin{\theta}}{\cos{\theta}} = F_u - F_v \frac{\sin{\theta}}{\cos{\theta}} \\
|
|
m \ddot{d_u} + 2m\dot{d_u}\dot{\theta}\frac{\cos{\theta}}{\sin{\theta}} + m d_u\ddot{\theta}\frac{\cos{\theta}}{\sin{\theta}} - m d_u\dot{\theta}^2
|
|
+ m \ddot{d_v} \frac{\cos{\theta}}{\sin{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} - m d_v\dot{\theta}^2 \frac{\cos{\theta}}{\sin{\theta}}
|
|
+ k d_u + k d_v \frac{\cos{\theta}}{\sin{\theta}} = F_u + F_v \frac{\cos{\theta}}{\sin{\theta}}
|
|
\end{align*}
|
|
|
|
<p>
|
|
We can then subtract and add the previous equations to obtain the following equations:
|
|
</p>
|
|
<div class="important">
|
|
\begin{equation}
|
|
\label{orgb43453a}
|
|
m \ddot{d_u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\label{org01f818e}
|
|
m \ddot{d_v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta}
|
|
\end{equation}
|
|
|
|
</div>
|
|
|
|
<p>
|
|
We obtain two differential equations that are coupled through:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Euler forces</b>: \(m d_v \ddot{\theta}\)</li>
|
|
<li><b>Coriolis forces</b>: \(2 m \dot{d_v} \dot{\theta}\)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass \(m\) and stiffness \(k- m\dot{\theta}^2\).
|
|
Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to <b>centrifugal forces</b>).
|
|
</p>
|
|
|
|
<p>
|
|
The forces induced by the rotating reference frame are independent of the stiffness of the actuator.
|
|
The resulting effect of those forces should then be higher when using softer actuators.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org23e861a" class="outline-3">
|
|
<h3 id="org23e861a"><span class="section-number-3">2.3</span> Numerical Values for the NASS</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
Let's define the parameters for the NASS.
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Light sample mass [kg]</td>
|
|
<td class="org-right">3.5e+01</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Heavy sample mass [kg]</td>
|
|
<td class="org-right">8.5e+01</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max rot. speed - light [rpm]</td>
|
|
<td class="org-right">6.0e+01</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max rot. speed - heavy [rpm]</td>
|
|
<td class="org-right">1.0e+00</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Voice Coil Stiffness [N/m]</td>
|
|
<td class="org-right">1.0e+03</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Piezo Stiffness [N/m]</td>
|
|
<td class="org-right">1.0e+08</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max rot. acceleration [rad/s2]</td>
|
|
<td class="org-right">1.0e+00</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max mass excentricity [m]</td>
|
|
<td class="org-right">1.0e-02</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max Horizontal speed [m/s]</td>
|
|
<td class="org-right">2.0e-01</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8834a4b" class="outline-3">
|
|
<h3 id="org8834a4b"><span class="section-number-3">2.4</span> Euler and Coriolis forces - Numerical Result</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
First we will determine the value for Euler and Coriolis forces during regular experiment.
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Euler forces</b>: \(m d_v \ddot{\theta}\)</li>
|
|
<li><b>Coriolis forces</b>: \(2 m \dot{d_v} \dot{\theta}\)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The obtained values are displayed in table <a href="#orgdbd5160">1</a>.
|
|
</p>
|
|
|
|
<table id="orgdbd5160" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 1:</span> Euler and Coriolis forces for the NASS</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-left">Light</th>
|
|
<th scope="col" class="org-left">Heavy</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Coriolis</td>
|
|
<td class="org-left">88.0N</td>
|
|
<td class="org-left">3.6N</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Euler</td>
|
|
<td class="org-left">0.4N</td>
|
|
<td class="org-left">0.8N</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3fc75f8" class="outline-3">
|
|
<h3 id="org3fc75f8"><span class="section-number-3">2.5</span> Negative Spring Effect - Numerical Result</h3>
|
|
<div class="outline-text-3" id="text-2-5">
|
|
<p>
|
|
The negative stiffness due to the rotation is equal to \(-m{\omega_0}^2\).
|
|
</p>
|
|
|
|
<p>
|
|
The values for the negative spring effect are displayed in table <a href="#org7c845ef">2</a>.
|
|
</p>
|
|
|
|
<p>
|
|
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
|
|
</p>
|
|
|
|
<table id="org7c845ef" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 2:</span> Negative Spring effect</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-left">Light</th>
|
|
<th scope="col" class="org-left">Heavy</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Neg. Spring</td>
|
|
<td class="org-left">1381.7[N/m]</td>
|
|
<td class="org-left">0.9[N/m]</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgca44f56" class="outline-3">
|
|
<h3 id="orgca44f56"><span class="section-number-3">2.6</span> Limitations due to coupling</h3>
|
|
<div class="outline-text-3" id="text-2-6">
|
|
<p>
|
|
To simplify, we consider a constant rotating speed \(\dot{\theta} = {\omega_0}\) and thus \(\ddot{\theta} = 0\).
|
|
</p>
|
|
|
|
<p>
|
|
From equations \eqref{orgb43453a} and \eqref{org01f818e}, we obtain:
|
|
</p>
|
|
\begin{align*}
|
|
(m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\
|
|
(m s^2 + (k - m{\omega_0}^2)) d_v &= F_v - 2 m {\omega_0} s d_u \\
|
|
\end{align*}
|
|
|
|
<p>
|
|
From second equation:
|
|
\[ d_v = \frac{1}{m s^2 + (k - m{\omega_0}^2)} F_v - \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} d_u \]
|
|
</p>
|
|
|
|
<p>
|
|
And we re-inject \(d_v\) into the first equation:
|
|
</p>
|
|
\begin{equation*}
|
|
(m s^2 + (k - m{\omega_0}^2)) d_u = F_u + \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} F_v - \frac{(2 m {\omega_0} s)^2}{m s^2 + (k - m{\omega_0}^2)} d_u
|
|
\end{equation*}
|
|
|
|
\begin{equation*}
|
|
\frac{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}{m s^2 + (k - m{\omega_0}^2)} d_u = F_u + \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} F_v
|
|
\end{equation*}
|
|
|
|
<p>
|
|
Finally we obtain \(d_u\) function of \(F_u\) and \(F_v\).
|
|
\[ d_u = \frac{m s^2 + (k - m{\omega_0}^2)}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_u + \frac{2 m {\omega_0} s}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_v \]
|
|
</p>
|
|
|
|
<p>
|
|
Similarly we can obtain \(d_v\) function of \(F_u\) and \(F_v\):
|
|
\[ d_v = \frac{m s^2 + (k - m{\omega_0}^2)}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_v - \frac{2 m {\omega_0} s}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_u \]
|
|
</p>
|
|
|
|
<p>
|
|
The two previous equations can be written in a matrix form:
|
|
</p>
|
|
<div class="important">
|
|
\begin{equation}
|
|
\label{org2b23e3b}
|
|
\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
|
|
\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}
|
|
\begin{bmatrix}
|
|
ms^2 + (k-m{\omega_0}^2) & 2 m \omega_0 s \\
|
|
-2 m \omega_0 s & ms^2 + (k-m{\omega_0}^2) \\
|
|
\end{bmatrix}
|
|
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
|
\end{equation}
|
|
|
|
</div>
|
|
|
|
<p>
|
|
Then, coupling is negligible if \(|-m \omega^2 + (k - m{\omega_0}^2)| \gg |2 m {\omega_0} \omega|\).
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org972ba28" class="outline-4">
|
|
<h4 id="org972ba28"><span class="section-number-4">2.6.1</span> Numerical Analysis</h4>
|
|
<div class="outline-text-4" id="text-2-6-1">
|
|
<p>
|
|
We plot on the same graph \(\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \omega_0 \omega|}\) for the voice coil and the piezo:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>with the light sample (figure <a href="#org2eaf004">2</a>).</li>
|
|
<li>with the heavy sample (figure <a href="#orge6601b9">3</a>).</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org2eaf004" class="figure">
|
|
<p><img src="Figures/coupling_light.png" alt="coupling_light.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Relative Coupling for light mass and high rotation speed</p>
|
|
</div>
|
|
|
|
|
|
<div id="orge6601b9" class="figure">
|
|
<p><img src="Figures/coupling_heavy.png" alt="coupling_heavy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Relative Coupling for heavy mass and low rotation speed</p>
|
|
</div>
|
|
|
|
<div class="important">
|
|
<p>
|
|
Coupling is higher for actuators with small stiffness.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org24a2547" class="outline-3">
|
|
<h3 id="org24a2547"><span class="section-number-3">2.7</span> Limitations due to negative stiffness effect</h3>
|
|
<div class="outline-text-3" id="text-2-7">
|
|
<p>
|
|
If \(\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}\), then the negative spring effect is negligible and \(k - m\dot{\theta}^2 \approx k\).
|
|
</p>
|
|
|
|
<p>
|
|
Let's estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table <a href="#orge84ae0f">3</a>.
|
|
</p>
|
|
<table id="orge84ae0f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 3:</span> Maximum rotation speed at which negative stiffness is negligible (\(0.1\sqrt{\frac{k}{m}}\))</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-left">Voice Coil</th>
|
|
<th scope="col" class="org-left">Piezo</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Light</td>
|
|
<td class="org-left">5[rpm]</td>
|
|
<td class="org-left">1614[rpm]</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Heavy</td>
|
|
<td class="org-left">3[rpm]</td>
|
|
<td class="org-left">1036[rpm]</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
The negative spring effect is proportional to the rotational speed \(\omega\).
|
|
The system dynamics will be much more affected when using soft actuator.
|
|
</p>
|
|
|
|
<div class="important">
|
|
<p>
|
|
Negative stiffness effect has very important effect when using soft actuators.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
The system can even goes unstable when \(m \omega^2 > k\), that is when the centrifugal forces are higher than the forces due to stiffness.
|
|
</p>
|
|
|
|
<p>
|
|
From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table <a href="#org94d23e2">4</a>)
|
|
</p>
|
|
|
|
<table id="org94d23e2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 4:</span> Minimum possible stiffness</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-right">Light</th>
|
|
<th scope="col" class="org-right">Heavy</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">k min [N/m]</td>
|
|
<td class="org-right">2199</td>
|
|
<td class="org-right">89</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org90bd4c5" class="outline-3">
|
|
<h3 id="org90bd4c5"><span class="section-number-3">2.8</span> Effect of rotation speed on the plant</h3>
|
|
<div class="outline-text-3" id="text-2-8">
|
|
<p>
|
|
As shown in equation \eqref{org2b23e3b}, the plant changes with the rotation speed \(\omega_0\).
|
|
</p>
|
|
|
|
<p>
|
|
Then, we compute the bode plot of the direct term and coupling term for multiple rotating speed.
|
|
</p>
|
|
|
|
<p>
|
|
Then we compare the result between voice coil and piezoelectric actuators.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb2a8b4a" class="outline-4">
|
|
<h4 id="orgb2a8b4a"><span class="section-number-4">2.8.1</span> Voice coil actuator</h4>
|
|
<div class="outline-text-4" id="text-2-8-1">
|
|
|
|
<div id="org0f9ed57" class="figure">
|
|
<p><img src="Figures/G_ws_vc.png" alt="G_ws_vc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Voice coil</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgb82c1d1" class="figure">
|
|
<p><img src="Figures/Gc_ws_vc.png" alt="Gc_ws_vc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Voice coil</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org34e6778" class="outline-4">
|
|
<h4 id="org34e6778"><span class="section-number-4">2.8.2</span> Piezoelectric actuator</h4>
|
|
<div class="outline-text-4" id="text-2-8-2">
|
|
|
|
<div id="org359d5f5" class="figure">
|
|
<p><img src="Figures/G_ws_pz.png" alt="G_ws_pz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Piezoelectric actuator</p>
|
|
</div>
|
|
|
|
|
|
<div id="org4f616e4" class="figure">
|
|
<p><img src="Figures/Gc_ws_pz.png" alt="Gc_ws_pz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Piezoelectric actuator</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org36cd742" class="outline-4">
|
|
<h4 id="org36cd742"><span class="section-number-4">2.8.3</span> Analysis</h4>
|
|
<div class="outline-text-4" id="text-2-8-3">
|
|
<p>
|
|
When the rotation speed is null, the coupling terms are equal to zero and the diagonal terms corresponds to one degree of freedom mass spring system.
|
|
</p>
|
|
|
|
<p>
|
|
When the rotation speed in not null, the resonance frequency is duplicated into two pairs of complex conjugate poles.
|
|
</p>
|
|
|
|
<p>
|
|
As the rotation speed increases, one of the two resonant frequency goes to lower frequencies as the other one goes to higher frequencies.
|
|
</p>
|
|
|
|
<p>
|
|
The poles of the coupling terms are the same as the poles of the diagonal terms. The magnitude of the coupling terms are increasing with the rotation speed.
|
|
</p>
|
|
|
|
<p>
|
|
As shown in the previous figures, the system with voice coil is much more sensitive to rotation speed.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org23ea4ed" class="outline-4">
|
|
<h4 id="org23ea4ed"><span class="section-number-4">2.8.4</span> Campbell diagram</h4>
|
|
<div class="outline-text-4" id="text-2-8-4">
|
|
<p>
|
|
The poles of the system are computed for multiple values of the rotation frequency. To simplify the computation of the poles, we add some damping to the system.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = mlight;
|
|
k = kvc;
|
|
c = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #6434A3;">*</span>sqrt<span style="color: #707183;">(</span>k<span style="color: #6434A3;">*</span>m<span style="color: #707183;">)</span>;
|
|
|
|
ws = linspace<span style="color: #707183;">(</span><span style="color: #D0372D;">0</span>, <span style="color: #D0372D;">10</span>, <span style="color: #D0372D;">100</span><span style="color: #707183;">)</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
|
|
|
|
polesvc = zeros<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span>, length<span style="color: #7388D6;">(</span>ws<span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
|
|
|
|
<span style="color: #0000FF;">for</span> <span style="color: #BA36A5;">i</span> = <span style="color: #D0372D;">1</span><span style="color: #D0372D;">:length</span><span style="color: #707183;">(</span><span style="color: #D0372D;">ws</span><span style="color: #707183;">)</span>
|
|
polei = pole<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">/</span><span style="color: #7388D6;">(</span><span style="color: #909183;">(</span>m<span style="color: #6434A3;">*</span>s<span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> c<span style="color: #6434A3;">*</span>s <span style="color: #6434A3;">+</span> <span style="color: #709870;">(</span>k <span style="color: #6434A3;">-</span> m<span style="color: #6434A3;">*</span>ws<span style="color: #907373;">(</span><span style="color: #D0372D;">i</span><span style="color: #907373;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #709870;">)</span><span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> <span style="color: #909183;">(</span><span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span>m<span style="color: #6434A3;">*</span>ws<span style="color: #709870;">(</span><span style="color: #D0372D;">i</span><span style="color: #709870;">)</span><span style="color: #6434A3;">*</span>s<span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
|
|
polesvc<span style="color: #707183;">(</span><span style="color: #6434A3;">:</span>, <span style="color: #D0372D;">i</span><span style="color: #707183;">)</span> = sort<span style="color: #707183;">(</span>polei<span style="color: #7388D6;">(</span>imag<span style="color: #909183;">(</span>polei<span style="color: #909183;">)</span> <span style="color: #6434A3;">></span> <span style="color: #D0372D;">0</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
|
|
<span style="color: #0000FF;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = mlight;
|
|
k = kpz;
|
|
c = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #6434A3;">*</span>sqrt<span style="color: #707183;">(</span>k<span style="color: #6434A3;">*</span>m<span style="color: #707183;">)</span>;
|
|
|
|
ws = linspace<span style="color: #707183;">(</span><span style="color: #D0372D;">0</span>, <span style="color: #D0372D;">1000</span>, <span style="color: #D0372D;">100</span><span style="color: #707183;">)</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
|
|
|
|
polespz = zeros<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span>, length<span style="color: #7388D6;">(</span>ws<span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
|
|
|
|
<span style="color: #0000FF;">for</span> <span style="color: #BA36A5;">i</span> = <span style="color: #D0372D;">1</span><span style="color: #D0372D;">:length</span><span style="color: #707183;">(</span><span style="color: #D0372D;">ws</span><span style="color: #707183;">)</span>
|
|
polei = pole<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">/</span><span style="color: #7388D6;">(</span><span style="color: #909183;">(</span>m<span style="color: #6434A3;">*</span>s<span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> c<span style="color: #6434A3;">*</span>s <span style="color: #6434A3;">+</span> <span style="color: #709870;">(</span>k <span style="color: #6434A3;">-</span> m<span style="color: #6434A3;">*</span>ws<span style="color: #907373;">(</span><span style="color: #D0372D;">i</span><span style="color: #907373;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #709870;">)</span><span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span> <span style="color: #6434A3;">+</span> <span style="color: #909183;">(</span><span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span>m<span style="color: #6434A3;">*</span>ws<span style="color: #709870;">(</span><span style="color: #D0372D;">i</span><span style="color: #709870;">)</span><span style="color: #6434A3;">*</span>s<span style="color: #909183;">)</span><span style="color: #6434A3;">^</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
|
|
polespz<span style="color: #707183;">(</span><span style="color: #6434A3;">:</span>, <span style="color: #D0372D;">i</span><span style="color: #707183;">)</span> = sort<span style="color: #707183;">(</span>polei<span style="color: #7388D6;">(</span>imag<span style="color: #909183;">(</span>polei<span style="color: #909183;">)</span> <span style="color: #6434A3;">></span> <span style="color: #D0372D;">0</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
|
|
<span style="color: #0000FF;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures <a href="#org0f74744">8</a> and <a href="#orgab3524b">9</a>).
|
|
</p>
|
|
|
|
<p>
|
|
When the real part of one pole becomes positive, the system goes unstable.
|
|
</p>
|
|
|
|
<p>
|
|
For the voice coil (figure <a href="#org0f74744">8</a>), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure <a href="#orgab3524b">9</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org0f74744" class="figure">
|
|
<p><img src="Figures/poles_w_vc.png" alt="poles_w_vc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="orgab3524b" class="figure">
|
|
<p><img src="Figures/poles_w_pz.png" alt="poles_w_pz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Piezoelectric actuator and light sample</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org89b80ab" class="outline-2">
|
|
<h2 id="org89b80ab"><span class="section-number-2">3</span> Control Strategies</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org786bfb0"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgbdd9948" class="outline-3">
|
|
<h3 id="orgbdd9948"><span class="section-number-3">3.1</span> Measurement in the fixed reference frame</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
First, let's consider a measurement in the fixed referenced frame.
|
|
</p>
|
|
|
|
<p>
|
|
The transfer function from actuator \([F_u, F_v]\) to sensor \([D_x, D_y]\) is then \(G(\theta)\).
|
|
</p>
|
|
|
|
<p>
|
|
Then the measurement is subtracted to the reference signal \([r_x, r_y]\) to obtain the position error in the fixed reference frame \([\epsilon_x, \epsilon_y]\).
|
|
</p>
|
|
|
|
<p>
|
|
The position error \([\epsilon_x, \epsilon_y]\) is then express in the rotating frame corresponding to the actuators \([\epsilon_u, \epsilon_v]\).
|
|
</p>
|
|
|
|
<p>
|
|
Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon_v]\) to the actuator forces \([F_u, F_v]\).
|
|
</p>
|
|
|
|
<p>
|
|
The block diagram is shown on figure <a href="#org4a8c2aa">10</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org4a8c2aa" class="figure">
|
|
<p><img src="./Figures/control_measure_fixed_2dof.png" alt="control_measure_fixed_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Control with a measure from fixed frame</p>
|
|
</div>
|
|
|
|
<p>
|
|
The loop gain is then \(L = G(\theta) K J(\theta)\).
|
|
</p>
|
|
|
|
<p>
|
|
One question we wish to answer is: is \(G(\theta) J(\theta) = G(\theta_0) J(\theta_0)\)?
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org724b218" class="outline-3">
|
|
<h3 id="org724b218"><span class="section-number-3">3.2</span> Measurement in the rotating frame</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
Let's consider that the measurement is made in the rotating reference frame.
|
|
</p>
|
|
|
|
<p>
|
|
The corresponding block diagram is shown figure <a href="#orge83e07d">11</a>
|
|
</p>
|
|
|
|
|
|
<div id="orge83e07d" class="figure">
|
|
<p><img src="./Figures/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Control with a measure from rotating frame</p>
|
|
</div>
|
|
|
|
<p>
|
|
The loop gain is \(L = G K\).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org30fbee8" class="outline-2">
|
|
<h2 id="org30fbee8"><span class="section-number-2">4</span> Multi Body Model - Simscape</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="orgfce2ea4"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orge1f000c" class="outline-3">
|
|
<h3 id="orge1f000c"><span class="section-number-3">4.1</span> Initialize</h3>
|
|
</div>
|
|
<div id="outline-container-org8b4df15" class="outline-3">
|
|
<h3 id="org8b4df15"><span class="section-number-3">4.2</span> Parameter for the Simscape simulations</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
First we define the parameters that must be defined in order to run the Simscape simulation.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>; <span style="color: #8D8D84; font-style: italic;">% Rotation speed [rad/s]</span>
|
|
|
|
theta_e = <span style="color: #D0372D;">0</span>; <span style="color: #8D8D84; font-style: italic;">% Static measurement error on the angle theta [rad]</span>
|
|
|
|
m = <span style="color: #D0372D;">5</span>; <span style="color: #8D8D84; font-style: italic;">% mass of the sample [kg]</span>
|
|
|
|
mTuv = <span style="color: #D0372D;">30</span>;<span style="color: #8D8D84; font-style: italic;">% Mass of the moving part of the Tuv stage [kg]</span>
|
|
kTuv = <span style="color: #D0372D;">1e8</span>; <span style="color: #8D8D84; font-style: italic;">% Stiffness of the Tuv stage [N/m]</span>
|
|
cTuv = <span style="color: #D0372D;">0</span>; <span style="color: #8D8D84; font-style: italic;">% Damping of the Tuv stage [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then, we defined parameters that will be used in the following analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">mlight = <span style="color: #D0372D;">5</span>; <span style="color: #8D8D84; font-style: italic;">% Mass for light sample [kg]</span>
|
|
mheavy = <span style="color: #D0372D;">55</span>; <span style="color: #8D8D84; font-style: italic;">% Mass for heavy sample [kg]</span>
|
|
|
|
wlight = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>; <span style="color: #8D8D84; font-style: italic;">% Max rot. speed for light sample [rad/s]</span>
|
|
wheavy = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">60</span>; <span style="color: #8D8D84; font-style: italic;">% Max rot. speed for heavy sample [rad/s]</span>
|
|
|
|
kvc = <span style="color: #D0372D;">1e3</span>; <span style="color: #8D8D84; font-style: italic;">% Voice Coil Stiffness [N/m]</span>
|
|
kpz = <span style="color: #D0372D;">1e8</span>; <span style="color: #8D8D84; font-style: italic;">% Piezo Stiffness [N/m]</span>
|
|
|
|
d = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">01</span>; <span style="color: #8D8D84; font-style: italic;">% Maximum excentricity from rotational axis [m]</span>
|
|
|
|
freqs = logspace<span style="color: #707183;">(</span><span style="color: #6434A3;">-</span><span style="color: #D0372D;">2</span>, <span style="color: #D0372D;">3</span>, <span style="color: #D0372D;">1000</span><span style="color: #707183;">)</span>; <span style="color: #8D8D84; font-style: italic;">% Frequency vector for analysis [Hz]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga3ac610" class="outline-3">
|
|
<h3 id="orga3ac610"><span class="section-number-3">4.3</span> Identification in the rotating referenced frame</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<p>
|
|
We initialize the inputs and outputs of the system to identify:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Inputs: \(f_u\) and \(f_v\)</li>
|
|
<li>Outputs: \(d_u\) and \(d_v\)</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #D0372D;">0</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #008000;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fu'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fv'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">3</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>du'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">4</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dv'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We start we identify the transfer functions at high speed with the light sample.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w = wlight; <span style="color: #8D8D84; font-style: italic;">% Rotation speed [rad/s]</span>
|
|
m = mlight; <span style="color: #8D8D84; font-style: italic;">% mass of the sample [kg]</span>
|
|
|
|
kTuv = kpz;
|
|
Gpz_light = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
Gpz_light.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gpz_light.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
|
|
kTuv = kvc;
|
|
Gvc_light = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
Gvc_light.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_light.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then we identify the system with an heavy mass and low speed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w = wheavy; <span style="color: #8D8D84; font-style: italic;">% Rotation speed [rad/s]</span>
|
|
m = mheavy; <span style="color: #8D8D84; font-style: italic;">% mass of the sample [kg]</span>
|
|
|
|
kTuv = kpz;
|
|
Gpz_heavy = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
Gpz_heavy.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gpz_heavy.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
|
|
kTuv = kvc;
|
|
Gvc_heavy = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
Gvc_heavy.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_heavy.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga381ded" class="outline-3">
|
|
<h3 id="orga381ded"><span class="section-number-3">4.4</span> Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<p>
|
|
From the previous identification, we plot the coupling ratio in both case (figure <a href="#org1359930">12</a>).
|
|
We obtain the same result than the analytical case (figures <a href="#org2eaf004">2</a> and <a href="#orge6601b9">3</a>).
|
|
</p>
|
|
|
|
<div id="org1359930" class="figure">
|
|
<p><img src="Figures/coupling_ration_light_heavy.png" alt="coupling_ration_light_heavy.png" />
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6b388ff" class="outline-3">
|
|
<h3 id="org6b388ff"><span class="section-number-3">4.5</span> Plant Control</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
<p>
|
|
The goal is the study control problems due to the coupling that appears because of the rotation.
|
|
</p>
|
|
|
|
<p>
|
|
First, we identify the system when the rotation speed is null and then when the rotation speed is equal to 60rpm.
|
|
</p>
|
|
|
|
<p>
|
|
The actuators are voice coil with some damping.
|
|
</p>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_simscape_light.png" alt="coupling_simscape_light.png" />
|
|
</p>
|
|
</div>
|
|
|
|
<p>
|
|
And then with the heavy sample.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = wheavy;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
m = mheavy;
|
|
|
|
k = kpz;
|
|
c = <span style="color: #D0372D;">1e3</span>;
|
|
Gpz_heavy = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
k = kvc;
|
|
c = <span style="color: #D0372D;">1e3</span>;
|
|
Gvc_heavy = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
Gpz_heavy.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gpz_heavy.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_heavy.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_heavy.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_simscape_heavy.png" alt="coupling_simscape_heavy.png" />
|
|
</p>
|
|
</div>
|
|
|
|
<p>
|
|
Plot the ratio between the main transfer function and the coupling term:
|
|
</p>
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_ration_simscape_light.png" alt="coupling_ration_simscape_light.png" />
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_ration_simscape_heavy.png" alt="coupling_ration_simscape_heavy.png" />
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgdb709bf" class="outline-4">
|
|
<h4 id="orgdb709bf"><span class="section-number-4">4.5.1</span> Low rotation speed and High rotation speed</h4>
|
|
<div class="outline-text-4" id="text-4-5-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">60</span>; angle_e = <span style="color: #D0372D;">0</span>;
|
|
G_low = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>; angle_e = <span style="color: #D0372D;">0</span>;
|
|
G_high = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
G_low.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G_low.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
G_high.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G_high.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>G_low, G_high<span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5822ce2" class="outline-3">
|
|
<h3 id="org5822ce2"><span class="section-number-3">4.6</span> Identification in the fixed frame</h3>
|
|
<div class="outline-text-3" id="text-4-6">
|
|
<p>
|
|
Let's define some options as well as the inputs and outputs for linearization.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #D0372D;">0</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #008000;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fx'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fy'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">3</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dx'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">4</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dy'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then define the error estimation of the error and the rotational speed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
rot_speed = <span style="color: #D0372D;">0</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we run the linearization.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output names</span>
|
|
G.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fx', 'Fy'</span><span style="color: #707183;">}</span>;
|
|
G.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
Gr = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output names</span>
|
|
Gr.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fx', 'Fy'</span><span style="color: #707183;">}</span>;
|
|
Gr.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #D0372D;">1</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">180</span>;
|
|
rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
Ge = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output names</span>
|
|
Ge.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fx', 'Fy'</span><span style="color: #707183;">}</span>;
|
|
Ge.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>G<span style="color: #707183;">)</span>;
|
|
<span style="color: #8D8D84; font-style: italic;">% exportFig('G_x_y', 'wide-tall');</span>
|
|
|
|
<span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>Ge<span style="color: #707183;">)</span>;
|
|
<span style="color: #8D8D84; font-style: italic;">% exportFig('G_x_y_e', 'normal-normal');</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfa9ed99" class="outline-3">
|
|
<h3 id="orgfa9ed99"><span class="section-number-3">4.7</span> Identification from actuator forces to displacement in the fixed frame</h3>
|
|
<div class="outline-text-3" id="text-4-7">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #D0372D;">0</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #008000;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fu'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fv'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">3</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dx'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">4</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dy'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
G = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
|
|
|
|
G.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
G1 = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">4</span><span style="color: #707183;">)</span>;
|
|
|
|
G1.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G1.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
G2 = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">8</span><span style="color: #707183;">)</span>;
|
|
|
|
G2.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G2.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>G, G1, G2<span style="color: #707183;">)</span>;
|
|
exportFig<span style="color: #707183;">(</span><span style="color: #008000;">'G_u_v_to_x_y', 'wide-tall'</span><span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbc833bb" class="outline-3">
|
|
<h3 id="orgbc833bb"><span class="section-number-3">4.8</span> Effect of the rotating Speed</h3>
|
|
<div class="outline-text-3" id="text-4-8">
|
|
<p>
|
|
<a id="org45bb7b1"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaf21bf8" class="outline-4">
|
|
<h4 id="orgaf21bf8"><span class="section-number-4">4.8.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h4>
|
|
</div>
|
|
<div id="outline-container-orgdd964cc" class="outline-4">
|
|
<h4 id="orgdd964cc"><span class="section-number-4">4.8.2</span> <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</h4>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgc30bae9" class="outline-3">
|
|
<h3 id="orgc30bae9"><span class="section-number-3">4.9</span> Effect of the X-Y stage stiffness</h3>
|
|
<div class="outline-text-3" id="text-4-9">
|
|
<p>
|
|
<a id="orge951cc4"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org3a4478a" class="outline-4">
|
|
<h4 id="org3a4478a"><span class="section-number-4">4.9.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h4>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org12e1d75" class="outline-2">
|
|
<h2 id="org12e1d75"><span class="section-number-2">5</span> Control Implementation</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org4a3b8a3"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org70652b4" class="outline-3">
|
|
<h3 id="org70652b4"><span class="section-number-3">5.1</span> Measurement in the fixed reference frame</h3>
|
|
</div>
|
|
</div>
|
|
<div id="bibliography">
|
|
<h2>References</h2>
|
|
|
|
<table>
|
|
|
|
<tr valign="top">
|
|
<td align="right" class="bibtexnumber">
|
|
[<a name="smith99_scien_engin_guide_digit_signal">1</a>]
|
|
</td>
|
|
<td class="bibtexitem">
|
|
Steven W. Smith.
|
|
<em>The Scientist and Engineer's Guide to Digital Signal Processing
|
|
- Second Edition</em>.
|
|
California Technical Publishing, 1999.
|
|
|
|
</td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Thomas Dehaeze</p>
|
|
<p class="date">Created: 2019-01-24 jeu. 15:17</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|