2019-01-18 17:18:02 +01:00
<?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 >
2019-01-18 17:46:54 +01:00
<!-- 2019 - 01 - 18 ven. 17:46 -->
2019-01-18 17:18:02 +01:00
< 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) 2012-2018 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 >
2019-01-18 17:46:54 +01:00
< li > < a href = "#org892e46e" > 1. Goal of this note< / a > < / li >
< li > < a href = "#orgbb4d730" > 2. System< / a >
2019-01-18 17:18:02 +01:00
< ul >
2019-01-18 17:46:54 +01:00
< li > < a href = "#orgf6286ea" > 2.1. System description< / a > < / li >
< li > < a href = "#org6517c3a" > 2.2. Equations< / a > < / li >
< li > < a href = "#org66b66d3" > 2.3. < span class = "todo TODO" > TODO< / span > Analysis< / a >
< ul >
< li > < a href = "#org1aee292" > 2.3.1. Stiff actuators< / a > < / li >
< li > < a href = "#org3d277ca" > 2.3.2. Negative Stiffness< / a > < / li >
2019-01-18 17:18:02 +01:00
< / ul >
< / li >
< / ul >
< / li >
2019-01-18 17:46:54 +01:00
< li > < a href = "#org1d7bfef" > 3. Analytical Computation of forces for the NASS< / a >
2019-01-18 17:18:02 +01:00
< ul >
2019-01-18 17:46:54 +01:00
< li > < a href = "#org9862d4d" > 3.1. Parameters< / a > < / li >
< li > < a href = "#orged72531" > 3.2. Euler and Coriolis forces< / a > < / li >
< li > < a href = "#org1ad22a2" > 3.3. Negative Spring Effect< / a > < / li >
2019-01-18 17:18:02 +01:00
< / ul >
< / li >
2019-01-18 17:46:54 +01:00
< li > < a href = "#orgb20d1e2" > 4. Control Strategies< / a >
< ul >
< li > < a href = "#org67681a1" > 4.1. Measurement in the fixed reference frame< / a > < / li >
< li > < a href = "#org358433f" > 4.2. Measurement in the rotating frame< / a > < / li >
2019-01-18 17:18:02 +01:00
< / ul >
< / li >
2019-01-18 17:46:54 +01:00
< li > < a href = "#org403dcc8" > 5. Effect of the rotating Speed< / a >
2019-01-18 17:18:02 +01:00
< ul >
2019-01-18 17:46:54 +01:00
< li > < a href = "#orga27aa6d" > 5.1. < 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 = "#org5b37262" > 5.2. < 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 >
2019-01-18 17:18:02 +01:00
< / ul >
< / li >
2019-01-18 17:46:54 +01:00
< li > < a href = "#org3eb9f54" > 6. Effect of the X-Y stage stiffness< / a >
2019-01-18 17:18:02 +01:00
< ul >
2019-01-18 17:46:54 +01:00
< li > < a href = "#org9fbd479" > 6.1. < span class = "todo TODO" > TODO< / span > At full speed, check how the coupling changes with the stiffness of the actuators< / a > < / li >
2019-01-18 17:18:02 +01:00
< / ul >
< / li >
< / ul >
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org892e46e" class = "outline-2" >
< h2 id = "org892e46e" > < span class = "section-number-2" > 1< / span > Goal of this note< / h2 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-2" id = "text-1" >
< p >
The control objective is to stabilize the position of a rotating object with respect to a non-rotating frame.
< / p >
< p >
The actuators are also rotating with the object.
< / p >
< p >
We want to compare the two different approach:
< / 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 >
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-orgbb4d730" class = "outline-2" >
< h2 id = "orgbb4d730" > < span class = "section-number-2" > 2< / span > System< / h2 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-2" id = "text-2" >
< p >
2019-01-18 17:46:54 +01:00
< a id = "orgfb8b8b0" > < / a >
2019-01-18 17:18:02 +01:00
< / p >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-orgf6286ea" class = "outline-3" >
< h3 id = "orgf6286ea" > < span class = "section-number-3" > 2.1< / span > System description< / h3 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-3" id = "text-2-1" >
< p >
2019-01-18 17:46:54 +01:00
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure < a href = "#org6527df8" > 1< / a > ).
2019-01-18 17:18:02 +01:00
< / p >
< p >
The control inputs are the forces applied in 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 actuators.
< / p >
2019-01-18 17:46:54 +01:00
< div id = "org6527df8" class = "figure" >
2019-01-18 17:18:02 +01:00
< 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 >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org6517c3a" class = "outline-3" >
< h3 id = "org6517c3a" > < span class = "section-number-3" > 2.2< / span > Equations< / h3 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-3" id = "text-2-2" >
< p >
2019-01-18 17:46:54 +01:00
< a id = "orga34f88d" > < / a >
2019-01-18 17:18:02 +01:00
< / p >
< p >
2019-01-18 17:46:54 +01:00
Based on the figure < a href = "#org6527df8" > 1< / a > , we can write the equations of motion of the system.
2019-01-18 17:18:02 +01:00
< / p >
< p >
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
< / p >
\begin{align}
2019-01-18 17:46:54 +01:00
\label{org4d9790f}
2019-01-18 17:18:02 +01:00
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}
2019-01-18 17:46:54 +01:00
\label{orgb67b4fc}
2019-01-18 17:18:02 +01:00
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*}
2019-01-18 17:46:54 +01:00
\label{orgcf126c0}
2019-01-18 17:18:02 +01:00
\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 >
2019-01-18 17:46:54 +01:00
By appling the Lagrangian equations, we obtain:
2019-01-18 17:18:02 +01:00
< / 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 >
2019-01-18 17:46:54 +01:00
By injecting the previous result into the Lagrangian equation, we obtain:
2019-01-18 17:18:02 +01:00
< / 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{align*}
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} \\
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{align*}
< / div >
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org66b66d3" class = "outline-3" >
< h3 id = "org66b66d3" > < span class = "section-number-3" > 2.3< / span > < span class = "todo TODO" > TODO< / span > Analysis< / h3 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-3" id = "text-2-3" >
< 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 \(k-d_u m\dot{\theta}^2\).
Thus, the term \(-d_u m\dot{\theta}^2\) acts like a negative stiffness (due to < b > centrifugal forces< / b > ).
< / p >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org1aee292" class = "outline-4" >
< h4 id = "org1aee292" > < span class = "section-number-4" > 2.3.1< / span > Stiff actuators< / h4 >
< div class = "outline-text-4" id = "text-2-3-1" >
< p >
Let's say we use stiff actuators such that \(m \ddot{d_u} + (k - m\dot{\theta}^2) d_u \approx k d_u\).
< / p >
< p >
Let's suppose that \(F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \approx F_u\).
< / p >
< p >
Then we obtain \(d_u = \frac{F_u}{k}\) that we can re inject in the other equation to obtain:
\[ m \ddot{d_v} + (k - m\dot{\theta}^2) d_v & = F_v - 2 m\frac{\dot{F_u}}{k}\dot{\theta} - m \frac{F_u}{k}\ddot{\theta} \]
< / p >
2019-01-18 17:18:02 +01:00
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org3d277ca" class = "outline-4" >
< h4 id = "org3d277ca" > < span class = "section-number-4" > 2.3.2< / span > Negative Stiffness< / h4 >
< div class = "outline-text-4" id = "text-2-3-2" >
2019-01-18 17:18:02 +01:00
< p >
2019-01-18 17:46:54 +01:00
If \(\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}\), then the negative spring effect is negligible and \(k - m\dot{\theta}^2 \approx k\).
2019-01-18 17:18:02 +01:00
< / p >
2019-01-18 17:46:54 +01:00
< / div >
< / div >
< / div >
< / div >
2019-01-18 17:18:02 +01:00
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org1d7bfef" class = "outline-2" >
< h2 id = "org1d7bfef" > < span class = "section-number-2" > 3< / span > Analytical Computation of forces for the NASS< / h2 >
< div class = "outline-text-2" id = "text-3" >
2019-01-18 17:18:02 +01:00
< p >
2019-01-18 17:46:54 +01:00
For the NASS, the Euler forces should be less of a problem as \(\ddot{\theta}\) should be very small when conducting an experiment.
2019-01-18 17:18:02 +01:00
< / p >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org9862d4d" class = "outline-3" >
< h3 id = "org9862d4d" > < span class = "section-number-3" > 3.1< / span > Parameters< / h3 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-3" id = "text-3-1" >
< p >
Let's define the parameters for the NASS.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > mlight = < span style = "color: #D0372D;" > 35< / span > ; < span style = "color: #8D8D84; font-style: italic;" > % [kg]< / span >
mheavy = < span style = "color: #D0372D;" > 85< / span > ; < span style = "color: #8D8D84; font-style: italic;" > % [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;" > % [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;" > % [rad/s]< / span >
wdot = < span style = "color: #D0372D;" > 1< / span > ; < span style = "color: #8D8D84; font-style: italic;" > % [rad/s2]< / span >
d = < span style = "color: #D0372D;" > 0< / span > .< span style = "color: #D0372D;" > 1< / span > ; < span style = "color: #8D8D84; font-style: italic;" > % [m]< / span >
ddot = < span style = "color: #D0372D;" > 0< / span > .< span style = "color: #D0372D;" > 2< / span > ; < span style = "color: #8D8D84; font-style: italic;" > % [m/s]< / span >
< / pre >
< / div >
2019-01-18 17:46:54 +01:00
< / div >
< / div >
< div id = "outline-container-orged72531" class = "outline-3" >
< h3 id = "orged72531" > < span class = "section-number-3" > 3.2< / span > Euler and Coriolis forces< / h3 >
< div class = "outline-text-3" id = "text-3-2" >
< p >
First we will determine the value for Euler and Coriolis forces during regular experiment.
< / p >
2019-01-18 17:18:02 +01:00
< p >
2019-01-18 17:46:54 +01:00
We then compute the corresponding values of the Coriolis and Euler forces, and the obtained values are displayed in table < a href = "#orgae713d9" > 1< / a > .
2019-01-18 17:18:02 +01:00
< / p >
2019-01-18 17:46:54 +01:00
< table id = "orgae713d9" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
2019-01-18 17:18:02 +01:00
< 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" > 44.0 N< / td >
< td class = "org-left" > 1.8 N< / td >
< / tr >
< tr >
< td class = "org-left" > Euler< / td >
< td class = "org-left" > 3.5 N< / td >
< td class = "org-left" > 8.5 N< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org1ad22a2" class = "outline-3" >
< h3 id = "org1ad22a2" > < span class = "section-number-3" > 3.3< / span > Negative Spring Effect< / h3 >
< div class = "outline-text-3" id = "text-3-3" >
2019-01-18 17:18:02 +01:00
< p >
2019-01-18 17:46:54 +01:00
The values for the negative spring effect are displayed in table < a href = "#org7244d2d" > 2< / a > .
2019-01-18 17:18:02 +01:00
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
< / p >
2019-01-18 17:46:54 +01:00
< table id = "org7244d2d" 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 >
2019-01-18 17:18:02 +01:00
< 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 >
2019-01-18 17:46:54 +01:00
< td class = "org-left" > Neg. Spring< / td >
2019-01-18 17:18:02 +01:00
< td class = "org-left" > 3.5 N/m< / td >
< td class = "org-left" > 8.5 N/m< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-orgb20d1e2" class = "outline-2" >
< h2 id = "orgb20d1e2" > < span class = "section-number-2" > 4< / span > Control Strategies< / h2 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-2" id = "text-4" >
< p >
2019-01-18 17:46:54 +01:00
< a id = "orgec63a1f" > < / a >
2019-01-18 17:18:02 +01:00
< / p >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org67681a1" class = "outline-3" >
< h3 id = "org67681a1" > < span class = "section-number-3" > 4.1< / span > Measurement in the fixed reference frame< / h3 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-3" id = "text-4-1" >
< 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 >
2019-01-18 17:46:54 +01:00
The block diagram is shown on figure < a href = "#org4869ac5" > 2< / a > .
2019-01-18 17:18:02 +01:00
< / p >
2019-01-18 17:46:54 +01:00
< div id = "org4869ac5" class = "figure" >
2019-01-18 17:18:02 +01:00
< p > < img src = "./Figures/control_measure_fixed_2dof.png" alt = "control_measure_fixed_2dof.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Control with a measure from fixed frame< / p >
< / div >
< p >
The loop gain is then \(L = G(\theta) K J(\theta)\).
< / p >
< p >
2019-01-18 17:46:54 +01:00
One question we wish to answer is: is \(G(\theta) J(\theta) = G(\theta_0) J(\theta_0)\)?
2019-01-18 17:18:02 +01:00
< / p >
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org358433f" class = "outline-3" >
< h3 id = "org358433f" > < span class = "section-number-3" > 4.2< / span > Measurement in the rotating frame< / h3 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-3" id = "text-4-2" >
< p >
2019-01-18 17:46:54 +01:00
Let's consider that the measurement is made in the rotating reference frame.
2019-01-18 17:18:02 +01:00
< / p >
< p >
2019-01-18 17:46:54 +01:00
The corresponding block diagram is shown figure < a href = "#org781b9ae" > 3< / a >
2019-01-18 17:18:02 +01:00
< / p >
2019-01-18 17:46:54 +01:00
< div id = "org781b9ae" class = "figure" >
2019-01-18 17:18:02 +01:00
< p > < img src = "./Figures/control_measure_rotating_2dof.png" alt = "control_measure_rotating_2dof.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Control with a measure from rotating frame< / p >
< / div >
< p >
The loop gain is \(L = G K\).
< / p >
< / div >
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org403dcc8" class = "outline-2" >
< h2 id = "org403dcc8" > < span class = "section-number-2" > 5< / span > Effect of the rotating Speed< / h2 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-2" id = "text-5" >
< p >
2019-01-18 17:46:54 +01:00
< a id = "org6624b66" > < / a >
2019-01-18 17:18:02 +01:00
< / p >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-orga27aa6d" class = "outline-3" >
< h3 id = "orga27aa6d" > < span class = "section-number-3" > 5.1< / span > < span class = "todo TODO" > TODO< / span > Use realistic parameters for the mass of the sample and stiffness of the X-Y stage< / h3 >
2019-01-18 17:18:02 +01:00
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org5b37262" class = "outline-3" >
< h3 id = "org5b37262" > < span class = "section-number-3" > 5.2< / 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)< / h3 >
2019-01-18 17:18:02 +01:00
< / div >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org3eb9f54" class = "outline-2" >
< h2 id = "org3eb9f54" > < span class = "section-number-2" > 6< / span > Effect of the X-Y stage stiffness< / h2 >
2019-01-18 17:18:02 +01:00
< div class = "outline-text-2" id = "text-6" >
< p >
2019-01-18 17:46:54 +01:00
< a id = "org8208f86" > < / a >
2019-01-18 17:18:02 +01:00
< / p >
< / div >
2019-01-18 17:46:54 +01:00
< div id = "outline-container-org9fbd479" class = "outline-3" >
< h3 id = "org9fbd479" > < span class = "section-number-3" > 6.1< / span > < span class = "todo TODO" > TODO< / span > At full speed, check how the coupling changes with the stiffness of the actuators< / h3 >
2019-01-18 17:18:02 +01:00
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Thomas Dehaeze< / p >
2019-01-18 17:46:54 +01:00
< p class = "date" > Created: 2019-01-18 ven. 17:46< / p >
2019-01-18 17:18:02 +01:00
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
< / div >
< / body >
< / html >
