nass-simscape/docs/control.html

725 lines
29 KiB
HTML
Raw Normal View History

2019-10-08 11:13:38 +02:00
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
2019-10-08 11:13:38 +02:00
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
2020-04-01 16:17:26 +02:00
<!-- 2020-04-01 mer. 16:16 -->
2019-10-08 11:13:38 +02:00
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
2020-03-23 10:05:32 +01:00
<title>Control of the Nano-Active-Stabilization-System</title>
2019-10-08 11:13:38 +02:00
<meta name="generator" content="Org mode" />
<meta name="author" content="Dehaeze Thomas" />
<style type="text/css">
<!--/*--><![CDATA[/*><!--*/
.title { text-align: center;
margin-bottom: .2em; }
.subtitle { text-align: center;
font-size: medium;
font-weight: bold;
margin-top:0; }
.todo { font-family: monospace; color: red; }
.done { font-family: monospace; color: green; }
.priority { font-family: monospace; color: orange; }
.tag { background-color: #eee; font-family: monospace;
padding: 2px; font-size: 80%; font-weight: normal; }
.timestamp { color: #bebebe; }
.timestamp-kwd { color: #5f9ea0; }
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
.underline { text-decoration: underline; }
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
p.verse { margin-left: 3%; }
pre {
border: 1px solid #ccc;
box-shadow: 3px 3px 3px #eee;
padding: 8pt;
font-family: monospace;
overflow: auto;
margin: 1.2em;
}
pre.src {
position: relative;
overflow: visible;
padding-top: 1.2em;
}
pre.src:before {
display: none;
position: absolute;
background-color: white;
top: -10px;
right: 10px;
padding: 3px;
border: 1px solid black;
}
pre.src:hover:before { display: inline;}
/* Languages per Org manual */
pre.src-asymptote:before { content: 'Asymptote'; }
pre.src-awk:before { content: 'Awk'; }
pre.src-C:before { content: 'C'; }
/* pre.src-C++ doesn't work in CSS */
pre.src-clojure:before { content: 'Clojure'; }
pre.src-css:before { content: 'CSS'; }
pre.src-D:before { content: 'D'; }
pre.src-ditaa:before { content: 'ditaa'; }
pre.src-dot:before { content: 'Graphviz'; }
pre.src-calc:before { content: 'Emacs Calc'; }
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
pre.src-fortran:before { content: 'Fortran'; }
pre.src-gnuplot:before { content: 'gnuplot'; }
pre.src-haskell:before { content: 'Haskell'; }
pre.src-hledger:before { content: 'hledger'; }
pre.src-java:before { content: 'Java'; }
pre.src-js:before { content: 'Javascript'; }
pre.src-latex:before { content: 'LaTeX'; }
pre.src-ledger:before { content: 'Ledger'; }
pre.src-lisp:before { content: 'Lisp'; }
pre.src-lilypond:before { content: 'Lilypond'; }
pre.src-lua:before { content: 'Lua'; }
pre.src-matlab:before { content: 'MATLAB'; }
pre.src-mscgen:before { content: 'Mscgen'; }
pre.src-ocaml:before { content: 'Objective Caml'; }
pre.src-octave:before { content: 'Octave'; }
pre.src-org:before { content: 'Org mode'; }
pre.src-oz:before { content: 'OZ'; }
pre.src-plantuml:before { content: 'Plantuml'; }
pre.src-processing:before { content: 'Processing.js'; }
pre.src-python:before { content: 'Python'; }
pre.src-R:before { content: 'R'; }
pre.src-ruby:before { content: 'Ruby'; }
pre.src-sass:before { content: 'Sass'; }
pre.src-scheme:before { content: 'Scheme'; }
pre.src-screen:before { content: 'Gnu Screen'; }
pre.src-sed:before { content: 'Sed'; }
pre.src-sh:before { content: 'shell'; }
pre.src-sql:before { content: 'SQL'; }
pre.src-sqlite:before { content: 'SQLite'; }
/* additional languages in org.el's org-babel-load-languages alist */
pre.src-forth:before { content: 'Forth'; }
pre.src-io:before { content: 'IO'; }
pre.src-J:before { content: 'J'; }
pre.src-makefile:before { content: 'Makefile'; }
pre.src-maxima:before { content: 'Maxima'; }
pre.src-perl:before { content: 'Perl'; }
pre.src-picolisp:before { content: 'Pico Lisp'; }
pre.src-scala:before { content: 'Scala'; }
pre.src-shell:before { content: 'Shell Script'; }
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
/* additional language identifiers per "defun org-babel-execute"
in ob-*.el */
pre.src-cpp:before { content: 'C++'; }
pre.src-abc:before { content: 'ABC'; }
pre.src-coq:before { content: 'Coq'; }
pre.src-groovy:before { content: 'Groovy'; }
/* additional language identifiers from org-babel-shell-names in
ob-shell.el: ob-shell is the only babel language using a lambda to put
the execution function name together. */
pre.src-bash:before { content: 'bash'; }
pre.src-csh:before { content: 'csh'; }
pre.src-ash:before { content: 'ash'; }
pre.src-dash:before { content: 'dash'; }
pre.src-ksh:before { content: 'ksh'; }
pre.src-mksh:before { content: 'mksh'; }
pre.src-posh:before { content: 'posh'; }
/* Additional Emacs modes also supported by the LaTeX listings package */
pre.src-ada:before { content: 'Ada'; }
pre.src-asm:before { content: 'Assembler'; }
pre.src-caml:before { content: 'Caml'; }
pre.src-delphi:before { content: 'Delphi'; }
pre.src-html:before { content: 'HTML'; }
pre.src-idl:before { content: 'IDL'; }
pre.src-mercury:before { content: 'Mercury'; }
pre.src-metapost:before { content: 'MetaPost'; }
pre.src-modula-2:before { content: 'Modula-2'; }
pre.src-pascal:before { content: 'Pascal'; }
pre.src-ps:before { content: 'PostScript'; }
pre.src-prolog:before { content: 'Prolog'; }
pre.src-simula:before { content: 'Simula'; }
pre.src-tcl:before { content: 'tcl'; }
pre.src-tex:before { content: 'TeX'; }
pre.src-plain-tex:before { content: 'Plain TeX'; }
pre.src-verilog:before { content: 'Verilog'; }
pre.src-vhdl:before { content: 'VHDL'; }
pre.src-xml:before { content: 'XML'; }
pre.src-nxml:before { content: 'XML'; }
/* add a generic configuration mode; LaTeX export needs an additional
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
pre.src-conf:before { content: 'Configuration File'; }
table { border-collapse:collapse; }
caption.t-above { caption-side: top; }
caption.t-bottom { caption-side: bottom; }
td, th { vertical-align:top; }
th.org-right { text-align: center; }
th.org-left { text-align: center; }
th.org-center { text-align: center; }
td.org-right { text-align: right; }
td.org-left { text-align: left; }
td.org-center { text-align: center; }
dt { font-weight: bold; }
.footpara { display: inline; }
.footdef { margin-bottom: 1em; }
.figure { padding: 1em; }
.figure p { text-align: center; }
.equation-container {
display: table;
text-align: center;
width: 100%;
}
.equation {
vertical-align: middle;
}
.equation-label {
display: table-cell;
text-align: right;
vertical-align: middle;
}
.inlinetask {
padding: 10px;
border: 2px solid gray;
margin: 10px;
background: #ffffcc;
}
#org-div-home-and-up
{ text-align: right; font-size: 70%; white-space: nowrap; }
textarea { overflow-x: auto; }
.linenr { font-size: smaller }
.code-highlighted { background-color: #ffff00; }
.org-info-js_info-navigation { border-style: none; }
#org-info-js_console-label
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
.org-info-js_search-highlight
{ background-color: #ffff00; color: #000000; font-weight: bold; }
.org-svg { width: 90%; }
/*]]>*/-->
</style>
2020-02-25 18:21:17 +01:00
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
<link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
<script type="text/javascript" src="./js/jquery.min.js"></script>
<script type="text/javascript" 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>
2019-10-08 11:13:38 +02:00
<script type="text/javascript">
2020-04-01 16:17:26 +02:00
// @license magnet:?xt=urn:btih:1f739d935676111cfff4b4693e3816e664797050&amp;dn=gpl-3.0.txt GPL-v3-or-Later
2019-10-08 11:13:38 +02:00
<!--/*--><![CDATA[/*><!--*/
2020-04-01 16:17:26 +02:00
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;
}
/*]]>*///-->
// @license-end
2019-10-08 11:13:38 +02:00
</script>
2020-03-23 10:05:32 +01:00
<script>
MathJax = {
tex: { macros: {
bm: ["\\boldsymbol{#1}",1],
}
}
};
</script>
<script type="text/javascript"
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
2019-10-08 11:13:38 +02:00
</head>
<body>
<div id="org-div-home-and-up">
2020-02-25 18:21:17 +01:00
<a accesskey="h" href="./index.html"> UP </a>
2019-10-08 11:13:38 +02:00
|
2020-02-25 18:21:17 +01:00
<a accesskey="H" href="./index.html"> HOME </a>
2019-10-08 11:13:38 +02:00
</div><div id="content">
2020-03-23 10:05:32 +01:00
<h1 class="title">Control of the Nano-Active-Stabilization-System</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org15699e9">1. Control Configuration - Introduction</a></li>
2020-03-26 17:25:43 +01:00
<li><a href="#org2be3166">2. Tracking Control in the Frame of the Nano-Hexapod - Basic Architectures</a>
2020-03-23 10:05:32 +01:00
<ul>
<li><a href="#org970ab39">2.1. Control in the frame of the Legs</a></li>
<li><a href="#org82193fb">2.2. Control in the Cartesian frame</a></li>
</ul>
</li>
2020-03-26 17:25:43 +01:00
<li><a href="#org7d7b7f4">3. Active Damping Architecture - Collocated Control (link)</a>
2020-03-23 10:05:32 +01:00
<ul>
<li><a href="#org3546873">3.1. Integral Force Feedback</a></li>
<li><a href="#org722b371">3.2. Direct Relative Velocity Feedback</a></li>
</ul>
</li>
2020-03-26 17:25:43 +01:00
<li><a href="#orgca70c79">4. HAC-LAC Architectures (link)</a>
2020-03-23 10:05:32 +01:00
<ul>
<li><a href="#orgd9c84f0">4.1. HAC-LAC using IFF and Tracking control in the frame of the Legs</a></li>
<li><a href="#orgeb80da1">4.2. HAC-LAC using IFF and Tracking control in the Cartesian frame</a></li>
2020-04-01 16:17:26 +02:00
<li><a href="#org8b2b21e">4.3. HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the task space</a></li>
<li><a href="#org1c04b26">4.4. HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</a></li>
2020-03-23 10:05:32 +01:00
</ul>
</li>
2020-03-26 17:25:43 +01:00
<li><a href="#orgab73896">5. Cascade Architectures (link)</a>
2020-03-23 10:05:32 +01:00
<ul>
<li><a href="#org3e5154f">5.1. Cascade Control with HAC-LAC Inner Loop and Primary Controller in the task space</a></li>
<li><a href="#org4353aca">5.2. Cascade Control with HAC-LAC Inner Loop and Primary Controller in the joint space</a></li>
</ul>
</li>
2020-03-26 17:25:43 +01:00
<li><a href="#org4ac6d11">6. Force Control (link)</a></li>
2020-03-23 10:05:32 +01:00
</ul>
</div>
</div>
<p>
The system consist of the following inputs and outputs (Figure <a href="#org2d9f6d0">1</a>):
</p>
<ul class="org-ul">
<li>\(\bm{\tau}\): Forces applied in each leg</li>
<li>\(\bm{\tau}_m\): Force sensor located in each leg</li>
<li>\(\bm{\mathcal{X}}\): Measurement of the payload position with respect to the granite</li>
<li>\(d\bm{\mathcal{L}}\): Measurement of the (small) relative motion of each leg</li>
</ul>
<div id="org2d9f6d0" class="figure">
<p><img src="figs/control_architecture_plant.png" alt="control_architecture_plant.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Block diagram with the inputs and outputs of the system</p>
</div>
<p>
In order to position the Sample with respect to the granite, we must use the measurement \(\bm{\mathcal{X}}\) in the control loop.
The wanted position of the sample with respect to the granite is represented by \(\bm{r}_\mathcal{X}\).
From \(\bm{r}_\mathcal{X}\) and \(\bm{\mathcal{X}}\), we can compute the required small change of pose of the nano-hexapod&rsquo;s top platform expressed in the frame of the nano-hexapod&rsquo;s base as shown in Figure <a href="#orgc4acef7">2</a>.
</p>
<p>
This can we considered as:
</p>
<ul class="org-ul">
<li>the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) expressed in a frame attach to the base of the nano-hexapod</li>
<li>the wanted (small) pose displacement \(\bm{r}_{d\mathcal{X}_n}\) of the nano-hexapod mobile platform with respect to its base</li>
</ul>
<div id="orgc4acef7" class="figure">
<p><img src="figs/control_architecture_pos_error.png" alt="control_architecture_pos_error.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Block diagram corresponding to the computation of the position error in the frame of the nano-hexapod</p>
</div>
<p>
In this document, we see how the different outputs of the system can be used to control of position \(\bm{\mathcal{X}}\).
</p>
<div id="outline-container-org15699e9" class="outline-2">
<h2 id="org15699e9"><span class="section-number-2">1</span> Control Configuration - Introduction</h2>
<div class="outline-text-2" id="text-1">
<p>
In this section, we discuss the control configuration for the NASS.
</p>
<p>
From <a class='org-ref-reference' href="#skogestad07_multiv_feedb_contr">skogestad07_multiv_feedb_contr</a>:
</p>
<blockquote>
<p>
We define the <b>control configuration</b> to be the restrictions imposed on the overall controller \(K\) by decomposing it into a set of <b>local controllers</b> with predetermined links and with a possibly predetermined design sequence where subcontrollers are designed locally.
</p>
<p>
Some elements used to build up a specific control configuration are:
</p>
<ul class="org-ul">
<li><b>Cascade controllers</b>. The output from one controller is the input to another</li>
<li><b>Decentralized controllers</b>. The control system consists of independent feedback controllers which interconnect a subset of the output measurements with a subset of the manipulated inputs.
These subsets should not be used by any other controller</li>
<li><b>Feedforward elements</b>. Link measured disturbances and manipulated inputs</li>
<li><b>Decoupling elements</b>. Link one set of manipulated inputs with another set of manipulated inputs.
They are used to improve the performance of decentralized control systems.</li>
</ul>
</blockquote>
<p>
Decoupling elements will be used to convert quantities from the joint space to the task space and vice-versa.
</p>
<p>
Decentralized controllers will be largely used both for Tracking control (Section <a href="#org251e3c9">2</a>) and for Active Damping techniques (Section <a href="#org1b3cc21">3</a>)
</p>
<p>
Combining both can be done in an HAC-LAC topology presented in Section <a href="#org31fa800">4</a>.
</p>
<p>
The use of decentralized controllers is proposed in Section <a href="#orga038762">5</a>.
</p>
</div>
</div>
2020-03-26 17:25:43 +01:00
<div id="outline-container-org2be3166" class="outline-2">
<h2 id="org2be3166"><span class="section-number-2">2</span> Tracking Control in the Frame of the Nano-Hexapod - Basic Architectures</h2>
2020-03-23 10:05:32 +01:00
<div class="outline-text-2" id="text-2">
<p>
<a id="org251e3c9"></a>
</p>
<p>
In this section, we suppose that we want to track some reference position \(\bm{r}_{\mathcal{X}_n}\) corresponding to the pose of the nano-hexapod&rsquo;s mobile platform with respect to its fixed base.
</p>
<p>
To do so, we have to the use the leg&rsquo;s length measurement \(d\bm{\mathcal{L}}\).
</p>
<p>
However, thanks to the forward and inverse kinematics, the controller can either be designed in the task space or in the joint space.
</p>
<p>
These to configuration are described in the next two sections.
</p>
</div>
<div id="outline-container-org970ab39" class="outline-3">
<h3 id="org970ab39"><span class="section-number-3">2.1</span> Control in the frame of the Legs</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org8583193"></a>
</p>
<p>
From the wanted small change in pose of the nano-hexapod&rsquo;s mobile platform \(\bm{r}_{d\mathcal{X}_n}\), we can use the Inverse Kinematics of the nano-hexapod to compute the corresponding small change of the leg length of the nano-hexapod \(\bm{r}_{d\mathcal{L}}\).
Then, this is subtracted by the measurement of the leg relative displacement \(d\bm{\mathcal{L}}\) to obtain to displacement error of each leg \(\bm{\epsilon}_{d\mathcal{L}}\).
Finally, a diagonal (Decentralized) controller \(\bm{K}_\mathcal{L}\) can be used.
</p>
<div id="org3211e10" class="figure">
<p><img src="figs/control_architecture_leg_frame.png" alt="control_architecture_leg_frame.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Control in the frame of the legs</p>
</div>
</div>
</div>
<div id="outline-container-org82193fb" class="outline-3">
<h3 id="org82193fb"><span class="section-number-3">2.2</span> Control in the Cartesian frame</h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="orgbd7e263"></a>
</p>
<p>
From the relative displacement of each leg \(d\bm{\mathcal{L}}\), the pose of the nano-hexapod&rsquo;s mobile platform \(\bm{\mathcal{X}_n}\) is estimated.
It is then subtracted from reference pose of the nano-hexapod \(\bm{r}_{\mathcal{X}_n}\) to obtain the pose error \(\bm{\epsilon}_{\mathcal{X}_n}\).
A diagonal controller \(\bm{K}_\mathcal{X}\) is used to generate forces and torques applied on the payload in a frame attached to the nano-hexapod&rsquo;s base.
These forces are then converted to forces applied in each of the nano-hexapod&rsquo;s actuators by the use of the Jacobian \(\bm{J}^{-T}\).
</p>
<div id="org81b6823" class="figure">
<p><img src="figs/control_architecture_cartesian_frame.png" alt="control_architecture_cartesian_frame.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Control in the cartesian Frame (rotating frame attached to the nano-hexapod&rsquo;s base)</p>
</div>
</div>
</div>
</div>
2020-03-26 17:25:43 +01:00
<div id="outline-container-org7d7b7f4" class="outline-2">
<h2 id="org7d7b7f4"><span class="section-number-2">3</span> Active Damping Architecture - Collocated Control (<a href="control_active_damping.html">link</a>)</h2>
2020-03-23 10:05:32 +01:00
<div class="outline-text-2" id="text-3">
<p>
<a id="org1b3cc21"></a>
</p>
<p>
From <a class='org-ref-reference' href="#preumont18_vibrat_contr_activ_struc_fourt_edition">preumont18_vibrat_contr_activ_struc_fourt_edition</a>:
</p>
<blockquote>
<p>
Active damping is very effective in reducing the settling time of transient disturbances and the effect of steady state disturbances near the resonance frequencies of the system; however, away from the resonances, the active damping is completely ineffective and leaves the closed-loop response essentially unchanged.
Such low-gain controllers are often called Low Authority Controllers (LAC), because they modify the poles of the system only slightly.
</p>
</blockquote>
<p>
Two very well known active damping techniques are <b>Integral Force Feedback</b> and <b>Direct Velocity Feedback</b>.
</p>
<p>
These two active damping techniques are collocated control techniques.
</p>
2020-03-26 17:25:43 +01:00
<p>
The active damping techniques are studied in <a href="control_active_damping.html">this</a> document.
</p>
2020-03-23 10:05:32 +01:00
</div>
<div id="outline-container-org3546873" class="outline-3">
<h3 id="org3546873"><span class="section-number-3">3.1</span> Integral Force Feedback</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="orgb398117"></a>
</p>
<p>
In this active damping technique, the force sensors in each leg is used.
</p>
<p>
The controller \(\bm{K}_\text{IFF}\) is a diagonal matrix, each of its diagonal element consists of:
</p>
<ul class="org-ul">
<li>an pure integrator</li>
<li>a gain \(g\) that can be tuned to achieve a maximum damping</li>
</ul>
\begin{equation}
\bm{K}_\text{IFF}(s) = \frac{g}{s} \bm{I}_{6}
\end{equation}
<p>
A lead-lag can also be used instead of a pure integrator.
</p>
<div id="org19b5f2d" class="figure">
<p><img src="figs/control_architecture_iff.png" alt="control_architecture_iff.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Integral Force Feedback</p>
</div>
</div>
</div>
<div id="outline-container-org722b371" class="outline-3">
<h3 id="org722b371"><span class="section-number-3">3.2</span> Direct Relative Velocity Feedback</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="orgfaf575b"></a>
</p>
<p>
The controller \(\bm{K}_\text{DVF}\) is a diagonal matrix.
Each diagonal element consists of:
</p>
<ul class="org-ul">
<li>a derivative action up to some frequency \(\omega_0\)</li>
<li>a gain \(g\) that can be tuned to achieve a maximum damping</li>
</ul>
\begin{equation}
\bm{K}_\text{DVF}(s) = \frac{g s}{\omega_0 + s} \bm{I}_{6}
\end{equation}
<div id="org402f972" class="figure">
<p><img src="figs/control_architecture_dvf.png" alt="control_architecture_dvf.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Direct Velocity Feedback</p>
</div>
</div>
</div>
</div>
2020-03-26 17:25:43 +01:00
<div id="outline-container-orgca70c79" class="outline-2">
<h2 id="orgca70c79"><span class="section-number-2">4</span> HAC-LAC Architectures (<a href="control_hac_lac.html">link</a>)</h2>
2020-03-23 10:05:32 +01:00
<div class="outline-text-2" id="text-4">
<p>
<a id="org31fa800"></a>
</p>
<p>
Here we can combine Active Damping Techniques (Low authority control) with a tracking controller (high authority control).
Usually, the low authority controller is designed first, and the high authority controller is designed based on the damped plant.
</p>
<p>
From <a class='org-ref-reference' href="#preumont18_vibrat_contr_activ_struc_fourt_edition">preumont18_vibrat_contr_activ_struc_fourt_edition</a>:
</p>
<blockquote>
<p>
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure <a href="#org1b2c5c7">7</a>.
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages:
</p>
<ul class="org-ul">
<li>The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth</li>
<li>The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)</li>
<li>The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)</li>
</ul>
</blockquote>
<div id="org1b2c5c7" class="figure">
<p><img src="figs/control_architecture_hac_lac.png" alt="control_architecture_hac_lac.png" />
</p>
<p><span class="figure-number">Figure 7: </span>HAC-LAC Control Architecture</p>
</div>
<p>
If there is only one input to the system, the HAC-LAC topology can be represented as depicted in Figure <a href="#org91828a2">8</a>.
Usually, the Low Authority Controller is first design, and then the High Authority Controller is designed based on the damped plant.
</p>
<div id="org91828a2" class="figure">
<p><img src="figs/control_architecture_hac_lac_one_input.png" alt="control_architecture_hac_lac_one_input.png" />
</p>
<p><span class="figure-number">Figure 8: </span>HAC-LAC Architecture with a system having only one input</p>
</div>
</div>
<div id="outline-container-orgd9c84f0" class="outline-3">
<h3 id="orgd9c84f0"><span class="section-number-3">4.1</span> HAC-LAC using IFF and Tracking control in the frame of the Legs</h3>
<div class="outline-text-3" id="text-4-1">
<div id="orgd235561" class="figure">
<p><img src="figs/control_architecture_hac_iff_L.png" alt="control_architecture_hac_iff_L.png" />
</p>
<p><span class="figure-number">Figure 9: </span>IFF + Control in the frame of the legs</p>
</div>
</div>
</div>
<div id="outline-container-orgeb80da1" class="outline-3">
<h3 id="orgeb80da1"><span class="section-number-3">4.2</span> HAC-LAC using IFF and Tracking control in the Cartesian frame</h3>
<div class="outline-text-3" id="text-4-2">
<div id="orgb89bca0" class="figure">
<p><img src="figs/control_architecture_hac_iff_X.png" alt="control_architecture_hac_iff_X.png" />
</p>
<p><span class="figure-number">Figure 10: </span>IFF + Control in the cartesian frame</p>
</div>
</div>
</div>
2020-03-26 17:25:43 +01:00
2020-04-01 16:17:26 +02:00
<div id="outline-container-org8b2b21e" class="outline-3">
<h3 id="org8b2b21e"><span class="section-number-3">4.3</span> HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the task space</h3>
2020-03-26 17:25:43 +01:00
<div class="outline-text-3" id="text-4-3">
<div class="figure">
<p><img src="figs/control_architecture_hac_iff_pos_X.png" alt="control_architecture_hac_iff_pos_X.png" />
</p>
</div>
</div>
</div>
2020-04-01 16:17:26 +02:00
<div id="outline-container-org1c04b26" class="outline-3">
<h3 id="org1c04b26"><span class="section-number-3">4.4</span> HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</h3>
<div class="outline-text-3" id="text-4-4">
<div class="figure">
<p><img src="figs/control_architecture_hac_iff_pos_L.png" alt="control_architecture_hac_iff_pos_L.png" />
</p>
</div>
</div>
</div>
2020-03-23 10:05:32 +01:00
</div>
2020-03-26 17:25:43 +01:00
<div id="outline-container-orgab73896" class="outline-2">
<h2 id="orgab73896"><span class="section-number-2">5</span> Cascade Architectures (<a href="control_cascade.html">link</a>)</h2>
2020-03-23 10:05:32 +01:00
<div class="outline-text-2" id="text-5">
<p>
<a id="orga038762"></a>
</p>
<p>
2020-04-01 16:17:26 +02:00
The principle of Cascade control is shown in Figure <a href="#org03ef231">13</a> and explained as follow:
2020-03-23 10:05:32 +01:00
</p>
<blockquote>
<p>
To follow <b>two objectives</b> with different properties in one control system, usually a <b>hierarchy</b> of two feedback loops is used in practice.
This kind of control topology is called <b>cascade control</b>, which is used when there are <b>several measurements and one prime control variable</b>.
2020-04-01 16:17:26 +02:00
Cascade control is implemented by <b>nesting</b> the control loops, as shown in Figure <a href="#org03ef231">13</a>.
2020-03-23 10:05:32 +01:00
The output control loop is called the <b>primary loop</b>, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system. &#x2013; <a class='org-ref-reference' href="#taghirad13_paral">taghirad13_paral</a>
</p>
</blockquote>
<div id="org03ef231" class="figure">
<p><img src="figs/control_architecture_cascade_control.png" alt="control_architecture_cascade_control.png" />
</p>
2020-04-01 16:17:26 +02:00
<p><span class="figure-number">Figure 13: </span>Cascade Control Architecture</p>
2020-03-23 10:05:32 +01:00
</div>
<p>
This control topology seems adapted for the NASS, as indeed we have more inputs than outputs
</p>
<p>
In the NASS&rsquo;s case:
</p>
<ul class="org-ul">
<li>The primary objective is to position the sample with respect to the granite, thus the outer loop (and primary controller) should corresponds to a motion control loop</li>
</ul>
<p>
The inner loop can be composed of the system controlled with the HAC-LAC topology.
</p>
</div>
<div id="outline-container-org3e5154f" class="outline-3">
<h3 id="org3e5154f"><span class="section-number-3">5.1</span> Cascade Control with HAC-LAC Inner Loop and Primary Controller in the task space</h3>
<div class="outline-text-3" id="text-5-1">
<div id="orgff7dfc6" class="figure">
<p><img src="figs/control_architecture_cascade_L.png" alt="control_architecture_cascade_L.png" />
</p>
2020-04-01 16:17:26 +02:00
<p><span class="figure-number">Figure 14: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
2020-03-23 10:05:32 +01:00
</div>
</div>
</div>
<div id="outline-container-org4353aca" class="outline-3">
<h3 id="org4353aca"><span class="section-number-3">5.2</span> Cascade Control with HAC-LAC Inner Loop and Primary Controller in the joint space</h3>
<div class="outline-text-3" id="text-5-2">
<div id="org4bc4c4c" class="figure">
<p><img src="figs/control_architecture_cascade_X.png" alt="control_architecture_cascade_X.png" />
</p>
2020-04-01 16:17:26 +02:00
<p><span class="figure-number">Figure 15: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the Cartesian Frame</p>
2020-03-23 10:05:32 +01:00
</div>
</div>
</div>
</div>
2020-03-26 17:25:43 +01:00
<div id="outline-container-org4ac6d11" class="outline-2">
<h2 id="org4ac6d11"><span class="section-number-2">6</span> Force Control (<a href="control_force.html">link</a>)</h2>
2020-03-23 10:05:32 +01:00
<div class="outline-text-2" id="text-6">
<p>
Signals:
</p>
<ul class="org-ul">
<li>\(\bm{r}_\mathcal{F}\) is the wanted total force/torque to be applied to the payload</li>
<li>\(\bm{\epsilon}_\mathcal{F}\) is the force/torque errors that should be applied to the payload</li>
<li>\(\bm{\tau}\) is the force applied in each actuator</li>
</ul>
<div class="figure">
<p><img src="figs/control_architecture_force.png" alt="control_architecture_force.png" />
</p>
</div>
</div>
</div>
2020-03-26 17:25:43 +01:00
2020-03-23 10:05:32 +01:00
<p>
<h1 class='org-ref-bib-h1'>Bibliography</h1>
<ul class='org-ref-bib'><li><a id="skogestad07_multiv_feedb_contr">[skogestad07_multiv_feedb_contr]</a> <a name="skogestad07_multiv_feedb_contr"></a>Skogestad & Postlethwaite, Multivariable Feedback Control: Analysis and Design, John Wiley (2007).</li>
<li><a id="preumont18_vibrat_contr_activ_struc_fourt_edition">[preumont18_vibrat_contr_activ_struc_fourt_edition]</a> <a name="preumont18_vibrat_contr_activ_struc_fourt_edition"></a>Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).</li>
<li><a id="taghirad13_paral">[taghirad13_paral]</a> <a name="taghirad13_paral"></a>Taghirad, Parallel robots : mechanics and control, CRC Press (2013).</li>
</ul>
</p>
2019-10-08 11:13:38 +02:00
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
2020-04-01 16:17:26 +02:00
<p class="date">Created: 2020-04-01 mer. 16:16</p>
2019-10-08 11:13:38 +02:00
</div>
</body>
</html>