1338 lines
62 KiB
HTML
1338 lines
62 KiB
HTML
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Active Damping</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org18c5fea">1. Undamped System</a>
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<ul>
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<li><a href="#org3e876e9">1.1. Init</a></li>
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<li><a href="#org5e8426f">1.2. Identification</a></li>
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<li><a href="#orgbd5ad99">1.3. Sensitivity to disturbances</a></li>
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<li><a href="#orge32a5c2">1.4. Undamped Plant</a></li>
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<li><a href="#org5812f24">1.5. Save</a></li>
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</ul>
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</li>
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<li><a href="#org6b0cb6e">2. Integral Force Feedback</a>
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<ul>
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<li><a href="#orgcd91088">2.1. One degree-of-freedom example</a>
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<ul>
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<li><a href="#org2e01f23">2.1.1. Equations</a></li>
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<li><a href="#org10a6bb4">2.1.2. Matlab Example</a></li>
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</ul>
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</li>
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<li><a href="#org896aad1">2.2. Control Design</a></li>
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<li><a href="#org2f66101">2.3. Sensitivity to disturbances</a></li>
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<li><a href="#org0d47de1">2.4. Damped Plant</a></li>
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<li><a href="#orgd43ddf5">2.5. Save</a></li>
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<li><a href="#org66221ae">2.6. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgb11e313">3. Relative Motion Control</a>
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<ul>
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<li><a href="#orga766aee">3.1. One degree-of-freedom example</a>
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<ul>
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<li><a href="#orga0bbe25">3.1.1. Equations</a></li>
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<li><a href="#org32b38d2">3.1.2. Matlab Example</a></li>
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</ul>
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</li>
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<li><a href="#org8dba3f6">3.2. Control Design</a></li>
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<li><a href="#orgc61326f">3.3. Sensitivity to disturbances</a></li>
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<li><a href="#orgc7efe56">3.4. Damped Plant</a></li>
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<li><a href="#org3981d3b">3.5. Save</a></li>
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<li><a href="#org4ff6cc5">3.6. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org44b3ec9">4. Direct Velocity Feedback</a>
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<ul>
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<li><a href="#orgdbca8bc">4.1. One degree-of-freedom example</a>
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<ul>
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<li><a href="#org88e6e40">4.1.1. Equations</a></li>
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<li><a href="#orgd273177">4.1.2. Matlab Example</a></li>
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</ul>
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</li>
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<li><a href="#orgbc6a859">4.2. Control Design</a></li>
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<li><a href="#orge803ae6">4.3. Conclusion</a></li>
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</ul>
|
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</li>
|
|
<li><a href="#org1b033b6">5. Comparison</a>
|
|
<ul>
|
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<li><a href="#org7da72db">5.1. Comparison</a></li>
|
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</ul>
|
|
</li>
|
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<li><a href="#org26ec68a">6. Conclusion</a></li>
|
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</ul>
|
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</div>
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</div>
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<p>
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|
First, in section <a href="#orgc482051">1</a>, we will looked at the undamped system.
|
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</p>
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|
|
|
<p>
|
|
Then, we will compare three active damping techniques:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>In section <a href="#org098dd4f">2</a>: the integral force feedback is used</li>
|
|
<li>In section <a href="#orgd55cd62">3</a>: the relative motion control is used</li>
|
|
<li>In section <a href="#orge9ebafd">4</a>: the direct velocity feedback is used</li>
|
|
</ul>
|
|
|
|
<p>
|
|
For each of the active damping technique, we will:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Compare the sensitivity from disturbances</li>
|
|
<li>Look at the damped plant</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The disturbances are:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Ground motion</li>
|
|
<li>Direct forces</li>
|
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<li>Motion errors of all the stages</li>
|
|
</ul>
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|
|
|
<div id="outline-container-org18c5fea" class="outline-2">
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<h2 id="org18c5fea"><span class="section-number-2">1</span> Undamped System</h2>
|
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<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
<a id="orgc482051"></a>
|
|
</p>
|
|
<p>
|
|
We first look at the undamped system.
|
|
The performance of this undamped system will be compared with the damped system using various techniques.
|
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</p>
|
|
</div>
|
|
<div id="outline-container-org3e876e9" class="outline-3">
|
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<h3 id="org3e876e9"><span class="section-number-3">1.1</span> Init</h3>
|
|
<div class="outline-text-3" id="text-1-1">
|
|
<p>
|
|
We initialize all the stages with the default parameters.
|
|
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
All the controllers are set to 0.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5e8426f" class="outline-3">
|
|
<h3 id="org5e8426f"><span class="section-number-3">1.2</span> Identification</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
<p>
|
|
We identify the various transfer functions of the system
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G = identifyPlant<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbd5ad99" class="outline-3">
|
|
<h3 id="orgbd5ad99"><span class="section-number-3">1.3</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<p>
|
|
The sensitivity to disturbances are shown on figure <a href="#orgba6dab5">1</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgba6dab5" class="figure">
|
|
<p><img src="figs/sensitivity_dist_undamped.png" alt="sensitivity_dist_undamped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Undamped sensitivity to disturbances (<a href="./figs/sensitivity_dist_undamped.png">png</a>, <a href="./figs/sensitivity_dist_undamped.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge32a5c2" class="outline-3">
|
|
<h3 id="orge32a5c2"><span class="section-number-3">1.4</span> Undamped Plant</h3>
|
|
<div class="outline-text-3" id="text-1-4">
|
|
<p>
|
|
The "plant" (transfer function from forces applied by the nano-hexapod to the measured displacement of the sample with respect to the granite) bode plot is shown on figure <a href="#orgba6dab5">1</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org3538011" class="figure">
|
|
<p><img src="figs/plant_undamped.png" alt="plant_undamped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Transfer Function from cartesian forces to displacement for the undamped plant (<a href="./figs/plant_undamped.png">png</a>, <a href="./figs/plant_undamped.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5812f24" class="outline-3">
|
|
<h3 id="org5812f24"><span class="section-number-3">1.5</span> Save</h3>
|
|
<div class="outline-text-3" id="text-1-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6b0cb6e" class="outline-2">
|
|
<h2 id="org6b0cb6e"><span class="section-number-2">2</span> Integral Force Feedback</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="org098dd4f"></a>
|
|
</p>
|
|
<p>
|
|
Integral Force Feedback is applied.
|
|
In section <a href="#orgfa2c8aa">2.1</a>, IFF is applied on a uni-axial system to understand its behavior.
|
|
Then, it is applied on the simscape model.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgcd91088" class="outline-3">
|
|
<h3 id="orgcd91088"><span class="section-number-3">2.1</span> One degree-of-freedom example</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
<a id="orgfa2c8aa"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org2e01f23" class="outline-4">
|
|
<h4 id="org2e01f23"><span class="section-number-4">2.1.1</span> Equations</h4>
|
|
<div class="outline-text-4" id="text-2-1-1">
|
|
|
|
<div id="org1249cd2" class="figure">
|
|
<p><img src="figs/iff_1dof.png" alt="iff_1dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Integral Force Feedback applied to a 1dof system</p>
|
|
</div>
|
|
|
|
<p>
|
|
The dynamic of the system is described by the following equation:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
<p>
|
|
The measured force \(F_m\) is:
|
|
</p>
|
|
\begin{align}
|
|
F_m &= F - kx - csx + kw + csw \\
|
|
&= ms^2 x - F_d
|
|
\end{align}
|
|
<p>
|
|
The Integral Force Feedback controller is \(K = -\frac{g}{s}\), and thus the applied force by this controller is:
|
|
</p>
|
|
\begin{equation}
|
|
F_{\text{IFF}} = -\frac{g}{s} F_m = -\frac{g}{s} (ms^2 x - F_d)
|
|
\end{equation}
|
|
<p>
|
|
Once the IFF is applied, the new dynamics of the system is:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d + F - kx - csx + kw + csw - \frac{g}{s} (ms^2x - F_d)
|
|
\end{equation}
|
|
|
|
<p>
|
|
And finally:
|
|
</p>
|
|
\begin{equation}
|
|
x = F_d \frac{1 + \frac{g}{s}}{ms^2 + (mg + c)s + k} + F \frac{1}{ms^2 + (mg + c)s + k} + w \frac{k + cs}{ms^2 + (mg + c)s + k}
|
|
\end{equation}
|
|
|
|
<p>
|
|
We can see that this:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>adds damping to the system by a value \(mg\)</li>
|
|
<li>lower the compliance as low frequency by a factor: \(1 + g/s\)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
If we want critical damping:
|
|
</p>
|
|
\begin{equation}
|
|
\xi = \frac{1}{2} \frac{c + gm}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
|
|
<p>
|
|
This is attainable if we have:
|
|
</p>
|
|
\begin{equation}
|
|
g = \frac{\sqrt{km} - c}{m}
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org10a6bb4" class="outline-4">
|
|
<h4 id="org10a6bb4"><span class="section-number-4">2.1.2</span> Matlab Example</h4>
|
|
<div class="outline-text-4" id="text-2-1-2">
|
|
<p>
|
|
Let define the system parameters.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = <span class="org-highlight-numbers-number">50</span>; <span class="org-comment">% [kg]</span>
|
|
k = <span class="org-highlight-numbers-number">1e6</span>; <span class="org-comment">% [N/m]</span>
|
|
c = <span class="org-highlight-numbers-number">1e3</span>; <span class="org-comment">% [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The state space model of the system is defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">A = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
B = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
C = <span class="org-rainbow-delimiters-depth-1">[</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-type">-</span>c <span class="org-type">-</span>k<span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
D = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
sys = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
sys.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'d'</span>, <span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.StateName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The controller \(K_\text{IFF}\) is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kiff = <span class="org-type">-</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>k<span class="org-type">*</span>m<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>c<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span>m<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span>s;
|
|
Kiff.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
Kiff.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the closed loop system is computed below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys_iff = feedback<span class="org-rainbow-delimiters-depth-1">(</span>sys, Kiff, <span class="org-string">'name'</span>, <span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgd023ec5" class="figure">
|
|
<p><img src="figs/iff_1dof_sensitivitiy.png" alt="iff_1dof_sensitivitiy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Sensitivity to disturbance when IFF is applied on the 1dof system (<a href="./figs/iff_1dof_sensitivitiy.png">png</a>, <a href="./figs/iff_1dof_sensitivitiy.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org896aad1" class="outline-3">
|
|
<h3 id="org896aad1"><span class="section-number-3">2.2</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
Let's load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure <a href="#org8eadf09">5</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org8eadf09" class="figure">
|
|
<p><img src="figs/iff_plant.png" alt="iff_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/iff_plant.png">png</a>, <a href="./figs/iff_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
The controller for each pair of actuator/sensor is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_iff = <span class="org-type">-</span><span class="org-highlight-numbers-number">1000</span><span class="org-type">/</span>s;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The corresponding loop gains are shown in figure <a href="#org5693406">6</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org5693406" class="figure">
|
|
<p><img src="figs/iff_open_loop.png" alt="iff_open_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/iff_open_loop.png">png</a>, <a href="./figs/iff_open_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2f66101" class="outline-3">
|
|
<h3 id="org2f66101"><span class="section-number-3">2.3</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
Let's initialize the system prior to identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
All the controllers are set to 0.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_iff = <span class="org-type">-</span>K_iff<span class="org-type">*</span>eye<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We identify the system dynamics now that the IFF controller is ON.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_iff = identifyPlant<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
As shown on figure <a href="#org153b020">7</a>:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>The top platform of the nano-hexapod how behaves as a "free-mass".</li>
|
|
<li>The transfer function from direct forces \(F_s\) to the relative displacement \(D\) is equivalent to the one of an isolated mass.</li>
|
|
<li>The transfer function from ground motion \(D_g\) to the relative displacement \(D\) tends to the transfer function from \(D_g\) to the displacement of the granite (the sample is being isolated thanks to IFF).
|
|
However, as the goal is to make the relative displacement \(D\) as small as possible (e.g. to make the sample motion follows the granite motion), this is not a good thing.</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org153b020" class="figure">
|
|
<p><img src="figs/sensitivity_dist_iff.png" alt="sensitivity_dist_iff.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Sensitivity to disturbance once the IFF controller is applied to the system (<a href="./figs/sensitivity_dist_iff.png">png</a>, <a href="./figs/sensitivity_dist_iff.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<div class="warning">
|
|
<p>
|
|
The order of the models are very high and thus the plots may be wrong.
|
|
For instance, the plots are not the same when using <code>minreal</code>.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
|
|
<div id="org186b636" class="figure">
|
|
<p><img src="figs/sensitivity_dist_stages_iff.png" alt="sensitivity_dist_stages_iff.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Sensitivity to force disturbances in various stages when IFF is applied (<a href="./figs/sensitivity_dist_stages_iff.png">png</a>, <a href="./figs/sensitivity_dist_stages_iff.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0d47de1" class="outline-3">
|
|
<h3 id="org0d47de1"><span class="section-number-3">2.4</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
Now, look at the new damped plant to control.
|
|
</p>
|
|
|
|
<p>
|
|
It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure <a href="#org03e6de1">9</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org03e6de1" class="figure">
|
|
<p><img src="figs/plant_iff_damped.png" alt="plant_iff_damped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Damped Plant after IFF is applied (<a href="./figs/plant_iff_damped.png">png</a>, <a href="./figs/plant_iff_damped.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
However, it increases coupling at low frequency (figure <a href="#org67bf738">10</a>).
|
|
</p>
|
|
|
|
<div id="org67bf738" class="figure">
|
|
<p><img src="figs/plant_iff_coupling.png" alt="plant_iff_coupling.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Coupling induced by IFF (<a href="./figs/plant_iff_coupling.png">png</a>, <a href="./figs/plant_iff_coupling.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd43ddf5" class="outline-3">
|
|
<h3 id="orgd43ddf5"><span class="section-number-3">2.5</span> Save</h3>
|
|
<div class="outline-text-3" id="text-2-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org66221ae" class="outline-3">
|
|
<h3 id="org66221ae"><span class="section-number-3">2.6</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-6">
|
|
<div class="important">
|
|
<p>
|
|
Integral Force Feedback:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Robust (guaranteed stability)</li>
|
|
<li>Acceptable Damping</li>
|
|
<li>Increase the sensitivity to disturbances at low frequencies</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb11e313" class="outline-2">
|
|
<h2 id="orgb11e313"><span class="section-number-2">3</span> Relative Motion Control</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgd55cd62"></a>
|
|
</p>
|
|
<p>
|
|
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orga766aee" class="outline-3">
|
|
<h3 id="orga766aee"><span class="section-number-3">3.1</span> One degree-of-freedom example</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
<a id="orgb0a118a"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orga0bbe25" class="outline-4">
|
|
<h4 id="orga0bbe25"><span class="section-number-4">3.1.1</span> Equations</h4>
|
|
<div class="outline-text-4" id="text-3-1-1">
|
|
|
|
<div id="org06fc4d7" class="figure">
|
|
<p><img src="figs/rmc_1dof.png" alt="rmc_1dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Relative Motion Control applied to a 1dof system</p>
|
|
</div>
|
|
|
|
<p>
|
|
The dynamic of the system is:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
<p>
|
|
In terms of the stage deformation \(d = x - w\):
|
|
</p>
|
|
\begin{equation}
|
|
(ms^2 + cs + k) d = -ms^2 w + F_d + F
|
|
\end{equation}
|
|
<p>
|
|
The relative motion control law is:
|
|
</p>
|
|
\begin{equation}
|
|
K = -g s
|
|
\end{equation}
|
|
<p>
|
|
Thus, the applied force is:
|
|
</p>
|
|
\begin{equation}
|
|
F = -g s d
|
|
\end{equation}
|
|
<p>
|
|
And the new dynamics will be:
|
|
</p>
|
|
\begin{equation}
|
|
d = w \frac{-ms^2}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
|
|
\end{equation}
|
|
|
|
<p>
|
|
And thus damping is added.
|
|
</p>
|
|
|
|
<p>
|
|
If critical damping is wanted:
|
|
</p>
|
|
\begin{equation}
|
|
\xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
<p>
|
|
This corresponds to a gain:
|
|
</p>
|
|
\begin{equation}
|
|
g = \sqrt{km} - c
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org32b38d2" class="outline-4">
|
|
<h4 id="org32b38d2"><span class="section-number-4">3.1.2</span> Matlab Example</h4>
|
|
<div class="outline-text-4" id="text-3-1-2">
|
|
<p>
|
|
Let define the system parameters.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = <span class="org-highlight-numbers-number">50</span>; <span class="org-comment">% [kg]</span>
|
|
k = <span class="org-highlight-numbers-number">1e6</span>; <span class="org-comment">% [N/m]</span>
|
|
c = <span class="org-highlight-numbers-number">1e3</span>; <span class="org-comment">% [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The state space model of the system is defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">A = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
B = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
C = <span class="org-rainbow-delimiters-depth-1">[</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-type">-</span>c <span class="org-type">-</span>k<span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
D = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
sys = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
sys.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'d'</span>, <span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.StateName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The controller \(K_\text{RMC}\) is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Krmc = <span class="org-type">-</span><span class="org-rainbow-delimiters-depth-1">(</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>k<span class="org-type">*</span>m<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">-</span>c<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span>s;
|
|
Krmc.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
Krmc.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the closed loop system is computed below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys_rmc = feedback<span class="org-rainbow-delimiters-depth-1">(</span>sys, Krmc, <span class="org-string">'name'</span>, <span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org79a04c1" class="figure">
|
|
<p><img src="figs/rmc_1dof_sensitivitiy.png" alt="rmc_1dof_sensitivitiy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Sensitivity to disturbance when RMC is applied on the 1dof system (<a href="./figs/rmc_1dof_sensitivitiy.png">png</a>, <a href="./figs/rmc_1dof_sensitivitiy.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8dba3f6" class="outline-3">
|
|
<h3 id="org8dba3f6"><span class="section-number-3">3.2</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
Let's load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure <a href="#org6c3e08a">13</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org6c3e08a" class="figure">
|
|
<p><img src="figs/rmc_plant.png" alt="rmc_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/rmc_plant.png">png</a>, <a href="./figs/rmc_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
The Relative Motion Controller is defined below.
|
|
A Low pass Filter is added to make the controller transfer function proper.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_rmc = s<span class="org-type">*</span><span class="org-highlight-numbers-number">50000</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">10000</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained loop gains are shown in figure <a href="#org732033d">14</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org732033d" class="figure">
|
|
<p><img src="figs/rmc_open_loop.png" alt="rmc_open_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/rmc_open_loop.png">png</a>, <a href="./figs/rmc_open_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc61326f" class="outline-3">
|
|
<h3 id="orgc61326f"><span class="section-number-3">3.3</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
Let's initialize the system prior to identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And initialize the controllers.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_rmc = <span class="org-type">-</span>K_rmc<span class="org-type">*</span>eye<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span>zeros<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We identify the system dynamics now that the RMC controller is ON.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_rmc = identifyPlant<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
As shown in figure <a href="#org835d7a1">15</a>, RMC control succeed in lowering the sensitivity to disturbances near resonance of the system.
|
|
</p>
|
|
|
|
|
|
<div id="org835d7a1" class="figure">
|
|
<p><img src="figs/sensitivity_dist_rmc.png" alt="sensitivity_dist_rmc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Sensitivity to disturbance once the RMC controller is applied to the system (<a href="./figs/sensitivity_dist_rmc.png">png</a>, <a href="./figs/sensitivity_dist_rmc.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgdd34d9c" class="figure">
|
|
<p><img src="figs/sensitivity_dist_stages_rmc.png" alt="sensitivity_dist_stages_rmc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Sensitivity to force disturbances in various stages when RMC is applied (<a href="./figs/sensitivity_dist_stages_rmc.png">png</a>, <a href="./figs/sensitivity_dist_stages_rmc.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc7efe56" class="outline-3">
|
|
<h3 id="orgc7efe56"><span class="section-number-3">3.4</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
|
|
<div id="org484efda" class="figure">
|
|
<p><img src="figs/plant_rmc_damped.png" alt="plant_rmc_damped.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Damped Plant after RMC is applied (<a href="./figs/plant_rmc_damped.png">png</a>, <a href="./figs/plant_rmc_damped.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3981d3b" class="outline-3">
|
|
<h3 id="org3981d3b"><span class="section-number-3">3.5</span> Save</h3>
|
|
<div class="outline-text-3" id="text-3-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4ff6cc5" class="outline-3">
|
|
<h3 id="org4ff6cc5"><span class="section-number-3">3.6</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-3-6">
|
|
<div class="important">
|
|
<p>
|
|
Relative Motion Control:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li></li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org44b3ec9" class="outline-2">
|
|
<h2 id="org44b3ec9"><span class="section-number-2">4</span> Direct Velocity Feedback</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="orge9ebafd"></a>
|
|
</p>
|
|
<p>
|
|
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgdbca8bc" class="outline-3">
|
|
<h3 id="orgdbca8bc"><span class="section-number-3">4.1</span> One degree-of-freedom example</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
<a id="orge6c5743"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org88e6e40" class="outline-4">
|
|
<h4 id="org88e6e40"><span class="section-number-4">4.1.1</span> Equations</h4>
|
|
<div class="outline-text-4" id="text-4-1-1">
|
|
|
|
<div id="org1905873" class="figure">
|
|
<p><img src="figs/dvf_1dof.png" alt="dvf_1dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Direct Velocity Feedback applied to a 1dof system</p>
|
|
</div>
|
|
|
|
<p>
|
|
The dynamic of the system is:
|
|
</p>
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
<p>
|
|
In terms of the stage deformation \(d = x - w\):
|
|
</p>
|
|
\begin{equation}
|
|
(ms^2 + cs + k) d = -ms^2 w + F_d + F
|
|
\end{equation}
|
|
<p>
|
|
The direct velocity feedback law shown in figure <a href="#org1905873">18</a> is:
|
|
</p>
|
|
\begin{equation}
|
|
K = -g
|
|
\end{equation}
|
|
<p>
|
|
Thus, the applied force is:
|
|
</p>
|
|
\begin{equation}
|
|
F = -g \dot{x}
|
|
\end{equation}
|
|
<p>
|
|
And the new dynamics will be:
|
|
</p>
|
|
\begin{equation}
|
|
d = w \frac{-ms^2 - gs}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
|
|
\end{equation}
|
|
|
|
<p>
|
|
And thus damping is added.
|
|
</p>
|
|
|
|
<p>
|
|
If critical damping is wanted:
|
|
</p>
|
|
\begin{equation}
|
|
\xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
<p>
|
|
This corresponds to a gain:
|
|
</p>
|
|
\begin{equation}
|
|
g = \sqrt{km} - c
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd273177" class="outline-4">
|
|
<h4 id="orgd273177"><span class="section-number-4">4.1.2</span> Matlab Example</h4>
|
|
<div class="outline-text-4" id="text-4-1-2">
|
|
<p>
|
|
Let define the system parameters.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = <span class="org-highlight-numbers-number">50</span>; <span class="org-comment">% [kg]</span>
|
|
k = <span class="org-highlight-numbers-number">1e6</span>; <span class="org-comment">% [N/m]</span>
|
|
c = <span class="org-highlight-numbers-number">1e3</span>; <span class="org-comment">% [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The state space model of the system is defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">A = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
|
<span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
B = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>m <span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
C = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
D = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span>;
|
|
<span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
sys = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
sys.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
sys.StateName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Because we need \(\dot{x}\) for feedback, we compute it from the outputs
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_xdot = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>s;
|
|
<span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
G_xdot.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'ddot'</span>, <span class="org-string">'d'</span>, <span class="org-string">'wddot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
G_xdot.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'xdot'</span>, <span class="org-string">'d'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, the system is described by <code>sys</code> as defined below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys = series<span class="org-rainbow-delimiters-depth-1">(</span>sys, G_xdot, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">2</span> <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">2</span> <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The controller \(K_\text{DVF}\) is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kdvf = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>k<span class="org-type">*</span>m<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>c<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
Kdvf.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'xdot'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
Kdvf.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the closed loop system is computed below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sys_dvf = feedback<span class="org-rainbow-delimiters-depth-1">(</span>sys, Kdvf, <span class="org-string">'name'</span>, <span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained sensitivity to disturbances is shown in figure <a href="#org2c541ab">19</a>.
|
|
</p>
|
|
|
|
<div id="org2c541ab" class="figure">
|
|
<p><img src="figs/dvf_1dof_sensitivitiy.png" alt="dvf_1dof_sensitivitiy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Sensitivity to disturbance when DVF is applied on the 1dof system (<a href="./figs/dvf_1dof_sensitivitiy.png">png</a>, <a href="./figs/dvf_1dof_sensitivitiy.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbc6a859" class="outline-3">
|
|
<h3 id="orgbc6a859"><span class="section-number-3">4.2</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
Let's load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure <a href="#org6220251">20</a>).
|
|
</p>
|
|
|
|
<p>
|
|
The plant looks to complicated to be reasonably controlled.
|
|
</p>
|
|
|
|
|
|
<div id="org6220251" class="figure">
|
|
<p><img src="figs/dvf_plant.png" alt="dvf_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/dvf_plant.png">png</a>, <a href="./figs/dvf_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge803ae6" class="outline-3">
|
|
<h3 id="orge803ae6"><span class="section-number-3">4.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="important">
|
|
<p>
|
|
Direct Velocity Feedback:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Not usable</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1b033b6" class="outline-2">
|
|
<h2 id="org1b033b6"><span class="section-number-2">5</span> Comparison</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org79856f6"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org7da72db" class="outline-3">
|
|
<h3 id="org7da72db"><span class="section-number-3">5.1</span> Comparison</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_rmc'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org26ec68a" class="outline-2">
|
|
<h2 id="org26ec68a"><span class="section-number-2">6</span> Conclusion</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="orgd7d6c31"></a>
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2019-10-18 ven. 17:33</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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</div>
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</body>
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</html>
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