1535 lines
53 KiB
HTML
1535 lines
53 KiB
HTML
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</head>
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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 with an uni-axial model</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="#org7b64a90">1. Undamped System</a>
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<ul>
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<li><a href="#org7409841">1.1. Init</a></li>
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<li><a href="#org7514f31">1.2. Identification</a></li>
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<li><a href="#org30a5b37">1.3. Sensitivity to disturbances</a></li>
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<li><a href="#orgdda82c0">1.4. Undamped Plant</a></li>
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</ul>
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</li>
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<li><a href="#org5a3389e">2. Integral Force Feedback</a>
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<ul>
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<li><a href="#orgf3a80f4">2.1. One degree-of-freedom example</a>
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<ul>
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<li><a href="#orgfba4e95">2.1.1. Equations</a></li>
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<li><a href="#org8487491">2.1.2. Matlab Example</a></li>
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</ul>
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</li>
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<li><a href="#orgf884e9a">2.2. Control Design</a></li>
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<li><a href="#org55e3bff">2.3. Identification of the damped plant</a></li>
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<li><a href="#org340a3d7">2.4. Sensitivity to disturbances</a></li>
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<li><a href="#orgba948f4">2.5. Damped Plant</a></li>
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<li><a href="#org8b5a4f4">2.6. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgc4ca1b5">3. Relative Motion Control</a>
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<ul>
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<li><a href="#org3a4a38f">3.1. One degree-of-freedom example</a>
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<ul>
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<li><a href="#orgc75e93d">3.1.1. Equations</a></li>
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<li><a href="#org6772235">3.1.2. Matlab Example</a></li>
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</ul>
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</li>
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<li><a href="#orgf74b07b">3.2. Control Design</a></li>
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<li><a href="#org2b50744">3.3. Identification of the damped plant</a></li>
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<li><a href="#orgfe16b4e">3.4. Sensitivity to disturbances</a></li>
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<li><a href="#org1d28b24">3.5. Damped Plant</a></li>
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<li><a href="#org69267d0">3.6. Conclusion</a></li>
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</ul>
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</li>
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|
|
<li><a href="#org3cc03b0">4. Direct Velocity Feedback</a>
|
||
|
|
<ul>
|
||
|
|
<li><a href="#org8fee25f">4.1. One degree-of-freedom example</a>
|
||
|
|
<ul>
|
||
|
|
<li><a href="#orge4d9f41">4.1.1. Equations</a></li>
|
||
|
|
<li><a href="#org3305ce8">4.1.2. Matlab Example</a></li>
|
||
|
|
</ul>
|
||
|
|
</li>
|
||
|
|
<li><a href="#orgc2ae0be">4.2. Control Design</a></li>
|
||
|
|
<li><a href="#org6eda033">4.3. Identification of the damped plant</a></li>
|
||
|
|
<li><a href="#orgbc3b2d2">4.4. Sensitivity to disturbances</a></li>
|
||
|
|
<li><a href="#org69d8b1b">4.5. Damped Plant</a></li>
|
||
|
|
<li><a href="#org0fffc79">4.6. Conclusion</a></li>
|
||
|
|
</ul>
|
||
|
|
</li>
|
||
|
|
<li><a href="#org21441bc">5. Comparison</a>
|
||
|
|
<ul>
|
||
|
|
<li><a href="#orgbe907b4">5.1. Load the plants</a></li>
|
||
|
|
<li><a href="#orgde6308d">5.2. Sensitivity to Disturbance</a></li>
|
||
|
|
<li><a href="#orgf8558bc">5.3. Damped Plant</a></li>
|
||
|
|
</ul>
|
||
|
|
</li>
|
||
|
|
<li><a href="#org7146202">6. Conclusion</a></li>
|
||
|
|
</ul>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
First, in section <a href="#orgbf3f2ef">1</a>, we will looked at the undamped system.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Then, we will compare three active damping techniques:
|
||
|
|
</p>
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li>In section <a href="#org5797b2c">2</a>: the integral force feedback is used</li>
|
||
|
|
<li>In section <a href="#org8dc40dc">3</a>: the relative motion control is used</li>
|
||
|
|
<li>In section <a href="#orge3322d7">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>
|
||
|
|
<li>Motion errors of all the stages</li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
<div id="outline-container-org7b64a90" class="outline-2">
|
||
|
|
<h2 id="org7b64a90"><span class="section-number-2">1</span> Undamped System</h2>
|
||
|
|
<div class="outline-text-2" id="text-1">
|
||
|
|
<p>
|
||
|
|
<a id="orgbf3f2ef"></a>
|
||
|
|
</p>
|
||
|
|
<div class="note">
|
||
|
|
<p>
|
||
|
|
All the files (data and Matlab scripts) are accessible <a href="data/undamped_system.zip">here</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
<p>
|
||
|
|
We first look at the undamped system.
|
||
|
|
The performance of this undamped system will be compared with the damped system using various techniques.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org7409841" class="outline-3">
|
||
|
|
<h3 id="org7409841"><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">initializeReferences();
|
||
|
|
initializeGround();
|
||
|
|
initializeGranite();
|
||
|
|
initializeTy();
|
||
|
|
initializeRy();
|
||
|
|
initializeRz();
|
||
|
|
initializeMicroHexapod();
|
||
|
|
initializeAxisc();
|
||
|
|
initializeMirror();
|
||
|
|
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||
|
|
initializeSample(<span class="org-string">'mass'</span>, 50);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
All the controllers are set to 0.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">K = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_iff = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_rmc = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_dvf = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org7514f31" class="outline-3">
|
||
|
|
<h3 id="org7514f31"><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();
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And we save it for further analysis.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping_uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org30a5b37" class="outline-3">
|
||
|
|
<h3 id="org30a5b37"><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="#orgcf7fa15">1</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgcf7fa15" 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 id="org5f406f3" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_dist_stages.png" alt="sensitivity_dist_stages.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 2: </span>Sensitivity to force disturbances in various stages (<a href="./figs/sensitivity_dist_stages.png">png</a>, <a href="./figs/sensitivity_dist_stages.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgdda82c0" class="outline-3">
|
||
|
|
<h3 id="orgdda82c0"><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="#orgcf7fa15">1</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgae47083" class="figure">
|
||
|
|
<p><img src="figs/plant_undamped.png" alt="plant_undamped.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 3: </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>
|
||
|
|
|
||
|
|
<div id="outline-container-org5a3389e" class="outline-2">
|
||
|
|
<h2 id="org5a3389e"><span class="section-number-2">2</span> Integral Force Feedback</h2>
|
||
|
|
<div class="outline-text-2" id="text-2">
|
||
|
|
<p>
|
||
|
|
<a id="org5797b2c"></a>
|
||
|
|
</p>
|
||
|
|
<div class="note">
|
||
|
|
<p>
|
||
|
|
All the files (data and Matlab scripts) are accessible <a href="data/iff.zip">here</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
<p>
|
||
|
|
Integral Force Feedback is applied.
|
||
|
|
In section <a href="#org7f37ded">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-orgf3a80f4" class="outline-3">
|
||
|
|
<h3 id="orgf3a80f4"><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="org7f37ded"></a>
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-orgfba4e95" class="outline-4">
|
||
|
|
<h4 id="orgfba4e95"><span class="section-number-4">2.1.1</span> Equations</h4>
|
||
|
|
<div class="outline-text-4" id="text-2-1-1">
|
||
|
|
|
||
|
|
<div id="org1acdc30" class="figure">
|
||
|
|
<p><img src="figs/iff_1dof.png" alt="iff_1dof.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 4: </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-org8487491" class="outline-4">
|
||
|
|
<h4 id="org8487491"><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 = 50; <span class="org-comment">% [kg]</span>
|
||
|
|
k = 1e6; <span class="org-comment">% [N/m]</span>
|
||
|
|
c = 1e3; <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-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
||
|
|
1 0];
|
||
|
|
|
||
|
|
B = [1<span class="org-type">/</span>m 1<span class="org-type">/</span>m <span class="org-type">-</span>1;
|
||
|
|
0 0 0];
|
||
|
|
|
||
|
|
C = [ 0 1;
|
||
|
|
<span class="org-type">-</span>c <span class="org-type">-</span>k];
|
||
|
|
|
||
|
|
D = [0 0 0;
|
||
|
|
1 0 0];
|
||
|
|
|
||
|
|
sys = ss(A, B, C, D);
|
||
|
|
sys.InputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span>};
|
||
|
|
sys.OutputName = {<span class="org-string">'d'</span>, <span class="org-string">'Fm'</span>};
|
||
|
|
sys.StateName = {<span class="org-string">'ddot'</span>, <span class="org-string">'d'</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>((sqrt(k<span class="org-type">*</span>m)<span class="org-type">-</span>c)<span class="org-type">/</span>m)<span class="org-type">/</span>s;
|
||
|
|
Kiff.InputName = {<span class="org-string">'Fm'</span>};
|
||
|
|
Kiff.OutputName = {<span class="org-string">'F'</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(sys, Kiff, <span class="org-string">'name'</span>, <span class="org-type">+</span>1);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org92c6747" class="figure">
|
||
|
|
<p><img src="figs/iff_1dof_sensitivitiy.png" alt="iff_1dof_sensitivitiy.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 5: </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-orgf884e9a" class="outline-3">
|
||
|
|
<h3 id="orgf884e9a"><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-string">'./active_damping_uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</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="#org644e078">6</a>).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org644e078" class="figure">
|
||
|
|
<p><img src="figs/iff_plant.png" alt="iff_plant.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 6: </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>1000<span class="org-type">/</span>s;
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The corresponding loop gains are shown in figure <a href="#org36e3a94">7</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org36e3a94" class="figure">
|
||
|
|
<p><img src="figs/iff_open_loop.png" alt="iff_open_loop.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 7: </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-org55e3bff" class="outline-3">
|
||
|
|
<h3 id="org55e3bff"><span class="section-number-3">2.3</span> Identification of the damped plant</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">initializeReferences();
|
||
|
|
initializeGround();
|
||
|
|
initializeGranite();
|
||
|
|
initializeTy();
|
||
|
|
initializeRy();
|
||
|
|
initializeRz();
|
||
|
|
initializeMicroHexapod();
|
||
|
|
initializeAxisc();
|
||
|
|
initializeMirror();
|
||
|
|
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||
|
|
initializeSample(<span class="org-string">'mass'</span>, 50);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
All the controllers are set to 0.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">K = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_iff = <span class="org-type">-</span>K_iff<span class="org-type">*</span>eye(6);
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_rmc = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_dvf = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</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();
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And we save the damped plant for further analysis
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping_uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org340a3d7" class="outline-3">
|
||
|
|
<h3 id="org340a3d7"><span class="section-number-3">2.4</span> Sensitivity to disturbances</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-4">
|
||
|
|
<p>
|
||
|
|
As shown on figure <a href="#org38217ee">8</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="org38217ee" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_dist_iff.png" alt="sensitivity_dist_iff.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 8: </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="orga9bd11d" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_dist_stages_iff.png" alt="sensitivity_dist_stages_iff.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 9: </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-orgba948f4" class="outline-3">
|
||
|
|
<h3 id="orgba948f4"><span class="section-number-3">2.5</span> Damped Plant</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-5">
|
||
|
|
<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="#orgd7333dd">10</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgd7333dd" class="figure">
|
||
|
|
<p><img src="figs/plant_iff_damped.png" alt="plant_iff_damped.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 10: </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="#org8017b2f">11</a>).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div id="org8017b2f" class="figure">
|
||
|
|
<p><img src="figs/plant_iff_coupling.png" alt="plant_iff_coupling.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 11: </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-org8b5a4f4" class="outline-3">
|
||
|
|
<h3 id="org8b5a4f4"><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-orgc4ca1b5" class="outline-2">
|
||
|
|
<h2 id="orgc4ca1b5"><span class="section-number-2">3</span> Relative Motion Control</h2>
|
||
|
|
<div class="outline-text-2" id="text-3">
|
||
|
|
<p>
|
||
|
|
<a id="org8dc40dc"></a>
|
||
|
|
</p>
|
||
|
|
<div class="note">
|
||
|
|
<p>
|
||
|
|
All the files (data and Matlab scripts) are accessible <a href="data/rmc.zip">here</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
<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-org3a4a38f" class="outline-3">
|
||
|
|
<h3 id="org3a4a38f"><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="org6f16e09"></a>
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-orgc75e93d" class="outline-4">
|
||
|
|
<h4 id="orgc75e93d"><span class="section-number-4">3.1.1</span> Equations</h4>
|
||
|
|
<div class="outline-text-4" id="text-3-1-1">
|
||
|
|
|
||
|
|
<div id="org64900ec" class="figure">
|
||
|
|
<p><img src="figs/rmc_1dof.png" alt="rmc_1dof.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 12: </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-org6772235" class="outline-4">
|
||
|
|
<h4 id="org6772235"><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 = 50; <span class="org-comment">% [kg]</span>
|
||
|
|
k = 1e6; <span class="org-comment">% [N/m]</span>
|
||
|
|
c = 1e3; <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-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
||
|
|
1 0];
|
||
|
|
|
||
|
|
B = [1<span class="org-type">/</span>m 1<span class="org-type">/</span>m <span class="org-type">-</span>1;
|
||
|
|
0 0 0];
|
||
|
|
|
||
|
|
C = [ 0 1;
|
||
|
|
<span class="org-type">-</span>c <span class="org-type">-</span>k];
|
||
|
|
|
||
|
|
D = [0 0 0;
|
||
|
|
1 0 0];
|
||
|
|
|
||
|
|
sys = ss(A, B, C, D);
|
||
|
|
sys.InputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span>};
|
||
|
|
sys.OutputName = {<span class="org-string">'d'</span>, <span class="org-string">'Fm'</span>};
|
||
|
|
sys.StateName = {<span class="org-string">'ddot'</span>, <span class="org-string">'d'</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>(sqrt(k<span class="org-type">*</span>m)<span class="org-type">-</span>c)<span class="org-type">*</span>s;
|
||
|
|
Krmc.InputName = {<span class="org-string">'d'</span>};
|
||
|
|
Krmc.OutputName = {<span class="org-string">'F'</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(sys, Krmc, <span class="org-string">'name'</span>, <span class="org-type">+</span>1);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgaedc5af" class="figure">
|
||
|
|
<p><img src="figs/rmc_1dof_sensitivitiy.png" alt="rmc_1dof_sensitivitiy.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 13: </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-orgf74b07b" class="outline-3">
|
||
|
|
<h3 id="orgf74b07b"><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-string">'./active_damping_uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</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="#orga93b3b1">14</a>).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orga93b3b1" class="figure">
|
||
|
|
<p><img src="figs/rmc_plant.png" alt="rmc_plant.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 14: </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>50000<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10000);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The obtained loop gains are shown in figure <a href="#orga5b8f12">15</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orga5b8f12" class="figure">
|
||
|
|
<p><img src="figs/rmc_open_loop.png" alt="rmc_open_loop.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 15: </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-org2b50744" class="outline-3">
|
||
|
|
<h3 id="org2b50744"><span class="section-number-3">3.3</span> Identification of the damped plant</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">initializeReferences();
|
||
|
|
initializeGround();
|
||
|
|
initializeGranite();
|
||
|
|
initializeTy();
|
||
|
|
initializeRy();
|
||
|
|
initializeRz();
|
||
|
|
initializeMicroHexapod();
|
||
|
|
initializeAxisc();
|
||
|
|
initializeMirror();
|
||
|
|
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||
|
|
initializeSample(<span class="org-string">'mass'</span>, 50);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And initialize the controllers.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">K = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_iff = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_rmc = <span class="org-type">-</span>K_rmc<span class="org-type">*</span>eye(6);
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_dvf = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</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();
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And we save the damped plant for further analysis.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping_uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgfe16b4e" class="outline-3">
|
||
|
|
<h3 id="orgfe16b4e"><span class="section-number-3">3.4</span> Sensitivity to disturbances</h3>
|
||
|
|
<div class="outline-text-3" id="text-3-4">
|
||
|
|
<p>
|
||
|
|
As shown in figure <a href="#org58aec78">16</a>, RMC control succeed in lowering the sensitivity to disturbances near resonance of the system.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org58aec78" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_dist_rmc.png" alt="sensitivity_dist_rmc.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 16: </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="org72cd54b" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_dist_stages_rmc.png" alt="sensitivity_dist_stages_rmc.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 17: </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-org1d28b24" class="outline-3">
|
||
|
|
<h3 id="org1d28b24"><span class="section-number-3">3.5</span> Damped Plant</h3>
|
||
|
|
<div class="outline-text-3" id="text-3-5">
|
||
|
|
|
||
|
|
<div id="org2267bd4" class="figure">
|
||
|
|
<p><img src="figs/plant_rmc_damped.png" alt="plant_rmc_damped.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 18: </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-org69267d0" class="outline-3">
|
||
|
|
<h3 id="org69267d0"><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-org3cc03b0" class="outline-2">
|
||
|
|
<h2 id="org3cc03b0"><span class="section-number-2">4</span> Direct Velocity Feedback</h2>
|
||
|
|
<div class="outline-text-2" id="text-4">
|
||
|
|
<p>
|
||
|
|
<a id="orge3322d7"></a>
|
||
|
|
</p>
|
||
|
|
<div class="note">
|
||
|
|
<p>
|
||
|
|
All the files (data and Matlab scripts) are accessible <a href="data/dvf.zip">here</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
<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-org8fee25f" class="outline-3">
|
||
|
|
<h3 id="org8fee25f"><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="org3a699cb"></a>
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-orge4d9f41" class="outline-4">
|
||
|
|
<h4 id="orge4d9f41"><span class="section-number-4">4.1.1</span> Equations</h4>
|
||
|
|
<div class="outline-text-4" id="text-4-1-1">
|
||
|
|
|
||
|
|
<div id="org93ae6e4" class="figure">
|
||
|
|
<p><img src="figs/dvf_1dof.png" alt="dvf_1dof.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 19: </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="#org93ae6e4">19</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-org3305ce8" class="outline-4">
|
||
|
|
<h4 id="org3305ce8"><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 = 50; <span class="org-comment">% [kg]</span>
|
||
|
|
k = 1e6; <span class="org-comment">% [N/m]</span>
|
||
|
|
c = 1e3; <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-type">-</span>c<span class="org-type">/</span>m <span class="org-type">-</span>k<span class="org-type">/</span>m;
|
||
|
|
1 0];
|
||
|
|
|
||
|
|
B = [1<span class="org-type">/</span>m 1<span class="org-type">/</span>m <span class="org-type">-</span>1;
|
||
|
|
0 0 0];
|
||
|
|
|
||
|
|
C = [1 0;
|
||
|
|
0 1;
|
||
|
|
0 0];
|
||
|
|
|
||
|
|
D = [0 0 0;
|
||
|
|
0 0 0;
|
||
|
|
0 0 1];
|
||
|
|
|
||
|
|
sys = ss(A, B, C, D);
|
||
|
|
sys.InputName = {<span class="org-string">'F'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'wddot'</span>};
|
||
|
|
sys.OutputName = {<span class="org-string">'ddot'</span>, <span class="org-string">'d'</span>, <span class="org-string">'wddot'</span>};
|
||
|
|
sys.StateName = {<span class="org-string">'ddot'</span>, <span class="org-string">'d'</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 = [1, 0, 1<span class="org-type">/</span>s;
|
||
|
|
0, 1, 0];
|
||
|
|
G_xdot.InputName = {<span class="org-string">'ddot'</span>, <span class="org-string">'d'</span>, <span class="org-string">'wddot'</span>};
|
||
|
|
G_xdot.OutputName = {<span class="org-string">'xdot'</span>, <span class="org-string">'d'</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(sys, G_xdot, [1 2 3], [1 2 3]);
|
||
|
|
</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-type">-</span>(sqrt(k<span class="org-type">*</span>m)<span class="org-type">-</span>c));
|
||
|
|
Kdvf.InputName = {<span class="org-string">'xdot'</span>};
|
||
|
|
Kdvf.OutputName = {<span class="org-string">'F'</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(sys, Kdvf, <span class="org-string">'name'</span>, <span class="org-type">+</span>1);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The obtained sensitivity to disturbances is shown in figure <a href="#org1c3277a">20</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div id="org1c3277a" class="figure">
|
||
|
|
<p><img src="figs/dvf_1dof_sensitivitiy.png" alt="dvf_1dof_sensitivitiy.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 20: </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-orgc2ae0be" class="outline-3">
|
||
|
|
<h3 id="orgc2ae0be"><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-string">'./active_damping_uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</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="#org0e1d7de">21</a>).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org0e1d7de" class="figure">
|
||
|
|
<p><img src="figs/dvf_plant.png" alt="dvf_plant.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 21: </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>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The controller is defined below and the obtained loop gain is shown in figure <a href="#org1e696e9">22</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">K_dvf = tf(3e4);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org1e696e9" class="figure">
|
||
|
|
<p><img src="figs/dvf_open_loop_gain.png" alt="dvf_open_loop_gain.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 22: </span>Loop Gain for DVF (<a href="./figs/dvf_open_loop_gain.png">png</a>, <a href="./figs/dvf_open_loop_gain.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org6eda033" class="outline-3">
|
||
|
|
<h3 id="org6eda033"><span class="section-number-3">4.3</span> Identification of the damped plant</h3>
|
||
|
|
<div class="outline-text-3" id="text-4-3">
|
||
|
|
<p>
|
||
|
|
Let’s initialize the system prior to identification.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">initializeReferences();
|
||
|
|
initializeGround();
|
||
|
|
initializeGranite();
|
||
|
|
initializeTy();
|
||
|
|
initializeRy();
|
||
|
|
initializeRz();
|
||
|
|
initializeMicroHexapod();
|
||
|
|
initializeAxisc();
|
||
|
|
initializeMirror();
|
||
|
|
initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||
|
|
initializeSample(<span class="org-string">'mass'</span>, 50);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And initialize the controllers.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">K = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_iff = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_rmc = tf(zeros(6));
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
K_dvf = <span class="org-type">-</span>K_dvf<span class="org-type">*</span>eye(6);
|
||
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</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_dvf = identifyPlant();
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And we save the damped plant for further analysis.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping_uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_dvf'</span>, <span class="org-string">'-append'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgbc3b2d2" class="outline-3">
|
||
|
|
<h3 id="orgbc3b2d2"><span class="section-number-3">4.4</span> Sensitivity to disturbances</h3>
|
||
|
|
<div class="outline-text-3" id="text-4-4">
|
||
|
|
|
||
|
|
<div id="org2558226" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_dist_dvf.png" alt="sensitivity_dist_dvf.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 23: </span>Sensitivity to disturbance once the DVF controller is applied to the system (<a href="./figs/sensitivity_dist_dvf.png">png</a>, <a href="./figs/sensitivity_dist_dvf.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org649f2c8" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_dist_stages_dvf.png" alt="sensitivity_dist_stages_dvf.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 24: </span>Sensitivity to force disturbances in various stages when DVF is applied (<a href="./figs/sensitivity_dist_stages_dvf.png">png</a>, <a href="./figs/sensitivity_dist_stages_dvf.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org69d8b1b" class="outline-3">
|
||
|
|
<h3 id="org69d8b1b"><span class="section-number-3">4.5</span> Damped Plant</h3>
|
||
|
|
<div class="outline-text-3" id="text-4-5">
|
||
|
|
|
||
|
|
<div id="org2fa3671" class="figure">
|
||
|
|
<p><img src="figs/plant_dvf_damped.png" alt="plant_dvf_damped.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 25: </span>Damped Plant after DVF is applied (<a href="./figs/plant_dvf_damped.png">png</a>, <a href="./figs/plant_dvf_damped.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org0fffc79" class="outline-3">
|
||
|
|
<h3 id="org0fffc79"><span class="section-number-3">4.6</span> Conclusion</h3>
|
||
|
|
<div class="outline-text-3" id="text-4-6">
|
||
|
|
<div class="important">
|
||
|
|
<p>
|
||
|
|
Direct Velocity Feedback:
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org21441bc" class="outline-2">
|
||
|
|
<h2 id="org21441bc"><span class="section-number-2">5</span> Comparison</h2>
|
||
|
|
<div class="outline-text-2" id="text-5">
|
||
|
|
<p>
|
||
|
|
<a id="org2dcea31"></a>
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-orgbe907b4" class="outline-3">
|
||
|
|
<h3 id="orgbe907b4"><span class="section-number-3">5.1</span> Load the plants</h3>
|
||
|
|
<div class="outline-text-3" id="text-5-1">
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping_uniaxial/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-string">'G_dvf'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgde6308d" class="outline-3">
|
||
|
|
<h3 id="orgde6308d"><span class="section-number-3">5.2</span> Sensitivity to Disturbance</h3>
|
||
|
|
<div class="outline-text-3" id="text-5-2">
|
||
|
|
|
||
|
|
<div id="orgfde858b" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_comp_ground_motion_z.png" alt="sensitivity_comp_ground_motion_z.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 26: </span>caption (<a href="./figs/sensitivity_comp_ground_motion_z.png">png</a>, <a href="./figs/sensitivity_comp_ground_motion_z.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org8aab19c" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_comp_direct_forces_z.png" alt="sensitivity_comp_direct_forces_z.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 27: </span>caption (<a href="./figs/sensitivity_comp_direct_forces_z.png">png</a>, <a href="./figs/sensitivity_comp_direct_forces_z.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgfffe60e" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_comp_spindle_z.png" alt="sensitivity_comp_spindle_z.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 28: </span>caption (<a href="./figs/sensitivity_comp_spindle_z.png">png</a>, <a href="./figs/sensitivity_comp_spindle_z.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orga94d0ef" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_comp_ty_z.png" alt="sensitivity_comp_ty_z.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 29: </span>caption (<a href="./figs/sensitivity_comp_ty_z.png">png</a>, <a href="./figs/sensitivity_comp_ty_z.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org94a317f" class="figure">
|
||
|
|
<p><img src="figs/sensitivity_comp_ty_x.png" alt="sensitivity_comp_ty_x.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 30: </span>caption (<a href="./figs/sensitivity_comp_ty_x.png">png</a>, <a href="./figs/sensitivity_comp_ty_x.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgf8558bc" class="outline-3">
|
||
|
|
<h3 id="orgf8558bc"><span class="section-number-3">5.3</span> Damped Plant</h3>
|
||
|
|
<div class="outline-text-3" id="text-5-3">
|
||
|
|
|
||
|
|
<div id="org043ecf3" class="figure">
|
||
|
|
<p><img src="figs/plant_comp_damping_z.png" alt="plant_comp_damping_z.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 31: </span>Plant for the \(z\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_z.png">png</a>, <a href="./figs/plant_comp_damping_z.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org20c1ba1" class="figure">
|
||
|
|
<p><img src="figs/plant_comp_damping_x.png" alt="plant_comp_damping_x.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 32: </span>Plant for the \(x\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_x.png">png</a>, <a href="./figs/plant_comp_damping_x.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org22f4195" class="figure">
|
||
|
|
<p><img src="figs/plant_comp_damping_coupling.png" alt="plant_comp_damping_coupling.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 33: </span>Comparison of one off-diagonal plant for different damping technique applied (<a href="./figs/plant_comp_damping_coupling.png">png</a>, <a href="./figs/plant_comp_damping_coupling.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org7146202" class="outline-2">
|
||
|
|
<h2 id="org7146202"><span class="section-number-2">6</span> Conclusion</h2>
|
||
|
|
<div class="outline-text-2" id="text-6">
|
||
|
|
<p>
|
||
|
|
<a id="org58549a4"></a>
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<div id="postamble" class="status">
|
||
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
||
|
|
<p class="date">Created: 2020-02-25 mar. 18:08</p>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|