1252 lines
49 KiB
HTML
1252 lines
49 KiB
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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">Control Requirements</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="#org4418235">1. Goal</a></li>
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<li><a href="#org0341df1">2. Simplify Model for the Nano-Hexapod</a>
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<ul>
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<li><a href="#org136c9af">2.1. Model of the nano-hexapod</a></li>
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<li><a href="#org2fbecfd">2.2. How to include Ground Motion in the model?</a></li>
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</ul>
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</li>
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<li><a href="#org8c1e462">3. Motion of the micro-station</a></li>
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<li><a href="#org6182074">4. Values and Plant</a>
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<ul>
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<li><a href="#org19b83b7">4.1. Definition of the values</a></li>
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</ul>
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</li>
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<li><a href="#org0e9811a">5. Control using \(d\)</a>
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<ul>
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<li><a href="#org6a29cc6">5.1. Control Architecture</a></li>
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<li><a href="#org5a120e1">5.2. Analytical Analysis</a></li>
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</ul>
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</li>
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<li><a href="#orga741e48">6. Control using \(F_m\)</a>
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<ul>
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<li><a href="#org02a7ab1">6.1. Control Architecture</a></li>
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<li><a href="#orgdd5134e">6.2. Pure Integrator</a></li>
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<li><a href="#org5011ab0">6.3. Low pass filter</a></li>
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</ul>
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</li>
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<li><a href="#org4fce174">7. Comparison</a></li>
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<li><a href="#org5e0585d">8. Control using \(x\)</a>
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<ul>
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<li><a href="#orgfab8395">8.1. Analytical analysis</a></li>
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<li><a href="#org625e3c2">8.2. Control implementation</a></li>
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<li><a href="#org8d34d7f">8.3. Results</a></li>
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</ul>
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</li>
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<li><a href="#orgd2547fd">9. Two degree of freedom control</a></li>
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<li><a href="#orgab31e9f">10. Soft nano-hexapod</a></li>
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<li><a href="#orgcb7520f">11. Compare Soft and Stiff nano-hexapods</a></li>
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<li><a href="#orgc0253c3">12. Estimate the level of vibration</a></li>
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<li><a href="#org764c4a9">13. Requirements on the norm of closed-loop transfer functions</a></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org4418235" class="outline-2">
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<h2 id="org4418235"><span class="section-number-2">1</span> Goal</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The goal here is to write clear specifications for the NASS.
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</p>
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<p>
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This can then be used for the control synthesis and for the design of the nano-hexapod.
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</p>
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<p>
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Ideal, specifications on the norm of closed loop transfer function should be written.
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</p>
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</div>
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</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org0341df1" class="outline-2">
|
||
|
|
<h2 id="org0341df1"><span class="section-number-2">2</span> Simplify Model for the Nano-Hexapod</h2>
|
||
|
|
<div class="outline-text-2" id="text-2">
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-org136c9af" class="outline-3">
|
||
|
|
<h3 id="org136c9af"><span class="section-number-3">2.1</span> Model of the nano-hexapod</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-1">
|
||
|
|
<p>
|
||
|
|
Let’s consider the simple mechanical system in Figure <a href="#orgfa3391a">1</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgfa3391a" class="figure">
|
||
|
|
<p><img src="figs/nass_simple_model.png" alt="nass_simple_model.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 1: </span>Simplified mechanical system for the nano-hexapod</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The signals are described in table <a href="#orgd89e830">1</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<table id="orgd89e830" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
|
<caption class="t-above"><span class="table-number">Table 1:</span> Signals definition for the generalized plant</caption>
|
||
|
|
|
||
|
|
<colgroup>
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
</colgroup>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th scope="col" class="org-left"> </th>
|
||
|
|
<th scope="col" class="org-left"><b>Symbol</b></th>
|
||
|
|
<th scope="col" class="org-left"><b>Meaning</b></th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><b>Exogenous Inputs</b></td>
|
||
|
|
<td class="org-left">\(x_\mu\)</td>
|
||
|
|
<td class="org-left">Motion of the $ν$-hexapod’s base</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"> </td>
|
||
|
|
<td class="org-left">\(F_d\)</td>
|
||
|
|
<td class="org-left">External Forces applied to the Payload</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"> </td>
|
||
|
|
<td class="org-left">\(r\)</td>
|
||
|
|
<td class="org-left">Reference signal for tracking</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><b>Exogenous Outputs</b></td>
|
||
|
|
<td class="org-left">\(x\)</td>
|
||
|
|
<td class="org-left">Absolute Motion of the Payload</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><b>Sensed Outputs</b></td>
|
||
|
|
<td class="org-left">\(F_m\)</td>
|
||
|
|
<td class="org-left">Force Sensors in each leg</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"> </td>
|
||
|
|
<td class="org-left">\(d\)</td>
|
||
|
|
<td class="org-left">Measured displacement of each leg</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"> </td>
|
||
|
|
<td class="org-left">\(x\)</td>
|
||
|
|
<td class="org-left">Absolute Motion of the Payload</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><b>Control Signals</b></td>
|
||
|
|
<td class="org-left">\(F\)</td>
|
||
|
|
<td class="org-left">Actuator Inputs</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
For the nano-hexapod alone, we have the following equations:
|
||
|
|
\[ \begin{align*}
|
||
|
|
x &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \\
|
||
|
|
F_m &= \frac{ms^2}{ms^2 + k} F - \frac{k}{ms^2 + k} F_d + \frac{k m s^2}{ms^2 + k} x_\mu \\
|
||
|
|
d &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d - \frac{ms^2}{ms^2 + k} x_\mu
|
||
|
|
\end{align*} \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
We can write the equations function of \(\omega_\nu = \sqrt{\frac{k}{m}}\):
|
||
|
|
\[ \begin{align*}
|
||
|
|
x &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
|
||
|
|
F_m &= \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} F - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{k \frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
|
||
|
|
d &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d - \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu
|
||
|
|
\end{align*} \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
<b>Assumptions</b>:
|
||
|
|
</p>
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li>the forces applied by the nano-hexapod have no influence on the micro-station, specifically on the displacement of the top platform of the micro-hexapod.</li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
This means that the nano-hexapod can be considered separately from the micro-station and that the motion \(x_\mu\) is imposed and considered as an external input.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The nano-hexapod can thus be represented as in Figure <a href="#orgb2d1168">2</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgb2d1168" class="figure">
|
||
|
|
<p><img src="figs/nano_station_inputs_outputs.png" alt="nano_station_inputs_outputs.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 2: </span>Block representation of the nano-hexapod</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org2fbecfd" class="outline-3">
|
||
|
|
<h3 id="org2fbecfd"><span class="section-number-3">2.2</span> How to include Ground Motion in the model?</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-2">
|
||
|
|
<p>
|
||
|
|
What we measure is not the absolute motion \(x\), but the relative motion \(x - w\) where \(w\) is the motion of the granite.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Also, \(w\) induces some motion \(x_\mu\) through the transmissibility of the micro-station.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div class="figure">
|
||
|
|
<p><img src="figs/nano_station_inputs_outputs_ground_motion.png" alt="nano_station_inputs_outputs_ground_motion.png" />
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org8c1e462" class="outline-2">
|
||
|
|
<h2 id="org8c1e462"><span class="section-number-2">3</span> Motion of the micro-station</h2>
|
||
|
|
<div class="outline-text-2" id="text-3">
|
||
|
|
<p>
|
||
|
|
As explained, we consider \(x_\mu\) as an external input (\(F\) has no influence on \(x_\mu\)).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\(x_\mu\) is the motion of the micro-station’s top platform due to the motion of each stage of the micro-station.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
We consider that \(x_\mu\) has the following form:
|
||
|
|
\[ x_\mu = T_\mu r + d_\mu \]
|
||
|
|
where:
|
||
|
|
</p>
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li>\(T_\mu r\) corresponds to the response of the stages due to the reference \(r\)</li>
|
||
|
|
<li>\(d_\mu\) is the motion of the hexapod due to all the vibrations of the stages</li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\(T_\mu\) can be considered to be a low pass filter with a bandwidth corresponding approximatively to the bandwidth of the micro-station’s stages.
|
||
|
|
To simplify, we can consider \(T_\mu\) to be a first order low pass filter:
|
||
|
|
\[ T_\mu = \frac{1}{1 + s/\omega_\mu} \]
|
||
|
|
where \(\omega_\mu\) corresponds to the tracking speed of the micro-station.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
What is important to note is that while \(x_\mu\) is viewed as a perturbation from the nano-hexapod point of view, \(x_\mu\) <b>does</b> depend on the reference signal \(r\).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Also, here, we suppose that the granite is not moving.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
If we now include the motion of the granite \(w\), we obtain the block diagram shown in Figure <a href="#org974c98f">4</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org974c98f" class="figure">
|
||
|
|
<p><img src="figs/nano_station_ground_motion.png" alt="nano_station_ground_motion.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 4: </span>Ground Motion \(w\) included</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\(T_w\) is the mechanical transmissibility of the micro-station.
|
||
|
|
We can approximate this transfer function by a second order low pass filter:
|
||
|
|
\[ T_w = \frac{1}{1 + 2 \xi s/\omega_0 + s^2/\omega_0^2} \]
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org6182074" class="outline-2">
|
||
|
|
<h2 id="org6182074"><span class="section-number-2">4</span> Values and Plant</h2>
|
||
|
|
<div class="outline-text-2" id="text-4">
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-org19b83b7" class="outline-3">
|
||
|
|
<h3 id="org19b83b7"><span class="section-number-3">4.1</span> Definition of the values</h3>
|
||
|
|
<div class="outline-text-3" id="text-4-1">
|
||
|
|
<p>
|
||
|
|
Let’s define the mass and stiffness of the nano-hexapod.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">m = 50; <span class="org-comment">% [kg]</span>
|
||
|
|
k = 1e7; <span class="org-comment">% [N/m]</span>
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Let’s define the Plant as shown in Figure <a href="#orgb2d1168">2</a>:
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Gn = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> k)<span class="org-type">*</span>[<span class="org-type">-</span>k, k<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, m<span class="org-type">*</span>s<span class="org-type">^</span>2; 1, <span class="org-type">-</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, 1; 1, k, 1];
|
||
|
|
Gn.InputName = {<span class="org-string">'Fd'</span>, <span class="org-string">'xmu'</span>, <span class="org-string">'F'</span>};
|
||
|
|
Gn.OutputName = {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'x'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Now, define the transmissibility transfer function \(T_\mu\) corresponding to the micro-station motion.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">wmu = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% [rad/s]</span>
|
||
|
|
|
||
|
|
Tmu = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wmu);
|
||
|
|
Tmu.InputName = {<span class="org-string">'r1'</span>};
|
||
|
|
Tmu.OutputName = {<span class="org-string">'ymu'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>40;
|
||
|
|
xi = 0.5;
|
||
|
|
Tw = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2);
|
||
|
|
Tw.InputName = {<span class="org-string">'w1'</span>};
|
||
|
|
Tw.OutputName = {<span class="org-string">'dw'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
We add the fact that \(x_\mu = d_\mu + T_\mu r + T_w w\):
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Wsplit = [tf(1); tf(1)];
|
||
|
|
Wsplit.InputName = {<span class="org-string">'w'</span>};
|
||
|
|
Wsplit.OutputName = {<span class="org-string">'w1'</span>, <span class="org-string">'w2'</span>};
|
||
|
|
|
||
|
|
S = sumblk(<span class="org-string">'xmu = ymu + dmu + dw'</span>);
|
||
|
|
|
||
|
|
Sw = sumblk(<span class="org-string">'y = x - w2'</span>);
|
||
|
|
|
||
|
|
Gpz = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {<span class="org-string">'Fd'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'r1'</span>, <span class="org-string">'F'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'y'</span>});
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org0e9811a" class="outline-2">
|
||
|
|
<h2 id="org0e9811a"><span class="section-number-2">5</span> Control using \(d\)</h2>
|
||
|
|
<div class="outline-text-2" id="text-5">
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-org6a29cc6" class="outline-3">
|
||
|
|
<h3 id="org6a29cc6"><span class="section-number-3">5.1</span> Control Architecture</h3>
|
||
|
|
<div class="outline-text-3" id="text-5-1">
|
||
|
|
<p>
|
||
|
|
Let’s consider a feedback loop using \(d\) as shown in Figure <a href="#orgb50386a">5</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgb50386a" class="figure">
|
||
|
|
<p><img src="figs/nano_station_control_d.png" alt="nano_station_control_d.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 5: </span>Feedback diagram using \(d\)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org5a120e1" class="outline-3">
|
||
|
|
<h3 id="org5a120e1"><span class="section-number-3">5.2</span> Analytical Analysis</h3>
|
||
|
|
<div class="outline-text-3" id="text-5-2">
|
||
|
|
<p>
|
||
|
|
Let’s apply a direct velocity feedback by deriving \(d\):
|
||
|
|
\[ F = F^\prime - g s d \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Thus:
|
||
|
|
\[ d = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d - \frac{ms^2}{ms^2 + gs + k} x_\mu \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\[ F = \frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
and
|
||
|
|
\[ x = \frac{1}{ms^2 + k} (\frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu) + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\[ x = \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F^\prime + \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F_d + \frac{mgs^3 + k(ms^2 + gs + k)}{(ms^2 + k) (ms^2 + gs + k)} x_\mu \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And we finally obtain:
|
||
|
|
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} x_\mu \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">K_dvf = 2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m)<span class="org-type">*</span>s;
|
||
|
|
K_dvf.InputName = {<span class="org-string">'d'</span>};
|
||
|
|
K_dvf.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
|
||
|
|
Gpz_dvf = feedback(Gpz, K_dvf, <span class="org-string">'name'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Now let’s consider that \(x_\mu = d_\mu + T_\mu r\)
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + T_\mu \frac{gs + k}{ms^2 + gs + k} r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And \(\epsilon = r - x\):
|
||
|
|
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 + gs + k - T_\mu (gs + k)}{ms^2 + gs + k} r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div class="important">
|
||
|
|
<p>
|
||
|
|
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 - S_\mu(gs + k)}{ms^2 + gs + k} r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orga741e48" class="outline-2">
|
||
|
|
<h2 id="orga741e48"><span class="section-number-2">6</span> Control using \(F_m\)</h2>
|
||
|
|
<div class="outline-text-2" id="text-6">
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-org02a7ab1" class="outline-3">
|
||
|
|
<h3 id="org02a7ab1"><span class="section-number-3">6.1</span> Control Architecture</h3>
|
||
|
|
<div class="outline-text-3" id="text-6-1">
|
||
|
|
<p>
|
||
|
|
Let’s consider a feedback loop using \(d\) as shown in Figure <a href="#orgb50386a">5</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org5012ef2" class="figure">
|
||
|
|
<p><img src="figs/nano_station_control_Fm.png" alt="nano_station_control_Fm.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 6: </span>Feedback diagram using \(F_m\)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgdd5134e" class="outline-3">
|
||
|
|
<h3 id="orgdd5134e"><span class="section-number-3">6.2</span> Pure Integrator</h3>
|
||
|
|
<div class="outline-text-3" id="text-6-2">
|
||
|
|
<p>
|
||
|
|
Let’s apply integral force feedback by integration \(F_m\):
|
||
|
|
\[ F = F^\prime - \frac{g}{s} F_m \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And we finally obtain:
|
||
|
|
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} x_\mu \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">K_iff = 2<span class="org-type">*</span>sqrt(k<span class="org-type">/</span>m)<span class="org-type">/</span>s;
|
||
|
|
K_iff.InputName = {<span class="org-string">'Fm'</span>};
|
||
|
|
K_iff.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
|
||
|
|
Gpz_iff = feedback(Gpz, K_iff, <span class="org-string">'name'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Now let’s consider that \(x_\mu = d_\mu + T_\mu r\)
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{T_\mu k}{ms^2 + mgs + k} r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And \(\epsilon = r - x\):
|
||
|
|
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + k - T_\mu k}{ms^2 + mgs + k} r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div class="important">
|
||
|
|
<p>
|
||
|
|
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + S_\mu k}{ms^2 + mgs + k} r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org5011ab0" class="outline-3">
|
||
|
|
<h3 id="org5011ab0"><span class="section-number-3">6.3</span> Low pass filter</h3>
|
||
|
|
<div class="outline-text-3" id="text-6-3">
|
||
|
|
<p>
|
||
|
|
Instead of a pure integrator, let’s use a low pass filter with a cut-off frequency above the bandwidth of the micro-station \(\omega_mu\)
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab"><span class="org-comment">% K_iff = (2*sqrt(k/m)/(2*wmu))*(1/(1 + s/(2*wmu)));</span>
|
||
|
|
<span class="org-comment">% K_iff.InputName = {'Fm'};</span>
|
||
|
|
<span class="org-comment">% K_iff.OutputName = {'F'};</span>
|
||
|
|
|
||
|
|
<span class="org-comment">% Gpz_iff = feedback(Gpz, K_iff, 'name');</span>
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org4fce174" class="outline-2">
|
||
|
|
<h2 id="org4fce174"><span class="section-number-2">7</span> Comparison</h2>
|
||
|
|
<div class="outline-text-2" id="text-7">
|
||
|
|
|
||
|
|
<div id="orgc10daac" class="figure">
|
||
|
|
<p><img src="figs/comp_iff_dvf_simplified.png" alt="comp_iff_dvf_simplified.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 7: </span>Obtained transfer functions for DVF and IFF (<a href="./figs/comp_iff_dvf_simplified.png">png</a>, <a href="./figs/comp_iff_dvf_simplified.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
|
|
||
|
|
|
||
|
|
<colgroup>
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
</colgroup>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th scope="col" class="org-left"> </th>
|
||
|
|
<th scope="col" class="org-left">\(d_\mu\)</th>
|
||
|
|
<th scope="col" class="org-left">\(F_d\)</th>
|
||
|
|
<th scope="col" class="org-left">\(w\)</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">IFF</td>
|
||
|
|
<td class="org-left">Better filtering of the vibrations</td>
|
||
|
|
<td class="org-left">More sensitive to External forces</td>
|
||
|
|
<td class="org-left"> </td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">DVF</td>
|
||
|
|
<td class="org-left">inverse</td>
|
||
|
|
<td class="org-left">inverse</td>
|
||
|
|
<td class="org-left">Little bit better at low frequencies</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org5e0585d" class="outline-2">
|
||
|
|
<h2 id="org5e0585d"><span class="section-number-2">8</span> Control using \(x\)</h2>
|
||
|
|
<div class="outline-text-2" id="text-8">
|
||
|
|
</div>
|
||
|
|
<div id="outline-container-orgfab8395" class="outline-3">
|
||
|
|
<h3 id="orgfab8395"><span class="section-number-3">8.1</span> Analytical analysis</h3>
|
||
|
|
<div class="outline-text-3" id="text-8-1">
|
||
|
|
<p>
|
||
|
|
Let’s first consider that only the output \(x\) is used for feedback (Figure <a href="#orgd366408">8</a>)
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="orgd366408" class="figure">
|
||
|
|
<p><img src="figs/nano_station_control_x.png" alt="nano_station_control_x.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 8: </span>Feedback diagram using \(x\)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
We then have:
|
||
|
|
\[ \epsilon &= r - G_{\frac{x}{F}} K \epsilon - G_{\frac{x}{F_d}} F_d - G_{\frac{x}{x_\mu}} d_\mu - G_{\frac{x}{x_\mu}} T_\mu r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
And then:
|
||
|
|
</p>
|
||
|
|
<div class="important">
|
||
|
|
<p>
|
||
|
|
\[ \epsilon = \frac{-G_{\frac{x}{F_d}}}{1 + G_{\frac{x}{F}}K} F_d + \frac{-G_{\frac{x}{x_\mu}}}{1 + G_{\frac{x}{F}}K} d_\mu + \frac{1 - G_{\frac{x}{x_\mu}} T_\mu}{1 + G_{\frac{x}{F}}K} r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
With \(S = \frac{1}{1 + G_{\frac{x}{F}} K}\), we have:
|
||
|
|
\[ \epsilon = - S G_{\frac{x}{F_d}} F_d - S G_{\frac{x}{x_\mu}} d_\mu + S (1 - G_{\frac{x}{x_\mu}} T_\mu) r \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
We have 3 terms that we would like to have small by design:
|
||
|
|
</p>
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li>\(G_{\frac{x}{F_d}} = \frac{1}{ms^2 + k}\): thus \(k\) and \(m\) should be high to lower the effect of direct forces \(F_d\)</li>
|
||
|
|
<li>\(G_{\frac{x}{x_\mu}} = \frac{k}{ms^2 + k} = \frac{1}{1 + \frac{s^2}{\omega_\nu^2}}\): \(\omega_\nu\) should be small enough such that it filters out the vibrations of the micro-station</li>
|
||
|
|
<li>\(1 - G_{\frac{x}{x_\mu}} T_\mu\)</li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
\[ 1 - G_{\frac{x}{x_\mu}} T_\mu = 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} T_\mu \]
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
We can approximate \(T_\mu \approx \frac{1}{1 + \frac{s}{\omega_\mu}}\) to have:
|
||
|
|
</p>
|
||
|
|
\begin{align*}
|
||
|
|
1 - G_{\frac{x}{x_\mu}} T_\mu &= 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} \frac{1}{1 + \frac{s}{\omega_\mu}} \\
|
||
|
|
&\approx \frac{\frac{s}{\omega_\mu}}{1 + \frac{s}{\omega_\mu}} = S_\mu \text{ if } \omega_\nu > \omega_\mu \\
|
||
|
|
&\approx \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} = \text{ if } \omega_\nu < \omega_\mu
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
<p>
|
||
|
|
In our case, we have \(\omega_\nu > \omega_\mu\) and thus we cannot lower this term.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Some implications on the design are summarized on table <a href="#orga5207fc">2</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<table id="orga5207fc" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
|
<caption class="t-above"><span class="table-number">Table 2:</span> Design recommendation</caption>
|
||
|
|
|
||
|
|
<colgroup>
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
</colgroup>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th scope="col" class="org-left">Exogenous Outputs</th>
|
||
|
|
<th scope="col" class="org-left">Design recommendation</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">\(F_d\)</td>
|
||
|
|
<td class="org-left">high \(k\), high \(m\)</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">\(d_\mu\)</td>
|
||
|
|
<td class="org-left">low \(k\), high \(m\)</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">\(r\)</td>
|
||
|
|
<td class="org-left">no influence if \(\omega_\nu > \omega_\mu\)</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org625e3c2" class="outline-3">
|
||
|
|
<h3 id="org625e3c2"><span class="section-number-3">8.2</span> Control implementation</h3>
|
||
|
|
<div class="outline-text-3" id="text-8-2">
|
||
|
|
<p>
|
||
|
|
Controller for the damped plant using DVF.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">wb = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% control bandwidth [rad/s]</span>
|
||
|
|
|
||
|
|
<span class="org-comment">% Lead</span>
|
||
|
|
h = 2.0;
|
||
|
|
wz = wb<span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
|
||
|
|
wp = wb<span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
|
||
|
|
|
||
|
|
H = 1<span class="org-type">/</span>h<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wz)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wp);
|
||
|
|
|
||
|
|
<span class="org-comment">% Integrator until 10Hz</span>
|
||
|
|
Hi = (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>10)<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>10);
|
||
|
|
|
||
|
|
K = Hi<span class="org-type">*</span>H<span class="org-type">*</span>(1<span class="org-type">/</span>s);
|
||
|
|
|
||
|
|
Kpz_dvf = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
|
||
|
|
Kpz_dvf.InputName = {<span class="org-string">'e'</span>};
|
||
|
|
Kpz_dvf.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Controller for the damped plant using IFF.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">wb = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% control bandwidth [rad/s]</span>
|
||
|
|
|
||
|
|
<span class="org-comment">% Lead</span>
|
||
|
|
h = 2.0;
|
||
|
|
wz = wb<span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
|
||
|
|
wp = wb<span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
|
||
|
|
|
||
|
|
H = 1<span class="org-type">/</span>h<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wz)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wp);
|
||
|
|
|
||
|
|
<span class="org-comment">% Integrator until 10Hz</span>
|
||
|
|
Hi = (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>10)<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>10);
|
||
|
|
|
||
|
|
K = Hi<span class="org-type">*</span>H<span class="org-type">*</span>(1<span class="org-type">/</span>s);
|
||
|
|
|
||
|
|
Kpz_iff = K<span class="org-type">/</span>abs(freqresp(K<span class="org-type">*</span>Gpz_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
|
||
|
|
Kpz_iff.InputName = {<span class="org-string">'e'</span>};
|
||
|
|
Kpz_iff.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Loop gain
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div id="org0d0fb80" class="figure">
|
||
|
|
<p><img src="figs/simple_loop_gain_pz.png" alt="simple_loop_gain_pz.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 9: </span>Loop Gain (<a href="./figs/simple_loop_gain_pz.png">png</a>, <a href="./figs/simple_loop_gain_pz.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Let’s connect all the systems as shown in Figure <a href="#orgd366408">8</a>.
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>);
|
||
|
|
|
||
|
|
R = [tf(1); tf(1)];
|
||
|
|
R.InputName = {<span class="org-string">'r'</span>};
|
||
|
|
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
|
||
|
|
|
||
|
|
Gpz_fb_dvf = connect(Gpz_dvf, Kpz_dvf, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>});
|
||
|
|
Gpz_fb_iff = connect(Gpz_iff, Kpz_iff, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>});
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org8d34d7f" class="outline-3">
|
||
|
|
<h3 id="org8d34d7f"><span class="section-number-3">8.3</span> Results</h3>
|
||
|
|
<div class="outline-text-3" id="text-8-3">
|
||
|
|
|
||
|
|
<div id="org2b4e783" class="figure">
|
||
|
|
<p><img src="figs/simple_hac_lac_results.png" alt="simple_hac_lac_results.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 10: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results.png">png</a>, <a href="./figs/simple_hac_lac_results.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgd2547fd" class="outline-2">
|
||
|
|
<h2 id="orgd2547fd"><span class="section-number-2">9</span> Two degree of freedom control</h2>
|
||
|
|
<div class="outline-text-2" id="text-9">
|
||
|
|
<p>
|
||
|
|
Let’s try to implement the control architecture shown in Figure <a href="#org7ce0167">11</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The pre-filter \(K_r\) is added in order to improve the reference tracking performances.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org7ce0167" class="figure">
|
||
|
|
<p><img src="figs/nano_station_control_2dof_x.png" alt="nano_station_control_2dof_x.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 11: </span>Two degrees of freedom feedback control</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
In order to design the pre-filter \(K_r\), the dynamics of the system should be known quite precisely (Dynamics of the nano-hexapod + \(T_\mu\)).
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Krpz = inv(Gpz_fb(<span class="org-string">'y'</span>, <span class="org-string">'r'</span>));
|
||
|
|
|
||
|
|
Krpz.InputName = {<span class="org-string">'r2'</span>};
|
||
|
|
Krpz.OutputName = {<span class="org-string">'r3'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r3 - y'</span>);
|
||
|
|
|
||
|
|
R = [tf(1); tf(1)];
|
||
|
|
R.InputName = {<span class="org-string">'r'</span>};
|
||
|
|
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
|
||
|
|
|
||
|
|
Gpz_2dof = connect(Gpz_dvf, Krpz, Kpz, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>});
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgab31e9f" class="outline-2">
|
||
|
|
<h2 id="orgab31e9f"><span class="section-number-2">10</span> Soft nano-hexapod</h2>
|
||
|
|
<div class="outline-text-2" id="text-10">
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">m = 50; <span class="org-comment">% [kg]</span>
|
||
|
|
k = 1e3; <span class="org-comment">% [N/m]</span>
|
||
|
|
|
||
|
|
Gn = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> k)<span class="org-type">*</span>[<span class="org-type">-</span>k, k<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, m<span class="org-type">*</span>s<span class="org-type">^</span>2; 1, <span class="org-type">-</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2, 1; 1, k, 1];
|
||
|
|
Gn.InputName = {<span class="org-string">'Fd'</span>, <span class="org-string">'xmu'</span>, <span class="org-string">'F'</span>};
|
||
|
|
Gn.OutputName = {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'x'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">wmu = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50; <span class="org-comment">% [rad/s]</span>
|
||
|
|
|
||
|
|
Tmu = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wmu);
|
||
|
|
Tmu.InputName = {<span class="org-string">'r1'</span>};
|
||
|
|
Tmu.OutputName = {<span class="org-string">'ymu'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>40;
|
||
|
|
xi = 0.5;
|
||
|
|
Tw = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2);
|
||
|
|
Tw.InputName = {<span class="org-string">'w1'</span>};
|
||
|
|
Tw.OutputName = {<span class="org-string">'dw'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Wsplit = [tf(1); tf(1)];
|
||
|
|
Wsplit.InputName = {<span class="org-string">'w'</span>};
|
||
|
|
Wsplit.OutputName = {<span class="org-string">'w1'</span>, <span class="org-string">'w2'</span>};
|
||
|
|
|
||
|
|
S = sumblk(<span class="org-string">'xmu = ymu + dmu + dw'</span>);
|
||
|
|
|
||
|
|
Sw = sumblk(<span class="org-string">'y = x - w2'</span>);
|
||
|
|
|
||
|
|
Gvc = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {<span class="org-string">'Fd'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'r1'</span>, <span class="org-string">'F'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>, <span class="org-string">'y'</span>});
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Kvc_dvf = 2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m)<span class="org-type">*</span>s;
|
||
|
|
Kvc_dvf.InputName = {<span class="org-string">'d'</span>};
|
||
|
|
Kvc_dvf.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
|
||
|
|
Gvc_dvf = feedback(Gvc, Kvc_dvf, <span class="org-string">'name'</span>);
|
||
|
|
|
||
|
|
Kvc_iff = 2<span class="org-type">*</span>sqrt(k<span class="org-type">/</span>m)<span class="org-type">/</span>s;
|
||
|
|
Kvc_iff.InputName = {<span class="org-string">'Fm'</span>};
|
||
|
|
Kvc_iff.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
|
||
|
|
Gvc_iff = feedback(Gvc, Kvc_iff, <span class="org-string">'name'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">wb = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100; <span class="org-comment">% control bandwidth [rad/s]</span>
|
||
|
|
|
||
|
|
<span class="org-comment">% Lead</span>
|
||
|
|
h = 2.0;
|
||
|
|
wz = wb<span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
|
||
|
|
wp = wb<span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
|
||
|
|
|
||
|
|
H = 1<span class="org-type">/</span>h<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wz)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wp);
|
||
|
|
|
||
|
|
Kvc_dvf = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_dvf(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
|
||
|
|
Kvc_dvf.InputName = {<span class="org-string">'e'</span>};
|
||
|
|
Kvc_dvf.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
|
||
|
|
Kvc_iff = H<span class="org-type">/</span>abs(freqresp(H<span class="org-type">*</span>Gvc_iff(<span class="org-string">'y'</span>, <span class="org-string">'F'</span>), wb));
|
||
|
|
Kvc_iff.InputName = {<span class="org-string">'e'</span>};
|
||
|
|
Kvc_iff.OutputName = {<span class="org-string">'F'</span>};
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">Sfb = sumblk(<span class="org-string">'e = r2 - y'</span>);
|
||
|
|
|
||
|
|
R = [tf(1); tf(1)];
|
||
|
|
R.InputName = {<span class="org-string">'r'</span>};
|
||
|
|
R.OutputName = {<span class="org-string">'r1'</span>, <span class="org-string">'r2'</span>};
|
||
|
|
|
||
|
|
Gvc_fb_dvf = connect(Gvc_dvf, Kvc_dvf, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>});
|
||
|
|
Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, {<span class="org-string">'r'</span>, <span class="org-string">'dmu'</span>, <span class="org-string">'Fd'</span>, <span class="org-string">'w'</span>}, {<span class="org-string">'y'</span>, <span class="org-string">'e'</span>, <span class="org-string">'Fm'</span>, <span class="org-string">'d'</span>});
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org3817d8a" class="figure">
|
||
|
|
<p><img src="figs/simple_hac_lac_results_soft.png" alt="simple_hac_lac_results_soft.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 12: </span>Obtained closed-loop transfer functions (<a href="./figs/simple_hac_lac_results_soft.png">png</a>, <a href="./figs/simple_hac_lac_results_soft.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgcb7520f" class="outline-2">
|
||
|
|
<h2 id="orgcb7520f"><span class="section-number-2">11</span> Compare Soft and Stiff nano-hexapods</h2>
|
||
|
|
<div class="outline-text-2" id="text-11">
|
||
|
|
|
||
|
|
<div id="org55e0fe2" class="figure">
|
||
|
|
<p><img src="figs/simple_comp_vc_pz.png" alt="simple_comp_vc_pz.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 13: </span>Comparison of the closed-loop transfer functions for Soft and Stiff nano-hexapod (<a href="./figs/simple_comp_vc_pz.png">png</a>, <a href="./figs/simple_comp_vc_pz.pdf">pdf</a>)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
|
|
||
|
|
|
||
|
|
<colgroup>
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-center" />
|
||
|
|
|
||
|
|
<col class="org-center" />
|
||
|
|
</colgroup>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th scope="col" class="org-left"> </th>
|
||
|
|
<th scope="col" class="org-center"><b>Soft</b></th>
|
||
|
|
<th scope="col" class="org-center"><b>Stiff</b></th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><b>Reference Tracking</b></td>
|
||
|
|
<td class="org-center">=</td>
|
||
|
|
<td class="org-center">=</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><b>Vibration Isolation</b></td>
|
||
|
|
<td class="org-center">+</td>
|
||
|
|
<td class="org-center">-</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><b>Compliance</b></td>
|
||
|
|
<td class="org-center">-</td>
|
||
|
|
<td class="org-center">+</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgc0253c3" class="outline-2">
|
||
|
|
<h2 id="orgc0253c3"><span class="section-number-2">12</span> Estimate the level of vibration</h2>
|
||
|
|
<div class="outline-text-2" id="text-12">
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">gm = load(<span class="org-string">'./mat/psd_gm.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'psd_gm'</span>);
|
||
|
|
rz = load(<span class="org-string">'./mat/pxsp_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxsp_r'</span>);
|
||
|
|
tyz = load(<span class="org-string">'./mat/pxz_ty_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxz_ty_r'</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">x_pz = sqrt(abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gpz_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm));
|
||
|
|
x_vc = sqrt(abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'dmu'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_rz <span class="org-type">+</span> psd_ty) <span class="org-type">+</span> abs(squeeze(freqresp(Gvc_fb_iff(<span class="org-string">'y'</span>, <span class="org-string">'w'</span>), f, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>(psd_gm));
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org764c4a9" class="outline-2">
|
||
|
|
<h2 id="org764c4a9"><span class="section-number-2">13</span> Requirements on the norm of closed-loop transfer functions</h2>
|
||
|
|
<div class="outline-text-2" id="text-13">
|
||
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
|
|
||
|
|
|
||
|
|
<colgroup>
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-left" />
|
||
|
|
</colgroup>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">reference tracking</td>
|
||
|
|
<td class="org-left">\(\epsilon/r\)</td>
|
||
|
|
<td class="org-left">-120dB at 1Hz</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">vibration isolation</td>
|
||
|
|
<td class="org-left">\(x/x_\mu\)</td>
|
||
|
|
<td class="org-left">-60dB above 10Hz</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left">compliance</td>
|
||
|
|
<td class="org-left">\(x/F_d\)</td>
|
||
|
|
<td class="org-left"> </td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<div id="postamble" class="status">
|
||
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
||
|
|
<p class="date">Created: 2020-03-06 ven. 15:10</p>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|