1054 lines
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1054 lines
34 KiB
HTML
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<!-- 2020-09-01 mar. 13:48 -->
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<title>Amplified Piezoelectric Stack Actuator</title>
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<meta name="author" content="Dehaeze Thomas" />
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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">Amplified Piezoelectric Stack Actuator</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="#org996fd7c">1. Simplified Model</a>
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<ul>
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<li><a href="#org76fe747">1.1. Parameters</a></li>
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<li><a href="#orgbe95c15">1.2. Identification</a></li>
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<li><a href="#orgeaad673">1.3. Root Locus</a></li>
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||
<li><a href="#orged9310d">1.4. Analytical Model</a></li>
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<li><a href="#org2f351a4">1.5. Analytical Analysis</a></li>
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</ul>
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</li>
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<li><a href="#orgf1a765f">2. Rotating X-Y platform</a>
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<ul>
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||
<li><a href="#orga31e2d2">2.1. Parameters</a></li>
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<li><a href="#orgb7a6747">2.2. Identification</a></li>
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||
<li><a href="#org290de6f">2.3. Root Locus</a></li>
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<li><a href="#org069f401">2.4. Analysis</a></li>
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</ul>
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||
</li>
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<li><a href="#org3c74f7f">3. Stewart Platform with Amplified Actuators</a>
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<ul>
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<li><a href="#org5a7c6dc">3.1. Initialization</a></li>
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||
<li><a href="#orgdfe4eeb">3.2. APA-100 Amplified Actuator</a>
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<ul>
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<li><a href="#org7798ce9">3.2.1. Identification</a></li>
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<li><a href="#orgb943e63">3.2.2. Controller Design</a></li>
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<li><a href="#org0bf79ae">3.2.3. Effect of the Low Authority Control on the Primary Plant</a></li>
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<li><a href="#org96f3a8d">3.2.4. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
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</ul>
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</li>
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<li><a href="#org297c2ad">3.3. Optimal Stiffnesses</a>
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<ul>
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||
<li><a href="#org5dc2ac7">3.3.1. Low Authority Controller</a>
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<ul>
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||
<li><a href="#orgf339f37">3.3.1.1. Identification</a></li>
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||
<li><a href="#orgfcd4ad7">3.3.1.2. Effect of the Low Authority Control on the Primary Plant</a></li>
|
||
<li><a href="#org32c3422">3.3.1.3. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
|
||
</ul>
|
||
</li>
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||
<li><a href="#orgce4f07c">3.3.2. High Authority Controller</a>
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||
<ul>
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||
<li><a href="#orgc16003c">3.3.2.1. Controller Design</a></li>
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<li><a href="#orgbcf7749">3.3.2.2. Sensibility to Disturbances and Noise Budget</a></li>
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<li><a href="#orga0e8915">3.3.2.3. Simulations of Tomography Experiment</a></li>
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<li><a href="#orgb6ec250">3.3.2.4. Results</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org1e2f810">3.4. Direct Velocity Feedback with Amplified Actuators</a></li>
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</ul>
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</li>
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<li><a href="#org936810c">4. APA300ML</a>
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<ul>
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<li><a href="#org797e3ca">4.1. Initialization</a></li>
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<li><a href="#org2300f00">4.2. Identification</a></li>
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||
<li><a href="#org4ec1cb7">4.3. Controller Design</a></li>
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<li><a href="#org1b67874">4.4. Effect of the Low Authority Control on the Primary Plant</a></li>
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||
<li><a href="#org655c28c">4.5. Control in the leg space</a></li>
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||
<li><a href="#org92df47e">4.6. Sensibility to Disturbances and Noise Budget</a></li>
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||
<li><a href="#orge84cedc">4.7. Simulations of Tomography Experiment</a></li>
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<li><a href="#org05ce41a">4.8. Results</a></li>
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||
</ul>
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||
</li>
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||
</ul>
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||
</div>
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||
</div>
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<p>
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||
The presented model is based on (<a href="#citeproc_bib_item_2">Souleille et al. 2018</a>).
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</p>
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<p>
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The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure <a href="#orgb707bbd">1</a>).
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The parameters are shown in the table below.
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</p>
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<div id="orgb707bbd" class="figure">
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<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Parameters used for the model of the APA 100M</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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<col class="org-left" />
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||
</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left"> </th>
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<th scope="col" class="org-left">Value</th>
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<th scope="col" class="org-left">Meaning</th>
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||
</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">\(m\)</td>
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<td class="org-left">\(1\,[kg]\)</td>
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<td class="org-left">Payload mass</td>
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</tr>
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<tr>
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<td class="org-left">\(k_e\)</td>
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<td class="org-left">\(4.8\,[N/\mu m]\)</td>
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<td class="org-left">Stiffness used to adjust the pole of the isolator</td>
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||
</tr>
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||
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<tr>
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<td class="org-left">\(k_1\)</td>
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<td class="org-left">\(0.96\,[N/\mu m]\)</td>
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||
<td class="org-left">Stiffness of the metallic suspension when the stack is removed</td>
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||
</tr>
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||
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||
<tr>
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<td class="org-left">\(k_a\)</td>
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<td class="org-left">\(65\,[N/\mu m]\)</td>
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<td class="org-left">Stiffness of the actuator</td>
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||
</tr>
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||
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||
<tr>
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||
<td class="org-left">\(c_1\)</td>
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<td class="org-left">\(10\,[N/(m/s)]\)</td>
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||
<td class="org-left">Added viscous damping</td>
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||
</tr>
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||
</tbody>
|
||
</table>
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||
|
||
<div id="outline-container-org996fd7c" class="outline-2">
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<h2 id="org996fd7c"><span class="section-number-2">1</span> Simplified Model</h2>
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||
<div class="outline-text-2" id="text-1">
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||
</div>
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||
<div id="outline-container-org76fe747" class="outline-3">
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<h3 id="org76fe747"><span class="section-number-3">1.1</span> Parameters</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="org-src-container">
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||
<pre class="src src-matlab">m = 1; % [kg]
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ke = 4.8e6; % [N/m]
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||
ce = 5; % [N/(m/s)]
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||
me = 0.001; % [kg]
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||
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||
k1 = 0.96e6; % [N/m]
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||
c1 = 10; % [N/(m/s)]
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||
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||
ka = 65e6; % [N/m]
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||
ca = 5; % [N/(m/s)]
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||
ma = 0.001; % [kg]
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||
|
||
h = 0.2; % [m]
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||
</pre>
|
||
</div>
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||
|
||
<p>
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||
IFF Controller:
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||
</p>
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||
<div class="org-src-container">
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||
<pre class="src src-matlab">Kiff = -8000/s;
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||
</pre>
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||
</div>
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||
</div>
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||
</div>
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||
|
||
<div id="outline-container-orgbe95c15" class="outline-3">
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||
<h3 id="orgbe95c15"><span class="section-number-3">1.2</span> Identification</h3>
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||
<div class="outline-text-3" id="text-1-2">
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||
<p>
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||
Identification in open-loop.
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||
</p>
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||
<div class="org-src-container">
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||
<pre class="src src-matlab">%% Name of the Simulink File
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||
mdl = 'amplified_piezo_model';
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||
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||
%% Input/Output definition
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||
clear io; io_i = 1;
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io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion
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io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force
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||
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||
io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
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||
io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
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||
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||
G = linearize(mdl, io, 0);
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||
G.InputName = {'w', 'f', 'F'};
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||
G.OutputName = {'Fs', 'x1'};
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||
</pre>
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||
</div>
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||
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||
<p>
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||
Identification in closed-loop.
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||
</p>
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||
<div class="org-src-container">
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||
<pre class="src src-matlab">%% Name of the Simulink File
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||
mdl = 'amplified_piezo_model';
|
||
|
||
%% Input/Output definition
|
||
clear io; io_i = 1;
|
||
io(io_i) = linio([mdl, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion
|
||
io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
|
||
io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force
|
||
|
||
io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
|
||
io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
|
||
|
||
Giff = linearize(mdl, io, 0);
|
||
Giff.InputName = {'w', 'f', 'F'};
|
||
Giff.OutputName = {'Fs', 'x1'};
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org55d1535" class="figure">
|
||
<p><img src="figs/amplified_piezo_tf_ol_and_cl.png" alt="amplified_piezo_tf_ol_and_cl.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 2: </span>Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)</p>
|
||
</div>
|
||
</div>
|
||
</div>
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||
|
||
<div id="outline-container-orgeaad673" class="outline-3">
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||
<h3 id="orgeaad673"><span class="section-number-3">1.3</span> Root Locus</h3>
|
||
<div class="outline-text-3" id="text-1-3">
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||
|
||
<div id="org85cd6e5" class="figure">
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||
<p><img src="figs/amplified_piezo_root_locus.png" alt="amplified_piezo_root_locus.png" />
|
||
</p>
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||
<p><span class="figure-number">Figure 3: </span>Root Locus</p>
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||
</div>
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||
</div>
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||
</div>
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||
|
||
<div id="outline-container-orged9310d" class="outline-3">
|
||
<h3 id="orged9310d"><span class="section-number-3">1.4</span> Analytical Model</h3>
|
||
<div class="outline-text-3" id="text-1-4">
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||
<p>
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||
If we apply the Newton’s second law of motion on the top mass, we obtain:
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||
\[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \]
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||
</p>
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||
|
||
<p>
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||
Then, we can write that the measured force \(F_s\) is equal to:
|
||
\[ F_s = k_a(w - x_e) + f = -k_e (x_1 - x_e) \]
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||
which gives:
|
||
\[ x_e = \frac{k_a}{k_e + k_a} w + \frac{1}{k_e + k_a} f + \frac{k_e}{k_e + k_a} x_1 \]
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||
</p>
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||
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||
<p>
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||
Re-injecting that into the previous equations gives:
|
||
\[ x_1 = F \frac{1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + f \frac{\frac{k_e}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \]
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||
\[ F_s = - F \frac{\frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_e k_a}{k_e + k_a} \Big( \frac{ms^2}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) - f \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
|
||
</p>
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||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ga = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * ...
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||
[ 1 , k1 + ke*ka/(ke + ka) , ke/(ke + ka) ;
|
||
-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke+ka)*(m*s^2 + k1)];
|
||
Ga.InputName = {'F', 'w', 'f'};
|
||
Ga.OutputName = {'x1', 'Fs'};
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org7970b34" class="figure">
|
||
<p><img src="figs/comp_simscape_analytical.png" alt="comp_simscape_analytical.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 4: </span>Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org2f351a4" class="outline-3">
|
||
<h3 id="org2f351a4"><span class="section-number-3">1.5</span> Analytical Analysis</h3>
|
||
<div class="outline-text-3" id="text-1-5">
|
||
<p>
|
||
For Integral Force Feedback Control, the plant is:
|
||
\[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
|
||
</p>
|
||
|
||
<p>
|
||
As high frequency, this converge to:
|
||
\[ \frac{F_s}{f} \underset{\omega\to\infty}{\longrightarrow} \frac{k_e}{k_e + k_a} \]
|
||
And at low frequency:
|
||
\[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
|
||
</p>
|
||
|
||
<p>
|
||
It has two complex conjugate zeros at:
|
||
\[ z = \pm j \sqrt{\frac{k_1}{m}} \]
|
||
And two complex conjugate poles at:
|
||
\[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \]
|
||
</p>
|
||
|
||
<p>
|
||
If maximal damping is to be attained with IFF, the distance between the zero and the pole is to be maximized.
|
||
Thus, we wish to maximize \(p/z\), which is equivalent as to minimize \(k_1\) and have \(k_e \approx k_a\) (supposing \(k_e + k_a \approx \text{cst}\)).
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 1;
|
||
k1 = 1e6;
|
||
ka = 1e6;
|
||
ke = 1e6;
|
||
|
||
Giff.InputName = {'f'};
|
||
Giff.OutputName = {'Fs'};
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgf1a765f" class="outline-2">
|
||
<h2 id="orgf1a765f"><span class="section-number-2">2</span> Rotating X-Y platform</h2>
|
||
<div class="outline-text-2" id="text-2">
|
||
<p>
|
||
This analysis gave rise to a paper (<a href="#citeproc_bib_item_1">Dehaeze and Collette 2020</a>).
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-orga31e2d2" class="outline-3">
|
||
<h3 id="orga31e2d2"><span class="section-number-3">2.1</span> Parameters</h3>
|
||
<div class="outline-text-3" id="text-2-1">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">m = 1; % [kg]
|
||
|
||
ke = 4.8e6; % [N/m]
|
||
ce = 5; % [N/(m/s)]
|
||
me = 0.001; % [kg]
|
||
|
||
k1 = 0.96e6; % [N/m]
|
||
c1 = 10; % [N/(m/s)]
|
||
|
||
ka = 65e6; % [N/m]
|
||
ca = 5; % [N/(m/s)]
|
||
ma = 0.001; % [kg]
|
||
|
||
h = 0.2; % [m]
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kiff = tf(0);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgb7a6747" class="outline-3">
|
||
<h3 id="orgb7a6747"><span class="section-number-3">2.2</span> Identification</h3>
|
||
<div class="outline-text-3" id="text-2-2">
|
||
<p>
|
||
Rotating speed in rad/s:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ws = 2*pi*[0, 1, 10, 100];
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Gs = {zeros(length(Ws), 1)};
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Identification in open-loop.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">%% Name of the Simulink File
|
||
mdl = 'amplified_piezo_xy_rotating_stage';
|
||
|
||
%% Input/Output definition
|
||
clear io; io_i = 1;
|
||
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
|
||
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
|
||
|
||
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
||
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
|
||
|
||
for i = 1:length(Ws)
|
||
ws = Ws(i);
|
||
G = linearize(mdl, io, 0);
|
||
G.InputName = {'fx', 'fy'};
|
||
G.OutputName = {'Fsx', 'Fsy'};
|
||
Gs(i) = {G};
|
||
end
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orga4fc975" class="figure">
|
||
<p><img src="figs/amplitifed_piezo_xy_rotation_plant_iff.png" alt="amplitifed_piezo_xy_rotation_plant_iff.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 5: </span>Transfer function matrix from forces to force sensors for multiple rotation speed</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org290de6f" class="outline-3">
|
||
<h3 id="org290de6f"><span class="section-number-3">2.3</span> Root Locus</h3>
|
||
<div class="outline-text-3" id="text-2-3">
|
||
|
||
<div id="orgccd3396" class="figure">
|
||
<p><img src="figs/amplified_piezo_xy_rotation_root_locus.png" alt="amplified_piezo_xy_rotation_root_locus.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 6: </span>Root locus for 3 rotating speed</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org069f401" class="outline-3">
|
||
<h3 id="org069f401"><span class="section-number-3">2.4</span> Analysis</h3>
|
||
<div class="outline-text-3" id="text-2-4">
|
||
<p>
|
||
The negative stiffness induced by the rotation is equal to \(m \omega_0^2\).
|
||
Thus, the maximum rotation speed where IFF can be applied is:
|
||
\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
|
||
</p>
|
||
|
||
<p>
|
||
Let’s verify that.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ws = 2*pi*[140, 160];
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Identification
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">%% Name of the Simulink File
|
||
mdl = 'amplified_piezo_xy_rotating_stage';
|
||
|
||
%% Input/Output definition
|
||
clear io; io_i = 1;
|
||
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
|
||
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
|
||
|
||
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
||
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
|
||
|
||
for i = 1:length(Ws)
|
||
ws = Ws(i);
|
||
G = linearize(mdl, io, 0);
|
||
G.InputName = {'fx', 'fy'};
|
||
G.OutputName = {'Fsx', 'Fsy'};
|
||
Gs(i) = {G};
|
||
end
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orgca23612" class="figure">
|
||
<p><img src="figs/amplified_piezo_xy_rotating_unstable_root_locus.png" alt="amplified_piezo_xy_rotating_unstable_root_locus.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 7: </span>Root Locus for the two considered rotation speed. For the red curve, the system is unstable.</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3c74f7f" class="outline-2">
|
||
<h2 id="org3c74f7f"><span class="section-number-2">3</span> Stewart Platform with Amplified Actuators</h2>
|
||
<div class="outline-text-2" id="text-3">
|
||
</div>
|
||
<div id="outline-container-org5a7c6dc" class="outline-3">
|
||
<h3 id="org5a7c6dc"><span class="section-number-3">3.1</span> Initialization</h3>
|
||
<div class="outline-text-3" id="text-3-1">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeGround();
|
||
initializeGranite();
|
||
initializeTy();
|
||
initializeRy();
|
||
initializeRz();
|
||
initializeMicroHexapod();
|
||
initializeAxisc();
|
||
initializeMirror();
|
||
|
||
initializeSimscapeConfiguration();
|
||
initializeDisturbances('enable', false);
|
||
initializeLoggingConfiguration('log', 'none');
|
||
|
||
initializeController('type', 'hac-iff');
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
We set the stiffness of the payload fixation:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kp = 1e8; % [N/m]
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgdfe4eeb" class="outline-3">
|
||
<h3 id="orgdfe4eeb"><span class="section-number-3">3.2</span> APA-100 Amplified Actuator</h3>
|
||
<div class="outline-text-3" id="text-3-2">
|
||
</div>
|
||
<div id="outline-container-org7798ce9" class="outline-4">
|
||
<h4 id="org7798ce9"><span class="section-number-4">3.2.1</span> Identification</h4>
|
||
<div class="outline-text-4" id="text-3-2-1">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = tf(zeros(6));
|
||
Kiff = tf(zeros(6));
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
We identify the system for the following payload masses:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ms = [1, 10, 50];
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The nano-hexapod has the following leg’s stiffness and damping.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeNanoHexapod('actuator', 'amplified');
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgb943e63" class="outline-4">
|
||
<h4 id="orgb943e63"><span class="section-number-4">3.2.2</span> Controller Design</h4>
|
||
<div class="outline-text-4" id="text-3-2-2">
|
||
<p>
|
||
The loop gain for IFF is shown in Figure <a href="#org0e2911a">8</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The corresponding root locus is shown in Figure <a href="#org5d7f6d3">9</a>.
|
||
</p>
|
||
|
||
<p>
|
||
Finally, the damping as function of the gain is display in Figure <a href="#org4743c83">10</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org0e2911a" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_loop_gain.png" alt="amplified_piezo_iff_loop_gain.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 8: </span>Dynamics for the Integral Force Feedback for three payload masses</p>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="org5d7f6d3" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_root_locus.png" alt="amplified_piezo_iff_root_locus.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 9: </span>Root Locus for the IFF control for three payload masses</p>
|
||
</div>
|
||
|
||
|
||
<div id="org4743c83" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_damping_gain.png" alt="amplified_piezo_iff_damping_gain.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 10: </span>Damping ratio of the poles as a function of the IFF gain</p>
|
||
</div>
|
||
|
||
<p>
|
||
The following controller for the Decentralized Integral Force Feedback is used:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kiff = -1e4/s*eye(6);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org0bf79ae" class="outline-4">
|
||
<h4 id="org0bf79ae"><span class="section-number-4">3.2.3</span> Effect of the Low Authority Control on the Primary Plant</h4>
|
||
<div class="outline-text-4" id="text-3-2-3">
|
||
|
||
<div id="org904efc3" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_plant_damped_X.png" alt="amplified_piezo_iff_plant_damped_X.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 11: </span>Primary plant in the task space with (dashed) and without (solid) IFF</p>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="orgddf3013" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_damped_plant_L.png" alt="amplified_piezo_iff_damped_plant_L.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 12: </span>Primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
|
||
</div>
|
||
|
||
<div id="orgd940ce9" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_damped_coupling_X.png" alt="amplified_piezo_iff_damped_coupling_X.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 13: </span>Coupling in the primary plant in the task with (dashed) and without (solid) IFF</p>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="org4278690" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_damped_coupling_L.png" alt="amplified_piezo_iff_damped_coupling_L.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 14: </span>Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org96f3a8d" class="outline-4">
|
||
<h4 id="org96f3a8d"><span class="section-number-4">3.2.4</span> Effect of the Low Authority Control on the Sensibility to Disturbances</h4>
|
||
<div class="outline-text-4" id="text-3-2-4">
|
||
|
||
<div id="org56179cd" class="figure">
|
||
<p><img src="figs/amplified_piezo_iff_disturbances.png" alt="amplified_piezo_iff_disturbances.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 15: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div class="outline-text-3" id="text-3-2">
|
||
<div class="important">
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-org297c2ad" class="outline-3">
|
||
<h3 id="org297c2ad"><span class="section-number-3">3.3</span> Optimal Stiffnesses</h3>
|
||
<div class="outline-text-3" id="text-3-3">
|
||
<p>
|
||
Based on the analytical analysis, we can determine the parameters of the amplified piezoelectric actuator in order to be able to add a lots of damping using IFF:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>\(k_1\) should be minimized.</li>
|
||
<li>\(k_e \approx k_a \approx 10^5 - 10^6\,[N/m]\)</li>
|
||
</ul>
|
||
|
||
<p>
|
||
However, this might not be realizable.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org5dc2ac7" class="outline-4">
|
||
<h4 id="org5dc2ac7"><span class="section-number-4">3.3.1</span> Low Authority Controller</h4>
|
||
<div class="outline-text-4" id="text-3-3-1">
|
||
</div>
|
||
<div id="outline-container-orgf339f37" class="outline-5">
|
||
<h5 id="orgf339f37"><span class="section-number-5">3.3.1.1</span> Identification</h5>
|
||
<div class="outline-text-5" id="text-3-3-1-1">
|
||
<p>
|
||
The nano-hexapod is initialized with the following parameters:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeNanoHexapod('actuator', 'amplified', ...
|
||
'k1', 1e4, ...
|
||
'ke', 1e6, ...
|
||
'ka', 1e6);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The obtain plan for the IFF control is shown in Figure <a href="#orgafcb4d0">16</a>.
|
||
The associated Root Locus is shown in Figure <a href="#org62c3e69">17</a>.
|
||
</p>
|
||
|
||
<p>
|
||
Based on that, the following IFF gain is chosen:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kiff = -1e3/s*eye(6);
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orgafcb4d0" class="figure">
|
||
<p><img src="figs/amplified_piezo_opt_stiff_iff_plant.png" alt="amplified_piezo_opt_stiff_iff_plant.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 16: </span>Plant dynamics for IFF with the amplified piezoelectric stack actuator</p>
|
||
</div>
|
||
|
||
|
||
<div id="org62c3e69" class="figure">
|
||
<p><img src="figs/amplified_piezo_opt_stiff_iff_root_locus.png" alt="amplified_piezo_opt_stiff_iff_root_locus.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 17: </span>Root Locus for IFF with the amplified piezoelectric stack actuator</p>
|
||
</div>
|
||
|
||
|
||
<div id="orgaf25c91" class="figure">
|
||
<p><img src="figs/amplified_piezo_opt_stiff_gain_damping.png" alt="amplified_piezo_opt_stiff_gain_damping.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 18: </span>Damping of the modes as a function of the IFF gain</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgfcd4ad7" class="outline-5">
|
||
<h5 id="orgfcd4ad7"><span class="section-number-5">3.3.1.2</span> Effect of the Low Authority Control on the Primary Plant</h5>
|
||
</div>
|
||
<div id="outline-container-org32c3422" class="outline-5">
|
||
<h5 id="org32c3422"><span class="section-number-5">3.3.1.3</span> Effect of the Low Authority Control on the Sensibility to Disturbances</h5>
|
||
<div class="outline-text-5" id="text-3-3-1-3">
|
||
|
||
<div id="orgf43f9ed" class="figure">
|
||
<p><img src="figs/amplified_piezo_opt_stiff_iff_dist.png" alt="amplified_piezo_opt_stiff_iff_dist.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 19: </span>Effect of disturbance with and without IFF</p>
|
||
</div>
|
||
<div class="important">
|
||
<p>
|
||
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgce4f07c" class="outline-4">
|
||
<h4 id="orgce4f07c"><span class="section-number-4">3.3.2</span> High Authority Controller</h4>
|
||
<div class="outline-text-4" id="text-3-3-2">
|
||
</div>
|
||
<div id="outline-container-orgc16003c" class="outline-5">
|
||
<h5 id="orgc16003c"><span class="section-number-5">3.3.2.1</span> Controller Design</h5>
|
||
<div class="outline-text-5" id="text-3-3-2-1">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">h = 2.5;
|
||
Kl = 5e6 * eye(6) * ...
|
||
1/h*(s/(2*pi*40/h) + 1)/(s/(2*pi*40*h) + 1) * ...
|
||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||
(s/2/pi/50 + 1)/(s/2/pi/50) * ...
|
||
(s/2/pi/10 + 1)/(s/2/pi/10) * ...
|
||
1/(1 + s/2/pi/200);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kl = 3e10 * eye(6) * ...
|
||
1/s * ...
|
||
(s+0.8)/s * ...
|
||
(s+50)/(s+0.01) * ...
|
||
(s+120)/(s+1000) * ...
|
||
(s+150)/(s+1000);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">load('mat/stages.mat', 'nano_hexapod');
|
||
K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgbcf7749" class="outline-5">
|
||
<h5 id="orgbcf7749"><span class="section-number-5">3.3.2.2</span> Sensibility to Disturbances and Noise Budget</h5>
|
||
<div class="outline-text-5" id="text-3-3-2-2">
|
||
<p>
|
||
We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-orga0e8915" class="outline-5">
|
||
<h5 id="orga0e8915"><span class="section-number-5">3.3.2.3</span> Simulations of Tomography Experiment</h5>
|
||
<div class="outline-text-5" id="text-3-3-2-3">
|
||
<p>
|
||
Let’s now simulate a tomography experiment.
|
||
To do so, we include all disturbances except vibrations of the translation stage.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeDisturbances();
|
||
initializeSimscapeConfiguration('gravity', false);
|
||
initializeLoggingConfiguration('log', 'all');
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
And we run the simulation for all three payload Masses.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-orgb6ec250" class="outline-5">
|
||
<h5 id="orgb6ec250"><span class="section-number-5">3.3.2.4</span> Results</h5>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-org1e2f810" class="outline-3">
|
||
<h3 id="org1e2f810"><span class="section-number-3">3.4</span> Direct Velocity Feedback with Amplified Actuators</h3>
|
||
<div class="outline-text-3" id="text-3-4">
|
||
<p>
|
||
Lack of collocation.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeController('type', 'hac-dvf');
|
||
K = tf(zeros(6));
|
||
Kdvf = tf(zeros(6));
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
We identify the system for the following payload masses:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ms = [1, 10, 50];
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeNanoHexapod('actuator', 'amplified', ...
|
||
'k1', 1e4, ...
|
||
'ke', 1e6, ...
|
||
'ka', 1e6);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org936810c" class="outline-2">
|
||
<h2 id="org936810c"><span class="section-number-2">4</span> APA300ML</h2>
|
||
<div class="outline-text-2" id="text-4">
|
||
</div>
|
||
<div id="outline-container-org797e3ca" class="outline-3">
|
||
<h3 id="org797e3ca"><span class="section-number-3">4.1</span> Initialization</h3>
|
||
<div class="outline-text-3" id="text-4-1">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeGround();
|
||
initializeGranite();
|
||
initializeTy();
|
||
initializeRy();
|
||
initializeRz();
|
||
initializeMicroHexapod();
|
||
initializeAxisc();
|
||
initializeMirror();
|
||
|
||
initializeSimscapeConfiguration();
|
||
initializeDisturbances('enable', false);
|
||
initializeLoggingConfiguration('log', 'none');
|
||
|
||
initializeController('type', 'hac-dvf');
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
We set the stiffness of the payload fixation:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kp = 1e8; % [N/m]
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org2300f00" class="outline-3">
|
||
<h3 id="org2300f00"><span class="section-number-3">4.2</span> Identification</h3>
|
||
<div class="outline-text-3" id="text-4-2">
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">K = tf(zeros(6));
|
||
Kdvf = tf(zeros(6));
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
We identify the system for the following payload masses:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Ms = [1, 10, 50];
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The nano-hexapod has the following leg’s stiffness and damping.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeNanoHexapod('actuator', 'amplified', 'k1', 0.4e6, 'ka', 43e6, 'ke', 1.5e6);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org4ec1cb7" class="outline-3">
|
||
<h3 id="org4ec1cb7"><span class="section-number-3">4.3</span> Controller Design</h3>
|
||
<div class="outline-text-3" id="text-4-3">
|
||
<p>
|
||
Damping as function of the gain
|
||
Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Kdvf = 5e5*s/(1+s/2/pi/1e3)*eye(6);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1b67874" class="outline-3">
|
||
<h3 id="org1b67874"><span class="section-number-3">4.4</span> Effect of the Low Authority Control on the Primary Plant</h3>
|
||
</div>
|
||
<div id="outline-container-org655c28c" class="outline-3">
|
||
<h3 id="org655c28c"><span class="section-number-3">4.5</span> Control in the leg space</h3>
|
||
<div class="outline-text-3" id="text-4-5">
|
||
<p>
|
||
We design a diagonal controller with all the same diagonal elements.
|
||
</p>
|
||
|
||
<p>
|
||
The requirements for the controller are:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>Crossover frequency of around 100Hz</li>
|
||
<li>Stable for all the considered payload masses</li>
|
||
<li>Sufficient phase and gain margin</li>
|
||
<li>Integral action at low frequency</li>
|
||
</ul>
|
||
|
||
<p>
|
||
The design controller is as follows:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>Lead centered around the crossover</li>
|
||
<li>An integrator below 10Hz</li>
|
||
<li>A low pass filter at 250Hz</li>
|
||
</ul>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">h = 2.0;
|
||
Kl = 1e9 * eye(6) * ...
|
||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||
1/h*(s/(2*pi*200/h) + 1)/(s/(2*pi*200*h) + 1) * ...
|
||
(s/2/pi/10 + 1)/(s/2/pi/10) * ...
|
||
1/(1 + s/2/pi/300);
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">load('mat/stages.mat', 'nano_hexapod');
|
||
K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org92df47e" class="outline-3">
|
||
<h3 id="org92df47e"><span class="section-number-3">4.6</span> Sensibility to Disturbances and Noise Budget</h3>
|
||
<div class="outline-text-3" id="text-4-6">
|
||
<p>
|
||
We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.
|
||
</p>
|
||
|
||
<div id="org44696dd" class="figure">
|
||
<p><img src="figs/opt_stiff_primary_control_L_psd_dist.png" alt="opt_stiff_primary_control_L_psd_dist.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 20: </span>Amplitude Spectral Density of the vertical position error of the sample when the HAC-DVF control is applied due to both the ground motion and spindle vibrations</p>
|
||
</div>
|
||
|
||
|
||
<div id="org15d4b96" class="figure">
|
||
<p><img src="figs/opt_stiff_primary_control_L_psd_tot.png" alt="opt_stiff_primary_control_L_psd_tot.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 21: </span>Amplitude Spectral Density of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
||
</div>
|
||
|
||
|
||
<div id="org7a31e40" class="figure">
|
||
<p><img src="figs/opt_stiff_primary_control_L_cas_tot.png" alt="opt_stiff_primary_control_L_cas_tot.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 22: </span>Cumulative Amplitude Spectrum of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orge84cedc" class="outline-3">
|
||
<h3 id="orge84cedc"><span class="section-number-3">4.7</span> Simulations of Tomography Experiment</h3>
|
||
<div class="outline-text-3" id="text-4-7">
|
||
<p>
|
||
Let’s now simulate a tomography experiment.
|
||
To do so, we include all disturbances except vibrations of the translation stage.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">initializeDisturbances();
|
||
initializeSimscapeConfiguration('gravity', false);
|
||
initializeLoggingConfiguration('log', 'all');
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
And we run the simulation for all three payload Masses.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-org05ce41a" class="outline-3">
|
||
<h3 id="org05ce41a"><span class="section-number-3">4.8</span> Results</h3>
|
||
<div class="outline-text-3" id="text-4-8">
|
||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
|
||
<div class="csl-bib-body">
|
||
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Dehaeze, T., and C. Collette. 2020. “Active Damping of Rotating Platforms Using Integral Force Feedback.” In <i>Proceedings of the International Conference on Modal Analysis Noise and Vibration Engineering (ISMA)</i>.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_2"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>CEAS Space Journal</i> 10 (2). Springer:157–65.</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="postamble" class="status">
|
||
<p class="author">Author: Dehaeze Thomas</p>
|
||
<p class="date">Created: 2020-09-01 mar. 13:48</p>
|
||
</div>
|
||
</body>
|
||
</html>
|