1573 lines
62 KiB
HTML
1573 lines
62 KiB
HTML
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Active Damping applied on the Simscape Model</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgdd48309">1. Undamped System</a>
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<ul>
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<li><a href="#orgf8565a4">1.1. Identification of the dynamics for Active Damping</a>
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<ul>
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<li><a href="#org8b450dc">1.1.1. Initialize the Simulation</a></li>
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<li><a href="#orge7f3d41">1.1.2. Identification</a></li>
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<li><a href="#orgad0525d">1.1.3. Obtained Plants for Active Damping</a></li>
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</ul>
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</li>
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<li><a href="#org77cdeea">1.2. Tomography Experiment</a>
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<ul>
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<li><a href="#orgc066249">1.2.1. Simulation</a></li>
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<li><a href="#org8188f8d">1.2.2. 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="#org35ace9a">2. Integral Force Feedback</a>
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<ul>
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<li><a href="#org64b724c">2.1. Control Design</a>
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<ul>
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<li><a href="#orga60342f">2.1.1. Plant</a></li>
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<li><a href="#org6e7a3b9">2.1.2. Control Design</a></li>
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<li><a href="#orgcffbce3">2.1.3. Diagonal Controller</a></li>
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<li><a href="#orgaac84c4">2.1.4. IFF with High Pass Filter</a></li>
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</ul>
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</li>
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<li><a href="#org641e0e1">2.2. Tomography Experiment</a>
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<ul>
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<li><a href="#org244d5ff">2.2.1. Simulation with IFF Controller</a></li>
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<li><a href="#org40a229f">2.2.2. Simulation with IFF Controller with added High Pass Filter</a></li>
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<li><a href="#org3982074">2.2.3. Compare with Undamped system</a></li>
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</ul>
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</li>
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<li><a href="#orgbb282ac">2.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgf20da86">3. Direct Velocity Feedback</a>
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<ul>
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<li><a href="#org7191e52">3.1. Control Design</a>
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<ul>
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<li><a href="#org56b88cd">3.1.1. Plant</a></li>
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<li><a href="#org8927c04">3.1.2. Control Design</a></li>
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<li><a href="#org27b066a">3.1.3. Diagonal Controller</a></li>
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</ul>
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</li>
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<li><a href="#org7b075f1">3.2. Tomography Experiment</a>
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<ul>
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<li><a href="#org29dea78">3.2.1. Initialize the Simulation</a></li>
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<li><a href="#org8b648f4">3.2.2. Simulation</a></li>
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<li><a href="#org0def86d">3.2.3. Compare with Undamped system</a></li>
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</ul>
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</li>
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<li><a href="#org857cf6a">3.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgad05b75">4. Inertial Control</a>
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<ul>
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<li><a href="#orgba4e94a">4.1. Control Design</a>
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<ul>
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<li><a href="#org5c18fbe">4.1.1. Plant</a></li>
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<li><a href="#orge529b8d">4.1.2. Control Design</a></li>
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<li><a href="#org6119dfa">4.1.3. Diagonal Controller</a></li>
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</ul>
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</li>
|
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<li><a href="#orgbe8a895">4.2. Tomography Experiment</a>
|
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<ul>
|
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<li><a href="#org8526a14">4.2.1. Initialize the Simulation</a></li>
|
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<li><a href="#orga670755">4.2.2. Simulation</a></li>
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<li><a href="#orgdc292f7">4.2.3. Compare with Undamped system</a></li>
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</ul>
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</li>
|
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<li><a href="#org4675ff3">4.3. Conclusion</a></li>
|
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</ul>
|
|
</li>
|
|
<li><a href="#org2cefbfb">5. Comparison</a>
|
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<ul>
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<li><a href="#org8cef845">5.1. Load the plants</a></li>
|
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<li><a href="#org4a8616e">5.2. Sensitivity to Disturbance</a></li>
|
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<li><a href="#orga89ca48">5.3. Damped Plant</a></li>
|
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<li><a href="#org7a74f40">5.4. Tomography Experiment</a>
|
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<ul>
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<li><a href="#org294c860">5.4.1. Load the Simulation Data</a></li>
|
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<li><a href="#org70e87cc">5.4.2. Frequency Domain Analysis</a></li>
|
|
</ul>
|
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</li>
|
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</ul>
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</li>
|
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<li><a href="#org952ed5d">6. Useful Functions</a>
|
|
<ul>
|
|
<li><a href="#orga9b2a73">6.1. prepareTomographyExperiment</a>
|
|
<ul>
|
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<li><a href="#orgdc8e362">Function Description</a></li>
|
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<li><a href="#org2d80135">Optional Parameters</a></li>
|
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<li><a href="#org342966f">Initialize the Simulation</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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</ul>
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</div>
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</div>
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<p>
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First, in section <a href="#org1e95733">1</a>, we will looked at the undamped system.
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</p>
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<p>
|
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Then, we will compare three active damping techniques:
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</p>
|
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<ul class="org-ul">
|
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<li>In section <a href="#org241a667">2</a>: the integral force feedback is used</li>
|
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<li>In section <a href="#orgc0638ac">3</a>: the direct velocity feedback is used</li>
|
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<li>In section <a href="#orgbea2985">4</a>: inertial control is used</li>
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</ul>
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<p>
|
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For each of the active damping technique, we will:
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</p>
|
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<ul class="org-ul">
|
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<li>Look at the damped plant</li>
|
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<li>Simulate tomography experiments</li>
|
|
<li>Compare the sensitivity from disturbances</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The disturbances are:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Ground motion</li>
|
|
<li>Motion errors of all the stages</li>
|
|
</ul>
|
|
|
|
<div id="outline-container-orgdd48309" class="outline-2">
|
|
<h2 id="orgdd48309"><span class="section-number-2">1</span> Undamped System</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
<a id="org1e95733"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/undamped_system.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<p>
|
|
We first look at the undamped system.
|
|
The performance of this undamped system will be compared with the damped system using various techniques.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf8565a4" class="outline-3">
|
|
<h3 id="orgf8565a4"><span class="section-number-3">1.1</span> Identification of the dynamics for Active Damping</h3>
|
|
<div class="outline-text-3" id="text-1-1">
|
|
</div>
|
|
<div id="outline-container-org8b450dc" class="outline-4">
|
|
<h4 id="org8b450dc"><span class="section-number-4">1.1.1</span> Initialize the Simulation</h4>
|
|
<div class="outline-text-4" id="text-1-1-1">
|
|
<p>
|
|
We initialize all the stages with the default parameters.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
|
initializeSample(<span class="org-string">'mass'</span>, 50);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We set the references to zero.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeReferences();
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And all the controllers are set to 0.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
|
|
K_ine = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>);
|
|
K_iff = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
|
K_dvf = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge7f3d41" class="outline-4">
|
|
<h4 id="orge7f3d41"><span class="section-number-4">1.1.2</span> Identification</h4>
|
|
<div class="outline-text-4" id="text-1-1-2">
|
|
<p>
|
|
First, we identify the dynamics of the system using the <code>linearize</code> function.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'sim_nass_active_damping'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Dnlm'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Fnlm'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Vlm'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
|
|
G.OutputName = {<span class="org-string">'Dnlm1'</span>, <span class="org-string">'Dnlm2'</span>, <span class="org-string">'Dnlm3'</span>, <span class="org-string">'Dnlm4'</span>, <span class="org-string">'Dnlm5'</span>, <span class="org-string">'Dnlm6'</span>, ...
|
|
<span class="org-string">'Fnlm1'</span>, <span class="org-string">'Fnlm2'</span>, <span class="org-string">'Fnlm3'</span>, <span class="org-string">'Fnlm4'</span>, <span class="org-string">'Fnlm5'</span>, <span class="org-string">'Fnlm6'</span>, ...
|
|
<span class="org-string">'Vnlm1'</span>, <span class="org-string">'Vnlm2'</span>, <span class="org-string">'Vnlm3'</span>, <span class="org-string">'Vnlm4'</span>, <span class="org-string">'Vnlm5'</span>, <span class="org-string">'Vnlm6'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then create transfer functions corresponding to the active damping plants.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_iff = minreal(G({<span class="org-string">'Fnlm1'</span>, <span class="org-string">'Fnlm2'</span>, <span class="org-string">'Fnlm3'</span>, <span class="org-string">'Fnlm4'</span>, <span class="org-string">'Fnlm5'</span>, <span class="org-string">'Fnlm6'</span>}, {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}));
|
|
G_dvf = minreal(G({<span class="org-string">'Dnlm1'</span>, <span class="org-string">'Dnlm2'</span>, <span class="org-string">'Dnlm3'</span>, <span class="org-string">'Dnlm4'</span>, <span class="org-string">'Dnlm5'</span>, <span class="org-string">'Dnlm6'</span>}, {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}));
|
|
G_ine = minreal(G({<span class="org-string">'Vnlm1'</span>, <span class="org-string">'Vnlm2'</span>, <span class="org-string">'Vnlm3'</span>, <span class="org-string">'Vnlm4'</span>, <span class="org-string">'Vnlm5'</span>, <span class="org-string">'Vnlm6'</span>}, {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}));
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we save them for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_dvf'</span>, <span class="org-string">'G_ine'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgad0525d" class="outline-4">
|
|
<h4 id="orgad0525d"><span class="section-number-4">1.1.3</span> Obtained Plants for Active Damping</h4>
|
|
<div class="outline-text-4" id="text-1-1-3">
|
|
|
|
<div id="org6f986cc" class="figure">
|
|
<p><img src="figs/nass_active_damping_iff_plant.png" alt="nass_active_damping_iff_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span><code>G_iff</code>: IFF Plant (<a href="./figs/nass_active_damping_iff_plant.png">png</a>, <a href="./figs/nass_active_damping_iff_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org5c7577f" class="figure">
|
|
<p><img src="figs/nass_active_damping_ine_plant.png" alt="nass_active_damping_ine_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span><code>G_dvf</code>: Plant for Direct Velocity Feedback (<a href="./figs/nass_active_damping_dvf_plant.png">png</a>, <a href="./figs/nass_active_damping_dvf_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org2be03d7" class="figure">
|
|
<p><img src="figs/nass_active_damping_inertial_plant.png" alt="nass_active_damping_inertial_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Inertial Feedback Plant (<a href="./figs/nass_active_damping_inertial_plant.png">png</a>, <a href="./figs/nass_active_damping_inertial_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org77cdeea" class="outline-3">
|
|
<h3 id="org77cdeea"><span class="section-number-3">1.2</span> Tomography Experiment</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
</div>
|
|
<div id="outline-container-orgc066249" class="outline-4">
|
|
<h4 id="orgc066249"><span class="section-number-4">1.2.1</span> Simulation</h4>
|
|
<div class="outline-text-4" id="text-1-2-1">
|
|
<p>
|
|
We initialize elements for the tomography experiment.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">prepareTomographyExperiment();
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We change the simulation stop time.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we save the simulation results for further analysis
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'Eg'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8188f8d" class="outline-4">
|
|
<h4 id="org8188f8d"><span class="section-number-4">1.2.2</span> Results</h4>
|
|
<div class="outline-text-4" id="text-1-2-2">
|
|
<p>
|
|
We load the results of tomography experiments.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>);
|
|
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org3e63333" class="figure">
|
|
<p><img src="figs/nass_act_damp_undamped_sim_tomo_trans.png" alt="nass_act_damp_undamped_sim_tomo_trans.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_undamped_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_undamped_sim_tomo_trans.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org2dc3b85" class="figure">
|
|
<p><img src="figs/nass_act_damp_undamped_sim_tomo_rot.png" alt="nass_act_damp_undamped_sim_tomo_rot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_undamped_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_undamped_sim_tomo_rot.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org35ace9a" class="outline-2">
|
|
<h2 id="org35ace9a"><span class="section-number-2">2</span> Integral Force Feedback</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="org241a667"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/iff.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<p>
|
|
Integral Force Feedback is applied on the simscape model.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org64b724c" class="outline-3">
|
|
<h3 id="org64b724c"><span class="section-number-3">2.1</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
</div>
|
|
<div id="outline-container-orga60342f" class="outline-4">
|
|
<h4 id="orga60342f"><span class="section-number-4">2.1.1</span> Plant</h4>
|
|
<div class="outline-text-4" id="text-2-1-1">
|
|
<p>
|
|
Let’s load the previously indentified undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_iff'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure <a href="#org627c128">6</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org627c128" class="figure">
|
|
<p><img src="figs/iff_plant.png" alt="iff_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/iff_plant.png">png</a>, <a href="./figs/iff_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6e7a3b9" class="outline-4">
|
|
<h4 id="org6e7a3b9"><span class="section-number-4">2.1.2</span> Control Design</h4>
|
|
<div class="outline-text-4" id="text-2-1-2">
|
|
<p>
|
|
The controller for each pair of actuator/sensor is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_iff = 1000<span class="org-type">/</span>s;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The corresponding loop gains are shown in figure <a href="#org79e2120">7</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org79e2120" class="figure">
|
|
<p><img src="figs/iff_open_loop.png" alt="iff_open_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/iff_open_loop.png">png</a>, <a href="./figs/iff_open_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcffbce3" class="outline-4">
|
|
<h4 id="orgcffbce3"><span class="section-number-4">2.1.3</span> Diagonal Controller</h4>
|
|
<div class="outline-text-4" id="text-2-1-3">
|
|
<p>
|
|
We create the diagonal controller and we add a minus sign as we have a positive
|
|
feedback architecture.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_iff = <span class="org-type">-</span>K_iff<span class="org-type">*</span>eye(6);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We save the controller for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_iff.mat'</span>, <span class="org-string">'K_iff'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaac84c4" class="outline-4">
|
|
<h4 id="orgaac84c4"><span class="section-number-4">2.1.4</span> IFF with High Pass Filter</h4>
|
|
<div class="outline-text-4" id="text-2-1-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w_hpf = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; <span class="org-comment">% Cut-off frequency for the high pass filter [rad/s]</span>
|
|
w_lpf = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>200; <span class="org-comment">% Cut-off frequency for the low pass filter [rad/s]</span>
|
|
|
|
K_iff = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>200<span class="org-type">/</span>s <span class="org-type">*</span> (s<span class="org-type">/</span>w_hpf)<span class="org-type">/</span>(s<span class="org-type">/</span>w_hpf <span class="org-type">+</span> 1) <span class="org-type">*</span> 1<span class="org-type">/</span>(s<span class="org-type">/</span>w_lpf <span class="org-type">+</span> 1);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The corresponding loop gains are shown in figure <a href="#org29c690e">8</a>.
|
|
</p>
|
|
|
|
<div id="org29c690e" class="figure">
|
|
<p><img src="figs/iff_hpf_open_loop.png" alt="iff_hpf_open_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Loop Gain for the Integral Force Feedback with an High pass filter (<a href="./figs/iff_hpf_open_loop.png">png</a>, <a href="./figs/iff_hpf_open_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
We create the diagonal controller and we add a minus sign as we have a positive
|
|
feedback architecture.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_iff = <span class="org-type">-</span>K_iff<span class="org-type">*</span>eye(6);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We save the controller for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_iff_hpf.mat'</span>, <span class="org-string">'K_iff'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org641e0e1" class="outline-3">
|
|
<h3 id="org641e0e1"><span class="section-number-3">2.2</span> Tomography Experiment</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
</div>
|
|
<div id="outline-container-org244d5ff" class="outline-4">
|
|
<h4 id="org244d5ff"><span class="section-number-4">2.2.1</span> Simulation with IFF Controller</h4>
|
|
<div class="outline-text-4" id="text-2-2-1">
|
|
<p>
|
|
We initialize elements for the tomography experiment.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">prepareTomographyExperiment();
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We set the IFF controller.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_iff.mat'</span>, <span class="org-string">'K_iff'</span>);
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We change the simulation stop time.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we save the simulation results for further analysis
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">En_iff = En;
|
|
Eg_iff = Eg;
|
|
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_iff'</span>, <span class="org-string">'Eg_iff'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org40a229f" class="outline-4">
|
|
<h4 id="org40a229f"><span class="section-number-4">2.2.2</span> Simulation with IFF Controller with added High Pass Filter</h4>
|
|
<div class="outline-text-4" id="text-2-2-2">
|
|
<p>
|
|
We initialize elements for the tomography experiment.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">prepareTomographyExperiment();
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We set the IFF controller with the High Pass Filter.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_iff_hpf.mat'</span>, <span class="org-string">'K_iff'</span>);
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We change the simulation stop time.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we save the simulation results for further analysis
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">En_iff_hpf = En;
|
|
Eg_iff_hpf = Eg;
|
|
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_iff_hpf'</span>, <span class="org-string">'Eg_iff_hpf'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3982074" class="outline-4">
|
|
<h4 id="org3982074"><span class="section-number-4">2.2.3</span> Compare with Undamped system</h4>
|
|
<div class="outline-text-4" id="text-2-2-3">
|
|
<p>
|
|
We load the results of tomography experiments.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_iff'</span>, <span class="org-string">'En_iff_hpf'</span>);
|
|
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org60cba32" class="figure">
|
|
<p><img src="figs/nass_act_damp_iff_sim_tomo_xy.png" alt="nass_act_damp_iff_sim_tomo_xy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Position Error during tomography experiment - XY Motion (<a href="./figs/nass_act_damp_iff_sim_tomo_xy.png">png</a>, <a href="./figs/nass_act_damp_iff_sim_tomo_xy.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orga362dc6" class="figure">
|
|
<p><img src="figs/nass_act_damp_iff_sim_tomo_trans.png" alt="nass_act_damp_iff_sim_tomo_trans.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_iff_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_iff_sim_tomo_trans.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgf19ed92" class="figure">
|
|
<p><img src="figs/nass_act_damp_iff_sim_tomo_rot.png" alt="nass_act_damp_iff_sim_tomo_rot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_iff_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_iff_sim_tomo_rot.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbb282ac" class="outline-3">
|
|
<h3 id="orgbb282ac"><span class="section-number-3">2.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<div class="important">
|
|
<p>
|
|
Integral Force Feedback:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Robust (guaranteed stability)</li>
|
|
<li>Acceptable Damping</li>
|
|
<li>Increase the sensitivity to disturbances at low frequencies</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf20da86" class="outline-2">
|
|
<h2 id="orgf20da86"><span class="section-number-2">3</span> Direct Velocity Feedback</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgc0638ac"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/dvf.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<p>
|
|
In the Direct Velocity Feedback (DVF), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
|
|
The actuator displacement can be measured with a capacitive sensor for instance.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org7191e52" class="outline-3">
|
|
<h3 id="org7191e52"><span class="section-number-3">3.1</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
</div>
|
|
<div id="outline-container-org56b88cd" class="outline-4">
|
|
<h4 id="org56b88cd"><span class="section-number-4">3.1.1</span> Plant</h4>
|
|
<div class="outline-text-4" id="text-3-1-1">
|
|
<p>
|
|
Let’s load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_dvf'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure <a href="#orgbb00993">12</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgbb00993" class="figure">
|
|
<p><img src="figs/dvf_plant.png" alt="dvf_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/dvf_plant.png">png</a>, <a href="./figs/dvf_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8927c04" class="outline-4">
|
|
<h4 id="org8927c04"><span class="section-number-4">3.1.2</span> Control Design</h4>
|
|
<div class="outline-text-4" id="text-3-1-2">
|
|
<p>
|
|
The Direct Velocity Feedback is defined below.
|
|
A Low pass Filter is added to make the controller transfer function proper.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_dvf = s<span class="org-type">*</span>20000<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10000);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained loop gains are shown in figure <a href="#orga64af99">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orga64af99" class="figure">
|
|
<p><img src="figs/dvf_open_loop.png" alt="dvf_open_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/dvf_open_loop.png">png</a>, <a href="./figs/dvf_open_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org27b066a" class="outline-4">
|
|
<h4 id="org27b066a"><span class="section-number-4">3.1.3</span> Diagonal Controller</h4>
|
|
<div class="outline-text-4" id="text-3-1-3">
|
|
<p>
|
|
We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_dvf = <span class="org-type">-</span>K_dvf<span class="org-type">*</span>eye(6);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We save the controller for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_dvf.mat'</span>, <span class="org-string">'K_dvf'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7b075f1" class="outline-3">
|
|
<h3 id="org7b075f1"><span class="section-number-3">3.2</span> Tomography Experiment</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
</div>
|
|
<div id="outline-container-org29dea78" class="outline-4">
|
|
<h4 id="org29dea78"><span class="section-number-4">3.2.1</span> Initialize the Simulation</h4>
|
|
<div class="outline-text-4" id="text-3-2-1">
|
|
<p>
|
|
We initialize elements for the tomography experiment.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">prepareTomographyExperiment();
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We set the DVF controller.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_dvf.mat'</span>, <span class="org-string">'K_dvf'</span>);
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8b648f4" class="outline-4">
|
|
<h4 id="org8b648f4"><span class="section-number-4">3.2.2</span> Simulation</h4>
|
|
<div class="outline-text-4" id="text-3-2-2">
|
|
<p>
|
|
We change the simulation stop time.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we save the simulation results for further analysis
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">En_dvf = En;
|
|
Eg_dvf = Eg;
|
|
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_dvf'</span>, <span class="org-string">'Eg_dvf'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0def86d" class="outline-4">
|
|
<h4 id="org0def86d"><span class="section-number-4">3.2.3</span> Compare with Undamped system</h4>
|
|
<div class="outline-text-4" id="text-3-2-3">
|
|
<p>
|
|
We load the results of tomography experiments.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_dvf'</span>);
|
|
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgdb79ac6" class="figure">
|
|
<p><img src="figs/nass_act_damp_dvf_sim_tomo_xy.png" alt="nass_act_damp_dvf_sim_tomo_xy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Position Error during tomography experiment - XY Motion (<a href="./figs/nass_act_damp_dvf_sim_tomo_xy.png">png</a>, <a href="./figs/nass_act_damp_dvf_sim_tomo_xy.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org9757afc" class="figure">
|
|
<p><img src="figs/nass_act_damp_dvf_sim_tomo_trans.png" alt="nass_act_damp_dvf_sim_tomo_trans.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_dvf_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_dvf_sim_tomo_trans.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orge7307ca" class="figure">
|
|
<p><img src="figs/nass_act_damp_dvf_sim_tomo_rot.png" alt="nass_act_damp_dvf_sim_tomo_rot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_dvf_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_dvf_sim_tomo_rot.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org857cf6a" class="outline-3">
|
|
<h3 id="org857cf6a"><span class="section-number-3">3.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<div class="important">
|
|
<p>
|
|
Direct Velocity Feedback:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li></li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgad05b75" class="outline-2">
|
|
<h2 id="orgad05b75"><span class="section-number-2">4</span> Inertial Control</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="orgbea2985"></a>
|
|
</p>
|
|
<div class="note">
|
|
<p>
|
|
All the files (data and Matlab scripts) are accessible <a href="data/ine.zip">here</a>.
|
|
</p>
|
|
|
|
</div>
|
|
<p>
|
|
In Inertial Control, a feedback is applied between the measured <b>absolute</b> motion (velocity or acceleration) of the platform to the actuator force input.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgba4e94a" class="outline-3">
|
|
<h3 id="orgba4e94a"><span class="section-number-3">4.1</span> Control Design</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
</div>
|
|
<div id="outline-container-org5c18fbe" class="outline-4">
|
|
<h4 id="org5c18fbe"><span class="section-number-4">4.1.1</span> Plant</h4>
|
|
<div class="outline-text-4" id="text-4-1-1">
|
|
<p>
|
|
Let’s load the undamped plant:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_ine'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure <a href="#orgbb536dd">17</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgbb536dd" class="figure">
|
|
<p><img src="figs/ine_plant.png" alt="ine_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/ine_plant.png">png</a>, <a href="./figs/ine_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge529b8d" class="outline-4">
|
|
<h4 id="orge529b8d"><span class="section-number-4">4.1.2</span> Control Design</h4>
|
|
<div class="outline-text-4" id="text-4-1-2">
|
|
<p>
|
|
The controller is defined below and the obtained loop gain is shown in figure <a href="#orgfbc4d0e">18</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_ine = 1e4<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgfbc4d0e" class="figure">
|
|
<p><img src="figs/ine_open_loop_gain.png" alt="ine_open_loop_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Loop Gain for Inertial Control (<a href="./figs/ine_open_loop_gain.png">png</a>, <a href="./figs/ine_open_loop_gain.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6119dfa" class="outline-4">
|
|
<h4 id="org6119dfa"><span class="section-number-4">4.1.3</span> Diagonal Controller</h4>
|
|
<div class="outline-text-4" id="text-4-1-3">
|
|
<p>
|
|
We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K_ine = <span class="org-type">-</span>K_ine<span class="org-type">*</span>eye(6);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We save the controller for further analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_ine.mat'</span>, <span class="org-string">'K_ine'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbe8a895" class="outline-3">
|
|
<h3 id="orgbe8a895"><span class="section-number-3">4.2</span> Tomography Experiment</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
</div>
|
|
<div id="outline-container-org8526a14" class="outline-4">
|
|
<h4 id="org8526a14"><span class="section-number-4">4.2.1</span> Initialize the Simulation</h4>
|
|
<div class="outline-text-4" id="text-4-2-1">
|
|
<p>
|
|
We initialize elements for the tomography experiment.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">prepareTomographyExperiment();
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We set the Inertial controller.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_ine.mat'</span>, <span class="org-string">'K_ine'</span>);
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga670755" class="outline-4">
|
|
<h4 id="orga670755"><span class="section-number-4">4.2.2</span> Simulation</h4>
|
|
<div class="outline-text-4" id="text-4-2-2">
|
|
<p>
|
|
We change the simulation stop time.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we save the simulation results for further analysis
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">En_ine = En;
|
|
Eg_ine = Eg;
|
|
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_ine'</span>, <span class="org-string">'Eg_ine'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgdc292f7" class="outline-4">
|
|
<h4 id="orgdc292f7"><span class="section-number-4">4.2.3</span> Compare with Undamped system</h4>
|
|
<div class="outline-text-4" id="text-4-2-3">
|
|
<p>
|
|
We load the results of tomography experiments.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_ine'</span>);
|
|
t = linspace(0, 3, length(En_ine(<span class="org-type">:</span>,1)));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org46006fa" class="figure">
|
|
<p><img src="figs/nass_act_damp_ine_sim_tomo_xy.png" alt="nass_act_damp_ine_sim_tomo_xy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Position Error during tomography experiment - XY Motion (<a href="./figs/nass_act_damp_ine_sim_tomo_xy.png">png</a>, <a href="./figs/nass_act_damp_ine_sim_tomo_xy.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgb258d75" class="figure">
|
|
<p><img src="figs/nass_act_damp_ine_sim_tomo_trans.png" alt="nass_act_damp_ine_sim_tomo_trans.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_ine_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_ine_sim_tomo_trans.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org93e0404" class="figure">
|
|
<p><img src="figs/nass_act_damp_ine_sim_tomo_rot.png" alt="nass_act_damp_ine_sim_tomo_rot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_ine_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_ine_sim_tomo_rot.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4675ff3" class="outline-3">
|
|
<h3 id="org4675ff3"><span class="section-number-3">4.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="important">
|
|
<p>
|
|
Inertial Control:
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2cefbfb" class="outline-2">
|
|
<h2 id="org2cefbfb"><span class="section-number-2">5</span> Comparison</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org96bfb90"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org8cef845" class="outline-3">
|
|
<h3 id="org8cef845"><span class="section-number-3">5.1</span> Load the plants</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_ine'</span>, <span class="org-string">'G_dvf'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4a8616e" class="outline-3">
|
|
<h3 id="org4a8616e"><span class="section-number-3">5.2</span> Sensitivity to Disturbance</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
|
|
<div id="orgd8c0d0b" class="figure">
|
|
<p><img src="figs/sensitivity_comp_ground_motion_z.png" alt="sensitivity_comp_ground_motion_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>caption (<a href="./figs/sensitivity_comp_ground_motion_z.png">png</a>, <a href="./figs/sensitivity_comp_ground_motion_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="orgd07cb74" class="figure">
|
|
<p><img src="figs/sensitivity_comp_direct_forces_z.png" alt="sensitivity_comp_direct_forces_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 23: </span>caption (<a href="./figs/sensitivity_comp_direct_forces_z.png">png</a>, <a href="./figs/sensitivity_comp_direct_forces_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org8e48ed2" class="figure">
|
|
<p><img src="figs/sensitivity_comp_spindle_z.png" alt="sensitivity_comp_spindle_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 24: </span>caption (<a href="./figs/sensitivity_comp_spindle_z.png">png</a>, <a href="./figs/sensitivity_comp_spindle_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgefa125c" class="figure">
|
|
<p><img src="figs/sensitivity_comp_ty_z.png" alt="sensitivity_comp_ty_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 25: </span>caption (<a href="./figs/sensitivity_comp_ty_z.png">png</a>, <a href="./figs/sensitivity_comp_ty_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org46a1428" class="figure">
|
|
<p><img src="figs/sensitivity_comp_ty_x.png" alt="sensitivity_comp_ty_x.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 26: </span>caption (<a href="./figs/sensitivity_comp_ty_x.png">png</a>, <a href="./figs/sensitivity_comp_ty_x.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga89ca48" class="outline-3">
|
|
<h3 id="orga89ca48"><span class="section-number-3">5.3</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
|
|
<div id="orge6466ce" class="figure">
|
|
<p><img src="figs/plant_comp_damping_z.png" alt="plant_comp_damping_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 27: </span>Plant for the \(z\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_z.png">png</a>, <a href="./figs/plant_comp_damping_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org48a88cc" class="figure">
|
|
<p><img src="figs/plant_comp_damping_x.png" alt="plant_comp_damping_x.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 28: </span>Plant for the \(x\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_x.png">png</a>, <a href="./figs/plant_comp_damping_x.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgeec8390" class="figure">
|
|
<p><img src="figs/plant_comp_damping_coupling.png" alt="plant_comp_damping_coupling.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 29: </span>Comparison of one off-diagonal plant for different damping technique applied (<a href="./figs/plant_comp_damping_coupling.png">png</a>, <a href="./figs/plant_comp_damping_coupling.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org7a74f40" class="outline-3">
|
|
<h3 id="org7a74f40"><span class="section-number-3">5.4</span> Tomography Experiment</h3>
|
|
<div class="outline-text-3" id="text-5-4">
|
|
</div>
|
|
<div id="outline-container-org294c860" class="outline-4">
|
|
<h4 id="org294c860"><span class="section-number-4">5.4.1</span> Load the Simulation Data</h4>
|
|
<div class="outline-text-4" id="text-5-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_iff_hpf'</span>, <span class="org-string">'En_dvf'</span>, <span class="org-string">'En_ine'</span>);
|
|
En_iff = En_iff_hpf;
|
|
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org70e87cc" class="outline-4">
|
|
<h4 id="org70e87cc"><span class="section-number-4">5.4.2</span> Frequency Domain Analysis</h4>
|
|
<div class="outline-text-4" id="text-5-4-2">
|
|
<p>
|
|
Window used for <code>pwelch</code> function.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">n_av = 8;
|
|
han_win = hanning(ceil(length(En(<span class="org-type">:</span>, 1))<span class="org-type">/</span>n_av));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org5597081" class="figure">
|
|
<p><img src="figs/act_damp_tomo_exp_comp_psd_trans.png" alt="act_damp_tomo_exp_comp_psd_trans.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 30: </span>PSD of the translation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_psd_trans.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_psd_trans.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org3145e09" class="figure">
|
|
<p><img src="figs/act_damp_tomo_exp_comp_psd_rot.png" alt="act_damp_tomo_exp_comp_psd_rot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 31: </span>PSD of the rotation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_psd_rot.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_psd_rot.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org4e62d44" class="figure">
|
|
<p><img src="figs/act_damp_tomo_exp_comp_cps_trans.png" alt="act_damp_tomo_exp_comp_cps_trans.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 32: </span>CPS of the translation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_cps_trans.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_cps_trans.pdf">pdf</a>)</p>
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</div>
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<div id="org634790f" class="figure">
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<p><img src="figs/act_damp_tomo_exp_comp_cps_rot.png" alt="act_damp_tomo_exp_comp_cps_rot.png" />
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</p>
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<p><span class="figure-number">Figure 33: </span>CPS of the rotation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_cps_rot.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_cps_rot.pdf">pdf</a>)</p>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org952ed5d" class="outline-2">
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<h2 id="org952ed5d"><span class="section-number-2">6</span> Useful Functions</h2>
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<div class="outline-text-2" id="text-6">
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</div>
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<div id="outline-container-orga9b2a73" class="outline-3">
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<h3 id="orga9b2a73"><span class="section-number-3">6.1</span> prepareTomographyExperiment</h3>
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<div class="outline-text-3" id="text-6-1">
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<p>
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<a id="org32c98de"></a>
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</p>
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<p>
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This Matlab function is accessible <a href="src/prepareTomographyExperiment.m">here</a>.
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</p>
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</div>
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<div id="outline-container-orgdc8e362" class="outline-4">
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<h4 id="orgdc8e362">Function Description</h4>
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<div class="outline-text-4" id="text-orgdc8e362">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[]</span> = <span class="org-function-name">prepareTomographyExperiment</span>(<span class="org-variable-name">args</span>)
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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-org2d80135" class="outline-4">
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<h4 id="org2d80135">Optional Parameters</h4>
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<div class="outline-text-4" id="text-org2d80135">
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<div class="org-src-container">
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<pre class="src src-matlab">arguments
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args.nass_actuator char {mustBeMember(args.nass_actuator,{<span class="org-string">'piezo'</span>, <span class="org-string">'lorentz'</span>})} = <span class="org-string">'piezo'</span>
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args.sample_mass (1,1) double {mustBeNumeric, mustBePositive} = 50
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args.Ry_period (1,1) double {mustBeNumeric, mustBePositive} = 1
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<span class="org-keyword">end</span>
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</pre>
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</div>
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</div>
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</div>
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|
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<div id="outline-container-org342966f" class="outline-4">
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<h4 id="org342966f">Initialize the Simulation</h4>
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<div class="outline-text-4" id="text-org342966f">
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<p>
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We initialize all the stages with the default parameters.
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</p>
|
|
<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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|
initializeMirror();
|
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</pre>
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</div>
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|
|
<p>
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The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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</p>
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|
<div class="org-src-container">
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<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, args.nass_actuator);
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initializeSample(<span class="org-string">'mass'</span>, args.sample_mass);
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</pre>
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</div>
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|
|
|
<p>
|
|
We set the references to zero.
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</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, args.Ry_period);
|
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</pre>
|
|
</div>
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|
|
|
<p>
|
|
And all the controllers are set to 0.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
|
|
K_ine = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>);
|
|
K_iff = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
|
|
K_dvf = tf(zeros(6));
|
|
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
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</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-01-20 lun. 17:36</p>
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</div>
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</body>
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</html>
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