1574 lines
62 KiB
HTML
1574 lines
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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="#org9342dde">1. Undamped System</a>
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<ul>
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<li><a href="#org8d4b4f6">1.1. Identification of the dynamics for Active Damping</a>
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<ul>
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<li><a href="#orgb855cd6">1.1.1. Initialize the Simulation</a></li>
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<li><a href="#orgb12b84c">1.1.2. Identification</a></li>
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<li><a href="#org34365e1">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="#orgdb2efe9">1.2. Tomography Experiment</a>
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<ul>
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<li><a href="#orgfb2c922">1.2.1. Simulation</a></li>
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<li><a href="#org264103d">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="#org67089f2">2. Integral Force Feedback</a>
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<ul>
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<li><a href="#orgf44db18">2.1. Control Design</a>
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<ul>
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<li><a href="#orgf1f220a">2.1.1. Plant</a></li>
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<li><a href="#org9e39d4f">2.1.2. Control Design</a></li>
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<li><a href="#org220ad45">2.1.3. Diagonal Controller</a></li>
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<li><a href="#org6f43c28">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="#orgbd702ce">2.2. Tomography Experiment</a>
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<ul>
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<li><a href="#org6c6f2d1">2.2.1. Simulation with IFF Controller</a></li>
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<li><a href="#orgad8bea8">2.2.2. Simulation with IFF Controller with added High Pass Filter</a></li>
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<li><a href="#orgedf02f4">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="#orgc5f3f9b">2.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org685023c">3. Direct Velocity Feedback</a>
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<ul>
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<li><a href="#org60b5b30">3.1. Control Design</a>
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<ul>
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<li><a href="#orgf848412">3.1.1. Plant</a></li>
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<li><a href="#org42b7fca">3.1.2. Control Design</a></li>
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<li><a href="#org2b8ee35">3.1.3. Diagonal Controller</a></li>
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</ul>
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</li>
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<li><a href="#org508cd3e">3.2. Tomography Experiment</a>
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<ul>
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<li><a href="#org8f01b99">3.2.1. Initialize the Simulation</a></li>
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<li><a href="#org57c8573">3.2.2. Simulation</a></li>
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<li><a href="#org3b80a89">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="#orgbfc0305">3.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org97c7099">4. Inertial Control</a>
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<ul>
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<li><a href="#org3d0c40e">4.1. Control Design</a>
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<ul>
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<li><a href="#org7d50913">4.1.1. Plant</a></li>
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<li><a href="#org3bd6be5">4.1.2. Control Design</a></li>
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<li><a href="#orgb77c3b3">4.1.3. Diagonal Controller</a></li>
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</ul>
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</li>
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<li><a href="#orgfa1ca32">4.2. Tomography Experiment</a>
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<ul>
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<li><a href="#orgf686c33">4.2.1. Initialize the Simulation</a></li>
|
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<li><a href="#org1a7fe63">4.2.2. Simulation</a></li>
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<li><a href="#orge8b8bf8">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="#orgde615df">4.3. Conclusion</a></li>
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</ul>
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</li>
|
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<li><a href="#orgada4654">5. Comparison</a>
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<ul>
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<li><a href="#org2976576">5.1. Load the plants</a></li>
|
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<li><a href="#org835d9db">5.2. Sensitivity to Disturbance</a></li>
|
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<li><a href="#orga94123d">5.3. Damped Plant</a></li>
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<li><a href="#org75eccc0">5.4. Tomography Experiment</a>
|
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<ul>
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<li><a href="#org341a139">5.4.1. Load the Simulation Data</a></li>
|
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<li><a href="#org267c98c">5.4.2. Frequency Domain Analysis</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="#org5f0f434">6. Useful Functions</a>
|
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<ul>
|
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<li><a href="#org6ecc4dc">6.1. prepareTomographyExperiment</a>
|
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<ul>
|
|
<li><a href="#org435d4b8">Function Description</a></li>
|
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<li><a href="#org45dbf99">Optional Parameters</a></li>
|
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<li><a href="#org80d47ea">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="#org02231b3">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="#orgd1ce2e4">2</a>: the integral force feedback is used</li>
|
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<li>In section <a href="#orgafbde82">3</a>: the direct velocity feedback is used</li>
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<li>In section <a href="#org431eadf">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-org9342dde" class="outline-2">
|
|
<h2 id="org9342dde"><span class="section-number-2">1</span> Undamped System</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
<a id="org02231b3"></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-org8d4b4f6" class="outline-3">
|
|
<h3 id="org8d4b4f6"><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-orgb855cd6" class="outline-4">
|
|
<h4 id="orgb855cd6"><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-orgb12b84c" class="outline-4">
|
|
<h4 id="orgb12b84c"><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-org34365e1" class="outline-4">
|
|
<h4 id="org34365e1"><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="orgac91a7a" 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="org97d6cca" 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="orgbb6a16f" 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-orgdb2efe9" class="outline-3">
|
|
<h3 id="orgdb2efe9"><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-orgfb2c922" class="outline-4">
|
|
<h4 id="orgfb2c922"><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-org264103d" class="outline-4">
|
|
<h4 id="org264103d"><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="org625b4f8" 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="orgd213d43" 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-org67089f2" class="outline-2">
|
|
<h2 id="org67089f2"><span class="section-number-2">2</span> Integral Force Feedback</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
<a id="orgd1ce2e4"></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-orgf44db18" class="outline-3">
|
|
<h3 id="orgf44db18"><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-orgf1f220a" class="outline-4">
|
|
<h4 id="orgf1f220a"><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="#orgac31951">6</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgac31951" 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-org9e39d4f" class="outline-4">
|
|
<h4 id="org9e39d4f"><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="#org4d8791f">7</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org4d8791f" 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-org220ad45" class="outline-4">
|
|
<h4 id="org220ad45"><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-org6f43c28" class="outline-4">
|
|
<h4 id="org6f43c28"><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="#orgee0b506">8</a>.
|
|
</p>
|
|
|
|
<div id="orgee0b506" 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-orgbd702ce" class="outline-3">
|
|
<h3 id="orgbd702ce"><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-org6c6f2d1" class="outline-4">
|
|
<h4 id="org6c6f2d1"><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-orgad8bea8" class="outline-4">
|
|
<h4 id="orgad8bea8"><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-orgedf02f4" class="outline-4">
|
|
<h4 id="orgedf02f4"><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="orgbe947f0" 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="orgc275297" 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="org03d0c7d" 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-orgc5f3f9b" class="outline-3">
|
|
<h3 id="orgc5f3f9b"><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-org685023c" class="outline-2">
|
|
<h2 id="org685023c"><span class="section-number-2">3</span> Direct Velocity Feedback</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgafbde82"></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-org60b5b30" class="outline-3">
|
|
<h3 id="org60b5b30"><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-orgf848412" class="outline-4">
|
|
<h4 id="orgf848412"><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="#org8ead4ee">12</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org8ead4ee" 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-org42b7fca" class="outline-4">
|
|
<h4 id="org42b7fca"><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="#orge5257d3">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orge5257d3" 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-org2b8ee35" class="outline-4">
|
|
<h4 id="org2b8ee35"><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-org508cd3e" class="outline-3">
|
|
<h3 id="org508cd3e"><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-org8f01b99" class="outline-4">
|
|
<h4 id="org8f01b99"><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-org57c8573" class="outline-4">
|
|
<h4 id="org57c8573"><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-org3b80a89" class="outline-4">
|
|
<h4 id="org3b80a89"><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="org9c27ffd" 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="org9e590a1" 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="orgf861835" 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-orgbfc0305" class="outline-3">
|
|
<h3 id="orgbfc0305"><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-org97c7099" class="outline-2">
|
|
<h2 id="org97c7099"><span class="section-number-2">4</span> Inertial Control</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org431eadf"></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-org3d0c40e" class="outline-3">
|
|
<h3 id="org3d0c40e"><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-org7d50913" class="outline-4">
|
|
<h4 id="org7d50913"><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="#org5d67443">17</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org5d67443" 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-org3bd6be5" class="outline-4">
|
|
<h4 id="org3bd6be5"><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="#org2b27a68">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="org2b27a68" 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-orgb77c3b3" class="outline-4">
|
|
<h4 id="orgb77c3b3"><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-orgfa1ca32" class="outline-3">
|
|
<h3 id="orgfa1ca32"><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-orgf686c33" class="outline-4">
|
|
<h4 id="orgf686c33"><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-org1a7fe63" class="outline-4">
|
|
<h4 id="org1a7fe63"><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-orge8b8bf8" class="outline-4">
|
|
<h4 id="orge8b8bf8"><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="orgab8bdcc" 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="org179b8c0" 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="org5050229" 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-orgde615df" class="outline-3">
|
|
<h3 id="orgde615df"><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-orgada4654" class="outline-2">
|
|
<h2 id="orgada4654"><span class="section-number-2">5</span> Comparison</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org98830ef"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org2976576" class="outline-3">
|
|
<h3 id="org2976576"><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-org835d9db" class="outline-3">
|
|
<h3 id="org835d9db"><span class="section-number-3">5.2</span> Sensitivity to Disturbance</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
|
|
<div id="orgdf47c67" 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>Sensitivity to ground motion in the Z direction on the Z motion error (<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="orgfc493eb" 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>Compliance in the Z direction: Sensitivity of direct forces applied on the sample in the Z direction on the Z motion error (<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="org31dee6d" 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>Sensitivity to forces applied in the Z direction by the Spindle on the Z motion error (<a href="./figs/sensitivity_comp_spindle_z.png">png</a>, <a href="./figs/sensitivity_comp_spindle_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgc54a0c8" 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>Sensitivity to forces applied in the Z direction by the Y translation stage on the Z motion error (<a href="./figs/sensitivity_comp_ty_z.png">png</a>, <a href="./figs/sensitivity_comp_ty_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="orgf732728" 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>Sensitivity to forces applied in the X direction by the Y translation stage on the X motion error (<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-orga94123d" class="outline-3">
|
|
<h3 id="orga94123d"><span class="section-number-3">5.3</span> Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
|
|
<div id="org7f947f5" 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="orgaf1a7e2" 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="org421e4e1" 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-org75eccc0" class="outline-3">
|
|
<h3 id="org75eccc0"><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-org341a139" class="outline-4">
|
|
<h4 id="org341a139"><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-org267c98c" class="outline-4">
|
|
<h4 id="org267c98c"><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="orga580e07" 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="orgdf9a147" 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>
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</div>
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<div id="orge199cbc" class="figure">
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<p><img src="figs/act_damp_tomo_exp_comp_cps_trans.png" alt="act_damp_tomo_exp_comp_cps_trans.png" />
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</p>
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<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="org4b2375c" 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-org5f0f434" class="outline-2">
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<h2 id="org5f0f434"><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-org6ecc4dc" class="outline-3">
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<h3 id="org6ecc4dc"><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="orgf80e6d6"></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-org435d4b8" class="outline-4">
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<h4 id="org435d4b8">Function Description</h4>
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<div class="outline-text-4" id="text-org435d4b8">
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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-org45dbf99" class="outline-4">
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<h4 id="org45dbf99">Optional Parameters</h4>
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<div class="outline-text-4" id="text-org45dbf99">
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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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<div id="outline-container-org80d47ea" class="outline-4">
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<h4 id="org80d47ea">Initialize the Simulation</h4>
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<div class="outline-text-4" id="text-org80d47ea">
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<p>
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We initialize all the stages with the default parameters.
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</p>
|
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<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();
|
|
</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>
|
|
<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.
|
|
</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);
|
|
</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>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-01-20 lun. 17:40</p>
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</div>
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</body>
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</html>
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