1298 lines
44 KiB
HTML
1298 lines
44 KiB
HTML
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<title>Control of the NASS with Voice coil actuators</title>
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<a accesskey="h" href="./index.html"> UP </a>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Control of the NASS with Voice coil actuators</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="#orge379987">1. HAC-LAC + Cascade Control Topology</a>
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<ul>
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<li><a href="#orgcaec167">1.1. Initialization</a></li>
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<li><a href="#orgf95b045">1.2. Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)</a>
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<ul>
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<li><a href="#org63f831b">1.2.1. Identification</a></li>
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<li><a href="#org203d651">1.2.2. Plant</a></li>
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<li><a href="#orgccc21d2">1.2.3. Root Locus</a></li>
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<li><a href="#org1a8ee8a">1.2.4. Controller and Loop Gain</a></li>
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</ul>
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</li>
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<li><a href="#org2a44e66">1.3. High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</a>
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<ul>
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<li><a href="#org989c2e9">1.3.1. Identification of the damped plant</a></li>
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<li><a href="#orgc3f6788">1.3.2. Obtained Plant</a></li>
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<li><a href="#orgd1632cf">1.3.3. Controller Design and Loop Gain</a></li>
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</ul>
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</li>
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<li><a href="#org0cd0c63">1.4. Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</a>
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<ul>
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<li><a href="#orga960106">1.4.1. Identification of the linearized plant</a></li>
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<li><a href="#orgce4f796">1.4.2. Obtained Plant</a></li>
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<li><a href="#org16f56fa">1.4.3. Controller Design</a></li>
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</ul>
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</li>
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<li><a href="#org9406388">1.5. Simulation</a></li>
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<li><a href="#org16024e0">1.6. Results</a>
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<ul>
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<li><a href="#org8e90e5a">1.6.1. Load the simulation results</a></li>
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<li><a href="#org0f974ff">1.6.2. Control effort</a></li>
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<li><a href="#org36679a6">1.6.3. Load the simulation results</a></li>
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</ul>
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</li>
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<li><a href="#org74d9dc7">1.7. Compliance of the nano-hexapod</a>
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<ul>
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<li><a href="#orgd768713">1.7.1. Identification</a></li>
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<li><a href="#org1a1ad20">1.7.2. Obtained Compliance</a></li>
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<li><a href="#org5b81db9">1.7.3. Comparison with Piezo</a></li>
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</ul>
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</li>
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<li><a href="#org3c7ca09">1.8. Robustness to Payload Variability</a>
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<ul>
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<li><a href="#orgf34ee18">1.8.1. Initialization</a></li>
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<li><a href="#org0d97c55">1.8.2. Low Authority Control</a></li>
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<li><a href="#orgab3ab98">1.8.3. High Authority Control</a></li>
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<li><a href="#org176695a">1.8.4. Primary Plant</a></li>
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<li><a href="#org678ff20">1.8.5. 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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<li><a href="#orgd649256">2. Other analysis</a>
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<ul>
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<li><a href="#org6cee30e">2.1. Robustness to Payload Variability</a></li>
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<li><a href="#org18b00fa">2.2. Direct HAC control in the task space - \(\bm{K}_\mathcal{X}\)</a>
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<ul>
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<li><a href="#org625b500">2.2.1. Identification</a></li>
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<li><a href="#org1387536">2.2.2. Obtained Plant in the Task Space</a></li>
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<li><a href="#org50b9f75">2.2.3. Obtained Plant in the Joint Space</a></li>
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<li><a href="#org19db2cd">2.2.4. Controller Design in the Joint Space</a></li>
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</ul>
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</li>
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<li><a href="#org5e26e70">2.3. On the usefulness of the High Authority Control loop / Linearization loop</a>
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<ul>
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<li><a href="#org7366662">2.3.1. Identification</a></li>
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<li><a href="#orgfab8847">2.3.2. Plant in the Task space</a></li>
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<li><a href="#org18aeea5">2.3.3. Plant in the Leg’s space</a></li>
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</ul>
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</li>
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<li><a href="#org015f992">2.4. DVF instead of IFF?</a>
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<ul>
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<li><a href="#org17cfb9d">2.4.1. Initialization and Identification</a></li>
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<li><a href="#orgd7117f7">2.4.2. Obtained Plant</a></li>
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<li><a href="#org51c8027">2.4.3. Controller</a></li>
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<li><a href="#org33637f1">2.4.4. HAC Identification</a></li>
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<li><a href="#orgec66083">2.4.5. Conclusion</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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<div id="outline-container-orge379987" class="outline-2">
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<h2 id="orge379987"><span class="section-number-2">1</span> HAC-LAC + Cascade Control Topology</h2>
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<div class="outline-text-2" id="text-1">
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<div id="org757d77b" class="figure">
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<p><img src="figs/cascade_control_architecture.png" alt="cascade_control_architecture.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
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</div>
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</div>
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<div id="outline-container-orgcaec167" class="outline-3">
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<h3 id="orgcaec167"><span class="section-number-3">1.1</span> Initialization</h3>
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<div class="outline-text-3" id="text-1-1">
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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();
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</pre>
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</div>
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<p>
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The nano-hexapod is a voice coil based hexapod and the sample has a mass of 1kg.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeNanoHexapod('actuator', 'lorentz');
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initializeSample('mass', 1);
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</pre>
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</div>
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<p>
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We set the references that corresponds to a tomography experiment.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeReferences('Rz_type', 'rotating', 'Rz_period', 1);
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeDisturbances();
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeController('type', 'cascade-hac-lac');
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeSimscapeConfiguration('gravity', true);
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</pre>
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</div>
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<p>
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We log the signals.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeLoggingConfiguration('log', 'all');
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">Kp = tf(zeros(6));
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Kl = tf(zeros(6));
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Kiff = tf(zeros(6));
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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-orgf95b045" class="outline-3">
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<h3 id="orgf95b045"><span class="section-number-3">1.2</span> Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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<a id="org224edef"></a>
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</p>
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</div>
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<div id="outline-container-org63f831b" class="outline-4">
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<h4 id="org63f831b"><span class="section-number-4">1.2.1</span> Identification</h4>
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<div class="outline-text-4" id="text-1-2-1">
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<p>
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Let’s first identify the plant for the IFF controller.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">%% Name of the Simulink File
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mdl = 'nass_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
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%% Run the linearization
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G_iff = linearize(mdl, io, 0);
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G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
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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-org203d651" class="outline-4">
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<h4 id="org203d651"><span class="section-number-4">1.2.2</span> Plant</h4>
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<div class="outline-text-4" id="text-1-2-2">
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<p>
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The obtained plant for IFF is shown in Figure <a href="#orga39f9fa">2</a>.
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</p>
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<div id="orga39f9fa" class="figure">
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<p><img src="figs/cascade_vc_iff_plant.png" alt="cascade_vc_iff_plant.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>IFF Plant (<a href="./figs/cascade_vc_iff_plant.png">png</a>, <a href="./figs/cascade_vc_iff_plant.pdf">pdf</a>)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgccc21d2" class="outline-4">
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<h4 id="orgccc21d2"><span class="section-number-4">1.2.3</span> Root Locus</h4>
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<div class="outline-text-4" id="text-1-2-3">
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<p>
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As seen in the root locus (Figure <a href="#org528b5f0">3</a>, no damping can be added to modes corresponding to the resonance of the micro-station.
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</p>
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<p>
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However, critical damping can be achieve for the resonances of the nano-hexapod as shown in the zoomed part of the root (Figure <a href="#org528b5f0">3</a>, left part).
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The maximum damping is obtained for a control gain of \(\approx 70\).
|
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</p>
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<div id="org528b5f0" class="figure">
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<p><img src="figs/cascade_vc_iff_root_locus.png" alt="cascade_vc_iff_root_locus.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Root Locus for the IFF control (<a href="./figs/cascade_vc_iff_root_locus.png">png</a>, <a href="./figs/cascade_vc_iff_root_locus.pdf">pdf</a>)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org1a8ee8a" class="outline-4">
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<h4 id="org1a8ee8a"><span class="section-number-4">1.2.4</span> Controller and Loop Gain</h4>
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<div class="outline-text-4" id="text-1-2-4">
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<p>
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We create the \(6 \times 6\) diagonal Integral Force Feedback controller.
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The obtained loop gain is shown in Figure <a href="#orgc890275">4</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Kiff = -70/s*eye(6);
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</pre>
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</div>
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<div id="orgc890275" class="figure">
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<p><img src="figs/cascade_vc_iff_loop_gain.png" alt="cascade_vc_iff_loop_gain.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Obtained Loop gain the IFF Control (<a href="./figs/cascade_vc_iff_loop_gain.png">png</a>, <a href="./figs/cascade_vc_iff_loop_gain.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 id="outline-container-org2a44e66" class="outline-3">
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<h3 id="org2a44e66"><span class="section-number-3">1.3</span> High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<p>
|
|
<a id="org1d54e1b"></a>
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</p>
|
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</div>
|
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<div id="outline-container-org989c2e9" class="outline-4">
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<h4 id="org989c2e9"><span class="section-number-4">1.3.1</span> Identification of the damped plant</h4>
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<div class="outline-text-4" id="text-1-3-1">
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<p>
|
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Let’s identify the dynamics from \(\bm{\tau}^\prime\) to \(d\bm{\mathcal{L}}\) as shown in Figure <a href="#org757d77b">1</a>.
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</p>
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|
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<div class="org-src-container">
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<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'output', [], 'Dnlm'); io_i = io_i + 1; % Leg Displacement
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|
|
|
%% Run the linearization
|
|
Gl = linearize(mdl, io, 0);
|
|
Gl.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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Gl.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
|
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</pre>
|
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</div>
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<p>
|
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There are some unstable poles in the Plant with very small imaginary parts.
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These unstable poles are probably not physical, and they disappear when taking the minimum realization of the plant.
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</p>
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<div class="org-src-container">
|
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<pre class="src src-matlab">isstable(Gl)
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|
Gl = minreal(Gl);
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isstable(Gl)
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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-orgc3f6788" class="outline-4">
|
|
<h4 id="orgc3f6788"><span class="section-number-4">1.3.2</span> Obtained Plant</h4>
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<div class="outline-text-4" id="text-1-3-2">
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<p>
|
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The obtained dynamics is shown in Figure <a href="#orgd1818fd">5</a>.
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</p>
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<p>
|
|
Few things can be said on the dynamics:
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</p>
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<ul class="org-ul">
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<li>the dynamics of the diagonal elements are almost all the same</li>
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<li>the system is well decoupled below the resonances of the nano-hexapod (1Hz)</li>
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<li>the dynamics of the diagonal elements are almost equivalent to a critically damped mass-spring-system with some spurious resonances above 50Hz corresponding to the resonances of the micro-station</li>
|
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</ul>
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|
|
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<div id="orgd1818fd" class="figure">
|
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<p><img src="figs/cascade_vc_hac_joint_plant.png" alt="cascade_vc_hac_joint_plant.png" />
|
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</p>
|
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<p><span class="figure-number">Figure 5: </span>Plant for the High Authority Control in the Joint Space (<a href="./figs/cascade_vc_hac_joint_plant.png">png</a>, <a href="./figs/cascade_vc_hac_joint_plant.pdf">pdf</a>)</p>
|
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</div>
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</div>
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</div>
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<div id="outline-container-orgd1632cf" class="outline-4">
|
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<h4 id="orgd1632cf"><span class="section-number-4">1.3.3</span> Controller Design and Loop Gain</h4>
|
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<div class="outline-text-4" id="text-1-3-3">
|
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<p>
|
|
As the plant is well decoupled, a diagonal plant is designed.
|
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</p>
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<div class="org-src-container">
|
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<pre class="src src-matlab">wc = 2*pi*10; % Bandwidth Bandwidth [rad/s]
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h = 2; % Lead parameter
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Kl = (s + 2*pi*1)/s;
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% Normalization of the gain of have a loop gain of 1 at frequency wc
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Kl = Kl.*diag(1./diag(abs(freqresp(Gl*Kl, wc))));
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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>
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<div id="outline-container-org0cd0c63" class="outline-3">
|
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<h3 id="org0cd0c63"><span class="section-number-3">1.4</span> Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</h3>
|
|
<div class="outline-text-3" id="text-1-4">
|
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<p>
|
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<a id="orga738520"></a>
|
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</p>
|
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</div>
|
|
<div id="outline-container-orga960106" class="outline-4">
|
|
<h4 id="orga960106"><span class="section-number-4">1.4.1</span> Identification of the linearized plant</h4>
|
|
<div class="outline-text-4" id="text-1-4-1">
|
|
<p>
|
|
We know identify the dynamics between \(\bm{r}_{\mathcal{X}_n}\) and \(\bm{r}_\mathcal{X}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller/Cascade-HAC-LAC/Kp'], 1, 'input'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
|
|
%% Run the linearization
|
|
Gp = linearize(mdl, io, 0);
|
|
Gp.InputName = {'rl1', 'rl2', 'rl3', 'rl4', 'rl5', 'rl6'};
|
|
Gp.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
A minus sign is added because the minus sign is already included in the plant identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(Gp)
|
|
Gp = -minreal(Gp);
|
|
isstable(Gp)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/stages.mat', 'nano_hexapod');
|
|
Gpx = Gp*inv(nano_hexapod.kinematics.J');
|
|
Gpx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
|
|
Gpl = nano_hexapod.kinematics.J*Gp;
|
|
Gpl.OutputName = {'El1', 'El2', 'El3', 'El4', 'El5', 'El6'};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgce4f796" class="outline-4">
|
|
<h4 id="orgce4f796"><span class="section-number-4">1.4.2</span> Obtained Plant</h4>
|
|
<div class="outline-text-4" id="text-1-4-2">
|
|
|
|
<div id="org8e042d5" class="figure">
|
|
<p><img src="figs/primary_plant_voice_coil_X.png" alt="primary_plant_voice_coil_X.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Obtained Primary plant in the Task space (<a href="./figs/primary_plant_voice_coil_X.png">png</a>, <a href="./figs/primary_plant_voice_coil_X.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org2cb4d6f" class="figure">
|
|
<p><img src="figs/primary_plant_voice_coil_L.png" alt="primary_plant_voice_coil_L.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Obtained Primary plant in the frame of the legs (<a href="./figs/primary_plant_voice_coil_L.png">png</a>, <a href="./figs/primary_plant_voice_coil_L.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org16f56fa" class="outline-4">
|
|
<h4 id="org16f56fa"><span class="section-number-4">1.4.3</span> Controller Design</h4>
|
|
<div class="outline-text-4" id="text-1-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wc = 2*pi*200; % Bandwidth Bandwidth [rad/s]
|
|
|
|
h = 2; % Lead parameter
|
|
|
|
Kp = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ...
|
|
(1/h) * (1 + s/wc*h)/(1 + s/wc/h); % For Piezo
|
|
% Kp = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * (s + 2*pi*10)/s * (s + 2*pi*1)/s ; % For voice coil
|
|
|
|
% Normalization of the gain of have a loop gain of 1 at frequency wc
|
|
Kp = Kp.*diag(1./diag(abs(freqresp(Gpx*Kp, wc))));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org6d0ebe2" class="figure">
|
|
<p><img src="figs/loop_gain_primary_voice_coil_X.png" alt="loop_gain_primary_voice_coil_X.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Obtained Loop gain for the primary controller in the Task space (<a href="./figs/loop_gain_primary_voice_coil_X.png">png</a>, <a href="./figs/loop_gain_primary_voice_coil_X.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
And now we include the Jacobian inside the controller.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kp = inv(nano_hexapod.kinematics.J')*Kp;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9406388" class="outline-3">
|
|
<h3 id="org9406388"><span class="section-number-3">1.5</span> Simulation</h3>
|
|
<div class="outline-text-3" id="text-1-5">
|
|
<p>
|
|
Let’s first save the 3 controllers for further analysis:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save('mat/hac_lac_cascade_vc_controllers.mat', 'Kiff', 'Kl', 'Kp')
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/conf_simulink.mat');
|
|
set_param(conf_simulink, 'StopTime', '2');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sim('nass_model');
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">cascade_hac_lac_lorentz = simout;
|
|
save('./mat/cascade_hac_lac.mat', 'cascade_hac_lac_lorentz', '-append');
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org16024e0" class="outline-3">
|
|
<h3 id="org16024e0"><span class="section-number-3">1.6</span> Results</h3>
|
|
<div class="outline-text-3" id="text-1-6">
|
|
</div>
|
|
<div id="outline-container-org8e90e5a" class="outline-4">
|
|
<h4 id="org8e90e5a"><span class="section-number-4">1.6.1</span> Load the simulation results</h4>
|
|
<div class="outline-text-4" id="text-1-6-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('./mat/experiment_tomography.mat', 'tomo_align_dist');
|
|
load('./mat/cascade_hac_lac.mat', 'cascade_hac_lac', 'cascade_hac_lac_lorentz');
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0f974ff" class="outline-4">
|
|
<h4 id="org0f974ff"><span class="section-number-4">1.6.2</span> Control effort</h4>
|
|
<div class="outline-text-4" id="text-1-6-2">
|
|
|
|
<div id="org8301604" class="figure">
|
|
<p><img src="figs/actuator_force_torques_tomography_voice_coil.png" alt="actuator_force_torques_tomography_voice_coil.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Actuator Action during a tomography experiment when using Voice Coil actuators (<a href="./figs/actuator_force_torques_tomography_voice_coil.png">png</a>, <a href="./figs/actuator_force_torques_tomography_voice_coil.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org36679a6" class="outline-4">
|
|
<h4 id="org36679a6"><span class="section-number-4">1.6.3</span> Load the simulation results</h4>
|
|
<div class="outline-text-4" id="text-1-6-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">n_av = 4;
|
|
han_win = hanning(ceil(length(cascade_hac_lac.Em.En.Data(:,1))/n_av));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">t = cascade_hac_lac.Em.En.Time;
|
|
Ts = t(2)-t(1);
|
|
|
|
[pxx_ol, f] = pwelch(tomo_align_dist.Em.En.Data, han_win, [], [], 1/Ts);
|
|
[pxx_ca, ~] = pwelch(cascade_hac_lac.Em.En.Data, han_win, [], [], 1/Ts);
|
|
[pxx_vc, ~] = pwelch(cascade_hac_lac_lorentz.Em.En.Data, han_win, [], [], 1/Ts);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org9a33ea7" class="figure">
|
|
<p><img src="figs/exp_tomography_voice_coil_psd_pos_error.png" alt="exp_tomography_voice_coil_psd_pos_error.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Power Spectral Density of the position error during a tomography experiment when using Voice Coil based nano-hexapod (<a href="./figs/exp_tomography_voice_coil_psd_pos_error.png">png</a>, <a href="./figs/exp_tomography_voice_coil_psd_pos_error.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orga0c2483" class="figure">
|
|
<p><img src="figs/exp_tomography_voice_coil_cap_pos_error.png" alt="exp_tomography_voice_coil_cap_pos_error.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Cumulative Amplitude Spectrum of the position error during a tomography experiment when using Voice Coil based nano-hexapod (<a href="./figs/exp_tomography_voice_coil_cap_pos_error.png">png</a>, <a href="./figs/exp_tomography_voice_coil_cap_pos_error.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org56aac09" class="figure">
|
|
<p><img src="figs/exp_tomography_voice_coil_time_domain.png" alt="exp_tomography_voice_coil_time_domain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Position error during a tomography experiment when using Voice Coil based nano-hexapod (<a href="./figs/exp_tomography_voice_coil_time_domain.png">png</a>, <a href="./figs/exp_tomography_voice_coil_time_domain.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org74d9dc7" class="outline-3">
|
|
<h3 id="org74d9dc7"><span class="section-number-3">1.7</span> Compliance of the nano-hexapod</h3>
|
|
<div class="outline-text-3" id="text-1-7">
|
|
</div>
|
|
<div id="outline-container-orgd768713" class="outline-4">
|
|
<h4 id="orgd768713"><span class="section-number-4">1.7.1</span> Identification</h4>
|
|
<div class="outline-text-4" id="text-1-7-1">
|
|
<p>
|
|
Let’s identify the Compliance of the NASS:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Disturbances/Fd'], 1, 'openinput'); io_i = io_i + 1; % Direct Forces/Torques applied on the sample
|
|
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
First in open-loop:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kp = tf(zeros(6));
|
|
Kl = tf(zeros(6));
|
|
Kiff = tf(zeros(6));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Run the linearization
|
|
Gc_ol = linearize(mdl, io, 0);
|
|
Gc_ol.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
Gc_ol.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then with the IFF control.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/hac_lac_cascade_vc_controllers.mat', 'Kiff')
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Run the linearization
|
|
Gc_iff = linearize(mdl, io, 0);
|
|
Gc_iff.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
Gc_iff.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
With the HAC control added
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/hac_lac_cascade_vc_controllers.mat', 'Kl')
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Run the linearization
|
|
Gc_hac = linearize(mdl, io, 0);
|
|
Gc_hac.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
Gc_hac.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally with the primary controller
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/hac_lac_cascade_vc_controllers.mat', 'Kp')
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Run the linearization
|
|
Gc_pri = linearize(mdl, io, 0);
|
|
Gc_pri.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
Gc_pri.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1a1ad20" class="outline-4">
|
|
<h4 id="org1a1ad20"><span class="section-number-4">1.7.2</span> Obtained Compliance</h4>
|
|
<div class="outline-text-4" id="text-1-7-2">
|
|
|
|
<div id="org3444b1d" class="figure">
|
|
<p><img src="figs/compliance_evolution_vc_cascade_control.png" alt="compliance_evolution_vc_cascade_control.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Evolution of the NASS compliance with each control loop added (<a href="./figs/compliance_evolution_vc_cascade_control.png">png</a>, <a href="./figs/compliance_evolution_vc_cascade_control.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5b81db9" class="outline-4">
|
|
<h4 id="org5b81db9"><span class="section-number-4">1.7.3</span> Comparison with Piezo</h4>
|
|
<div class="outline-text-4" id="text-1-7-3">
|
|
<p>
|
|
Let’s initialize a nano-hexapod with piezoelectric actuators.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeNanoHexapod('actuator', 'piezo');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We don’t use any controller.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kp = tf(zeros(6));
|
|
Kl = tf(zeros(6));
|
|
Kiff = tf(zeros(6));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Run the linearization
|
|
Gc_pz = linearize(mdl, io, 0);
|
|
Gc_pz.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
Gc_pz.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgc7cba59" class="figure">
|
|
<p><img src="figs/compliance_comp_pz_vc_cascade.png" alt="compliance_comp_pz_vc_cascade.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Comparison of the compliance using the open-loop piezo-nano hexapod and the voice coil nano-hexapod with the cascade control topology (<a href="./figs/compliance_comp_pz_vc_cascade.png">png</a>, <a href="./figs/compliance_comp_pz_vc_cascade.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org3c7ca09" class="outline-3">
|
|
<h3 id="org3c7ca09"><span class="section-number-3">1.8</span> Robustness to Payload Variability</h3>
|
|
<div class="outline-text-3" id="text-1-8">
|
|
</div>
|
|
<div id="outline-container-orgf34ee18" class="outline-4">
|
|
<h4 id="orgf34ee18"><span class="section-number-4">1.8.1</span> Initialization</h4>
|
|
<div class="outline-text-4" id="text-1-8-1">
|
|
<p>
|
|
Let’s change the payload mass, and see if the controller design for a payload mass of 1 still gives good performance.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeSample('mass', 50);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kp = tf(zeros(6));
|
|
Kl = tf(zeros(6));
|
|
Kiff = tf(zeros(6));
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0d97c55" class="outline-4">
|
|
<h4 id="org0d97c55"><span class="section-number-4">1.8.2</span> Low Authority Control</h4>
|
|
<div class="outline-text-4" id="text-1-8-2">
|
|
<p>
|
|
Let’s first identify the transfer function for the Low Authority control.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
|
|
|
|
%% Run the linearization
|
|
G_iff_m = linearize(mdl, io, 0);
|
|
G_iff_m.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G_iff_m.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained dynamics is compared when using a payload of 1Kg in Figure <a href="#orgda72d72">15</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgda72d72" class="figure">
|
|
<p><img src="figs/voice_coil_variability_mass_iff.png" alt="voice_coil_variability_mass_iff.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Dynamics of the LAC plant when using a 50Kg payload (dashed) and when using a 1Kg payload (solid) (<a href="./figs/voice_coil_variability_mass_iff.png">png</a>, <a href="./figs/voice_coil_variability_mass_iff.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
A gain of 50 will here suffice to obtain critical damping of the nano-hexapod modes.
|
|
</p>
|
|
|
|
<p>
|
|
Let’s load the IFF controller designed when the payload has a mass of 1Kg.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/hac_lac_cascade_vc_controllers.mat', 'Kiff')
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org2d34f99" class="figure">
|
|
<p><img src="figs/voice_coil_variability_mass_iff_loop_gain.png" alt="voice_coil_variability_mass_iff_loop_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Loop gain for the IFF Control when using a 50Kg payload (dashed) and when using a 1Kg payload (solid) (<a href="./figs/voice_coil_variability_mass_iff_loop_gain.png">png</a>, <a href="./figs/voice_coil_variability_mass_iff_loop_gain.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgab3ab98" class="outline-4">
|
|
<h4 id="orgab3ab98"><span class="section-number-4">1.8.3</span> High Authority Control</h4>
|
|
<div class="outline-text-4" id="text-1-8-3">
|
|
<p>
|
|
We use the Integral Force Feedback developed with a mass of 1Kg and we identify the dynamics for the High Authority Controller in the case of the 50Kg payload
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'output', [], 'Dnlm'); io_i = io_i + 1; % Leg Displacement
|
|
|
|
%% Run the linearization
|
|
Gl_m = linearize(mdl, io, 0);
|
|
Gl_m.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
Gl_m.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
|
|
|
|
isstable(Gl_m)
|
|
Gl_m = minreal(Gl_m);
|
|
isstable(Gl_m)
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org4a13006" class="figure">
|
|
<p><img src="figs/voice_coil_variability_mass_hac_plant.png" alt="voice_coil_variability_mass_hac_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Dynamics of the HAC plant when using a 50Kg payload (dashed) and when using a 1Kg payload (solid) (<a href="./figs/voice_coil_variability_mass_hac_plant.png">png</a>, <a href="./figs/voice_coil_variability_mass_hac_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
We load the HAC controller design when the payload has a mass of 1Kg.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/hac_lac_cascade_vc_controllers.mat', 'Kl')
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgbc6b4d5" class="figure">
|
|
<p><img src="figs/voice_coil_variability_mass_hac_lool_gain.png" alt="voice_coil_variability_mass_hac_lool_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Loop Gain of the HAC when using a 50Kg payload (dashed) and when using a 1Kg payload (solid) (<a href="./figs/voice_coil_variability_mass_hac_lool_gain.png">png</a>, <a href="./figs/voice_coil_variability_mass_hac_lool_gain.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org176695a" class="outline-4">
|
|
<h4 id="org176695a"><span class="section-number-4">1.8.4</span> Primary Plant</h4>
|
|
<div class="outline-text-4" id="text-1-8-4">
|
|
<p>
|
|
We use the Low Authority Controller developed with a mass of 1Kg and we identify the dynamics for the Primary controller in the case of the 50Kg payload.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller/Cascade-HAC-LAC/Kp'], 1, 'input'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
|
|
%% Run the linearization
|
|
Gp_m = linearize(mdl, io, 0);
|
|
Gp_m.InputName = {'rl1', 'rl2', 'rl3', 'rl4', 'rl5', 'rl6'};
|
|
Gp_m.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
A minus sign is added to cancel the minus sign already included in the identified plant.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(Gp_m)
|
|
Gp_m = -minreal(Gp_m);
|
|
isstable(Gp_m)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/stages.mat', 'nano_hexapod');
|
|
Gpx_m = Gp_m*inv(nano_hexapod.kinematics.J');
|
|
Gpx_m.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
|
|
Gpl_m = nano_hexapod.kinematics.J*Gp_m;
|
|
Gpl_m.OutputName = {'El1', 'El2', 'El3', 'El4', 'El5', 'El6'};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="important">
|
|
<p>
|
|
There are two zeros with positive real part for the plant in the y direction at about 100Hz.
|
|
This is problematic as it limits the bandwidth to be less than \(\approx 50\ \text{Hz}\).
|
|
</p>
|
|
|
|
<p>
|
|
It is important here to physically understand why such “positive” zero appears.
|
|
</p>
|
|
|
|
<p>
|
|
If we make a “rigid” 50kg paylaod, the positive zero disappears.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
|
|
<div id="orgef87d4d" class="figure">
|
|
<p><img src="figs/voice_coil_variability_mass_primary_plant.png" alt="voice_coil_variability_mass_primary_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Dynamics of the Primary plant when using a 50Kg payload (dashed) and when using a 1Kg payload (solid) (<a href="./figs/voice_coil_variability_mass_primary_plant.png">png</a>, <a href="./figs/voice_coil_variability_mass_primary_plant.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
We load the primary controller that was design when the payload has a mass of 1Kg.
|
|
</p>
|
|
|
|
<p>
|
|
We load the HAC controller design when the payload has a mass of 1Kg.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/hac_lac_cascade_vc_controllers.mat', 'Kp')
|
|
Kp_x = nano_hexapod.kinematics.J'*Kp;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wc = 2*pi*50; % Bandwidth Bandwidth [rad/s]
|
|
|
|
h = 2; % Lead parameter
|
|
|
|
Kp = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ...
|
|
(1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ...
|
|
(s + 2*pi*1)/s * ...
|
|
1/(1+s/2/wc); % For Piezo
|
|
|
|
% Normalization of the gain of have a loop gain of 1 at frequency wc
|
|
Kp = Kp.*diag(1./diag(abs(freqresp(Gpx_m*Kp, wc))));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orga08b4d8" class="figure">
|
|
<p><img src="figs/voice_coil_variability_mass_primary_lool_gain.png" alt="voice_coil_variability_mass_primary_lool_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Loop Gain of the Primary loop when using a 50Kg payload (dashed) and when using a 1Kg payload (solid) (<a href="./figs/voice_coil_variability_mass_primary_lool_gain.png">png</a>, <a href="./figs/voice_coil_variability_mass_primary_lool_gain.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org678ff20" class="outline-4">
|
|
<h4 id="org678ff20"><span class="section-number-4">1.8.5</span> Simulation</h4>
|
|
<div class="outline-text-4" id="text-1-8-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/conf_simulink.mat');
|
|
set_param(conf_simulink, 'StopTime', '2');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we simulate the system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sim('nass_model');
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">cascade_hac_lac_lorentz_high_mass = simout;
|
|
save('./mat/cascade_hac_lac.mat', 'cascade_hac_lac_lorentz_high_mass', '-append');
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('./mat/experiment_tomography.mat', 'tomo_align_dist');
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd649256" class="outline-2">
|
|
<h2 id="orgd649256"><span class="section-number-2">2</span> Other analysis</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
</div>
|
|
<div id="outline-container-org6cee30e" class="outline-3">
|
|
<h3 id="org6cee30e"><span class="section-number-3">2.1</span> Robustness to Payload Variability</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<ul class="org-ul">
|
|
<li class="off"><code>[ ]</code> For 3/masses (1kg, 10kg, 50kg), plot each of the 3 plants</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org18b00fa" class="outline-3">
|
|
<h3 id="org18b00fa"><span class="section-number-3">2.2</span> Direct HAC control in the task space - \(\bm{K}_\mathcal{X}\)</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
|
|
<div id="org5ded988" class="figure">
|
|
<p><img src="figs/control_architecture_hac_iff_pos_X.png" alt="control_architecture_hac_iff_pos_X.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>Control Architecture containing an IFF controller and a Controller in the task space</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org625b500" class="outline-4">
|
|
<h4 id="org625b500"><span class="section-number-4">2.2.1</span> Identification</h4>
|
|
<div class="outline-text-4" id="text-2-2-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeController('type', 'hac-iff');
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller/HAC-IFF/Kx'], 1, 'input'); io_i = io_i + 1; % Control input
|
|
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, 0);
|
|
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(G)
|
|
G = -minreal(G);
|
|
isstable(G)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/stages.mat', 'nano_hexapod');
|
|
Gx = G*inv(nano_hexapod.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
|
|
Gl = nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'El1', 'El2', 'El3', 'El4', 'El5', 'El6'};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1387536" class="outline-4">
|
|
<h4 id="org1387536"><span class="section-number-4">2.2.2</span> Obtained Plant in the Task Space</h4>
|
|
</div>
|
|
<div id="outline-container-org50b9f75" class="outline-4">
|
|
<h4 id="org50b9f75"><span class="section-number-4">2.2.3</span> Obtained Plant in the Joint Space</h4>
|
|
</div>
|
|
<div id="outline-container-org19db2cd" class="outline-4">
|
|
<h4 id="org19db2cd"><span class="section-number-4">2.2.4</span> Controller Design in the Joint Space</h4>
|
|
<div class="outline-text-4" id="text-2-2-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wc = 2*pi*200; % Bandwidth Bandwidth [rad/s]
|
|
|
|
h = 2; % Lead parameter
|
|
|
|
Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ... % Lead
|
|
(1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ... % Lead
|
|
(s + 2*pi*10)/s * ... % Pseudo Integrator
|
|
1/(1+s/2/pi/500); % Low pass Filter
|
|
|
|
% Normalization of the gain of have a loop gain of 1 at frequency wc
|
|
Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wc = 2*pi*200; % Bandwidth Bandwidth [rad/s]
|
|
|
|
h = 2; % Lead parameter
|
|
|
|
Kl = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ... % Lead
|
|
(1/h) * (1 + s/wc*h)/(1 + s/wc/h) * ... % Lead
|
|
(s + 2*pi*1)/s * ... % Pseudo Integrator
|
|
(s + 2*pi*10)/s * ... % Pseudo Integrator
|
|
1/(1+s/2/pi/500); % Low pass Filter
|
|
|
|
% Normalization of the gain of have a loop gain of 1 at frequency wc
|
|
Kl = Kl.*diag(1./diag(abs(freqresp(Gl*Kl, wc))));
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5e26e70" class="outline-3">
|
|
<h3 id="org5e26e70"><span class="section-number-3">2.3</span> On the usefulness of the High Authority Control loop / Linearization loop</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
Let’s see what happens is we omit the HAC loop and we directly try to control the damped plant using the measurement of the sample with respect to the granite \(\bm{\mathcal{X}}\).
|
|
</p>
|
|
|
|
<p>
|
|
We can do that in two different ways:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>in the task space as shown in Figure <a href="#orge366d0b">22</a></li>
|
|
<li>in the space of the legs as shown in Figure <a href="#orgd23329e">23</a></li>
|
|
</ul>
|
|
|
|
|
|
<div id="orge366d0b" class="figure">
|
|
<p><img src="figs/control_architecture_iff_X.png" alt="control_architecture_iff_X.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>IFF control + primary controller in the task space</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgd23329e" class="figure">
|
|
<p><img src="figs/control_architecture_iff_L.png" alt="control_architecture_iff_L.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 23: </span>HAC-LAC control architecture in the frame of the legs</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7366662" class="outline-4">
|
|
<h4 id="org7366662"><span class="section-number-4">2.3.1</span> Identification</h4>
|
|
<div class="outline-text-4" id="text-2-3-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeController('type', 'hac-iff');
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller/HAC-IFF/Kx'], 1, 'input'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, 0);
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|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
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|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(G)
|
|
G = -minreal(G);
|
|
isstable(G)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfab8847" class="outline-4">
|
|
<h4 id="orgfab8847"><span class="section-number-4">2.3.2</span> Plant in the Task space</h4>
|
|
<div class="outline-text-4" id="text-2-3-2">
|
|
<p>
|
|
The obtained plant is shown in Figure
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gx = G*inv(nano_hexapod.kinematics.J');
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org18aeea5" class="outline-4">
|
|
<h4 id="org18aeea5"><span class="section-number-4">2.3.3</span> Plant in the Leg’s space</h4>
|
|
<div class="outline-text-4" id="text-2-3-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gl = nano_hexapod.kinematics.J*G;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org015f992" class="outline-3">
|
|
<h3 id="org015f992"><span class="section-number-3">2.4</span> DVF instead of IFF?</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
</div>
|
|
<div id="outline-container-org17cfb9d" class="outline-4">
|
|
<h4 id="org17cfb9d"><span class="section-number-4">2.4.1</span> Initialization and Identification</h4>
|
|
<div class="outline-text-4" id="text-2-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeController('type', 'hac-dvf');
|
|
Kdvf = tf(zeros(6));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Displacement Sensors
|
|
|
|
%% Run the linearization
|
|
G_dvf = linearize(mdl, io, 0);
|
|
G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G_dvf.OutputName = {'Dlm1', 'Dlm2', 'Dlm3', 'Dlm4', 'Dlm5', 'Dlm6'};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd7117f7" class="outline-4">
|
|
<h4 id="orgd7117f7"><span class="section-number-4">2.4.2</span> Obtained Plant</h4>
|
|
</div>
|
|
<div id="outline-container-org51c8027" class="outline-4">
|
|
<h4 id="org51c8027"><span class="section-number-4">2.4.3</span> Controller</h4>
|
|
<div class="outline-text-4" id="text-2-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kdvf = -850*s/(1+s/2/pi/1000)*eye(6);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org33637f1" class="outline-4">
|
|
<h4 id="org33637f1"><span class="section-number-4">2.4.4</span> HAC Identification</h4>
|
|
<div class="outline-text-4" id="text-2-4-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller/HAC-DVF/Kx'], 1, 'input'); io_i = io_i + 1; % Control input
|
|
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, 0);
|
|
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable(G)
|
|
G = -minreal(G);
|
|
isstable(G)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/stages.mat', 'nano_hexapod');
|
|
Gx = G*inv(nano_hexapod.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
|
|
Gl = nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'El1', 'El2', 'El3', 'El4', 'El5', 'El6'};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgec66083" class="outline-4">
|
|
<h4 id="orgec66083"><span class="section-number-4">2.4.5</span> Conclusion</h4>
|
|
<div class="outline-text-4" id="text-2-4-5">
|
|
<div class="important">
|
|
<p>
|
|
DVF can be used instead of IFF.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-05-05 mar. 10:34</p>
|
|
</div>
|
|
</body>
|
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</html>
|