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808 lines
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<title>Control of the NASS with optimal stiffness</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 optimal stiffness</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="#orgdb6e4f3">1. Low Authority Control - Decentralized Direct Velocity Feedback</a>
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<ul>
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<li><a href="#org9319182">1.1. Initialization</a></li>
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<li><a href="#org8b02bfa">1.2. Identification</a></li>
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<li><a href="#org6f820ae">1.3. Controller Design</a></li>
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<li><a href="#org5434213">1.4. Effect of the Low Authority Control on the Primary Plant</a></li>
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<li><a href="#orgc187d77">1.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
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<li><a href="#org9539bfe">1.6. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org2797edc">2. Primary Control in the leg space</a>
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<ul>
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<li><a href="#org8c682ad">2.1. Plant in the leg space</a></li>
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<li><a href="#org8308914">2.2. Control in the leg space</a></li>
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<li><a href="#org5a96746">2.3. Sensibility to Disturbances and Noise Budget</a></li>
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<li><a href="#org370df82">2.4. Simulations of Tomography Experiment</a></li>
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<li><a href="#orgd19d198">2.5. Results</a></li>
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<li><a href="#org4bc7412">2.6. Actuator Stroke and Forces</a></li>
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<li><a href="#orgead2d0d">2.7. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgbb6ed0e">3. Further More complex simulations</a>
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<ul>
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<li><a href="#orgbe6d807">3.1. Simulation with Micro-Hexapod Offset</a>
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<ul>
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<li><a href="#orgd36f8c4">3.1.1. Simulation</a></li>
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<li><a href="#org8fe5f58">3.1.2. Results</a></li>
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</ul>
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</li>
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<li><a href="#org9753678">3.2. Simultaneous Translation scans and Spindle’s rotation</a>
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<ul>
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<li><a href="#org2619c3f">3.2.1. Simulation</a></li>
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<li><a href="#org8a2b562">3.2.2. 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="#org26c4ef6">4. Primary Control in the task space</a>
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<ul>
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<li><a href="#org48bda71">4.1. Plant in the task space</a></li>
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<li><a href="#org4251f93">4.2. Control in the task space</a>
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<ul>
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<li><a href="#org226215f">4.2.1. Stability</a></li>
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</ul>
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</li>
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<li><a href="#org0aa9f05">4.3. Simulation</a></li>
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<li><a href="#org4a860c5">4.4. Conclusion</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgdb6e4f3" class="outline-2">
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<h2 id="orgdb6e4f3"><span class="section-number-2">1</span> Low Authority Control - Decentralized Direct Velocity Feedback</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org8d1c1a1"></a>
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</p>
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<div id="org26ba32b" class="figure">
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<p><img src="figs/control_architecture_dvf.png" alt="control_architecture_dvf.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Low Authority Control: Decentralized Direct Velocity Feedback</p>
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</div>
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</div>
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<div id="outline-container-org9319182" class="outline-3">
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<h3 id="org9319182"><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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<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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initializeSimscapeConfiguration();
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initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
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initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
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initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
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</pre>
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</div>
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<p>
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We set the stiffness of the payload fixation:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> Kp = 1e8; <span class="org-comment">% [N/m]</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-org8b02bfa" class="outline-3">
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<h3 id="org8b02bfa"><span class="section-number-3">1.2</span> Identification</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab"> K = tf(zeros(6));
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Kdvf = tf(zeros(6));
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</pre>
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</div>
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<p>
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We identify the system for the following payload masses:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> Ms = [1, 10, 50];
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</pre>
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</div>
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<p>
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The nano-hexapod has the following leg’s stiffness and damping.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> initializeNanoHexapod(<span class="org-string">'k'</span>, 1e5, <span class="org-string">'c'</span>, 2e2);
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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-org6f820ae" class="outline-3">
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<h3 id="org6f820ae"><span class="section-number-3">1.3</span> Controller Design</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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The obtain dynamics from actuators forces \(\tau_i\) to the relative motion of the legs \(d\mathcal{L}_i\) is shown in Figure <a href="#org3f64a5c">2</a> for the three considered payload masses.
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</p>
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<p>
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The Root Locus is shown in Figure <a href="#org4342572">3</a> and wee see that we have unconditional stability.
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</p>
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<p>
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In order to choose the gain such that we obtain good damping for all the three payload masses, we plot the damping ration of the modes as a function of the gain for all three payload masses in Figure <a href="#org79a7504">4</a>.
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</p>
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<div id="org3f64a5c" class="figure">
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<p><img src="figs/opt_stiff_dvf_plant.png" alt="opt_stiff_dvf_plant.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Dynamics for the Direct Velocity Feedback active damping for three payload masses</p>
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</div>
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<div id="org4342572" class="figure">
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<p><img src="figs/opt_stiff_dvf_root_locus.png" alt="opt_stiff_dvf_root_locus.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Root Locus for the DVF control for three payload masses</p>
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</div>
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<p>
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Damping as function of the gain
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</p>
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<div id="org79a7504" class="figure">
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<p><img src="figs/opt_stiff_dvf_damping_gain.png" alt="opt_stiff_dvf_damping_gain.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Damping ratio of the poles as a function of the DVF gain</p>
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</div>
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<p>
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Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> Kdvf = 5e3<span class="org-type">*</span>s<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>1e3)<span class="org-type">*</span>eye(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-org5434213" class="outline-3">
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<h3 id="org5434213"><span class="section-number-3">1.4</span> Effect of the Low Authority Control on the Primary Plant</h3>
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<div class="outline-text-3" id="text-1-4">
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<p>
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Let’s identify the dynamics from actuator forces \(\bm{\tau}\) to displacement as measured by the metrology \(\bm{\mathcal{X}}\):
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\[ \bm{G}(s) = \frac{\bm{\mathcal{X}}}{\bm{\tau}} \]
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We do so both when the DVF is applied and when it is not applied.
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</p>
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<p>
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Then, we compute the transfer function from forces applied by the actuators \(\bm{\mathcal{F}}\) to the measured position error in the frame of the nano-hexapod \(\bm{\epsilon}_{\mathcal{X}_n}\):
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\[ \bm{G}_\mathcal{X}(s) = \frac{\bm{\epsilon}_{\mathcal{X}_n}}{\bm{\mathcal{F}}} = \bm{G}(s) \bm{J}^{-T} \]
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The obtained dynamics is shown in Figure <a href="#org4b0b413">5</a>.
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</p>
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<div class="important" id="orgd787228">
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<p>
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A zero with a positive real part is introduced in the transfer function from \(\mathcal{F}_y\) to \(\mathcal{X}_y\) after Decentralized Direct Velocity Feedback is applied.
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</p>
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</div>
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<p>
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And we compute the transfer function from actuator forces \(\bm{\tau}\) to position error of each leg \(\bm{\epsilon}_\mathcal{L}\):
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\[ \bm{G}_\mathcal{L} = \frac{\bm{\epsilon}_\mathcal{L}}{\bm{\tau}} = \bm{J} \bm{G}(s) \]
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The obtained dynamics is shown in Figure <a href="#org0a1130b">6</a>.
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</p>
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<div id="org4b0b413" class="figure">
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<p><img src="figs/opt_stiff_primary_plant_damped_X.png" alt="opt_stiff_primary_plant_damped_X.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Primary plant in the task space with (dashed) and without (solid) Direct Velocity Feedback</p>
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</div>
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<div id="org0a1130b" class="figure">
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<p><img src="figs/opt_stiff_primary_plant_damped_L.png" alt="opt_stiff_primary_plant_damped_L.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback</p>
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</div>
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<p>
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The coupling (off diagonal elements) of \(\bm{G}_\mathcal{X}\) are shown in Figure <a href="#org67cb6cd">7</a> both when DVF is applied and when it is not.
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</p>
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<p>
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The coupling does not change a lot with DVF.
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</p>
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<p>
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The coupling in the space of the legs \(\bm{G}_\mathcal{L}\) are shown in Figure <a href="#org6331a29">8</a>.
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</p>
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<div class="important" id="orgc16e5dd">
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<p>
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The magnitude of the coupling between \(\tau_i\) and \(d\mathcal{L}_j\) (Figure <a href="#org6331a29">8</a>) around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
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</p>
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</div>
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<div id="org67cb6cd" class="figure">
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<p><img src="figs/opt_stiff_primary_plant_damped_coupling_X.png" alt="opt_stiff_primary_plant_damped_coupling_X.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Coupling in the primary plant in the task with (dashed) and without (solid) Direct Velocity Feedback</p>
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</div>
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<div id="org6331a29" class="figure">
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<p><img src="figs/opt_stiff_primary_plant_damped_coupling_L.png" alt="opt_stiff_primary_plant_damped_coupling_L.png" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Coupling in the primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgc187d77" class="outline-3">
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<h3 id="orgc187d77"><span class="section-number-3">1.5</span> Effect of the Low Authority Control on the Sensibility to Disturbances</h3>
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<div class="outline-text-3" id="text-1-5">
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<p>
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We may now see how Decentralized Direct Velocity Feedback changes the sensibility to disturbances, namely:
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</p>
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<ul class="org-ul">
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<li>Ground motion</li>
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<li>Spindle and Translation stage vibrations</li>
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<li>Direct forces applied to the sample</li>
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</ul>
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<p>
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To simplify the analysis, we here only consider the vertical direction, thus, we will look at the transfer functions:
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</p>
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<ul class="org-ul">
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<li>from vertical ground motion \(D_{w,z}\) to the vertical position error of the sample \(E_z\)</li>
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<li>from vertical vibration forces of the spindle \(F_{R_z,z}\) to \(E_z\)</li>
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<li>from vertical vibration forces of the translation stage \(F_{T_y,z}\) to \(E_z\)</li>
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<li>from vertical direct forces (such as cable forces) \(F_{d,z}\) to \(E_z\)</li>
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</ul>
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<p>
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The norm of these transfer functions are shown in Figure <a href="#orgc177556">9</a>.
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</p>
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<div id="orgc177556" class="figure">
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<p><img src="figs/opt_stiff_sensibility_dist_dvf.png" alt="opt_stiff_sensibility_dist_dvf.png" />
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</p>
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<p><span class="figure-number">Figure 9: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied</p>
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</div>
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<div class="important" id="orgb9e4f65">
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<p>
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Decentralized Direct Velocity Feedback is shown to increase the effect of stages vibrations at high frequency and to reduce the effect of ground motion and direct forces at low frequency.
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</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org9539bfe" class="outline-3">
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<h3 id="org9539bfe"><span class="section-number-3">1.6</span> Conclusion</h3>
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<div class="outline-text-3" id="text-1-6">
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<div class="important" id="orgf2d2cca">
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<p>
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</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-org2797edc" class="outline-2">
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<h2 id="org2797edc"><span class="section-number-2">2</span> Primary Control in the leg space</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orge570650"></a>
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</p>
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<p>
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In this section we implement the control architecture shown in Figure <a href="#orga22fd47">10</a> consisting of:
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</p>
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<ul class="org-ul">
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<li>an inner loop with a decentralized direct velocity feedback control</li>
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<li>an outer loop where the controller \(\bm{K}_\mathcal{L}\) is designed in the frame of the legs</li>
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</ul>
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<div id="orga22fd47" class="figure">
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<p><img src="figs/control_architecture_hac_dvf_pos_L.png" alt="control_architecture_hac_dvf_pos_L.png" />
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</p>
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<p><span class="figure-number">Figure 10: </span>Cascade Control Architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg’s space</p>
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</div>
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<p>
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The controller for decentralized direct velocity feedback is the one designed in Section <a href="#org8d1c1a1">1</a>.
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</p>
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</div>
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<div id="outline-container-org8c682ad" class="outline-3">
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<h3 id="org8c682ad"><span class="section-number-3">2.1</span> Plant in the leg space</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We now look at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\).
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</p>
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<p>
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The diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses are shown in Figure <a href="#org15749d9">11</a>.
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</p>
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<p>
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The plant dynamics below \(100\ [Hz]\) is only slightly dependent on the payload mass.
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</p>
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<div id="org15749d9" class="figure">
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<p><img src="figs/opt_stiff_primary_plant_L.png" alt="opt_stiff_primary_plant_L.png" />
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</p>
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<p><span class="figure-number">Figure 11: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org8308914" class="outline-3">
|
|
<h3 id="org8308914"><span class="section-number-3">2.2</span> Control in the leg space</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
We design a diagonal controller with all the same diagonal elements.
|
|
</p>
|
|
|
|
<p>
|
|
The requirements for the controller are:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Crossover frequency of around 100Hz</li>
|
|
<li>Stable for all the considered payload masses</li>
|
|
<li>Sufficient phase and gain margin</li>
|
|
<li>Integral action at low frequency</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The design controller is as follows:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Lead centered around the crossover</li>
|
|
<li>An integrator below 10Hz</li>
|
|
<li>A low pass filter at 250Hz</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The loop gain is shown in Figure <a href="#org05096f1">12</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> h = 2.0;
|
|
Kl = 2e7 <span class="org-type">*</span> eye(6) <span class="org-type">*</span> ...
|
|
1<span class="org-type">/</span>h<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<span class="org-type">/</span>h) <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<span class="org-type">*</span>h) <span class="org-type">+</span> 1) <span class="org-type">*</span> ...
|
|
1<span class="org-type">/</span>h<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>200<span class="org-type">/</span>h) <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>200<span class="org-type">*</span>h) <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>10 <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>10) <span class="org-type">*</span> ...
|
|
1<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>300);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org05096f1" class="figure">
|
|
<p><img src="figs/opt_stiff_primary_loop_gain_L.png" alt="opt_stiff_primary_loop_gain_L.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Loop gain for the primary plant</p>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
|
|
K = Kl<span class="org-type">*</span>nano_hexapod.kinematics.J<span class="org-type">*</span>diag([1, 1, 1, 1, 1, 0]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5a96746" class="outline-3">
|
|
<h3 id="org5a96746"><span class="section-number-3">2.3</span> Sensibility to Disturbances and Noise Budget</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.
|
|
</p>
|
|
<p>
|
|
We compare the norm of these transfer function for the vertical direction when no control is applied and when HAC-LAC control is applied: Figure <a href="#orgc3a7b3a">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgc3a7b3a" class="figure">
|
|
<p><img src="figs/opt_stiff_primary_control_L_senbility_dist.png" alt="opt_stiff_primary_control_L_senbility_dist.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Sensibility to disturbances when the HAC-LAC control is applied</p>
|
|
</div>
|
|
<p>
|
|
Then, we load the Power Spectral Density of the perturbations and we look at the obtained PSD of the displacement error in the vertical direction due to the disturbances:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Figure <a href="#org7ae258c">14</a>: Amplitude Spectral Density of the vertical position error due to both the vertical ground motion and the vertical vibrations of the spindle</li>
|
|
<li>Figure <a href="#org72bd654">15</a>: Comparison of the Amplitude Spectral Density of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
|
<li>Figure <a href="#org08b6134">16</a>: Comparison of the Cumulative Amplitude Spectrum of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org7ae258c" class="figure">
|
|
<p><img src="figs/opt_stiff_primary_control_L_psd_dist.png" alt="opt_stiff_primary_control_L_psd_dist.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Amplitude Spectral Density of the vertical position error of the sample when the HAC-DVF control is applied due to both the ground motion and spindle vibrations</p>
|
|
</div>
|
|
|
|
|
|
<div id="org72bd654" class="figure">
|
|
<p><img src="figs/opt_stiff_primary_control_L_psd_tot.png" alt="opt_stiff_primary_control_L_psd_tot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Amplitude Spectral Density of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
|
</div>
|
|
|
|
|
|
<div id="org08b6134" class="figure">
|
|
<p><img src="figs/opt_stiff_primary_control_L_cas_tot.png" alt="opt_stiff_primary_control_L_cas_tot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Cumulative Amplitude Spectrum of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org370df82" class="outline-3">
|
|
<h3 id="org370df82"><span class="section-number-3">2.4</span> Simulations of Tomography Experiment</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
Let’s now simulate a tomography experiment.
|
|
To do so, we include all disturbances except vibrations of the translation stage.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> initializeDisturbances();
|
|
initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
|
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we run the simulation for all three payload Masses.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgd19d198" class="outline-3">
|
|
<h3 id="orgd19d198"><span class="section-number-3">2.5</span> Results</h3>
|
|
<div class="outline-text-3" id="text-2-5">
|
|
<p>
|
|
Let’s now see how this controller performs.
|
|
</p>
|
|
|
|
<p>
|
|
First, we compute the Power Spectral Density of the sample’s position error and we compare it with the open loop case in Figure <a href="#orgeb9afe3">17</a>.
|
|
</p>
|
|
|
|
<p>
|
|
Similarly, the Cumulative Amplitude Spectrum is shown in Figure <a href="#org6b85652">18</a>.
|
|
</p>
|
|
|
|
<p>
|
|
Finally, the time domain position error signals are shown in Figure <a href="#orge2cc533">19</a>.
|
|
</p>
|
|
|
|
<div id="orgeb9afe3" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_L_psd_disp_error.png" alt="opt_stiff_hac_dvf_L_psd_disp_error.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller</p>
|
|
</div>
|
|
|
|
|
|
<div id="org6b85652" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_L_cas_disp_error.png" alt="opt_stiff_hac_dvf_L_cas_disp_error.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller</p>
|
|
</div>
|
|
|
|
|
|
<div id="orge2cc533" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_L_pos_error.png" alt="opt_stiff_hac_dvf_L_pos_error.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4bc7412" class="outline-3">
|
|
<h3 id="org4bc7412"><span class="section-number-3">2.6</span> Actuator Stroke and Forces</h3>
|
|
<div class="outline-text-3" id="text-2-6">
|
|
|
|
<div id="orgd89d4d1" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_L_act_force.png" alt="opt_stiff_hac_dvf_L_act_force.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Force applied by the actuator during the simulation</p>
|
|
</div>
|
|
|
|
|
|
<div id="org78753f1" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_L_act_stroke.png" alt="opt_stiff_hac_dvf_L_act_stroke.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>Leg’s stroke during the simulation</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgead2d0d" class="outline-3">
|
|
<h3 id="orgead2d0d"><span class="section-number-3">2.7</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-7">
|
|
<div class="important" id="orgd04ff34">
|
|
<p>
|
|
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbb6ed0e" class="outline-2">
|
|
<h2 id="orgbb6ed0e"><span class="section-number-2">3</span> Further More complex simulations</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
</div>
|
|
<div id="outline-container-orgbe6d807" class="outline-3">
|
|
<h3 id="orgbe6d807"><span class="section-number-3">3.1</span> Simulation with Micro-Hexapod Offset</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
</div>
|
|
<div id="outline-container-orgd36f8c4" class="outline-4">
|
|
<h4 id="orgd36f8c4"><span class="section-number-4">3.1.1</span> Simulation</h4>
|
|
<div class="outline-text-4" id="text-3-1-1">
|
|
<p>
|
|
The micro-hexapod is inducing a 10mm offset of the sample center of mass with the rotation axis.
|
|
A tomography experiment is then simulated.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> initializeDisturbances();
|
|
initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
|
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
|
|
|
initializeSample(<span class="org-string">'mass'</span>, 1, <span class="org-string">'freq'</span>, 200);
|
|
initializeMicroHexapod(<span class="org-string">'AP'</span>, [10e<span class="org-type">-</span>3 0 0]);
|
|
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, 1, ...
|
|
<span class="org-string">'Dh_pos'</span>, [10e<span class="org-type">-</span>3; 0; 0; 0; 0; 0]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simulink.mat'</span>);
|
|
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
|
|
<span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8fe5f58" class="outline-4">
|
|
<h4 id="org8fe5f58"><span class="section-number-4">3.1.2</span> Results</h4>
|
|
<div class="outline-text-4" id="text-3-1-2">
|
|
|
|
<div id="org776d205" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_Dh_offset_disp_error.png" alt="opt_stiff_hac_dvf_Dh_offset_disp_error.png" />
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="org26c6131" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_Dh_offset_F.png" alt="opt_stiff_hac_dvf_Dh_offset_F.png" />
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="org176d06f" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_Dh_offset_dL.png" alt="opt_stiff_hac_dvf_Dh_offset_dL.png" />
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9753678" class="outline-3">
|
|
<h3 id="org9753678"><span class="section-number-3">3.2</span> Simultaneous Translation scans and Spindle’s rotation</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
</div>
|
|
<div id="outline-container-org2619c3f" class="outline-4">
|
|
<h4 id="org2619c3f"><span class="section-number-4">3.2.1</span> Simulation</h4>
|
|
<div class="outline-text-4" id="text-3-2-1">
|
|
<p>
|
|
A simulation is now performed with translation scans and spindle rotation at the same time.
|
|
</p>
|
|
|
|
<p>
|
|
The sample has a mass of 1kg, the spindle rotation speed is 60rpm and the translation scans have a period of 4s and a triangular shape.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> initializeDisturbances();
|
|
initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
|
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
|
|
|
initializeSample(<span class="org-string">'mass'</span>, 1, <span class="org-string">'freq'</span>, 200);
|
|
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, 1, ...
|
|
<span class="org-string">'Dy_type'</span>, <span class="org-string">'triangular'</span>, <span class="org-string">'Dy_amplitude'</span>, 5e<span class="org-type">-</span>3, <span class="org-string">'Dy_period'</span>, 4);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8a2b562" class="outline-4">
|
|
<h4 id="org8a2b562"><span class="section-number-4">3.2.2</span> Results</h4>
|
|
<div class="outline-text-4" id="text-3-2-2">
|
|
|
|
<div id="orgc2e9b19" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_Dy_scans_disp_error.png" alt="opt_stiff_hac_dvf_Dy_scans_disp_error.png" />
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="org35d233c" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_Dy_scans_F.png" alt="opt_stiff_hac_dvf_Dy_scans_F.png" />
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="org46908f8" class="figure">
|
|
<p><img src="figs/opt_stiff_hac_dvf_Dy_scans_dL.png" alt="opt_stiff_hac_dvf_Dy_scans_dL.png" />
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org26c4ef6" class="outline-2">
|
|
<h2 id="org26c4ef6"><span class="section-number-2">4</span> Primary Control in the task space</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org06af957"></a>
|
|
</p>
|
|
<p>
|
|
In this section, the control architecture shown in Figure <a href="#org9be312d">28</a> is applied and consists of:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>an inner Low Authority Control loop consisting of a decentralized direct velocity control controller</li>
|
|
<li>an outer loop with the primary controller \(\bm{K}_\mathcal{X}\) designed in the task space</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org9be312d" class="figure">
|
|
<p><img src="figs/control_architecture_hac_dvf_pos_X.png" alt="control_architecture_hac_dvf_pos_X.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 28: </span>HAC-LAC architecture</p>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org48bda71" class="outline-3">
|
|
<h3 id="org48bda71"><span class="section-number-3">4.1</span> Plant in the task space</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
Let’s look \(\bm{G}_\mathcal{X}(s)\).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4251f93" class="outline-3">
|
|
<h3 id="org4251f93"><span class="section-number-3">4.2</span> Control in the task space</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> Kx = tf(zeros(6));
|
|
|
|
h = 2.5;
|
|
Kx<span class="org-type">(1,1) </span>= 3e7 <span class="org-type">*</span> ...
|
|
1<span class="org-type">/</span>h<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<span class="org-type">/</span>h) <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<span class="org-type">*</span>h) <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>1 <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>1);
|
|
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|
Kx<span class="org-type">(2,2) </span>= Kx(1,1);
|
|
|
|
h = 2.5;
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|
Kx<span class="org-type">(3,3) </span>= 3e7 <span class="org-type">*</span> ...
|
|
1<span class="org-type">/</span>h<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<span class="org-type">/</span>h) <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<span class="org-type">*</span>h) <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>1 <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>1);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> h = 1.5;
|
|
Kx<span class="org-type">(4,4) </span>= 5e5 <span class="org-type">*</span> ...
|
|
1<span class="org-type">/</span>h<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<span class="org-type">/</span>h) <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<span class="org-type">*</span>h) <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>1 <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>1);
|
|
|
|
Kx<span class="org-type">(5,5) </span>= Kx(4,4);
|
|
|
|
h = 1.5;
|
|
Kx<span class="org-type">(6,6) </span>= 5e4 <span class="org-type">*</span> ...
|
|
1<span class="org-type">/</span>h<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>30<span class="org-type">/</span>h) <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>30<span class="org-type">*</span>h) <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>1 <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>1);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org226215f" class="outline-4">
|
|
<h4 id="org226215f"><span class="section-number-4">4.2.1</span> Stability</h4>
|
|
<div class="outline-text-4" id="text-4-2-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> <span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ms)</span>
|
|
isstable(feedback(Gm_x{<span class="org-constant">i</span>}<span class="org-type">*</span>Kx, eye(6), <span class="org-type">-</span>1))
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0aa9f05" class="outline-3">
|
|
<h3 id="org0aa9f05"><span class="section-number-3">4.3</span> Simulation</h3>
|
|
</div>
|
|
<div id="outline-container-org4a860c5" class="outline-3">
|
|
<h3 id="org4a860c5"><span class="section-number-3">4.4</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<div class="important" id="org4bf4213">
|
|
<p>
|
|
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2021-02-20 sam. 23:09</p>
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</div>
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</body>
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</html>
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