804 lines
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804 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="#org99c5b6d">1. Low Authority Control - Decentralized Direct Velocity Feedback</a>
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<ul>
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<li><a href="#orgf3f8aed">1.1. Initialization</a></li>
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<li><a href="#orgc5a1e81">1.2. Identification</a></li>
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<li><a href="#orgfef1a3f">1.3. Controller Design</a></li>
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<li><a href="#org3c73014">1.4. Effect of the Low Authority Control on the Primary Plant</a></li>
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<li><a href="#orgee5dbee">1.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
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<li><a href="#org8c0882d">1.6. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org81dc0a8">2. Primary Control in the leg space</a>
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<ul>
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<li><a href="#org1e7a412">2.1. Plant in the leg space</a></li>
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<li><a href="#orgf39520c">2.2. Control in the leg space</a></li>
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<li><a href="#org16d192f">2.3. Sensibility to Disturbances and Noise Budget</a></li>
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<li><a href="#org8f34c09">2.4. Simulations of Tomography Experiment</a></li>
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<li><a href="#orgbeadec8">2.5. Results</a></li>
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<li><a href="#orgf709759">2.6. Actuator Stroke and Forces</a></li>
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<li><a href="#orgb0f5db9">2.7. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org56b28cd">3. Further More complex simulations</a>
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<ul>
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<li><a href="#org6c1ddb5">3.1. Simulation with Micro-Hexapod Offset</a>
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<ul>
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<li><a href="#org57e2cfd">3.1.1. Simulation</a></li>
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<li><a href="#org2c93370">3.1.2. Results</a></li>
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</ul>
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</li>
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<li><a href="#org5cb899b">3.2. Simultaneous Translation scans and Spindle’s rotation</a>
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<ul>
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<li><a href="#org6710f28">3.2.1. Simulation</a></li>
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<li><a href="#org035df39">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="#org9bd2bf8">4. Primary Control in the task space</a>
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<ul>
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<li><a href="#org07b4a9d">4.1. Plant in the task space</a></li>
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<li><a href="#org7d888f9">4.2. Control in the task space</a>
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<ul>
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<li><a href="#orgb28634b">4.2.1. Stability</a></li>
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</ul>
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</li>
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<li><a href="#org9ea6a0a">4.3. Simulation</a></li>
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<li><a href="#org21304f7">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-org99c5b6d" class="outline-2">
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<h2 id="org99c5b6d"><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="orgfec42cb"></a>
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</p>
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<div id="org7f11a74" 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-orgf3f8aed" class="outline-3">
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<h3 id="orgf3f8aed"><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('enable', false);
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initializeLoggingConfiguration('log', 'none');
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initializeController('type', 'hac-dvf');
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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; % [N/m]
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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-orgc5a1e81" class="outline-3">
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<h3 id="orgc5a1e81"><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('k', 1e5, 'c', 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-orgfef1a3f" class="outline-3">
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<h3 id="orgfef1a3f"><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="#orgdb7af3b">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="#org5814b4f">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="#orgb16b2c3">4</a>.
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</p>
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<div id="orgdb7af3b" 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="org5814b4f" 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 controll 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="orgb16b2c3" 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*s/(1+s/2/pi/1e3)*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-org3c73014" class="outline-3">
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<h3 id="org3c73014"><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="#org45c1265">5</a>.
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</p>
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<div class="important">
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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="#org069e296">6</a>.
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</p>
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<div id="org45c1265" 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="org069e296" 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="#orgbb4e497">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="#orgc43d759">8</a>.
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</p>
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<div class="important">
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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="#orgc43d759">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="orgbb4e497" 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="orgc43d759" 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-orgee5dbee" class="outline-3">
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<h3 id="orgee5dbee"><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="#org199898b">9</a>.
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</p>
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<div id="org199898b" 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">
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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-org8c0882d" class="outline-3">
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<h3 id="org8c0882d"><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">
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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-org81dc0a8" class="outline-2">
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<h2 id="org81dc0a8"><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="orgd0beb6a"></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="#org7d5c8bc">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="org7d5c8bc" 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="#orgfec42cb">1</a>.
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</p>
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</div>
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<div id="outline-container-org1e7a412" class="outline-3">
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<h3 id="org1e7a412"><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="#org23d23ae">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="org23d23ae" 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-orgf39520c" class="outline-3">
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<h3 id="orgf39520c"><span class="section-number-3">2.2</span> Control in the leg space</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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We design a diagonal controller with all the same diagonal elements.
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</p>
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<p>
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The requirements for the controller are:
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</p>
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<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="#orgbcc0acb">12</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h = 2.0;
|
|
Kl = 2e7 * eye(6) * ...
|
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
|
1/h*(s/(2*pi*200/h) + 1)/(s/(2*pi*200*h) + 1) * ...
|
|
(s/2/pi/10 + 1)/(s/2/pi/10) * ...
|
|
1/(1 + s/2/pi/300);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgbcc0acb" 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('mat/stages.mat', 'nano_hexapod');
|
|
K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org16d192f" class="outline-3">
|
|
<h3 id="org16d192f"><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="#org9650e03">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org9650e03" 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="#org32928e0">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="#org7fda8f7">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="#org073608b">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="org32928e0" 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="org7fda8f7" 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="org073608b" 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-org8f34c09" class="outline-3">
|
|
<h3 id="org8f34c09"><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('gravity', false);
|
|
initializeLoggingConfiguration('log', 'all');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we run the simulation for all three payload Masses.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgbeadec8" class="outline-3">
|
|
<h3 id="orgbeadec8"><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="#org6cab6ef">17</a>.
|
|
</p>
|
|
|
|
<p>
|
|
Similarly, the Cumulative Amplitude Spectrum is shown in Figure <a href="#org33e9f1a">18</a>.
|
|
</p>
|
|
|
|
<p>
|
|
Finally, the time domain position error signals are shown in Figure <a href="#orgf0f1950">19</a>.
|
|
</p>
|
|
|
|
<div id="org6cab6ef" 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="org33e9f1a" 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="orgf0f1950" 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-orgf709759" class="outline-3">
|
|
<h3 id="orgf709759"><span class="section-number-3">2.6</span> Actuator Stroke and Forces</h3>
|
|
<div class="outline-text-3" id="text-2-6">
|
|
|
|
<div id="orgf9d6367" 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="org11b8730" 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-orgb0f5db9" class="outline-3">
|
|
<h3 id="orgb0f5db9"><span class="section-number-3">2.7</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-7">
|
|
<div class="important">
|
|
<p>
|
|
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org56b28cd" class="outline-2">
|
|
<h2 id="org56b28cd"><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-org6c1ddb5" class="outline-3">
|
|
<h3 id="org6c1ddb5"><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-org57e2cfd" class="outline-4">
|
|
<h4 id="org57e2cfd"><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('gravity', false);
|
|
initializeLoggingConfiguration('log', 'all');
|
|
|
|
initializeSample('mass', 1, 'freq', 200);
|
|
initializeMicroHexapod('AP', [10e-3 0 0]);
|
|
initializeReferences('Rz_type', 'rotating', 'Rz_period', 1, ...
|
|
'Dh_pos', [10e-3; 0; 0; 0; 0; 0]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load('mat/conf_simulink.mat');
|
|
set_param(conf_simulink, 'StopTime', '3');
|
|
sim('nass_model');
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2c93370" class="outline-4">
|
|
<h4 id="org2c93370"><span class="section-number-4">3.1.2</span> Results</h4>
|
|
<div class="outline-text-4" id="text-3-1-2">
|
|
|
|
<div id="org6be7e46" 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="org07fa12d" 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="orga4d03c5" 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-org5cb899b" class="outline-3">
|
|
<h3 id="org5cb899b"><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-org6710f28" class="outline-4">
|
|
<h4 id="org6710f28"><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 one 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('gravity', false);
|
|
initializeLoggingConfiguration('log', 'all');
|
|
|
|
initializeSample('mass', 1, 'freq', 200);
|
|
initializeReferences('Rz_type', 'rotating', 'Rz_period', 1, ...
|
|
'Dy_type', 'triangular', 'Dy_amplitude', 5e-3, 'Dy_period', 4);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org035df39" class="outline-4">
|
|
<h4 id="org035df39"><span class="section-number-4">3.2.2</span> Results</h4>
|
|
<div class="outline-text-4" id="text-3-2-2">
|
|
|
|
<div id="orgbfa1d02" 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="org760b96c" 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="orgae36e3d" 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-org9bd2bf8" class="outline-2">
|
|
<h2 id="org9bd2bf8"><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="orge9c2f9a"></a>
|
|
</p>
|
|
<p>
|
|
In this section, the control architecture shown in Figure <a href="#org7e70ccc">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="org7e70ccc" 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-org07b4a9d" class="outline-3">
|
|
<h3 id="org07b4a9d"><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-org7d888f9" class="outline-3">
|
|
<h3 id="org7d888f9"><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(1,1) = 3e7 * ...
|
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
|
(s/2/pi/1 + 1)/(s/2/pi/1);
|
|
|
|
Kx(2,2) = Kx(1,1);
|
|
|
|
h = 2.5;
|
|
Kx(3,3) = 3e7 * ...
|
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
|
(s/2/pi/1 + 1)/(s/2/pi/1);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h = 1.5;
|
|
Kx(4,4) = 5e5 * ...
|
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
|
(s/2/pi/1 + 1)/(s/2/pi/1);
|
|
|
|
Kx(5,5) = Kx(4,4);
|
|
|
|
h = 1.5;
|
|
Kx(6,6) = 5e4 * ...
|
|
1/h*(s/(2*pi*30/h) + 1)/(s/(2*pi*30*h) + 1) * ...
|
|
(s/2/pi/1 + 1)/(s/2/pi/1);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb28634b" class="outline-4">
|
|
<h4 id="orgb28634b"><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">for i = 1:length(Ms)
|
|
isstable(feedback(Gm_x{i}*Kx, eye(6), -1))
|
|
end
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9ea6a0a" class="outline-3">
|
|
<h3 id="org9ea6a0a"><span class="section-number-3">4.3</span> Simulation</h3>
|
|
</div>
|
|
<div id="outline-container-org21304f7" class="outline-3">
|
|
<h3 id="org21304f7"><span class="section-number-3">4.4</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<div class="important">
|
|
<p>
|
|
|
|
</p>
|
|
|
|
</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>
|
|
</html>
|