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1201 lines
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<title>NASS - Uniaxial Model</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" class="content">
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<h1 class="title">NASS - Uniaxial Model</h1>
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<div id="table-of-contents" role="doc-toc">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents" role="doc-toc">
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<ul>
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<li><a href="#org5d8ba75">1. Micro Station Model</a>
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<ul>
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<li><a href="#org06407a9">1.1. Measured dynamics</a></li>
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<li><a href="#org0846d69">1.2. Uniaxial Model</a></li>
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<li><a href="#org1789ecb">1.3. Comparison of the model and measurements</a></li>
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</ul>
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</li>
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<li><a href="#org0cf1174">2. Nano-Hexapod Model</a>
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<ul>
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<li><a href="#org1088c4b">2.1. Nano-Hexapod Parameters</a></li>
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<li><a href="#orgb983a1e">2.2. Obtained Dynamics</a></li>
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</ul>
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</li>
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<li><a href="#org56c9136">3. Disturbance Identification</a>
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<ul>
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<li><a href="#org6b04518">3.1. Ground Motion</a></li>
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<li><a href="#org9aac15b">3.2. Stage Vibration</a></li>
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</ul>
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</li>
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<li><a href="#org446c255">4. Open-Loop Dynamic Noise Budgeting</a>
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<ul>
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<li><a href="#org26ad26a">4.1. Sensitivity to disturbances</a></li>
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<li><a href="#orga2938ae">4.2. Open-Loop Dynamic Noise Budgeting</a></li>
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<li><a href="#org79864c7">4.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org226ab26">5. Active Damping</a>
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<ul>
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<li><a href="#org78584ca">5.1. Active Damping Strategies</a></li>
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<li><a href="#orgd89363e">5.2. Plant Dynamics for Active Damping</a></li>
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<li><a href="#orgdd9b4b6">5.3. Achievable Damping - Root Locus</a></li>
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<li><a href="#org85dd4e6">5.4. Change of sensitivity to disturbances</a></li>
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<li><a href="#orgc1f3a3e">5.5. Noise Budgeting after Active Damping</a></li>
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<li><a href="#org669e6ff">5.6. Obtained Damped Plant</a></li>
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<li><a href="#org957bd1d">5.7. Robustness to change of payload’s mass</a></li>
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<li><a href="#org8a3c2c8">5.8. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org97eb0e2">6. Position Feedback Controller</a>
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<ul>
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<li><a href="#org9a6dfd4">6.1. Damped Plant Dynamics</a></li>
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<li><a href="#org03f8665">6.2. Position Feedback Controller</a></li>
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<li><a href="#orgab1ced7">6.3. Closed-Loop Noise Budgeting</a></li>
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</ul>
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</li>
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<li><a href="#org90d30b4">7. Conclusion</a></li>
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</ul>
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</div>
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</div>
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<hr>
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<p>This report is also available as a <a href="./nass-uniaxial-model.pdf">pdf</a>.</p>
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<hr>
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<p>
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In this report, a uniaxial model of the Nano Active Stabilization System (NASS) is developed and used to have a first idea of the challenges involved in this complex system.
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Note that in this document, only the vertical direction is considered (which is the most stiff), but other directions were considered as well and yields similar conclusions.
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The model is schematically shown in Figure <a href="fig:uniaxial_overview_model_sections">fig:uniaxial_overview_model_sections</a> where the colors are representing the studied parts in different sections.
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</p>
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<p>
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In order to have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section <a href="sec:micro_station_model">sec:micro_station_model</a>).
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Then, a model of the nano-hexapod is added on top of the micro-station.
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With added sample and sensors, this gives a uniaxial dynamical model of the NASS that will be used for further analysis (Section <a href="sec:nano_station_model">sec:nano_station_model</a>).
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</p>
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<p>
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The disturbances affecting the position accuracy are identified experimentally (Section <a href="sec:uniaxial_disturbances">sec:uniaxial_disturbances</a>) and included in the model for dynamical noise budgeting (Section <a href="sec:uniaxial_noise_budgeting">sec:uniaxial_noise_budgeting</a>).
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In all the following analysis, there nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate stiffness.
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Three sample masses are also considered to verify the robustness of the applied control strategies to a change of sample.
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</p>
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<p>
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Three active damping techniques are then applied on the nano-hexapod.
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This helps to reduce the effect of disturbances as well as render the system easier to control afterwards (Section <a href="sec:uniaxial_active_damping">sec:uniaxial_active_damping</a>).
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</p>
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<p>
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Once the system is well damped, a feedback position controller is applied, and the obtained performances are compared (Section <a href="sec:uniaxial_position_control">sec:uniaxial_position_control</a>).
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</p>
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<p>
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Conclusion remarks are given in Section <a href="sec:conclusion">sec:conclusion</a>.
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</p>
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<div id="org3647f41" class="figure">
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<p><img src="figs/uniaxial_overview_model_sections.png" alt="uniaxial_overview_model_sections.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Uniaxial Micro-Station model in blue (Section <a href="sec:micro_station_model">sec:micro_station_model</a>), Nano-Hexapod and sample models in red (Section <a href="sec:nano_station_model">sec:nano_station_model</a>), Disturbances in yellow (Section <a href="sec:uniaxial_disturbances">sec:uniaxial_disturbances</a>), Active Damping in green (Section <a href="sec:uniaxial_active_damping">sec:uniaxial_active_damping</a>) and Position control in purple (Section <a href="sec:uniaxial_position_control">sec:uniaxial_position_control</a>)</p>
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</div>
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<div id="outline-container-org5d8ba75" class="outline-2">
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<h2 id="org5d8ba75"><span class="section-number-2">1.</span> Micro Station Model</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="orgebdb9da"></a>
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</p>
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<p>
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In this section, a uni-axial model of the micro-station is tuned in order to match measurements made on the micro-station
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The measurement setup is shown in Figure <a href="fig:micro_station_first_meas_dynamics">fig:micro_station_first_meas_dynamics</a> where several geophones are fixed to the micro-station and an instrumented hammer is used to inject forces on different stages of the micro-station.
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</p>
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<p>
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From the measured frequency response functions (FRF), the model can be tuned to approximate the uniaxial dynamics of the micro-station.
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</p>
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<div id="org43f1dcd" class="figure">
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<p><img src="figs/micro_station_first_meas_dynamics.jpg" alt="micro_station_first_meas_dynamics.jpg" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Experimental Setup for the first dynamical measurements on the Micro-Station. Geophones are fixed to the micro-station, and the granite as well as the micro-hexapod’s top platform are impact with an instrumented hammer</p>
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</div>
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</div>
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<div id="outline-container-org06407a9" class="outline-3">
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<h3 id="org06407a9"><span class="section-number-3">1.1.</span> Measured dynamics</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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The measurement setup is schematically shown in Figure <a href="fig:micro_station_meas_dynamics_schematic">fig:micro_station_meas_dynamics_schematic</a> where:
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</p>
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<ul class="org-ul">
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<li>Two hammer hits are performed, one on the Granite (force \(F_g\)), and one on the micro-hexapod’s top platform (force \(F_h\))</li>
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<li>The inertial motion of the granite \(x_g\) and the micro-hexapod’s top platform \(x_h\) are measured using geophones.</li>
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</ul>
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<p>
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From the forces applied by the instrumented hammer and the responses of the geophones, the following frequency response functions can be computed:
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</p>
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<ul class="org-ul">
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<li>from \(F_h\) to \(d_h\) (i.e. the compliance of the micro-station)</li>
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<li>from \(F_g\) to \(d_h\) (or from \(F_h\) to \(d_g\))</li>
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<li>from \(F_g\) to \(d_g\)</li>
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</ul>
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<div id="org6ac9ebf" class="figure">
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<p><img src="figs/micro_station_meas_dynamics_schematic.png" alt="micro_station_meas_dynamics_schematic.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Measurement setup - Schematic</p>
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</div>
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<p>
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Due to the bad coherence at low frequency, the frequency response functions are only shown between 20 and 200Hz (Figure <a href="fig:uniaxial_measured_frf_vertical">fig:uniaxial_measured_frf_vertical</a>).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Load measured FRF</span>
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load(<span class="org-string">'meas_microstation_frf.mat'</span>);
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</pre>
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</div>
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<div id="org89c7469" class="figure">
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<p><img src="figs/uniaxial_measured_frf_vertical.png" alt="uniaxial_measured_frf_vertical.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Measured Frequency Response Functions in the vertical direction</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org0846d69" class="outline-3">
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<h3 id="org0846d69"><span class="section-number-3">1.2.</span> Uniaxial Model</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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The uni-axial model of the micro-station is shown in Figure <a href="fig:uniaxial_comp_frf_meas_model">fig:uniaxial_comp_frf_meas_model</a>, with:
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</p>
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<ul class="org-ul">
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<li>Disturbances:
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<ul class="org-ul">
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<li>\(x_f\): Floor motion</li>
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<li>\(f_t\): Stage vibrations</li>
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</ul></li>
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<li>Hammer impacts: \(F_h\) and \(F_g\).</li>
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<li>Geophones: \(x_h\) and \(x_g\)</li>
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</ul>
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<div id="org478a573" class="figure">
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<p><img src="figs/uniaxial_model_micro_station.png" alt="uniaxial_model_micro_station.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Uniaxial model of the micro-station</p>
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</div>
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<p>
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Masses are estimated from the CAD.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Parameters - Mass</span>
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mh = 15; <span class="org-comment-delimiter">% </span><span class="org-comment">Micro Hexapod [kg]</span>
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mt = 1200; <span class="org-comment-delimiter">% </span><span class="org-comment">Ty + Ry + Rz [kg]</span>
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mg = 2500; <span class="org-comment-delimiter">% </span><span class="org-comment">Granite [kg]</span>
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</pre>
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</div>
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<p>
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And stiffnesses from the data-sheet of stage manufacturers.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Parameters - Stiffnesses</span>
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kh = 6.11e<span class="org-builtin">+</span>07; <span class="org-comment-delimiter">% </span><span class="org-comment">[N/m]</span>
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kt = 5.19e<span class="org-builtin">+</span>08; <span class="org-comment-delimiter">% </span><span class="org-comment">[N/m]</span>
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kg = 9.50e<span class="org-builtin">+</span>08; <span class="org-comment-delimiter">% </span><span class="org-comment">[N/m]</span>
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</pre>
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</div>
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<p>
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The damping coefficients are tuned to match the identified damping from the measurements.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Parameters - damping</span>
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ch = 2<span class="org-builtin">*</span>0.05<span class="org-builtin">*</span>sqrt(kh<span class="org-builtin">*</span>mh); <span class="org-comment-delimiter">% </span><span class="org-comment">[N/(m/s)]</span>
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ct = 2<span class="org-builtin">*</span>0.05<span class="org-builtin">*</span>sqrt(kt<span class="org-builtin">*</span>mt); <span class="org-comment-delimiter">% </span><span class="org-comment">[N/(m/s)]</span>
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cg = 2<span class="org-builtin">*</span>0.08<span class="org-builtin">*</span>sqrt(kg<span class="org-builtin">*</span>mg); <span class="org-comment-delimiter">% </span><span class="org-comment">[N/(m/s)]</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-org1789ecb" class="outline-3">
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<h3 id="org1789ecb"><span class="section-number-3">1.3.</span> Comparison of the model and measurements</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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The comparison between the measurements and the model is done in Figure <a href="fig:uniaxial_comp_frf_meas_model">fig:uniaxial_comp_frf_meas_model</a>.
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</p>
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<p>
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As the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation.
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More accurate models will be used later on.
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</p>
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<div id="org00917d8" class="figure">
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<p><img src="figs/uniaxial_comp_frf_meas_model.png" alt="uniaxial_comp_frf_meas_model.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Comparison of the measured FRF and identified ones from the uni-axial model</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-org0cf1174" class="outline-2">
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<h2 id="org0cf1174"><span class="section-number-2">2.</span> Nano-Hexapod Model</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org71aab0f"></a>
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</p>
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<p>
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A model of the nano-hexapod and sample is now added on top of the uni-axial model of the micro-station (Figure <a href="fig:uniaxial_model_micro_station-nass">fig:uniaxial_model_micro_station-nass</a>).
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</p>
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<p>
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Disturbances (shown in red) are:
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</p>
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<ul class="org-ul">
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<li>\(f_s\): direct forces applied to the sample (for instance cable forces)</li>
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<li>\(f_t\): disturbances coming from the imperfect stage scanning performances</li>
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<li>\(x_f\): floor motion</li>
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</ul>
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<p>
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The control signal is the force applied by the nano-hexapod \(f\) and the measurement is the relative motion between the sample and the granite \(d\).
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</p>
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<p>
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The sample is here considered as a rigid body and rigidly fixed to the nano-hexapod.
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The effect of having resonances between the sample’s point of interest and the nano-hexapod actuator will be considered in further analysis.
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</p>
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<div id="orge658374" class="figure">
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<p><img src="figs/uniaxial_model_micro_station-nass.png" alt="uniaxial_model_micro_station-nass.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Uni-axial model of the micro-station with added nano-hexapod (represented in blue) and sample (represented in green)</p>
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</div>
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</div>
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<div id="outline-container-org1088c4b" class="outline-3">
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<h3 id="org1088c4b"><span class="section-number-3">2.1.</span> Nano-Hexapod Parameters</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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The parameters for the nano-hexapod and sample are:
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</p>
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<ul class="org-ul">
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<li>\(m_s\) the sample mass that can vary from 1kg up to 50kg</li>
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<li>\(m_n\) the nano-hexapod mass which is set to 15kg</li>
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<li>\(k_n\) the nano-hexapod stiffness, which can vary depending on the chosen architecture/technology</li>
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</ul>
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<p>
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As a first example, let’s choose a nano-hexapod stiffness of \(10\,N/\mu m\) and a sample mass of 10kg.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Nano-Hexapod Parameters</span>
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mn = 15; <span class="org-comment-delimiter">% </span><span class="org-comment">[kg]</span>
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kn = 1e7; <span class="org-comment-delimiter">% </span><span class="org-comment">[N/m]</span>
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cn = 2<span class="org-builtin">*</span>0.01<span class="org-builtin">*</span>sqrt(mn<span class="org-builtin">*</span>kn); <span class="org-comment-delimiter">% </span><span class="org-comment">[N/(m/s)]</span>
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<span class="org-matlab-cellbreak">%% Sample Mass</span>
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ms = 10; <span class="org-comment-delimiter">% </span><span class="org-comment">[kg]</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-orgb983a1e" class="outline-3">
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<h3 id="orgb983a1e"><span class="section-number-3">2.2.</span> Obtained Dynamics</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
|
|
The sensitivity to disturbances (i.e. \(x_f\), \(f_t\) and \(f_s\)) are shown in Figure <a href="fig:uniaxial_sensitivity_dist_first_params">fig:uniaxial_sensitivity_dist_first_params</a>.
|
|
The <i>plant</i> (i.e. the transfer function from actuator force \(f\) to measured displacement \(d\)) is shown in Figure <a href="fig:uniaxial_plant_first_params">fig:uniaxial_plant_first_params</a>.
|
|
</p>
|
|
|
|
<p>
|
|
For further analysis, 9 configurations are considered: three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\), \(k_n = 1\,N/\mu m\) and \(k_n = 100\,N/\mu m\)) combined with three sample’s masses (\(m_s = 1\,kg\), \(m_s = 25\,kg\) and \(m_s = 50\,kg\)).
|
|
</p>
|
|
|
|
|
|
<div id="org91c466d" class="figure">
|
|
<p><img src="figs/uniaxial_sensitivity_dist_first_params.png" alt="uniaxial_sensitivity_dist_first_params.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Sensitivity to disturbances</p>
|
|
</div>
|
|
|
|
|
|
<div id="org54fe4b4" class="figure">
|
|
<p><img src="figs/uniaxial_plant_first_params.png" alt="uniaxial_plant_first_params.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Bode Plot of the transfer function from actuator forces to measured displacement by the metrology</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org56c9136" class="outline-2">
|
|
<h2 id="org56c9136"><span class="section-number-2">3.</span> Disturbance Identification</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgd13c200"></a>
|
|
</p>
|
|
<p>
|
|
In order to measure disturbances, two geophones are used, on located on the floor and on on the micro-hexapod’s top platform (see Figure <a href="fig:micro_station_meas_disturbances">fig:micro_station_meas_disturbances</a>).
|
|
</p>
|
|
|
|
<p>
|
|
The geophone on the floor is used to measured the floor motion \(x_f\) while the geophone on the micro-hexapod is used to measure vibrations introduced by scanning of the \(T_y\) stage and \(R_z\) stage.
|
|
</p>
|
|
|
|
|
|
<div id="org1f6bb07" class="figure">
|
|
<p><img src="figs/micro_station_meas_disturbances.png" alt="micro_station_meas_disturbances.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Disturbance measurement setup - Schematic</p>
|
|
</div>
|
|
|
|
|
|
<div id="org9b22047" class="figure">
|
|
<p><img src="figs/micro_station_dynamical_id_setup.jpg" alt="micro_station_dynamical_id_setup.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Two geophones are used to measure the micro-station vibrations induced by the scanning of the \(T_y\) and \(R_z\) stages</p>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org6b04518" class="outline-3">
|
|
<h3 id="org6b04518"><span class="section-number-3">3.1.</span> Ground Motion</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
The geophone fixed to the floor to measure the floor motion.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Load floor motion data</span>
|
|
<span class="org-comment-delimiter">% </span><span class="org-comment">t: time in [s]</span>
|
|
<span class="org-comment-delimiter">% </span><span class="org-comment">V: measured voltage genrated by the geophone and amplified by a 60dB gain voltage amplifier [V]</span>
|
|
load(<span class="org-string">'ground_motion_measurement.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'V'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The voltage generated by each geophone is amplified using a voltage amplifier (gain of 60dB) before going to the ADC.
|
|
The sensitivity of the geophone as well as the gain of the voltage amplifier are then taken into account to reconstruct the floor displacement.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensitivity of the geophone</span>
|
|
S0 = 88; <span class="org-comment-delimiter">% </span><span class="org-comment">Sensitivity [V/(m/s)]</span>
|
|
f0 = 2; <span class="org-comment-delimiter">% </span><span class="org-comment">Cut-off frequency [Hz]</span>
|
|
|
|
S = S0<span class="org-builtin">*</span>(s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>f0)<span class="org-builtin">/</span>(1<span class="org-builtin">+</span>s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>f0); <span class="org-comment-delimiter">% </span><span class="org-comment">Geophone's transfer function [V/(m/s)]</span>
|
|
|
|
<span class="org-matlab-cellbreak">%% Gain of the voltage amplifier</span>
|
|
G0_db = 60; <span class="org-comment-delimiter">% </span><span class="org-comment">[dB]</span>
|
|
G0 = 10<span class="org-builtin">^</span>(G0_db<span class="org-builtin">/</span>20); <span class="org-comment-delimiter">% </span><span class="org-comment">[abs]</span>
|
|
|
|
<span class="org-matlab-cellbreak">%% Transfer function from measured voltage to displacement</span>
|
|
G_geo = 1<span class="org-builtin">/</span>S<span class="org-builtin">/</span>G0<span class="org-builtin">/</span>s; <span class="org-comment-delimiter">% </span><span class="org-comment">[m/V]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The PSD \(S_{V_f}\) of the measured voltage \(V_f\) is computed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Compute measured voltage PSD</span>
|
|
Fs = 1<span class="org-builtin">/</span>(t(2)<span class="org-builtin">-</span>t(1)); <span class="org-comment-delimiter">% </span><span class="org-comment">Sampling Frequency [Hz]</span>
|
|
win = hanning(ceil(2<span class="org-builtin">*</span>Fs)); <span class="org-comment-delimiter">% </span><span class="org-comment">Hanning window</span>
|
|
|
|
[psd_V, f] = pwelch(V, win, [], [], Fs); <span class="org-comment-delimiter">% </span><span class="org-comment">[V^2/Hz]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The PSD of the corresponding displacement can be computed as follows:
|
|
</p>
|
|
\begin{equation}
|
|
S_{x_f}(\omega) = \frac{S_{V_f}(\omega)}{|S_{\text{geo}}(j\omega)| \cdot G_{\text{amp}} \cdot \omega}
|
|
\end{equation}
|
|
<p>
|
|
with:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(S_{\text{geo}}\) the sensitivity of the Geophone in \([Vs/m]\)</li>
|
|
<li>\(G_{\text{amp}}\) the gain of the voltage amplifier</li>
|
|
<li>\(\omega\) is here to integrate and have the displacement instead of the velocity</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Ground Motion ASD</span>
|
|
psd_xf = psd_V<span class="org-builtin">.*</span>abs(squeeze(freqresp(G_geo, f, <span class="org-string">'Hz'</span>)))<span class="org-builtin">.^</span>2; <span class="org-comment-delimiter">% </span><span class="org-comment">[m^2/Hz]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The amplitude spectral density \(\Gamma_{x_f}\) of the measured displacement \(x_f\) is shown in Figure <a href="fig:asd_floor_motion_id31">fig:asd_floor_motion_id31</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org7bc44c6" class="figure">
|
|
<p><img src="figs/asd_floor_motion_id31.png" alt="asd_floor_motion_id31.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Measured Amplitude Spectral Density of the Floor motion on ID31</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9aac15b" class="outline-3">
|
|
<h3 id="org9aac15b"><span class="section-number-3">3.2.</span> Stage Vibration</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
During Spindle rotation (here at 6rpm), the granite velocity and micro-hexapod’s top platform velocity are measured with the geophones.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Measured velocity of granite and hexapod during spindle rotation</span>
|
|
<span class="org-comment-delimiter">% </span><span class="org-comment">t: time in [s]</span>
|
|
<span class="org-comment-delimiter">% </span><span class="org-comment">vg: measured granite velocity [m/s]</span>
|
|
<span class="org-comment-delimiter">% </span><span class="org-comment">vg: measured micro-hexapod's top platform velocity [m/s]</span>
|
|
load(<span class="org-string">'meas_spindle_on.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'vg'</span>, <span class="org-string">'vh'</span>);
|
|
spindle_off = load(<span class="org-string">'meas_spindle_off.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'vg'</span>, <span class="org-string">'vh'</span>); <span class="org-comment-delimiter">% </span><span class="org-comment">No Rotation</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The Power Spectral Density of the relative velocity between the hexapod and the granite is computed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Compute Power Spectral Density of the relative velocity between granite and hexapod during spindle rotation</span>
|
|
Fs = 1<span class="org-builtin">/</span>(t(2)<span class="org-builtin">-</span>t(1)); <span class="org-comment-delimiter">% </span><span class="org-comment">Sampling Frequency [Hz]</span>
|
|
win = hanning(ceil(2<span class="org-builtin">*</span>Fs)); <span class="org-comment-delimiter">% </span><span class="org-comment">Hanning window</span>
|
|
|
|
[psd_vft, f] = pwelch(vh<span class="org-builtin">-</span>vg, win, [], [], Fs); <span class="org-comment-delimiter">% </span><span class="org-comment">[(m/s)^2/Hz]</span>
|
|
[psd_off, <span class="org-builtin">~</span>] = pwelch(spindle_off.vh<span class="org-builtin">-</span>spindle_off.vg, win, [], [], Fs); <span class="org-comment-delimiter">% </span><span class="org-comment">[(m/s)^2/Hz]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
It is then integrated to obtain the Amplitude Spectral Density of the relative motion which is compared with a non-rotating case (Figure <a href="fig:asd_vibration_spindle_rotation">fig:asd_vibration_spindle_rotation</a>).
|
|
It is shown that the spindle rotation induces vibrations in a wide frequency spectrum.
|
|
</p>
|
|
|
|
|
|
<div id="orgea3ea0b" class="figure">
|
|
<p><img src="figs/asd_vibration_spindle_rotation.png" alt="asd_vibration_spindle_rotation.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod’s top platform during Spindle rotating</p>
|
|
</div>
|
|
|
|
<p>
|
|
In order to compute the equivalent disturbance force \(f_t\) that induces such motion, the transfer function from \(f_t\) to the relative motion of the hexapod’s top platform and the granite is extracted from the model.
|
|
The power spectral density \(\Gamma_{f_{t}}\) of the disturbance force can be computed as follows:
|
|
</p>
|
|
\begin{equation}
|
|
\Gamma_{f_{t}}(\omega) = \frac{\Gamma_{v_{t}}(\omega)}{|G_{\text{model}}(j\omega)|^2}
|
|
\end{equation}
|
|
<p>
|
|
with:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\Gamma_{v_{t}}\) the measured power spectral density of the relative motion between the micro-hexapod’s top platform and the granite during the spindle’s rotation</li>
|
|
<li>\(G_{\text{model}}\) the transfer function (extracted from the uniaxial model) from \(f_t\) to the relative motion between the micro-hexapod’s top platform and the granite</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The obtained amplitude spectral density of the disturbance force \(f_t\) is shown in Figure <a href="fig:asd_disturbance_force">fig:asd_disturbance_force</a>.
|
|
</p>
|
|
|
|
<div id="org8cb4c81" class="figure">
|
|
<p><img src="figs/asd_disturbance_force.png" alt="asd_disturbance_force.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Estimated disturbance force ft from measurement and uniaxial model</p>
|
|
</div>
|
|
|
|
<p>
|
|
The vibrations induced by the \(T_y\) stage are not considered here as:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>the induced vibrations have less amplitude than the vibrations induced by the \(R_z\) stage</li>
|
|
<li>it can be scanned at lower velocities if the induced vibrations are an issue</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org446c255" class="outline-2">
|
|
<h2 id="org446c255"><span class="section-number-2">4.</span> Open-Loop Dynamic Noise Budgeting</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org36aefd3"></a>
|
|
</p>
|
|
<p>
|
|
Now that we have a model of the NASS and an estimation of the power spectral density of the disturbances, it is possible to perform an <i>open-loop dynamic noise budgeting</i>.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org26ad26a" class="outline-3">
|
|
<h3 id="org26ad26a"><span class="section-number-3">4.1.</span> Sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
From the Uni-axial model, the transfer function from the disturbances (\(f_s\), \(x_f\) and \(f_t\)) to the displacement \(d\) are computed.
|
|
</p>
|
|
|
|
<p>
|
|
This is done for <b>two extreme sample masses</b> \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\) and <b>three nano-hexapod stiffnesses</b>:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(k_n = 0.01\,N/\mu m\) that could represent a voice coil actuator with soft flexible guiding</li>
|
|
<li>\(k_n = 1\,N/\mu m\) that could represent a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator</li>
|
|
<li>\(k_n = 100\,N/\mu m\) that could represent a stiff piezoelectric stack actuator</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure <a href="fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses">fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses</a> for the light sample (same conclusions can be drawn with the heavy one).
|
|
</p>
|
|
|
|
<div class="important" id="org5e0fc42">
|
|
<p>
|
|
From Figure <a href="fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses">fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses</a>, following can be concluded for the <b>soft nano-hexapod</b>:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>It is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to the lower stiffness</li>
|
|
<li>Between the suspension mode of the nano-hexapod (here at 5Hz) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out.</li>
|
|
<li>Above the suspension mode of the nano-hexapod, the sample’s motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is almost \(1\).</li>
|
|
</ul>
|
|
|
|
</div>
|
|
|
|
|
|
<div id="orgeff9fdf" class="figure">
|
|
<p><img src="figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses.png" alt="uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Sensitivity to disturbances for three different nano-hexpod stiffnesses</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga2938ae" class="outline-3">
|
|
<h3 id="orga2938ae"><span class="section-number-3">4.2.</span> Open-Loop Dynamic Noise Budgeting</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
Now, the power spectral density of the disturbances is taken into account to estimate the residual motion \(d\) in each case.
|
|
</p>
|
|
|
|
<p>
|
|
The Cumulative Amplitude Spectrum of the relative motion \(d\) due to both the floor motion \(x_f\) and the stage vibrations \(f_t\) are shown in Figure <a href="fig:uniaxial_cas_d_disturbances_stiffnesses">fig:uniaxial_cas_d_disturbances_stiffnesses</a> for the three nano-hexapod stiffnesses.
|
|
</p>
|
|
|
|
<p>
|
|
It is shown that the effect of the floor motion is much less than the stage vibrations, except for the soft nano-hexapod below 5Hz.
|
|
</p>
|
|
|
|
|
|
<div id="orga2a41bd" class="figure">
|
|
<p><img src="figs/uniaxial_cas_d_disturbances_stiffnesses.png" alt="uniaxial_cas_d_disturbances_stiffnesses.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Cumulative Amplitude Spectrum of the relative motion d, due to both the floor motion and the stage vibrations (light sample)</p>
|
|
</div>
|
|
|
|
<p>
|
|
The total cumulative amplitude spectrum for the three nano-hexapod stiffnesses and for the two sample’s masses are shown in Figure <a href="fig:uniaxial_cas_d_disturbances_payload_masses">fig:uniaxial_cas_d_disturbances_payload_masses</a>.
|
|
The conclusion is that the sample’s mass has little effect on the cumulative amplitude spectrum of the relative motion \(d\).
|
|
</p>
|
|
|
|
|
|
<div id="org97fa868" class="figure">
|
|
<p><img src="figs/uniaxial_cas_d_disturbances_payload_masses.png" alt="uniaxial_cas_d_disturbances_payload_masses.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Cumulative Amplitude Spectrum of the relative motion d due to all disturbances, for two sample masses</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org79864c7" class="outline-3">
|
|
<h3 id="org79864c7"><span class="section-number-3">4.3.</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="important" id="orgb63097a">
|
|
<p>
|
|
The conclusion is that in order to have a closed-loop residual vibration \(d \approx 20\,nm\text{ rms}\), if a simple feedback controller is used, the required closed-loop bandwidth would be:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\))</li>
|
|
<li>\(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\))</li>
|
|
<li>\(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\))</li>
|
|
</ul>
|
|
|
|
<p>
|
|
This can be explain by the fact that above the suspension mode of the nano-hexapod, the stage vibrations are filtered out (see Figure <a href="fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses">fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses</a>).
|
|
</p>
|
|
|
|
<p>
|
|
This gives a first advantage to having a soft nano-hexapod.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org226ab26" class="outline-2">
|
|
<h2 id="org226ab26"><span class="section-number-2">5.</span> Active Damping</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org44d5a3d"></a>
|
|
</p>
|
|
<p>
|
|
In this section, three active damping are applied on the nano-hexapod (see Figure <a href="fig:uniaxial_active_damping_strategies">fig:uniaxial_active_damping_strategies</a>): Integral Force Feedback (IFF), Relative Damping Control (RDC) and Direct Velocity Feedback (DVF).
|
|
</p>
|
|
|
|
<p>
|
|
These active damping techniques are compared in terms of:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Reduction of the effect of disturbances (i.e. \(x_f\), \(f_t\) and \(f_s\)) on the displacement \(d\)</li>
|
|
<li>Achievable damping</li>
|
|
<li>Robustness to a change of sample’s mass</li>
|
|
</ul>
|
|
|
|
|
|
<div id="orge95bee1" class="figure">
|
|
<p><img src="figs/uniaxial_active_damping_strategies.png" alt="uniaxial_active_damping_strategies.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Three active damping strategies: Integral Force Feedback (IFF) using a force sensor, Relative Damping Control (RDC) using a relative displacement sensor, and Direct Velocity Feedback (DVF) using a geophone</p>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org78584ca" class="outline-3">
|
|
<h3 id="org78584ca"><span class="section-number-3">5.1.</span> Active Damping Strategies</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
The Integral Force Feedback strategy consists of using a force sensor in series with the actuator (see Figure <a href="fig:uniaxial_active_damping_iff_equiv">fig:uniaxial_active_damping_iff_equiv</a>, left).
|
|
</p>
|
|
|
|
<p>
|
|
The control strategy consists of integrating the measured force and feeding it back to the actuator:
|
|
</p>
|
|
\begin{equation}
|
|
K_{\text{IFF}}(s) = \frac{g}{s}
|
|
\end{equation}
|
|
|
|
<p>
|
|
The mechanical equivalent of this strategy is to add a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain \(k/g\) (see Figure <a href="fig:uniaxial_active_damping_iff_equiv">fig:uniaxial_active_damping_iff_equiv</a>, right).
|
|
</p>
|
|
|
|
|
|
<div id="orgdb12a77" class="figure">
|
|
<p><img src="figs/uniaxial_active_damping_iff_equiv.png" alt="uniaxial_active_damping_iff_equiv.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Integral Force Feedback is equivalent as to add a damper in series with the stiffness (the initial damping is here neglected for simplicity)</p>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
For the Relative Damping Control strategy, a relative motion sensor that measures the motion of the actuator is used (see Figure <a href="fig:uniaxial_active_damping_rdc_equiv">fig:uniaxial_active_damping_rdc_equiv</a>, left).
|
|
</p>
|
|
|
|
<p>
|
|
The derivative of this relative motion is used for the feedback signal:
|
|
</p>
|
|
\begin{equation}
|
|
K_{\text{RDC}}(s) = - g \cdot s
|
|
\end{equation}
|
|
|
|
<p>
|
|
The mechanical equivalent is to add a dashpot in parallel with the actuator with a damping coefficient equal to the controller gain \(g\) (see Figure <a href="fig:uniaxial_active_damping_rdc_equiv">fig:uniaxial_active_damping_rdc_equiv</a>, right).
|
|
</p>
|
|
|
|
|
|
<div id="org7725789" class="figure">
|
|
<p><img src="figs/uniaxial_active_damping_rdc_equiv.png" alt="uniaxial_active_damping_rdc_equiv.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Relative Damping Control is equivalent as adding a damper in parallel with the actuator/relative motion sensor</p>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, the Direct Velocity Feedback strategy consists of using an inertial sensor (usually a geophone), that measured the “absolute” velocity of the body fixed on top of the actuator (se Figure <a href="fig:uniaxial_active_damping_dvf_equiv">fig:uniaxial_active_damping_dvf_equiv</a>, left).
|
|
</p>
|
|
|
|
<p>
|
|
The measured velocity is then fed back to the actuator:
|
|
</p>
|
|
\begin{equation}
|
|
K_{\text{DVF}}(s) = - g
|
|
\end{equation}
|
|
|
|
<p>
|
|
This is equivalent as to fix a dashpot (with a damping coefficient equal to the controller gain \(g\)) between the body (one which the inertial sensor is fixed) and an inertial reference frame (see Figure <a href="fig:uniaxial_active_damping_dvf_equiv">fig:uniaxial_active_damping_dvf_equiv</a>, right).
|
|
This is usually refers to as “<i>sky hook damper</i>”.
|
|
</p>
|
|
|
|
|
|
<div id="org440966f" class="figure">
|
|
<p><img src="figs/uniaxial_active_damping_dvf_equiv.png" alt="uniaxial_active_damping_dvf_equiv.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>Direct velocity Feedback using an inertial sensor is equivalent to a “sky hook damper”</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd89363e" class="outline-3">
|
|
<h3 id="orgd89363e"><span class="section-number-3">5.2.</span> Plant Dynamics for Active Damping</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
<p>
|
|
The plant dynamics for all three active damping techniques are shown in Figure <a href="fig:uniaxial_plant_active_damping_techniques">fig:uniaxial_plant_active_damping_techniques</a>.
|
|
All have <b>alternating poles and zeros</b> meaning that the phase is bounded to \(\pm 90\,\text{deg}\) which makes the controller very robust.
|
|
</p>
|
|
|
|
<p>
|
|
When the nano-hexapod’s suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure <a href="fig:uniaxial_plant_active_damping_techniques">fig:uniaxial_plant_active_damping_techniques</a>), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.
|
|
</p>
|
|
|
|
<p>
|
|
For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure <a href="fig:uniaxial_plant_active_damping_techniques">fig:uniaxial_plant_active_damping_techniques</a>.
|
|
Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.
|
|
</p>
|
|
|
|
|
|
<div id="org97defdd" class="figure">
|
|
<p><img src="figs/uniaxial_plant_active_damping_techniques.png" alt="uniaxial_plant_active_damping_techniques.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>Plant dynamics for the three active damping techniques (IFF: right, RDC: middle, DVF: left), for three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\) in blue, \(k_n = 1\,N/\mu m\) in red and \(k_n = 100\,N/\mu m\) in yellow) and three sample’s masses (\(m_s = 1\,kg\): solid curves, \(m_s = 25\,kg\): dot-dashed curves, and \(m_s = 50\,kg\): dashed curves).</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgdd9b4b6" class="outline-3">
|
|
<h3 id="orgdd9b4b6"><span class="section-number-3">5.3.</span> Achievable Damping - Root Locus</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
<p>
|
|
The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
|
|
They are shown in Figure <a href="fig:uniaxial_root_locus_damping_techniques">fig:uniaxial_root_locus_damping_techniques</a>.
|
|
</p>
|
|
|
|
<p>
|
|
All three active damping approach can lead to <b>critical damping</b> of the nano-hexapod suspension mode.
|
|
</p>
|
|
|
|
<p>
|
|
There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure <a href="fig:uniaxial_root_locus_damping_techniques">fig:uniaxial_root_locus_damping_techniques</a>, right) and with RDC for a soft nano-hexapod (Figure <a href="fig:uniaxial_root_locus_damping_techniques_micro_station_mode">fig:uniaxial_root_locus_damping_techniques_micro_station_mode</a>).
|
|
This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).
|
|
</p>
|
|
|
|
|
|
<div id="org14fd23b" class="figure">
|
|
<p><img src="figs/uniaxial_root_locus_damping_techniques.png" alt="uniaxial_root_locus_damping_techniques.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 23: </span>Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses.</p>
|
|
</div>
|
|
|
|
|
|
<div id="org83e9551" class="figure">
|
|
<p><img src="figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png" alt="uniaxial_root_locus_damping_techniques_micro_station_mode.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 24: </span>Root Locus for the three damping techniques. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org85dd4e6" class="outline-3">
|
|
<h3 id="org85dd4e6"><span class="section-number-3">5.4.</span> Change of sensitivity to disturbances</h3>
|
|
<div class="outline-text-3" id="text-5-4">
|
|
<p>
|
|
The sensitivity to disturbances (direct forces \(f_s\), stage vibrations \(f_t\) and floor motion \(x_f\)) for all three active damping techniques are compared in Figure <a href="fig:uniaxial_sensitivity_dist_active_damping">fig:uniaxial_sensitivity_dist_active_damping</a>.
|
|
The comparison is done with the nano-hexapod having a stiffness \(k_n = 1\,N/\mu m\).
|
|
</p>
|
|
|
|
<div class="important" id="org9ef3f80">
|
|
<p>
|
|
Conclusions from Figure <a href="fig:uniaxial_sensitivity_dist_active_damping">fig:uniaxial_sensitivity_dist_active_damping</a> are:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>IFF degrades the sensitivity to direct forces on the sample (i.e. the compliance) below the resonance of the nano-hexapod</li>
|
|
<li>RDC degrades the sensitivity to stage vibrations around the nano-hexapod’s resonance as compared to the other two methods</li>
|
|
<li>both IFF and DVF degrades the sensitivity to floor motion below the resonance of the nano-hexapod</li>
|
|
</ul>
|
|
|
|
</div>
|
|
|
|
|
|
<div id="orgc4e8bac" class="figure">
|
|
<p><img src="figs/uniaxial_sensitivity_dist_active_damping.png" alt="uniaxial_sensitivity_dist_active_damping.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 25: </span>Change of sensitivity to disturbance with all three active damping strategies</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc1f3a3e" class="outline-3">
|
|
<h3 id="orgc1f3a3e"><span class="section-number-3">5.5.</span> Noise Budgeting after Active Damping</h3>
|
|
<div class="outline-text-3" id="text-5-5">
|
|
<p>
|
|
Cumulative Amplitude Spectrum of the distance \(d\) with all three active damping techniques are compared in Figure <a href="fig:uniaxial_cas_active_damping">fig:uniaxial_cas_active_damping</a>.
|
|
All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
|
|
</p>
|
|
|
|
<p>
|
|
Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
|
|
</p>
|
|
|
|
|
|
<div id="org24f815c" class="figure">
|
|
<p><img src="figs/uniaxial_cas_active_damping.png" alt="uniaxial_cas_active_damping.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 26: </span>Comparison of the cumulative amplitude spectrum (CAS) of the distance \(d\) for all three active damping techniques (OL in black, IFF in blue, RDC in red and DVF in yellow).</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org669e6ff" class="outline-3">
|
|
<h3 id="org669e6ff"><span class="section-number-3">5.6.</span> Obtained Damped Plant</h3>
|
|
<div class="outline-text-3" id="text-5-6">
|
|
<p>
|
|
The transfer functions from the plant input \(f\) to the relative displacement \(d\) while the active damping is implemented are shown in Figure <a href="fig:uniaxial_damped_plant_three_active_damping_techniques">fig:uniaxial_damped_plant_three_active_damping_techniques</a>.
|
|
All three active damping techniques yield similar damped plants.
|
|
</p>
|
|
|
|
|
|
<div id="org9512dd4" class="figure">
|
|
<p><img src="figs/uniaxial_damped_plant_three_active_damping_techniques.png" alt="uniaxial_damped_plant_three_active_damping_techniques.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 27: </span>Obtained damped transfer function from f to d for the three damping techniques</p>
|
|
</div>
|
|
|
|
<p>
|
|
The damped plants are shown in Figure <a href="fig:uniaxial_damped_plant_change_sample_mass">fig:uniaxial_damped_plant_change_sample_mass</a> for all three techniques, with the three considered nano-hexapod stiffnesses and sample’s masses.
|
|
</p>
|
|
|
|
<div id="org3269f2d" class="figure">
|
|
<p><img src="figs/uniaxial_damped_plant_change_sample_mass.png" alt="uniaxial_damped_plant_change_sample_mass.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 28: </span>Damped plant \(d/f\) - Robustness to change of sample’s mass for all three active damping techniques. Grey curves are the open-loop (i.e. undamped) plants.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org957bd1d" class="outline-3">
|
|
<h3 id="org957bd1d"><span class="section-number-3">5.7.</span> Robustness to change of payload’s mass</h3>
|
|
<div class="outline-text-3" id="text-5-7">
|
|
<p>
|
|
The Root Locus for the three damping techniques are shown in Figure <a href="fig:uniaxial_active_damping_robustness_mass_root_locus">fig:uniaxial_active_damping_robustness_mass_root_locus</a> for three sample’s mass (1kg, 25kg and 50kg).
|
|
The closed-loop poles are shown by the squares for a specific gain.
|
|
</p>
|
|
|
|
<p>
|
|
We can see that having heavier samples yields larger damping for IFF and smaller damping for RDC and DVF.
|
|
</p>
|
|
|
|
|
|
<div id="orgf5e0834" class="figure">
|
|
<p><img src="figs/uniaxial_active_damping_robustness_mass_root_locus.png" alt="uniaxial_active_damping_robustness_mass_root_locus.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 29: </span>Active Damping Robustness to change of sample’s mass - Root Locus for all three damping techniques with 3 different sample’s masses</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8a3c2c8" class="outline-3">
|
|
<h3 id="org8a3c2c8"><span class="section-number-3">5.8.</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-5-8">
|
|
<div class="important" id="orgb87f7bd">
|
|
<p>
|
|
Conclusions for Active Damping:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>All three active damping techniques yields good damping (Figure <a href="fig:uniaxial_root_locus_damping_techniques">fig:uniaxial_root_locus_damping_techniques</a>) and similar remaining vibrations (Figure <a href="fig:uniaxial_cas_active_damping">fig:uniaxial_cas_active_damping</a>)</li>
|
|
<li>The obtained damped plants (Figure <a href="fig:uniaxial_damped_plant_change_sample_mass">fig:uniaxial_damped_plant_change_sample_mass</a>) are equivalent for the three active damping techniques</li>
|
|
<li>Which one to be used will be determined with the use of more accurate models and will depend on which is the easiest to implement in practice</li>
|
|
</ul>
|
|
|
|
</div>
|
|
|
|
<table id="org573fe8e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 1:</span> Comparison of active damping strategies</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-left"><b>IFF</b></th>
|
|
<th scope="col" class="org-left"><b>RDC</b></th>
|
|
<th scope="col" class="org-left"><b>DVF</b></th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left"><b>Sensor</b></td>
|
|
<td class="org-left">Force sensor</td>
|
|
<td class="org-left">Relative motion sensor</td>
|
|
<td class="org-left">Inertial sensor</td>
|
|
</tr>
|
|
</tbody>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left"><b>Damping</b></td>
|
|
<td class="org-left">Up to critical</td>
|
|
<td class="org-left">Up to critical</td>
|
|
<td class="org-left">Up to Critical</td>
|
|
</tr>
|
|
</tbody>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left"><b>Robustness</b></td>
|
|
<td class="org-left">Requires collocation</td>
|
|
<td class="org-left">Requires collocation</td>
|
|
<td class="org-left">Impacted by geophone resonances</td>
|
|
</tr>
|
|
</tbody>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">\(f_s\) <b>Disturbance</b></td>
|
|
<td class="org-left">\(\nearrow\) at low frequency</td>
|
|
<td class="org-left">\(\searrow\) near resonance</td>
|
|
<td class="org-left">\(\searrow\) near resonance</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">\(f_t\) <b>Disturbance</b></td>
|
|
<td class="org-left">\(\searrow\) near resonance</td>
|
|
<td class="org-left">\(\nearrow\) near resonance</td>
|
|
<td class="org-left">\(\searrow\) near resonance</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">\(x_f\) <b>Disturbance</b></td>
|
|
<td class="org-left">\(\nearrow\) at low frequency</td>
|
|
<td class="org-left">\(\searrow\) near resonance</td>
|
|
<td class="org-left">\(\nearrow\) at low frequency</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org97eb0e2" class="outline-2">
|
|
<h2 id="org97eb0e2"><span class="section-number-2">6.</span> Position Feedback Controller</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="orgcfddbb2"></a>
|
|
</p>
|
|
<p>
|
|
The High Authority Control - Low Authority Control (HAC-LAC) architecture is shown in Figure <a href="fig:uniaxial_hac_lac_architecture">fig:uniaxial_hac_lac_architecture</a>.
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It corresponds to a <i>two step</i> control strategy:
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</p>
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|
<ul class="org-ul">
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|
<li>First, an active damping controller \(\bm{K}_{\textsc{LAC}}\) is implemented (see Section <a href="sec:uniaxial_active_damping">sec:uniaxial_active_damping</a>).
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|
It allows to reduce the vibration level, and it also makes the damped plant (transfer function from \(u^{\prime}\) to \(y\)) easier to control than the undamped plant (transfer function from \(u\) to \(y\)).</li>
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<li>Then, a position controller \(\bm{K}_{\textsc{HAC}}\) is implemented.</li>
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</ul>
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<p>
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|
Combined with the uniaxial model, it is shown in Figure <a href="fig:uniaxial_hac_lac_model">fig:uniaxial_hac_lac_model</a>.
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|
</p>
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|
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<div class="figure" id="org1d763c8">
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<div class="subfigure" id="org523c9eb">
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|
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<div id="org106c003" class="figure">
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<p><img src="figs/uniaxial_hac_lac_architecture.png" alt="uniaxial_hac_lac_architecture.png" />
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</p>
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</div>
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</div>
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<div class="subfigure" id="org200951b">
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|
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|
<div id="org28e9946" class="figure">
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<p><img src="figs/uniaxial_hac_lac_model.png" alt="uniaxial_hac_lac_model.png" />
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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-org9a6dfd4" class="outline-3">
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<h3 id="org9a6dfd4"><span class="section-number-3">6.1.</span> Damped Plant Dynamics</h3>
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|
<div class="outline-text-3" id="text-6-1">
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<p>
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|
As was shown in Section <a href="sec:uniaxial_active_damping">sec:uniaxial_active_damping</a>, all three proposed active damping techniques yield similar damping plants.
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|
Therefore, <i>Integral Force Feedback</i> will be used in this section to study the HAC-LAC performances.
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|
</p>
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|
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|
<p>
|
|
The obtained damped plants for the three nano-hexapod stiffnesses are shown in Figure <a href="fig:uniaxial_hac_iff_damped_plants_masses">fig:uniaxial_hac_iff_damped_plants_masses</a>.
|
|
</p>
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|
|
|
|
|
<div id="org13960bd" class="figure">
|
|
<p><img src="figs/uniaxial_hac_iff_damped_plants_masses.png" alt="uniaxial_hac_iff_damped_plants_masses.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 30: </span>Obtained damped plant using Integral Force Feedback for three sample’s 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-org03f8665" class="outline-3">
|
|
<h3 id="org03f8665"><span class="section-number-3">6.2.</span> Position Feedback Controller</h3>
|
|
<div class="outline-text-3" id="text-6-2">
|
|
<p>
|
|
The objective now is to design a position feedback controller for each of the three nano-hexapods that are robust to the change of sample’s mass.
|
|
</p>
|
|
|
|
<p>
|
|
The required feedback bandwidth was approximately determined un Section <a href="sec:uniaxial_noise_budgeting">sec:uniaxial_noise_budgeting</a>:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)).
|
|
Near this frequency, the plants are equivalent to a mass line.
|
|
The gain of the mass line can vary up to a fact \(\approx 5\) (suspended mass from \(16\,kg\) up to \(65\,kg\)).
|
|
This mean that the designed controller will need to have large gain margins to be robust to the change of sample’s mass.</li>
|
|
<li>\(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)).
|
|
Similarly to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line.
|
|
It will be probably easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode.</li>
|
|
<li>\(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)).
|
|
Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the wanted crossover frequency.
|
|
The micro-station is not stiff enough to have a clear stiffness line at this frequency.
|
|
Therefore, there are both a change of phase and gain depending on the sample’s mass.
|
|
This makes the robust design of the controller a little bit more complicated.</li>
|
|
</ul>
|
|
|
|
|
|
<p>
|
|
Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure <a href="fig:uniaxial_nyquist_hac">fig:uniaxial_nyquist_hac</a>).
|
|
These high authority controllers are generally composed of a two integrators at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
|
|
The loop gains for the three nano-hexapod are shown in Figure <a href="fig:uniaxial_loop_gain_hac">fig:uniaxial_loop_gain_hac</a>.
|
|
We can see that:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>for the soft and moderately stiff nano-hexapod, the crossover frequency varies a lot with the sample mass.
|
|
This is due to the fact that the crossover frequency corresponds to the mass line of the plant.</li>
|
|
<li>for the stiff nano-hexapod, the obtained crossover frequency is not at high as what was estimated necessary.
|
|
The crossover frequency in that case is close to the stiffness line of the plant, which makes the robust design of the controller easier.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Note that these controller were quickly tuned by hand and not designed using any optimization methods.
|
|
The goal is just to have a first estimation of the attainable performances.
|
|
</p>
|
|
|
|
|
|
<div id="org4022997" class="figure">
|
|
<p><img src="figs/uniaxial_nyquist_hac.png" alt="uniaxial_nyquist_hac.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 31: </span>Nyquist Plot - Hight Authority Controller for all three nano-hexapod stiffnesses (soft one in blue, moderately stiff in red and very stiff in yellow) and all sample masses (corresponding to the three curves of each color)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org82207e9" class="figure">
|
|
<p><img src="figs/uniaxial_loop_gain_hac.png" alt="uniaxial_loop_gain_hac.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 32: </span>Loop Gain - High Authority Controller for all three nano-hexapod stiffnesses (soft one in blue, moderately stiff in red and very stiff in yellow) and all sample masses (corresponding to the three curves of each color)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgab1ced7" class="outline-3">
|
|
<h3 id="orgab1ced7"><span class="section-number-3">6.3.</span> Closed-Loop Noise Budgeting</h3>
|
|
<div class="outline-text-3" id="text-6-3">
|
|
<p>
|
|
The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed.
|
|
These are compared with the open-loop and damped plants cases in Figure <a href="fig:uniaxial_sensitivity_dist_hac_lac">fig:uniaxial_sensitivity_dist_hac_lac</a> for just one configuration (moderately stiff nano-hexapod with 25kg sample’s mass).
|
|
As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.
|
|
</p>
|
|
|
|
|
|
<div id="org2169532" class="figure">
|
|
<p><img src="figs/uniaxial_sensitivity_dist_hac_lac.png" alt="uniaxial_sensitivity_dist_hac_lac.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 33: </span>Change of sensitivity to disturbances with LAC and with HAC-LAC</p>
|
|
</div>
|
|
|
|
<p>
|
|
The cumulative amplitude spectrum of the motion \(d\) is computed for all nano-hexapod configurations, all sample masses and in the open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases.
|
|
The results are shown in Figure <a href="fig:uniaxial_cas_hac_lac">fig:uniaxial_cas_hac_lac</a>.
|
|
Obtained root mean square values of the distance \(d\) are better for the soft nano-hexapod (\(\approx 25\,nm\) to \(\approx 35\,nm\) depending on the sample’s mass) than for the stiffer nano-hexapod (from \(\approx 30\,nm\) to \(\approx 70\,nm\)).
|
|
</p>
|
|
|
|
|
|
<div id="org6ff9216" class="figure">
|
|
<p><img src="figs/uniaxial_cas_hac_lac.png" alt="uniaxial_cas_hac_lac.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 34: </span>Cumulative Amplitude Spectrum for all three nano-hexapod stiffnesses - Comparison of OL, IFF and HAC-LAC cases</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org90d30b4" class="outline-2">
|
|
<h2 id="org90d30b4"><span class="section-number-2">7.</span> Conclusion</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
<p>
|
|
<a id="orgff48972"></a>
|
|
</p>
|
|
|
|
<p>
|
|
In this study, a uniaxial model of the nano-active-stabilization-system has been tuned both from dynamical measurements (Section <a href="sec:micro_station_model">sec:micro_station_model</a>) and from disturbances measurements (Section <a href="sec:uniaxial_disturbances">sec:uniaxial_disturbances</a>).
|
|
</p>
|
|
|
|
<p>
|
|
It has been shown that three active damping techniques can be used to critically damp the nano-hexapod resonances (Section <a href="sec:uniaxial_active_damping">sec:uniaxial_active_damping</a>).
|
|
However, this model does not allows to determine which one is most suited to this application.
|
|
</p>
|
|
|
|
<p>
|
|
Finally, position feedback controllers have been developed for three considered nano-hexapod stiffnesses.
|
|
These controllers were shown to be robust to the change of sample’s masses, and to provide good rejection of disturbances.
|
|
It has been found that having a soft nano-hexapod makes the plant dynamics easier to control (because decoupled from the micro-station dynamics) and requires less position feedback bandwidth to fulfill the requirements.
|
|
The moderately stiff nano-hexapod (\(k_n = 1\,N/\mu m\)) is requiring a bit more position feedback bandwidth, but it still seems to give acceptable results.
|
|
However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performances.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2023-02-17 Fri 11:29</p>
|
|
</div>
|
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|
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