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if(elem.cacheClassElem) elem.className = elem.cacheClassElem; if(elem.cacheClassTarget) target.className = elem.cacheClassTarget; } /*]]>*///--> // @license-end </script> <script> MathJax = { tex: { macros: { bm: ["\\boldsymbol{#1}",1], } } }; </script> <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> </head> <body> <div id="org-div-home-and-up"> <a accesskey="h" href="./index.html"> UP </a> | <a accesskey="H" href="./index.html"> HOME </a> </div><div id="content"> <h1 class="title">Simscape Uniaxial Model</h1> <div id="table-of-contents"> <h2>Table of Contents</h2> <div id="text-table-of-contents"> <ul> <li><a href="#org227ba84">1. Simscape Model</a></li> <li><a href="#org6bfb35d">2. Undamped System</a> <ul> <li><a href="#org006e24f">2.1. Init</a></li> <li><a href="#org4fbbce9">2.2. Identification</a></li> <li><a href="#org49ca9ab">2.3. Sensitivity to Disturbances</a></li> <li><a href="#org346ebf3">2.4. Noise Budget</a></li> <li><a href="#orgdb3535a">2.5. Plant</a></li> </ul> </li> <li><a href="#org33c3829">3. Integral Force Feedback</a> <ul> <li><a href="#org326d925">3.1. Control Design</a></li> <li><a href="#org86f3473">3.2. Identification</a></li> <li><a href="#org68c471e">3.3. Sensitivity to Disturbance</a></li> <li><a href="#org5edf015">3.4. Damped Plant</a></li> <li><a href="#orgfc93a1c">3.5. Conclusion</a></li> </ul> </li> <li><a href="#org07ff58f">4. Relative Motion Control</a> <ul> <li><a href="#org5704583">4.1. Control Design</a></li> <li><a href="#orga436aa7">4.2. Identification</a></li> <li><a href="#org133268a">4.3. Sensitivity to Disturbance</a></li> <li><a href="#org2f974d4">4.4. Damped Plant</a></li> <li><a href="#orgfdbd543">4.5. Conclusion</a></li> </ul> </li> <li><a href="#orgc9b3622">5. Direct Velocity Feedback</a> <ul> <li><a href="#org2050b01">5.1. Control Design</a></li> <li><a href="#orgc946d88">5.2. Identification</a></li> <li><a href="#orgde4b14e">5.3. Sensitivity to Disturbance</a></li> <li><a href="#org640c7d9">5.4. Damped Plant</a></li> <li><a href="#org94e9d5f">5.5. Conclusion</a></li> </ul> </li> <li><a href="#org5ac7dda">6. With Cedrat Piezo-electric Actuators</a> <ul> <li><a href="#org1bfaed7">6.1. Identification</a></li> <li><a href="#orgf32651c">6.2. Control Design</a></li> <li><a href="#orgc7383cd">6.3. Identification</a></li> <li><a href="#org34de1fd">6.4. Sensitivity to Disturbance</a></li> <li><a href="#org609c873">6.5. Damped Plant</a></li> <li><a href="#org3c4f6ff">6.6. Conclusion</a></li> </ul> </li> <li><a href="#org77a79e6">7. Comparison of Active Damping Techniques</a> <ul> <li><a href="#orgb0afe4f">7.1. Load the plants</a></li> <li><a href="#orge08556d">7.2. Sensitivity to Disturbance</a></li> <li><a href="#orgfff33b9">7.3. Noise Budget</a></li> <li><a href="#orgc582849">7.4. Damped Plant</a></li> <li><a href="#orgde3c3e7">7.5. Conclusion</a></li> </ul> </li> <li><a href="#org15965e0">8. Voice Coil</a> <ul> <li><a href="#orge0ae68a">8.1. Init</a></li> <li><a href="#orgd0cdd7a">8.2. Identification</a></li> <li><a href="#orgea2fb03">8.3. Sensitivity to Disturbances</a></li> <li><a href="#org88de5a5">8.4. Noise Budget</a></li> <li><a href="#org241513b">8.5. Integral Force Feedback</a></li> <li><a href="#org2a655f7">8.6. Identification of the Damped Plant</a></li> <li><a href="#orgb7cef4c">8.7. Noise Budget</a></li> <li><a href="#orga9d9dce">8.8. Conclusion</a></li> </ul> </li> </ul> </div> </div> <p> The idea is to use the same model as the full Simscape Model but to restrict the motion only in the vertical direction. </p> <p> This is done in order to more easily study the system and evaluate control techniques. </p> <div id="outline-container-org227ba84" class="outline-2"> <h2 id="org227ba84"><span class="section-number-2">1</span> Simscape Model</h2> <div class="outline-text-2" id="text-1"> <p> <a id="orge4f893b"></a> </p> <p> A schematic of the uniaxial model used for simulations is represented in figure <a href="#org4f6d475">1</a>. </p> <p> The perturbations \(w\) are: </p> <ul class="org-ul"> <li>\(F_s\): direct forces applied to the sample such as inertia forces and cable forces</li> <li>\(F_{rz}\): parasitic forces due to the rotation of the spindle</li> <li>\(F_{ty}\): parasitic forces due to scans with the translation stage</li> <li>\(D_w\): ground motion</li> </ul> <p> The quantity to \(z\) to control is: </p> <ul class="org-ul"> <li>\(D\): the position of the sample with respect to the granite</li> </ul> <p> The measured quantities \(v\) are: </p> <ul class="org-ul"> <li>\(D\): the position of the sample with respect to the granite</li> </ul> <p> We study the use of an additional sensor: </p> <ul class="org-ul"> <li>\(F_n\): a force sensor located in the nano-hexapod</li> <li>\(v_n\): an absolute velocity sensor located on the top platform of the nano-hexapod</li> <li>\(d_r\): a relative motion sensor located in the nano-hexapod</li> </ul> <p> The control signal \(u\) is: </p> <ul class="org-ul"> <li>\(F\) the force applied by the nano-hexapod actuator</li> </ul> <div id="org4f6d475" class="figure"> <p><img src="figs/uniaxial-model-nass-flexible.png" alt="uniaxial-model-nass-flexible.png" /> </p> <p><span class="figure-number">Figure 1: </span>Schematic of the uniaxial model used</p> </div> <p> Few active damping techniques will be compared in order to decide which sensor is to be included in the system. Schematics of the active damping techniques are displayed in figure <a href="#orgc460f0d">2</a>. </p> <div id="orgc460f0d" class="figure"> <p><img src="figs/uniaxial-model-nass-flexible-active-damping.png" alt="uniaxial-model-nass-flexible-active-damping.png" /> </p> <p><span class="figure-number">Figure 2: </span>Comparison of used active damping techniques</p> </div> </div> </div> <div id="outline-container-org6bfb35d" class="outline-2"> <h2 id="org6bfb35d"><span class="section-number-2">2</span> Undamped System</h2> <div class="outline-text-2" id="text-2"> <p> <a id="org83e5eca"></a> </p> <p> Let’s start by study the undamped system. </p> </div> <div id="outline-container-org006e24f" class="outline-3"> <h3 id="org006e24f"><span class="section-number-3">2.1</span> Init</h3> <div class="outline-text-3" id="text-2-1"> <p> We initialize all the stages with the default parameters. The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg. </p> <p> All the controllers are set to 0 (Open Loop). </p> </div> </div> <div id="outline-container-org4fbbce9" class="outline-3"> <h3 id="org4fbbce9"><span class="section-number-3">2.2</span> Identification</h3> <div class="outline-text-3" id="text-2-2"> <p> We identify the dynamics of the system. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>; </pre> </div> <p> The inputs and outputs are defined below and corresponds to the name of simulink blocks. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <p> Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G = linearize(mdl, io, options); G.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> <p> Finally, we save the identified system dynamics for further analysis. </p> <div class="org-src-container"> <pre class="src src-matlab">save(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G'</span>); </pre> </div> </div> </div> <div id="outline-container-org49ca9ab" class="outline-3"> <h3 id="org49ca9ab"><span class="section-number-3">2.3</span> Sensitivity to Disturbances</h3> <div class="outline-text-3" id="text-2-3"> <p> We show several plots representing the sensitivity to disturbances: </p> <ul class="org-ul"> <li>in figure <a href="#orgbfaf2d3">3</a> the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown</li> <li>in figure <a href="#orge6ba7ee">4</a>, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown</li> </ul> <div id="orgbfaf2d3" class="figure"> <p><img src="figs/uniaxial-sensitivity-disturbances.png" alt="uniaxial-sensitivity-disturbances.png" /> </p> <p><span class="figure-number">Figure 3: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-disturbances.png">png</a>, <a href="./figs/uniaxial-sensitivity-disturbances.pdf">pdf</a>)</p> </div> <div id="orge6ba7ee" class="figure"> <p><img src="figs/uniaxial-sensitivity-force-dist.png" alt="uniaxial-sensitivity-force-dist.png" /> </p> <p><span class="figure-number">Figure 4: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-force-dist.png">png</a>, <a href="./figs/uniaxial-sensitivity-force-dist.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org346ebf3" class="outline-3"> <h3 id="org346ebf3"><span class="section-number-3">2.4</span> Noise Budget</h3> <div class="outline-text-3" id="text-2-4"> <p> We first load the measured PSD of the disturbance. </p> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/disturbances_dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>); </pre> </div> <p> The effect of these disturbances on the distance \(D\) is computed below. The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure <a href="#org1c1d9b8">5</a> and the Cumulative Amplitude Spectrum is shown in figure <a href="#org2f678d5">6</a>. </p> <p> The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure <a href="#org2f678d5">6</a>. </p> <pre class="example"> 3.3793e-06 </pre> <div id="org1c1d9b8" class="figure"> <p><img src="figs/uniaxial-psd-dist.png" alt="uniaxial-psd-dist.png" /> </p> <p><span class="figure-number">Figure 5: </span>PSD of the effect of disturbances on \(D\) (<a href="./figs/uniaxial-psd-dist.png">png</a>, <a href="./figs/uniaxial-psd-dist.pdf">pdf</a>)</p> </div> <div id="org2f678d5" class="figure"> <p><img src="figs/uniaxial-cas-dist.png" alt="uniaxial-cas-dist.png" /> </p> <p><span class="figure-number">Figure 6: </span>CAS of the effect of disturbances on \(D\) (<a href="./figs/uniaxial-cas-dist.png">png</a>, <a href="./figs/uniaxial-cas-dist.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orgdb3535a" class="outline-3"> <h3 id="orgdb3535a"><span class="section-number-3">2.5</span> Plant</h3> <div class="outline-text-3" id="text-2-5"> <p> The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure <a href="#orgaaf3fcb">7</a>. It corresponds to the plant to control. </p> <div id="orgaaf3fcb" class="figure"> <p><img src="figs/uniaxial-plant.png" alt="uniaxial-plant.png" /> </p> <p><span class="figure-number">Figure 7: </span>Bode plot of the Plant (<a href="./figs/uniaxial-plant.png">png</a>, <a href="./figs/uniaxial-plant.pdf">pdf</a>)</p> </div> </div> </div> </div> <div id="outline-container-org33c3829" class="outline-2"> <h2 id="org33c3829"><span class="section-number-2">3</span> Integral Force Feedback</h2> <div class="outline-text-2" id="text-3"> <p> <a id="org37f1b7d"></a> </p> <div id="org884322f" class="figure"> <p><img src="figs/uniaxial-model-nass-flexible-iff.png" alt="uniaxial-model-nass-flexible-iff.png" /> </p> <p><span class="figure-number">Figure 8: </span>Uniaxial IFF Control Schematic</p> </div> </div> <div id="outline-container-org326d925" class="outline-3"> <h3 id="org326d925"><span class="section-number-3">3.1</span> Control Design</h3> <div class="outline-text-3" id="text-3-1"> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G'</span>); </pre> </div> <p> Let’s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor. </p> <div id="orgde62a50" class="figure"> <p><img src="figs/uniaxial_iff_plant.png" alt="uniaxial_iff_plant.png" /> </p> <p><span class="figure-number">Figure 9: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_iff_plant.png">png</a>, <a href="./figs/uniaxial_iff_plant.pdf">pdf</a>)</p> </div> <p> The controller for each pair of actuator/sensor is: </p> <div class="org-src-container"> <pre class="src src-matlab">K_iff = <span class="org-type">-</span>1000<span class="org-type">/</span>s; </pre> </div> <div id="orgf0a3805" class="figure"> <p><img src="figs/uniaxial_iff_open_loop.png" alt="uniaxial_iff_open_loop.png" /> </p> <p><span class="figure-number">Figure 10: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_iff_open_loop.png">png</a>, <a href="./figs/uniaxial_iff_open_loop.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org86f3473" class="outline-3"> <h3 id="org86f3473"><span class="section-number-3">3.2</span> Identification</h3> <div class="outline-text-3" id="text-3-2"> <p> Let’s initialize the system prior to identification. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> All the controllers are set to 0. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_iff = <span class="org-type">-</span>K_iff; save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_rmc = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>); K_dvf = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G_iff = linearize(mdl, io, options); G_iff.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G_iff.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab">save(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'-append'</span>); </pre> </div> </div> </div> <div id="outline-container-org68c471e" class="outline-3"> <h3 id="org68c471e"><span class="section-number-3">3.3</span> Sensitivity to Disturbance</h3> <div class="outline-text-3" id="text-3-3"> <div id="orge528565" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_iff.png" alt="uniaxial_sensitivity_dist_iff.png" /> </p> <p><span class="figure-number">Figure 11: </span>Sensitivity to disturbance once the IFF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_iff.pdf">pdf</a>)</p> </div> <div id="org01074d3" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_stages_iff.png" alt="uniaxial_sensitivity_dist_stages_iff.png" /> </p> <p><span class="figure-number">Figure 12: </span>Sensitivity to force disturbances in various stages when IFF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_iff.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org5edf015" class="outline-3"> <h3 id="org5edf015"><span class="section-number-3">3.4</span> Damped Plant</h3> <div class="outline-text-3" id="text-3-4"> <div id="org57280cb" class="figure"> <p><img src="figs/uniaxial_plant_iff_damped.png" alt="uniaxial_plant_iff_damped.png" /> </p> <p><span class="figure-number">Figure 13: </span>Damped Plant after IFF is applied (<a href="./figs/uniaxial_plant_iff_damped.png">png</a>, <a href="./figs/uniaxial_plant_iff_damped.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orgfc93a1c" class="outline-3"> <h3 id="orgfc93a1c"><span class="section-number-3">3.5</span> Conclusion</h3> <div class="outline-text-3" id="text-3-5"> <div class="important"> <p> Integral Force Feedback: </p> </div> </div> </div> </div> <div id="outline-container-org07ff58f" class="outline-2"> <h2 id="org07ff58f"><span class="section-number-2">4</span> Relative Motion Control</h2> <div class="outline-text-2" id="text-4"> <p> <a id="org7d87e9d"></a> </p> <p> In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input. </p> <div id="orgd2f9465" class="figure"> <p><img src="figs/uniaxial-model-nass-flexible-rmc.png" alt="uniaxial-model-nass-flexible-rmc.png" /> </p> <p><span class="figure-number">Figure 14: </span>Uniaxial RMC Control Schematic</p> </div> </div> <div id="outline-container-org5704583" class="outline-3"> <h3 id="org5704583"><span class="section-number-3">4.1</span> Control Design</h3> <div class="outline-text-3" id="text-4-1"> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G'</span>); </pre> </div> <p> Let’s look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor. </p> <div id="org8260ad7" class="figure"> <p><img src="figs/uniaxial_rmc_plant.png" alt="uniaxial_rmc_plant.png" /> </p> <p><span class="figure-number">Figure 15: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/uniaxial_rmc_plant.png">png</a>, <a href="./figs/uniaxial_rmc_plant.pdf">pdf</a>)</p> </div> <p> The Relative Motion Controller is defined below. A Low pass Filter is added to make the controller transfer function proper. </p> <div class="org-src-container"> <pre class="src src-matlab">K_rmc = s<span class="org-type">*</span>50000<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10000); </pre> </div> <div id="orga5e2583" class="figure"> <p><img src="figs/uniaxial_rmc_open_loop.png" alt="uniaxial_rmc_open_loop.png" /> </p> <p><span class="figure-number">Figure 16: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_rmc_open_loop.png">png</a>, <a href="./figs/uniaxial_rmc_open_loop.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orga436aa7" class="outline-3"> <h3 id="orga436aa7"><span class="section-number-3">4.2</span> Identification</h3> <div class="outline-text-3" id="text-4-2"> <p> Let’s initialize the system prior to identification. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> And initialize the controllers. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_iff = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_rmc = <span class="org-type">-</span>K_rmc; save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>); K_dvf = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G_rmc = linearize(mdl, io, options); G_rmc.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G_rmc.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab">save(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'-append'</span>); </pre> </div> </div> </div> <div id="outline-container-org133268a" class="outline-3"> <h3 id="org133268a"><span class="section-number-3">4.3</span> Sensitivity to Disturbance</h3> <div class="outline-text-3" id="text-4-3"> <div id="org9e0dde4" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_rmc.png" alt="uniaxial_sensitivity_dist_rmc.png" /> </p> <p><span class="figure-number">Figure 17: </span>Sensitivity to disturbance once the RMC controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_rmc.pdf">pdf</a>)</p> </div> <div id="org38d5224" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_stages_rmc.png" alt="uniaxial_sensitivity_dist_stages_rmc.png" /> </p> <p><span class="figure-number">Figure 18: </span>Sensitivity to force disturbances in various stages when RMC is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_rmc.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org2f974d4" class="outline-3"> <h3 id="org2f974d4"><span class="section-number-3">4.4</span> Damped Plant</h3> <div class="outline-text-3" id="text-4-4"> <div id="org18786e8" class="figure"> <p><img src="figs/uniaxial_plant_rmc_damped.png" alt="uniaxial_plant_rmc_damped.png" /> </p> <p><span class="figure-number">Figure 19: </span>Damped Plant after RMC is applied (<a href="./figs/uniaxial_plant_rmc_damped.png">png</a>, <a href="./figs/uniaxial_plant_rmc_damped.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orgfdbd543" class="outline-3"> <h3 id="orgfdbd543"><span class="section-number-3">4.5</span> Conclusion</h3> <div class="outline-text-3" id="text-4-5"> <div class="important"> <p> Relative Motion Control: </p> </div> </div> </div> </div> <div id="outline-container-orgc9b3622" class="outline-2"> <h2 id="orgc9b3622"><span class="section-number-2">5</span> Direct Velocity Feedback</h2> <div class="outline-text-2" id="text-5"> <p> <a id="orgdffe2c2"></a> </p> <p> In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input. </p> <div id="orgba6ec08" class="figure"> <p><img src="figs/uniaxial-model-nass-flexible-dvf.png" alt="uniaxial-model-nass-flexible-dvf.png" /> </p> <p><span class="figure-number">Figure 20: </span>Uniaxial DVF Control Schematic</p> </div> </div> <div id="outline-container-org2050b01" class="outline-3"> <h3 id="org2050b01"><span class="section-number-3">5.1</span> Control Design</h3> <div class="outline-text-3" id="text-5-1"> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G'</span>); </pre> </div> <div id="org7b83a4a" class="figure"> <p><img src="figs/uniaxial_dvf_plant.png" alt="uniaxial_dvf_plant.png" /> </p> <p><span class="figure-number">Figure 21: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_plant.png">png</a>, <a href="./figs/uniaxial_dvf_plant.pdf">pdf</a>)</p> </div> <div class="org-src-container"> <pre class="src src-matlab">K_dvf = tf(5e4); </pre> </div> <div id="org25567d5" class="figure"> <p><img src="figs/uniaxial_dvf_loop_gain.png" alt="uniaxial_dvf_loop_gain.png" /> </p> <p><span class="figure-number">Figure 22: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_loop_gain.png">png</a>, <a href="./figs/uniaxial_dvf_loop_gain.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orgc946d88" class="outline-3"> <h3 id="orgc946d88"><span class="section-number-3">5.2</span> Identification</h3> <div class="outline-text-3" id="text-5-2"> <p> Let’s initialize the system prior to identification. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> And initialize the controllers. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_iff = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_rmc = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>); K_dvf = <span class="org-type">-</span>K_dvf; save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G_dvf = linearize(mdl, io, options); G_dvf.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G_dvf.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab">save(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> </div> </div> <div id="outline-container-orgde4b14e" class="outline-3"> <h3 id="orgde4b14e"><span class="section-number-3">5.3</span> Sensitivity to Disturbance</h3> <div class="outline-text-3" id="text-5-3"> <div id="orgf9d4052" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_dvf.png" alt="uniaxial_sensitivity_dist_dvf.png" /> </p> <p><span class="figure-number">Figure 23: </span>Sensitivity to disturbance once the DVF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_dvf.pdf">pdf</a>)</p> </div> <div id="org98ddadb" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_stages_dvf.png" alt="uniaxial_sensitivity_dist_stages_dvf.png" /> </p> <p><span class="figure-number">Figure 24: </span>Sensitivity to force disturbances in various stages when DVF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_dvf.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org640c7d9" class="outline-3"> <h3 id="org640c7d9"><span class="section-number-3">5.4</span> Damped Plant</h3> <div class="outline-text-3" id="text-5-4"> <div id="orgbc9c953" class="figure"> <p><img src="figs/uniaxial_plant_dvf_damped.png" alt="uniaxial_plant_dvf_damped.png" /> </p> <p><span class="figure-number">Figure 25: </span>Damped Plant after DVF is applied (<a href="./figs/uniaxial_plant_dvf_damped.png">png</a>, <a href="./figs/uniaxial_plant_dvf_damped.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org94e9d5f" class="outline-3"> <h3 id="org94e9d5f"><span class="section-number-3">5.5</span> Conclusion</h3> <div class="outline-text-3" id="text-5-5"> <div class="important"> <p> Direct Velocity Feedback: </p> </div> </div> </div> </div> <div id="outline-container-org5ac7dda" class="outline-2"> <h2 id="org5ac7dda"><span class="section-number-2">6</span> With Cedrat Piezo-electric Actuators</h2> <div class="outline-text-2" id="text-6"> <p> <a id="org604af95"></a> </p> <p> The model used for the Cedrat actuator is shown in figure <a href="#org83591fa">26</a>. </p> <div id="org83591fa" class="figure"> <p><img src="figs/cedrat-uniaxial-actuator.png" alt="cedrat-uniaxial-actuator.png" /> </p> <p><span class="figure-number">Figure 26: </span>Schematic of the model used for the Cedrat Actuator</p> </div> </div> <div id="outline-container-org1bfaed7" class="outline-3"> <h3 id="org1bfaed7"><span class="section-number-3">6.1</span> Identification</h3> <div class="outline-text-3" id="text-6-1"> <p> Let’s initialize the system prior to identification. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>); initializeCedratPiezo(); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> And initialize the controllers. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_iff = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_rmc = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>); K_dvf = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> <p> We identify the dynamics of the system. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial_cedrat_bis'</span>; </pre> </div> <p> The inputs and outputs are defined below and corresponds to the name of simulink blocks. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <p> Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G = linearize(mdl, io, options); G.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> </div> </div> <div id="outline-container-orgf32651c" class="outline-3"> <h3 id="orgf32651c"><span class="section-number-3">6.2</span> Control Design</h3> <div class="outline-text-3" id="text-6-2"> <p> Let’s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor. </p> <div id="org2a02461" class="figure"> <p><img src="figs/uniaxial_cedrat_plant.png" alt="uniaxial_cedrat_plant.png" /> </p> <p><span class="figure-number">Figure 27: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_cedrat_plant.png">png</a>, <a href="./figs/uniaxial_cedrat_plant.pdf">pdf</a>)</p> </div> <p> The controller for each pair of actuator/sensor is: </p> <div class="org-src-container"> <pre class="src src-matlab">K_cedrat = <span class="org-type">-</span>5000<span class="org-type">/</span>s; </pre> </div> <div id="org7da2401" class="figure"> <p><img src="figs/uniaxial_cedrat_open_loop.png" alt="uniaxial_cedrat_open_loop.png" /> </p> <p><span class="figure-number">Figure 28: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_cedrat_open_loop.png">png</a>, <a href="./figs/uniaxial_cedrat_open_loop.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orgc7383cd" class="outline-3"> <h3 id="orgc7383cd"><span class="section-number-3">6.3</span> Identification</h3> <div class="outline-text-3" id="text-6-3"> <p> Let’s initialize the system prior to identification. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>); initializeCedratPiezo(); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> All the controllers are set to 0. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_iff = <span class="org-type">-</span>K_cedrat; save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_rmc = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>); K_dvf = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial_cedrat_bis'</span>; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G_cedrat = linearize(mdl, io, options); G_cedrat.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G_cedrat.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-comment">% save('./mat/uniaxial_plants.mat', 'G_cedrat', '-append');</span> </pre> </div> </div> </div> <div id="outline-container-org34de1fd" class="outline-3"> <h3 id="org34de1fd"><span class="section-number-3">6.4</span> Sensitivity to Disturbance</h3> <div class="outline-text-3" id="text-6-4"> <div id="org90f5c2b" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_cedrat.png" alt="uniaxial_sensitivity_dist_cedrat.png" /> </p> <p><span class="figure-number">Figure 29: </span>Sensitivity to disturbance once the CEDRAT controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_cedrat.pdf">pdf</a>)</p> </div> <div id="org5acd68e" class="figure"> <p><img src="figs/uniaxial_sensitivity_dist_stages_cedrat.png" alt="uniaxial_sensitivity_dist_stages_cedrat.png" /> </p> <p><span class="figure-number">Figure 30: </span>Sensitivity to force disturbances in various stages when CEDRAT is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org609c873" class="outline-3"> <h3 id="org609c873"><span class="section-number-3">6.5</span> Damped Plant</h3> <div class="outline-text-3" id="text-6-5"> <div id="org0c4e61a" class="figure"> <p><img src="figs/uniaxial_plant_cedrat_damped.png" alt="uniaxial_plant_cedrat_damped.png" /> </p> <p><span class="figure-number">Figure 31: </span>Damped Plant after CEDRAT is applied (<a href="./figs/uniaxial_plant_cedrat_damped.png">png</a>, <a href="./figs/uniaxial_plant_cedrat_damped.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org3c4f6ff" class="outline-3"> <h3 id="org3c4f6ff"><span class="section-number-3">6.6</span> Conclusion</h3> <div class="outline-text-3" id="text-6-6"> <div class="important"> <p> This gives similar results than with a classical force sensor. </p> </div> </div> </div> </div> <div id="outline-container-org77a79e6" class="outline-2"> <h2 id="org77a79e6"><span class="section-number-2">7</span> Comparison of Active Damping Techniques</h2> <div class="outline-text-2" id="text-7"> <p> <a id="orgd5640a5"></a> </p> </div> <div id="outline-container-orgb0afe4f" class="outline-3"> <h3 id="orgb0afe4f"><span class="section-number-3">7.1</span> Load the plants</h3> <div class="outline-text-3" id="text-7-1"> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'G_dvf'</span>); </pre> </div> </div> </div> <div id="outline-container-orge08556d" class="outline-3"> <h3 id="orge08556d"><span class="section-number-3">7.2</span> Sensitivity to Disturbance</h3> <div class="outline-text-3" id="text-7-2"> <div id="org8c59440" class="figure"> <p><img src="figs/uniaxial_sensitivity_ground_motion.png" alt="uniaxial_sensitivity_ground_motion.png" /> </p> <p><span class="figure-number">Figure 32: </span>Sensitivity to Ground Motion - Comparison (<a href="./figs/uniaxial_sensitivity_ground_motion.png">png</a>, <a href="./figs/uniaxial_sensitivity_ground_motion.pdf">pdf</a>)</p> </div> <div id="orgf1c26c0" class="figure"> <p><img src="figs/uniaxial_sensitivity_direct_force.png" alt="uniaxial_sensitivity_direct_force.png" /> </p> <p><span class="figure-number">Figure 33: </span>Sensitivity to disturbance - Comparison (<a href="./figs/uniaxial_sensitivity_direct_force.png">png</a>, <a href="./figs/uniaxial_sensitivity_direct_force.pdf">pdf</a>)</p> </div> <div id="orga77aca7" class="figure"> <p><img src="figs/uniaxial_sensitivity_fty.png" alt="uniaxial_sensitivity_fty.png" /> </p> <p><span class="figure-number">Figure 34: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_fty.png">png</a>, <a href="./figs/uniaxial_sensitivity_fty.pdf">pdf</a>)</p> </div> <div id="orga6c1630" class="figure"> <p><img src="figs/uniaxial_sensitivity_frz.png" alt="uniaxial_sensitivity_frz.png" /> </p> <p><span class="figure-number">Figure 35: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_frz.png">png</a>, <a href="./figs/uniaxial_sensitivity_frz.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orgfff33b9" class="outline-3"> <h3 id="orgfff33b9"><span class="section-number-3">7.3</span> Noise Budget</h3> <div class="outline-text-3" id="text-7-3"> <p> We first load the measured PSD of the disturbance. </p> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/disturbances_dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>); </pre> </div> <p> The effect of these disturbances on the distance \(D\) is computed for all active damping techniques. We then compute the Cumulative Amplitude Spectrum (figure <a href="#org68d1f1c">36</a>). </p> <div id="org68d1f1c" class="figure"> <p><img src="figs/uniaxial-comp-cas-dist.png" alt="uniaxial-comp-cas-dist.png" /> </p> <p><span class="figure-number">Figure 36: </span>Comparison of the Cumulative Amplitude Spectrum of \(D\) for different active damping techniques (<a href="./figs/uniaxial-comp-cas-dist.png">png</a>, <a href="./figs/uniaxial-comp-cas-dist.pdf">pdf</a>)</p> </div> <p> The obtained Root Mean Square Value for each active damping technique is shown below. </p> <table id="org63699f5" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <caption class="t-above"><span class="table-number">Table 1:</span> Obtain Root Mean Square value of \(D\) for each Active Damping Technique applied</caption> <colgroup> <col class="org-left" /> <col class="org-right" /> </colgroup> <thead> <tr> <th scope="col" class="org-left"> </th> <th scope="col" class="org-right">D [m rms]</th> </tr> </thead> <tbody> <tr> <td class="org-left">OL</td> <td class="org-right">3.38e-06</td> </tr> <tr> <td class="org-left">IFF</td> <td class="org-right">3.40e-06</td> </tr> <tr> <td class="org-left">RMC</td> <td class="org-right">3.37e-06</td> </tr> <tr> <td class="org-left">DVF</td> <td class="org-right">3.38e-06</td> </tr> </tbody> </table> <p> It is important to note that the effect of direct forces applied to the sample are not taken into account here. </p> </div> </div> <div id="outline-container-orgc582849" class="outline-3"> <h3 id="orgc582849"><span class="section-number-3">7.4</span> Damped Plant</h3> <div class="outline-text-3" id="text-7-4"> <div id="orga0c1298" class="figure"> <p><img src="figs/uniaxial_plant_damped_comp.png" alt="uniaxial_plant_damped_comp.png" /> </p> <p><span class="figure-number">Figure 37: </span>Damped Plant - Comparison (<a href="./figs/uniaxial_plant_damped_comp.png">png</a>, <a href="./figs/uniaxial_plant_damped_comp.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orgde3c3e7" class="outline-3"> <h3 id="orgde3c3e7"><span class="section-number-3">7.5</span> Conclusion</h3> <div class="outline-text-3" id="text-7-5"> <table id="org46e95c2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <caption class="t-above"><span class="table-number">Table 2:</span> Comparison of proposed active damping techniques</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">IFF</th> <th scope="col" class="org-left">RMC</th> <th scope="col" class="org-left">DVF</th> </tr> </thead> <tbody> <tr> <td class="org-left">Sensor Type</td> <td class="org-left">Force sensor</td> <td class="org-left">Relative Motion</td> <td class="org-left">Inertial</td> </tr> <tr> <td class="org-left">Guaranteed Stability</td> <td class="org-left">+</td> <td class="org-left">+</td> <td class="org-left">-</td> </tr> <tr> <td class="org-left">Sensitivity (\(D_w\))</td> <td class="org-left">-</td> <td class="org-left">+</td> <td class="org-left">-</td> </tr> <tr> <td class="org-left">Sensitivity (\(F_s\))</td> <td class="org-left">- (at low freq)</td> <td class="org-left">+</td> <td class="org-left">+</td> </tr> <tr> <td class="org-left">Sensitivity (\(F_{ty,rz}\))</td> <td class="org-left">+</td> <td class="org-left">-</td> <td class="org-left">+</td> </tr> <tr> <td class="org-left">Overall RMS of \(D\)</td> <td class="org-left">=</td> <td class="org-left">=</td> <td class="org-left">=</td> </tr> </tbody> </table> </div> </div> </div> <div id="outline-container-org15965e0" class="outline-2"> <h2 id="org15965e0"><span class="section-number-2">8</span> Voice Coil</h2> <div class="outline-text-2" id="text-8"> <p> <a id="orgf9e9d51"></a> </p> </div> <div id="outline-container-orge0ae68a" class="outline-3"> <h3 id="orge0ae68a"><span class="section-number-3">8.1</span> Init</h3> <div class="outline-text-3" id="text-8-1"> <p> We initialize all the stages with the default parameters. The nano-hexapod is an hexapod with voice coils and the sample has a mass of 50kg. </p> <p> All the controllers are set to 0 (Open Loop). </p> </div> </div> <div id="outline-container-orgd0cdd7a" class="outline-3"> <h3 id="orgd0cdd7a"><span class="section-number-3">8.2</span> Identification</h3> <div class="outline-text-3" id="text-8-2"> <p> We identify the dynamics of the system. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>; </pre> </div> <p> The inputs and outputs are defined below and corresponds to the name of simulink blocks. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <p> Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model. </p> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G_vc = linearize(mdl, io, options); G_vc.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G_vc.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> <p> Finally, we save the identified system dynamics for further analysis. </p> <div class="org-src-container"> <pre class="src src-matlab">save(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G_vc'</span>, <span class="org-string">'-append'</span>); </pre> </div> </div> </div> <div id="outline-container-orgea2fb03" class="outline-3"> <h3 id="orgea2fb03"><span class="section-number-3">8.3</span> Sensitivity to Disturbances</h3> <div class="outline-text-3" id="text-8-3"> <p> We load the dynamics when using a piezo-electric nano hexapod to compare the results. </p> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/uniaxial_plants.mat'</span>, <span class="org-string">'G'</span>); </pre> </div> <p> We show several plots representing the sensitivity to disturbances: </p> <ul class="org-ul"> <li>in figure <a href="#orgc80c94c">38</a> the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown</li> <li>in figure <a href="#org96743f2">39</a>, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown</li> </ul> <div id="orgc80c94c" class="figure"> <p><img src="figs/uniaxial-sensitivity-vc-disturbances.png" alt="uniaxial-sensitivity-vc-disturbances.png" /> </p> <p><span class="figure-number">Figure 38: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-vc-disturbances.png">png</a>, <a href="./figs/uniaxial-sensitivity-vc-disturbances.pdf">pdf</a>)</p> </div> <div id="org96743f2" class="figure"> <p><img src="figs/uniaxial-sensitivity-vc-force-dist.png" alt="uniaxial-sensitivity-vc-force-dist.png" /> </p> <p><span class="figure-number">Figure 39: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-vc-force-dist.png">png</a>, <a href="./figs/uniaxial-sensitivity-vc-force-dist.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org88de5a5" class="outline-3"> <h3 id="org88de5a5"><span class="section-number-3">8.4</span> Noise Budget</h3> <div class="outline-text-3" id="text-8-4"> <p> We first load the measured PSD of the disturbance. </p> <div class="org-src-container"> <pre class="src src-matlab">load(<span class="org-string">'./mat/disturbances_dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>); </pre> </div> <p> The effect of these disturbances on the distance \(D\) is computed below. The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure <a href="#orgd3fd39b">40</a> and the Cumulative Amplitude Spectrum is shown in figure <a href="#org2ccc8a7">41</a>. </p> <p> The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure <a href="#org2ccc8a7">41</a>. </p> <pre class="example"> 4.8793e-06 </pre> <div id="orgd3fd39b" class="figure"> <p><img src="figs/uniaxial-vc-psd-dist.png" alt="uniaxial-vc-psd-dist.png" /> </p> <p><span class="figure-number">Figure 40: </span>PSD of the displacement \(D\) due to disturbances (<a href="./figs/uniaxial-vc-psd-dist.png">png</a>, <a href="./figs/uniaxial-vc-psd-dist.pdf">pdf</a>)</p> </div> <div id="org2ccc8a7" class="figure"> <p><img src="figs/uniaxial-vc-cas-dist.png" alt="uniaxial-vc-cas-dist.png" /> </p> <p><span class="figure-number">Figure 41: </span>CAS of the displacement \(D\) due the disturbances (<a href="./figs/uniaxial-vc-cas-dist.png">png</a>, <a href="./figs/uniaxial-vc-cas-dist.pdf">pdf</a>)</p> </div> <div class="important"> <p> Even though the RMS value of the displacement \(D\) is lower when using a piezo-electric actuator, the motion is mainly due to high frequency disturbances which are more difficult to control (an higher control bandwidth is required). </p> <p> Thus, it may be desirable to use voice coil actuators. </p> </div> </div> </div> <div id="outline-container-org241513b" class="outline-3"> <h3 id="org241513b"><span class="section-number-3">8.5</span> Integral Force Feedback</h3> <div class="outline-text-3" id="text-8-5"> <div class="org-src-container"> <pre class="src src-matlab">K_iff = <span class="org-type">-</span>20<span class="org-type">/</span>s; </pre> </div> <div id="orgb6bf4fa" class="figure"> <p><img src="figs/uniaxial_iff_vc_open_loop.png" alt="uniaxial_iff_vc_open_loop.png" /> </p> <p><span class="figure-number">Figure 42: </span>Open Loop Transfer Function for IFF control when using a voice coil actuator (<a href="./figs/uniaxial_iff_vc_open_loop.png">png</a>, <a href="./figs/uniaxial_iff_vc_open_loop.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-org2a655f7" class="outline-3"> <h3 id="org2a655f7"><span class="section-number-3">8.6</span> Identification of the Damped Plant</h3> <div class="outline-text-3" id="text-8-6"> <p> Let’s initialize the system prior to identification. </p> <div class="org-src-container"> <pre class="src src-matlab">initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeAxisc(); initializeMirror(); initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'lorentz'</span>); initializeSample(<span class="org-string">'mass'</span>, 50); </pre> </div> <p> All the controllers are set to 0. </p> <div class="org-src-container"> <pre class="src src-matlab">K = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>); K_iff = <span class="org-type">-</span>K_iff; save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>); K_rmc = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span>); K_dvf = tf(0); save(<span class="org-string">'./mat/controllers_uniaxial.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>); </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> options = linearizeOptions; options.SampleTime = 0; <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span> mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>; </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span> io<span class="org-type">(1) </span>= linio([mdl, <span class="org-string">'/Dw'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Ground Motion</span> io<span class="org-type">(2) </span>= linio([mdl, <span class="org-string">'/Fs'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied on the sample</span> io<span class="org-type">(3) </span>= linio([mdl, <span class="org-string">'/Fnl'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Force applied by the NASS</span> io<span class="org-type">(4) </span>= linio([mdl, <span class="org-string">'/Fdty'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Ty</span> io<span class="org-type">(5) </span>= linio([mdl, <span class="org-string">'/Fdrz'</span>], 1, <span class="org-string">'input'</span>); <span class="org-comment">% Parasitic force Rz</span> io<span class="org-type">(6) </span>= linio([mdl, <span class="org-string">'/Dsm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of the sample</span> io<span class="org-type">(7) </span>= linio([mdl, <span class="org-string">'/Fnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Force sensor in NASS's legs</span> io<span class="org-type">(8) </span>= linio([mdl, <span class="org-string">'/Dnlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Displacement of NASS's legs</span> io<span class="org-type">(9) </span>= linio([mdl, <span class="org-string">'/Dgm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Absolute displacement of the granite</span> io<span class="org-type">(10) </span>= linio([mdl, <span class="org-string">'/Vlm'</span>], 1, <span class="org-string">'output'</span>); <span class="org-comment">% Measured absolute velocity of the top NASS platform</span> </pre> </div> <div class="org-src-container"> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> G_vc_iff = linearize(mdl, io, options); G_vc_iff.InputName = {<span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span> <span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span> <span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span> <span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span> <span class="org-string">'Frz'</span>}; <span class="org-comment">% Parasitic Force Rz [N]</span> G_vc_iff.OutputName = {<span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span> <span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span> <span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span> <span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span> <span class="org-string">'Vlm'</span>}; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span> </pre> </div> </div> </div> <div id="outline-container-orgb7cef4c" class="outline-3"> <h3 id="orgb7cef4c"><span class="section-number-3">8.7</span> Noise Budget</h3> <div class="outline-text-3" id="text-8-7"> <p> We compute the obtain PSD of the displacement \(D\) when using IFF. </p> <div id="org60accee" class="figure"> <p><img src="figs/uniaxial-cas-iff-vc.png" alt="uniaxial-cas-iff-vc.png" /> </p> <p><span class="figure-number">Figure 43: </span>CAS of the displacement \(D\) (<a href="./figs/uniaxial-cas-iff-vc.png">png</a>, <a href="./figs/uniaxial-cas-iff-vc.pdf">pdf</a>)</p> </div> </div> </div> <div id="outline-container-orga9d9dce" class="outline-3"> <h3 id="orga9d9dce"><span class="section-number-3">8.8</span> Conclusion</h3> <div class="outline-text-3" id="text-8-8"> <div class="important"> <p> The use of voice coil actuators would allow a better disturbance rejection for a fixed bandwidth compared with a piezo-electric hexapod. </p> <p> Similarly, it would require much lower bandwidth to attain the same level of disturbance rejection for \(D\). </p> </div> </div> </div> </div> </div> <div id="postamble" class="status"> <p class="author">Author: Dehaeze Thomas</p> <p class="date">Created: 2020-03-13 ven. 17:39</p> </div> </body> </html>