1090 lines
46 KiB
HTML
1090 lines
46 KiB
HTML
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<a accesskey="H" href="index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org9e9f810">1. Disturbances</a></li>
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<li><a href="#orgc44cf7e">2. Effect of disturbances on the position error</a>
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<ul>
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<li><a href="#org524df41">2.1. Initialization</a></li>
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<li><a href="#orgaf88c9f">2.2. Identification</a></li>
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<li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
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<li><a href="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
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<li><a href="#org0448746">2.5. Direct Forces (Compliance).</a></li>
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<li><a href="#orge784867">2.6. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org6527e58">3. Effect of granite stiffness</a>
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<ul>
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<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a>
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<ul>
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<li><a href="#orgbc34a65">3.1.1. Simple mass-spring-damper model</a></li>
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<li><a href="#org4ddec32">3.1.2. General Case</a></li>
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</ul>
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</li>
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<li><a href="#org9215f81">3.2. Soft Granite</a></li>
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<li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
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<li><a href="#org4b4fa39">3.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
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<ul>
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<li><a href="#org6bd588f">4.1. Noise Budgeting - Theory</a></li>
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<li><a href="#orgcc86f59">4.2. Power Spectral Densities</a></li>
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<li><a href="#orgef96b89">4.3. Cumulative Amplitude Spectrum</a></li>
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<li><a href="#org2b9df24">4.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org34c0f38">5. Closed Loop Budget Error</a>
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<ul>
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<li><a href="#orgdfef0eb">5.1. Approximation of the effect of feedback on the motion error</a></li>
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<li><a href="#orgf2d36a1">5.2. Reduction thanks to feedback - Required bandwidth</a></li>
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</ul>
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</li>
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<li><a href="#orgbf0fb63">6. Conclusion</a></li>
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</ul>
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</div>
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</div>
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<p>
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In this document is studied how the stiffness of the nano-hexapod will impact the effect of disturbances on the position error of the sample.
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</p>
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<p>
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It is divided in the following sections:
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</p>
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<ul class="org-ul">
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<li>Section <a href="#org17d3d6a">1</a>: the disturbances are listed and their Power Spectral Densities (PSD) are shown</li>
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<li>Section <a href="#orgf9e4300">2</a>: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses</li>
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<li>Section <a href="#orgd4105b6">3</a>:</li>
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<li>Section <a href="#org5d05990">4</a>: from both the PSD of the disturbances and the transfer function from disturbances to sample’s position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)</li>
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<li>Section <a href="#orgd3503fb">5</a>: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values</li>
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</ul>
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<div id="outline-container-org9e9f810" class="outline-2">
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<h2 id="org9e9f810"><span class="section-number-2">1</span> Disturbances</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org17d3d6a"></a>
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</p>
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<p>
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The main disturbances considered here are:
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</p>
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<ul class="org-ul">
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<li>\(D_w\): Ground displacement in the \(x\), \(y\) and \(z\) directions</li>
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<li>\(F_{ty}\): Forces applied by the Translation stage in the \(x\) and \(z\) directions</li>
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<li>\(F_{rz}\): Forces applied by the Spindle in the \(z\) direction</li>
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<li>\(F_d\): Direct forces applied at the center of mass of the Payload</li>
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</ul>
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<p>
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The level of these disturbances has been identified form experiments which are detailed in <a href="disturbances.html">this</a> document.
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</p>
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<p>
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The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures <a href="#org6b4e47c">1</a> and <a href="#orgb7b8e77">2</a>.
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</p>
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<p>
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In this study, the expected frequency content of the direct forces applied to the payload is not considered.
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</p>
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<div id="org6b4e47c" class="figure">
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<p><img src="figs/opt_stiff_dist_gm.png" alt="opt_stiff_dist_gm.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Amplitude Spectral Density of the Ground Displacement (<a href="./figs/opt_stiff_dist_gm.png">png</a>, <a href="./figs/opt_stiff_dist_gm.pdf">pdf</a>)</p>
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</div>
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<div id="orgb7b8e77" class="figure">
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<p><img src="figs/opt_stiff_dist_fty_frz.png" alt="opt_stiff_dist_fty_frz.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Amplitude Spectral Density of the “parasitic” forces comming from the Translation stage and the spindle (<a href="./figs/opt_stiff_dist_fty_frz.png">png</a>, <a href="./figs/opt_stiff_dist_fty_frz.pdf">pdf</a>)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgc44cf7e" class="outline-2">
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<h2 id="orgc44cf7e"><span class="section-number-2">2</span> Effect of disturbances on the position error</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orgf9e4300"></a>
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</p>
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<p>
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In this section, we use the Simscape model to identify the transfer function from disturbances to the position error of the sample.
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We do that for a wide range of nano-hexapod stiffnesses and we compare the obtained results.
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</p>
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</div>
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<div id="outline-container-org524df41" class="outline-3">
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<h3 id="org524df41"><span class="section-number-3">2.1</span> Initialization</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We initialize all the stages with the default parameters.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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</pre>
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</div>
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<p>
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We use 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">initializeSample(<span class="org-string">'mass'</span>, 10);
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</pre>
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</div>
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<p>
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We include gravity, and we use no controller.
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</p>
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<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
|
initializeController();
|
|
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
|
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaf88c9f" class="outline-3">
|
|
<h3 id="orgaf88c9f"><span class="section-number-3">2.2</span> Identification</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
The considered inputs are:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><code>Dwx</code>: Ground displacement in the \(x\) direction</li>
|
|
<li><code>Dwy</code>: Ground displacement in the \(y\) direction</li>
|
|
<li><code>Dwz</code>: Ground displacement in the \(z\) direction</li>
|
|
<li><code>Fty_x</code>: Forces applied by the Translation stage in the \(x\) direction</li>
|
|
<li><code>Fty_z</code>: Forces applied by the Translation stage in the \(z\) direction</li>
|
|
<li><code>Frz_z</code>: Forces applied by the Spindle in the \(z\) direction</li>
|
|
<li><code>Fd</code>: Direct forces applied at the center of mass of the Payload</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The outputs are <code>Ex</code>, <code>Ey</code>, <code>Ez</code>, <code>Erx</code>, <code>Ery</code>, <code>Erz</code> which are the 3 positions and 3 orientations errors of the sample.
|
|
</p>
|
|
|
|
<p>
|
|
We initialize the set of the nano-hexapod stiffnesses, and for each of them, we identify the dynamics from defined inputs to defined outputs.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org78dd34d" class="outline-3">
|
|
<h3 id="org78dd34d"><span class="section-number-3">2.3</span> Sensitivity to Stages vibration (Filtering)</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
The sensitivity the stage vibrations are displayed:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Figure <a href="#orgf55ba1b">3</a>: sensitivity to vertical spindle vibrations</li>
|
|
<li>Figure <a href="#orgce1ac2c">4</a>: sensitivity to vertical translation stage vibrations</li>
|
|
<li>Figure <a href="#org1a24ee2">5</a>: sensitivity to horizontal (x) translation stage vibrations</li>
|
|
</ul>
|
|
|
|
|
|
<div id="orgf55ba1b" class="figure">
|
|
<p><img src="figs/opt_stiff_sensitivity_Frz.png" alt="opt_stiff_sensitivity_Frz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Sensitivity to Spindle vertical motion error (\(F_{rz}\)) to the vertical error position of the sample (\(E_z\)) (<a href="./figs/opt_stiff_sensitivity_Frz.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Frz.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgce1ac2c" class="figure">
|
|
<p><img src="figs/opt_stiff_sensitivity_Fty_z.png" alt="opt_stiff_sensitivity_Fty_z.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Sensitivity to Translation stage vertical motion error (\(F_{ty,z}\)) to the vertical error position of the sample (\(E_z\)) (<a href="./figs/opt_stiff_sensitivity_Fty_z.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Fty_z.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org1a24ee2" class="figure">
|
|
<p><img src="figs/opt_stiff_sensitivity_Fty_x.png" alt="opt_stiff_sensitivity_Fty_x.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Sensitivity to Translation stage \(x\) motion error (\(F_{ty,x}\)) to the error position of the sample in the \(x\) direction (\(E_x\)) (<a href="./figs/opt_stiff_sensitivity_Fty_x.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Fty_x.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd4ea2f4" class="outline-3">
|
|
<h3 id="orgd4ea2f4"><span class="section-number-3">2.4</span> Effect of Ground motion (Transmissibility).</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
The effect of Ground motion on the position error of the sample is shown in Figure <a href="#org212587b">6</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org212587b" class="figure">
|
|
<p><img src="figs/opt_stiff_sensitivity_Dw.png" alt="opt_stiff_sensitivity_Dw.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Sensitivity to Ground motion (\(D_{w}\)) to the position error of the sample (\(E_y\) and \(E_z\)) (<a href="./figs/opt_stiff_sensitivity_Dw.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Dw.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0448746" class="outline-3">
|
|
<h3 id="org0448746"><span class="section-number-3">2.5</span> Direct Forces (Compliance).</h3>
|
|
<div class="outline-text-3" id="text-2-5">
|
|
<p>
|
|
The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure <a href="#orga33395f">7</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orga33395f" class="figure">
|
|
<p><img src="figs/opt_stiff_sensitivity_Fd.png" alt="opt_stiff_sensitivity_Fd.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Sensitivity to Direct forces and torques applied to the sample (\(F_d\), \(M_d\)) to the position error of the sample (<a href="./figs/opt_stiff_sensitivity_Fd.png">png</a>, <a href="./figs/opt_stiff_sensitivity_Fd.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge784867" class="outline-3">
|
|
<h3 id="orge784867"><span class="section-number-3">2.6</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-6">
|
|
<div class="important">
|
|
<p>
|
|
Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
|
|
</p>
|
|
|
|
<p>
|
|
In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section <a href="#org5d05990">4</a>.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6527e58" class="outline-2">
|
|
<h2 id="org6527e58"><span class="section-number-2">3</span> Effect of granite stiffness</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgd4105b6"></a>
|
|
</p>
|
|
<p>
|
|
In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgd3e5fe1" class="outline-3">
|
|
<h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
</div>
|
|
<div id="outline-container-orgbc34a65" class="outline-4">
|
|
<h4 id="orgbc34a65"><span class="section-number-4">3.1.1</span> Simple mass-spring-damper model</h4>
|
|
<div class="outline-text-4" id="text-3-1-1">
|
|
<p>
|
|
Let’s consider the system shown in Figure <a href="#org8fb9606">8</a> consisting of two stacked mass-spring-damper systems.
|
|
The bottom one represents the granite, and the top one all the positioning stages.
|
|
We want the smallest stage “deformation” \(d = x^\prime - x\) due to ground motion \(w\).
|
|
</p>
|
|
|
|
|
|
<div id="org8fb9606" class="figure">
|
|
<p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Mass Spring Damper system consisting of a granite and a positioning stage</p>
|
|
</div>
|
|
|
|
<p>
|
|
If we write the equation of motion of the system in Figure <a href="#org8fb9606">8</a>, we obtain:
|
|
</p>
|
|
\begin{align}
|
|
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
|
|
ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (w - x)
|
|
\end{align}
|
|
|
|
<p>
|
|
If we note \(d = x^\prime - x\), we obtain:
|
|
</p>
|
|
\begin{equation}
|
|
\frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4ddec32" class="outline-4">
|
|
<h4 id="org4ddec32"><span class="section-number-4">3.1.2</span> General Case</h4>
|
|
<div class="outline-text-4" id="text-3-1-2">
|
|
<p>
|
|
Let’s now considering a general positioning stage defined by:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(G^\prime(s) = \frac{F}{x}\): its mechanical “impedance”</li>
|
|
<li>\(D^\prime(s) = \frac{d}{x}\): its “deformation” transfer function</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org9702e0f" class="figure">
|
|
<p><img src="figs/general_system_granite_stiffness.png" alt="general_system_granite_stiffness.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Mass Spring Damper representing the granite and a general representation of positioning stages</p>
|
|
</div>
|
|
|
|
<p>
|
|
The equation of motion are:
|
|
</p>
|
|
\begin{align}
|
|
ms^2 x &= (cs + k) (x - w) - F \\
|
|
F &= G^\prime(s) x \\
|
|
d &= D^\prime(s) x
|
|
\end{align}
|
|
<p>
|
|
where:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(F\) is the force applied by the position stages on the granite mass</li>
|
|
</ul>
|
|
|
|
<div class="important">
|
|
<p>
|
|
We can express \(d\) as a function of \(w\):
|
|
</p>
|
|
\begin{equation}
|
|
\frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)}
|
|
\end{equation}
|
|
|
|
<p>
|
|
This is the transfer function that we would like to minimize.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
Let’s verify this formula for a simple mass/spring/damper positioning stage.
|
|
In that case, we have:
|
|
</p>
|
|
\begin{align*}
|
|
D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\
|
|
G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
|
|
\end{align*}
|
|
|
|
<p>
|
|
And finally:
|
|
</p>
|
|
\begin{equation}
|
|
\frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
|
|
\end{equation}
|
|
<p>
|
|
which is the same as the previously derived equation.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9215f81" class="outline-3">
|
|
<h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
Let’s initialize a soft granite and see how the sensitivity to disturbances will change.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">initializeGranite(<span class="org-string">'K'</span>, 5e5<span class="org-type">*</span>ones(3,1), <span class="org-string">'C'</span>, 5e3<span class="org-type">*</span>ones(3,1));
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8878556" class="outline-3">
|
|
<h3 id="org8878556"><span class="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
From Figure <a href="#org38146da">10</a>, we can see that having a “soft” granite suspension greatly lowers the sensitivity to ground motion.
|
|
The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
|
|
</p>
|
|
|
|
<p>
|
|
From Figures <a href="#orgc4c14fb">11</a> and <a href="#org533cc4b">12</a>, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
|
|
</p>
|
|
|
|
|
|
<div id="org38146da" class="figure">
|
|
<p><img src="figs/opt_stiff_soft_granite_Dw.png" alt="opt_stiff_soft_granite_Dw.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Dw.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Dw.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgc4c14fb" class="figure">
|
|
<p><img src="figs/opt_stiff_soft_granite_Frz.png" alt="opt_stiff_soft_granite_Frz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Frz.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Frz.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="org533cc4b" class="figure">
|
|
<p><img src="figs/opt_stiff_soft_granite_Fd.png" alt="opt_stiff_soft_granite_Fd.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Fd.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Fd.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4b4fa39" class="outline-3">
|
|
<h3 id="org4b4fa39"><span class="section-number-3">3.4</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<div class="important">
|
|
<p>
|
|
Having a soft granite suspension could greatly improve the sensitivity the ground motion and thus the level of sample vibration if it is found that ground motion is the limiting factor.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8a88fb0" class="outline-2">
|
|
<h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org5d05990"></a>
|
|
</p>
|
|
<p>
|
|
Now that the frequency content of disturbances have been estimated (Section <a href="#org17d3d6a">1</a>) and the transfer functions from disturbances to the position error of the sample have been identified (Section <a href="#orgf9e4300">2</a>), we can compute the level of sample vibration due to the disturbances.
|
|
</p>
|
|
|
|
<p>
|
|
We then can conclude and the nano-hexapod stiffness that will lower the sample position error.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org6bd588f" class="outline-3">
|
|
<h3 id="org6bd588f"><span class="section-number-3">4.1</span> Noise Budgeting - Theory</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
Let’s consider Figure <a href="#org7ff50a0">13</a> there \(G_d(s)\) is the transfer function from a signal \(d\) (the perturbation) to a signal \(y\) (the sample’s position error).
|
|
</p>
|
|
|
|
|
|
<div id="org7ff50a0" class="figure">
|
|
<p><img src="figs/psd_change_tf.png" alt="psd_change_tf.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Figure caption</p>
|
|
</div>
|
|
|
|
<p>
|
|
We can compute the Power Spectral Density (PSD) of signal \(y\) from the PSD of \(d\) and the norm of \(G_d(s)\):
|
|
</p>
|
|
\begin{equation}
|
|
S_{y}(\omega) = \left|G_d(j\omega)\right|^2 S_{d}(\omega) \label{eq:psd_transfer_function}
|
|
\end{equation}
|
|
|
|
<p>
|
|
If we now consider multiple disturbances \(d_1, \dots, d_n\) as shown in Figure <a href="#orgc24bdf6">14</a>, we have that:
|
|
</p>
|
|
\begin{equation}
|
|
S_{y}(\omega) = \left|G_{d_1}(j\omega)\right|^2 S_{d_1}(\omega) + \dots + \left|G_{d_n}(j\omega)\right|^2 S_{d_n}(\omega) \label{eq:sum_psd}
|
|
\end{equation}
|
|
|
|
<p>
|
|
Sometimes, we prefer to compute the <b>Amplitude</b> Spectral Density (ASD) which is related to the PSD by:
|
|
\[ \Gamma_y(\omega) = \sqrt{S_y(\omega)} \]
|
|
</p>
|
|
|
|
|
|
<div id="orgc24bdf6" class="figure">
|
|
<p><img src="figs/psd_change_tf_multiple_pert.png" alt="psd_change_tf_multiple_pert.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Figure caption</p>
|
|
</div>
|
|
|
|
<p>
|
|
The Cumulative Power Spectrum (CPS) is here defined as:
|
|
</p>
|
|
\begin{equation}
|
|
\Phi_y(\omega) = \int_\omega^\infty S_y(\nu) d\nu
|
|
\end{equation}
|
|
|
|
<p>
|
|
And the Cumulative Amplitude Spectrum (CAS):
|
|
</p>
|
|
\begin{equation}
|
|
\Psi(\omega) = \sqrt{\Phi(\omega)} = \sqrt{\int_\omega^\infty S_y(\nu) d\nu}
|
|
\end{equation}
|
|
|
|
<p>
|
|
The CAS evaluation for all frequency corresponds to the rms value of the considered quantity:
|
|
\[ y_{\text{rms}} = \Psi(\omega = 0) = \sqrt{\int_0^\infty S_y(\nu) d\nu} \]
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcc86f59" class="outline-3">
|
|
<h3 id="orgcc86f59"><span class="section-number-3">4.2</span> Power Spectral Densities</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
We compute the effect of perturbations on the motion error thanks to Eq. \eqref{eq:psd_transfer_function}.
|
|
</p>
|
|
|
|
<p>
|
|
The result is shown in:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Figure <a href="#orgd3d7b28">15</a>: PSD of the vertical sample’s motion error due to vertical ground motion</li>
|
|
<li>Figure <a href="#orgd8e87cd">16</a>: PSD of the vertical sample’s motion error due to vertical vibrations of the Spindle</li>
|
|
</ul>
|
|
|
|
|
|
<div id="orgd3d7b28" class="figure">
|
|
<p><img src="figs/opt_stiff_psd_dz_gm.png" alt="opt_stiff_psd_dz_gm.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Amplitude Spectral Density of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_psd_dz_gm.png">png</a>, <a href="./figs/opt_stiff_psd_dz_gm.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgd8e87cd" class="figure">
|
|
<p><img src="figs/opt_stiff_psd_dz_rz.png" alt="opt_stiff_psd_dz_rz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_psd_dz_rz.png">png</a>, <a href="./figs/opt_stiff_psd_dz_rz.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
We compute the effect of all perturbations on the vertical position error using Eq. \eqref{eq:sum_psd} and the resulting PSD is shown in Figure <a href="#orgdbfb5e0">17</a>.
|
|
</p>
|
|
|
|
<div id="orgdbfb5e0" class="figure">
|
|
<p><img src="figs/opt_stiff_psd_dz_tot.png" alt="opt_stiff_psd_dz_tot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Amplitude Spectral Density of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_psd_dz_tot.png">png</a>, <a href="./figs/opt_stiff_psd_dz_tot.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgef96b89" class="outline-3">
|
|
<h3 id="orgef96b89"><span class="section-number-3">4.3</span> Cumulative Amplitude Spectrum</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<p>
|
|
Similarly, the Cumulative Amplitude Spectrum of the sample vibrations are shown:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Figure <a href="#org488d65f">18</a>: due to vertical ground motion</li>
|
|
<li>Figure <a href="#orge5458c6">19</a>: due to vertical vibrations of the Spindle</li>
|
|
<li>Figure <a href="#orgf6888f0">20</a>: due to all considered perturbations</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The black dashed line corresponds to the performance objective of a sample vibration equal to \(10\ nm [rms]\).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Dwz'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
|
xlim([1, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org488d65f" class="figure">
|
|
<p><img src="figs/opt_stiff_cas_dz_gm.png" alt="opt_stiff_cas_dz_gm.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_dz_gm.png">png</a>, <a href="./figs/opt_stiff_cas_dz_gm.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_rz<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Frz_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
|
xlim([1, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orge5458c6" class="figure">
|
|
<p><img src="figs/opt_stiff_cas_dz_rz.png" alt="opt_stiff_cas_dz_rz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_dz_rz.png">png</a>, <a href="./figs/opt_stiff_cas_dz_rz.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_z$ $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
|
xlim([1, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgf6888f0" class="figure">
|
|
<p><img src="figs/opt_stiff_cas_dz_tot.png" alt="opt_stiff_cas_dz_tot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_dz_tot.png">png</a>, <a href="./figs/opt_stiff_cas_dz_tot.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2b9df24" class="outline-3">
|
|
<h3 id="org2b9df24"><span class="section-number-3">4.4</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<div class="important">
|
|
<p>
|
|
From Figure <a href="#orgf6888f0">20</a>, we can see that a soft nano-hexapod \(k<10^6\ [N/m]\) significantly reduces the effect of perturbations from 20Hz to 300Hz.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org34c0f38" class="outline-2">
|
|
<h2 id="org34c0f38"><span class="section-number-2">5</span> Closed Loop Budget Error</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="orgd3503fb"></a>
|
|
</p>
|
|
<p>
|
|
From the total open-loop power spectral density of the sample’s motion error, we can estimate what is the required control bandwidth for the sample’s motion error to be reduced down to \(10nm\).
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgdfef0eb" class="outline-3">
|
|
<h3 id="orgdfef0eb"><span class="section-number-3">5.1</span> Approximation of the effect of feedback on the motion error</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
Let’s consider Figure <a href="#org6308d80">21</a> where a controller \(K\) is used to reduce the effect of the disturbance \(d\) on the position error \(y\).
|
|
</p>
|
|
|
|
|
|
<div id="org6308d80" class="figure">
|
|
<p><img src="figs/effect_feedback_disturbance_diagram.png" alt="effect_feedback_disturbance_diagram.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>Feedback System</p>
|
|
</div>
|
|
|
|
<p>
|
|
The reduction of the impact of \(d\) on \(y\) thanks to feedback is described by the following equation:
|
|
</p>
|
|
\begin{equation}
|
|
\frac{y}{d} = \frac{G_d}{1 + KG}
|
|
\end{equation}
|
|
<p>
|
|
The transfer functions corresponding to \(G_d\) are those identified in Section <a href="#orgf9e4300">2</a>.
|
|
</p>
|
|
|
|
|
|
<p>
|
|
As a first approximation, we can consider that the controller \(K\) is designed in such a way that the loop gain \(KG\) is a pure integrator:
|
|
\[ L_1(s) = K_1(s) G(s) = \frac{\omega_c}{s} \]
|
|
where \(\omega_c\) is the crossover frequency.
|
|
</p>
|
|
|
|
|
|
<p>
|
|
We may then consider another controller in such a way that the loop gain corresponds to a double integrator with a lead centered with the crossover frequency \(\omega_c\):
|
|
\[ L_2(s) = K_2(s) G(s) = \left( \frac{\omega_c}{s} \right)^2 \cdot \frac{1 + \frac{s}{\omega_c/2}}{1 + \frac{s}{2\omega_c}} \]
|
|
</p>
|
|
|
|
|
|
<p>
|
|
In the next section, we see how the power spectral density of \(y\) is reduced as a function of the control bandwidth \(\omega_c\).
|
|
This will help to determine what is the approximate control bandwidth required such that the rms value of \(y\) is below \(10nm\).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf2d36a1" class="outline-3">
|
|
<h3 id="orgf2d36a1"><span class="section-number-3">5.2</span> Reduction thanks to feedback - Required bandwidth</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
<p>
|
|
Let’s first see how does the Cumulative Amplitude Spectrum of the sample’s motion error is modified by the control.
|
|
</p>
|
|
|
|
<p>
|
|
In Figure <a href="#orgcbef465">22</a> is shown the Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses.
|
|
The controller used in this simulation is \(K_1\). The loop gain is then a pure integrator.
|
|
</p>
|
|
|
|
<p>
|
|
In Figure <a href="#orgd677910">23</a> is shown the expected RMS value of the sample’s position error as a function of the control bandwidth, both for \(K_1\) (left plot) and \(K_2\) (right plot).
|
|
As expected, it is shown that \(K_2\) performs better than \(K_1\).
|
|
This Figure tells us how much control bandwidth is required to attain a certain level of performance, and that for all the considered nano-hexapod stiffnesses.
|
|
</p>
|
|
|
|
<p>
|
|
The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample’s motion error less than 10nm rms are gathered in Table <a href="#org5ab4860">1</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgcbef465" class="figure">
|
|
<p><img src="figs/opt_stiff_cas_closed_loop.png" alt="opt_stiff_cas_closed_loop.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth and 4 different nano-hexapod stiffnesses (<a href="./figs/opt_stiff_cas_closed_loop.png">png</a>, <a href="./figs/opt_stiff_cas_closed_loop.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgd677910" class="figure">
|
|
<p><img src="figs/opt_stiff_req_bandwidth_K1_K2.png" alt="opt_stiff_req_bandwidth_K1_K2.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 23: </span>Expected RMS value of the sample’s motion error \(E_z\) as a function of the control bandwidth when using \(K_1\) and \(K_2\) (<a href="./figs/opt_stiff_req_bandwidth_K1_K2.png">png</a>, <a href="./figs/opt_stiff_req_bandwidth_K1_K2.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<table id="org5ab4860" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 1:</span> Approximate required control bandwidth such that the motion error is below \(10nm\)</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left">Nano-Hexapod stiffness [N/m]</th>
|
|
<th scope="col" class="org-right">10<sup>3</sup></th>
|
|
<th scope="col" class="org-right">10<sup>4</sup></th>
|
|
<th scope="col" class="org-right">10<sup>5</sup></th>
|
|
<th scope="col" class="org-right">10<sup>6</sup></th>
|
|
<th scope="col" class="org-right">10<sup>7</sup></th>
|
|
<th scope="col" class="org-right">10<sup>8</sup></th>
|
|
<th scope="col" class="org-right">10<sup>9</sup></th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Required wc with L1 [Hz]</td>
|
|
<td class="org-right">152</td>
|
|
<td class="org-right">305</td>
|
|
<td class="org-right">1000</td>
|
|
<td class="org-right">870</td>
|
|
<td class="org-right">933</td>
|
|
<td class="org-right">870</td>
|
|
<td class="org-right">870</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Required wc with L2 [Hz]</td>
|
|
<td class="org-right">57</td>
|
|
<td class="org-right">66</td>
|
|
<td class="org-right">152</td>
|
|
<td class="org-right">152</td>
|
|
<td class="org-right">248</td>
|
|
<td class="org-right">266</td>
|
|
<td class="org-right">248</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbf0fb63" class="outline-2">
|
|
<h2 id="orgbf0fb63"><span class="section-number-2">6</span> Conclusion</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<div class="important">
|
|
<p>
|
|
From Figure <a href="#orgd677910">23</a> and Table <a href="#org5ab4860">1</a>, we can clearly see three different results depending on the nano-hexapod stiffness:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>For a soft nano-hexapod (\(k < 10^4\ [N/m]\)), the required bandwidth is \(\omega_c \approx 50-100\ Hz\)</li>
|
|
<li>For a nano-hexapods with \(10^5 < k < 10^6\ [N/m]\)), the required bandwidth is \(\omega_c \approx 150-300\ Hz\)</li>
|
|
<li>For a stiff nano-hexapods (\(k > 10^7\ [N/m]\)), the required bandwidth is \(\omega_c \approx 250-500\ Hz\)</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-04-08 mer. 12:12</p>
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</div>
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</body>
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</html>
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