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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">Identification of the 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="#orga5c4454">1. Identification</a></li>
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<li><a href="#org0c8db89">2. Sensitivity to Disturbances</a></li>
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<li><a href="#org000560b">3. Power Spectral Density of the effect of the disturbances</a></li>
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<li><a href="#orgf4292db">4. Compute the Power Spectral Density of the disturbance force</a></li>
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<li><a href="#org40a18b0">5. Noise Budget</a></li>
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<li><a href="#org0562d36">6. Save</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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The goal here is to extract the Power Spectral Density of the sources of perturbation.
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</p>
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<p>
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The sources of perturbations are (schematically shown in figure <a href="#org1a5776c">1</a>):
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</p>
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<ul class="org-ul">
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<li>\(D_w\): Ground Motion</li>
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<li>Parasitic forces applied in the system when scanning with the Translation Stage and the Spindle (\(F_{rz}\) and \(F_{ty}\)).
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These forces can be due to imperfect guiding for instance.</li>
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</ul>
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<p>
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Because we cannot measure directly the perturbation forces, we have the measure the effect of those perturbations on the system (in terms of velocity for instance using geophones, \(D\) on figure <a href="#org1a5776c">1</a>) and then, using a model, compute the forces that induced such velocity.
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</p>
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<div id="org1a5776c" class="figure">
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<p><img src="figs/uniaxial-model-micro-station.png" alt="uniaxial-model-micro-station.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Schematic of the Micro Station and the sources of disturbance</p>
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</div>
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<p>
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This file 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="#org6e1c6df">1</a>: transfer functions from the disturbance forces to the relative velocity of the hexapod with respect to the granite are computed using the Simscape Model representing the experimental setup</li>
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<li>Section <a href="#org122fad2">2</a>: the bode plot of those transfer functions are shown</li>
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<li>Section <a href="#org19ee725">3</a>: the measured PSD of the effect of the disturbances are shown</li>
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<li>Section <a href="#org7c3b0b5">4</a>: from the model and the measured PSD, the PSD of the disturbance forces are computed</li>
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<li>Section <a href="#org7e2fa27">5</a>: with the computed PSD, the noise budget of the system is done</li>
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</ul>
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<div id="outline-container-orga5c4454" class="outline-2">
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<h2 id="orga5c4454"><span class="section-number-2">1</span> Identification</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org6e1c6df"></a>
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</p>
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<p>
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The transfer functions from the disturbance forces to the relative velocity of the hexapod with respect to the granite are computed using the Simscape Model representing the experimental setup with the code below.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
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options = linearizeOptions;
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options.SampleTime = <span class="org-highlight-numbers-number">0</span>;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
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mdl = <span class="org-string">'sim_micro_station_disturbances'</span>;
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Micro-Hexapod</span></span>
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<span class="org-comment">% Input/Output definition</span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Ground Motion</span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fty'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Ty</span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Frz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Rz</span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dgm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute motion - Granite</span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dhm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute Motion - Hexapod</span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Vm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Relative Velocity hexapod/granite</span>
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|
</pre>
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|
</div>
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<div class="org-src-container">
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|
<pre class="src src-matlab"><span class="org-comment">% Run the linearization</span>
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|
G = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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|
|
|
<span class="org-comment">% Input/Output names</span>
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|
G.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dw'</span>, <span class="org-string">'Fty'</span>, <span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
G.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dgm'</span>, <span class="org-string">'Dhm'</span>, <span class="org-string">'Vm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
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|
</div>
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|
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|
<div id="outline-container-org0c8db89" class="outline-2">
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|
<h2 id="org0c8db89"><span class="section-number-2">2</span> Sensitivity to Disturbances</h2>
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|
<div class="outline-text-2" id="text-2">
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|
<p>
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|
<a id="org122fad2"></a>
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|
</p>
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|
|
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|
<div id="org8f38398" class="figure">
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<p><img src="figs/sensitivity_dist_gm.png" alt="sensitivity_dist_gm.png" />
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|
</p>
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|
<p><span class="figure-number">Figure 2: </span>Sensitivity to Ground Motion (<a href="./figs/sensitivity_dist_gm.png">png</a>, <a href="./figs/sensitivity_dist_gm.pdf">pdf</a>)</p>
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</div>
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|
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|
<div id="orga020f23" class="figure">
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|
<p><img src="figs/sensitivity_dist_fty.png" alt="sensitivity_dist_fty.png" />
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|
</p>
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|
<p><span class="figure-number">Figure 3: </span>Sensitivity to vertical forces applied by the Ty stage (<a href="./figs/sensitivity_dist_fty.png">png</a>, <a href="./figs/sensitivity_dist_fty.pdf">pdf</a>)</p>
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</div>
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<div id="org0dea65f" class="figure">
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<p><img src="figs/sensitivity_dist_frz.png" alt="sensitivity_dist_frz.png" />
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|
</p>
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|
<p><span class="figure-number">Figure 4: </span>Sensitivity to vertical forces applied by the Rz stage (<a href="./figs/sensitivity_dist_frz.png">png</a>, <a href="./figs/sensitivity_dist_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-org000560b" class="outline-2">
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|
<h2 id="org000560b"><span class="section-number-2">3</span> Power Spectral Density of the effect of the disturbances</h2>
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|
<div class="outline-text-2" id="text-3">
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|
<p>
|
|
<a id="org19ee725"></a>
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|
The PSD of the relative velocity between the hexapod and the marble in \([(m/s)^2/Hz]\) are loaded for the following sources of disturbance:
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|
</p>
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|
<ul class="org-ul">
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|
<li>Slip Ring Rotation</li>
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|
<li>Scan of the translation stage (effect in the vertical direction and in the horizontal direction)</li>
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|
</ul>
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|
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|
<p>
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|
Also, the Ground Motion is measured.
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|
</p>
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|
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|
<div class="org-src-container">
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|
<pre class="src src-matlab">gm = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/psd_gm.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'psd_gm'</span>, <span class="org-string">'psd_gv'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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|
rz = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/pxsp_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxsp_r'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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|
tyz = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/pxz_ty_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxz_ty_r'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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|
tyx = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/pxe_ty_r.mat'</span>, <span class="org-string">'f'</span>, <span class="org-string">'pxe_ty_r'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
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|
|
|
<p>
|
|
We now compute the relative velocity between the hexapod and the granite due to ground motion.
|
|
</p>
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|
<div class="org-src-container">
|
|
<pre class="src src-matlab">gm.psd_rv = gm.psd_gm<span class="org-type">.*</span>abs<span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>freqresp<span class="org-rainbow-delimiters-depth-3">(</span>G<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-string">'Vm'</span>, <span class="org-string">'Dw'</span><span class="org-rainbow-delimiters-depth-4">)</span>, gm.f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">.^</span><span class="org-highlight-numbers-number">2</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The Power Spectral Density of the relative motion/velocity of the hexapod with respect to the granite are shown in figures <a href="#orgdd6d206">5</a> and <a href="#org67d7e50">6</a>.
|
|
</p>
|
|
|
|
<p>
|
|
The Cumulative Amplitude Spectrum of the relative motion is shown in figure <a href="#orgd8c990c">7</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgdd6d206" class="figure">
|
|
<p><img src="figs/dist_effect_relative_velocity.png" alt="dist_effect_relative_velocity.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Amplitude Spectral Density of the relative velocity of the hexapod with respect to the granite due to different sources of perturbation (<a href="./figs/dist_effect_relative_velocity.png">png</a>, <a href="./figs/dist_effect_relative_velocity.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org67d7e50" class="figure">
|
|
<p><img src="figs/dist_effect_relative_motion.png" alt="dist_effect_relative_motion.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Amplitude Spectral Density of the relative displacement of the hexapod with respect to the granite due to different sources of perturbation (<a href="./figs/dist_effect_relative_motion.png">png</a>, <a href="./figs/dist_effect_relative_motion.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgd8c990c" class="figure">
|
|
<p><img src="figs/dist_effect_relative_motion_cas.png" alt="dist_effect_relative_motion_cas.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Cumulative Amplitude Spectrum of the relative motion due to different sources of perturbation (<a href="./figs/dist_effect_relative_motion_cas.png">png</a>, <a href="./figs/dist_effect_relative_motion_cas.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf4292db" class="outline-2">
|
|
<h2 id="orgf4292db"><span class="section-number-2">4</span> Compute the Power Spectral Density of the disturbance force</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org7c3b0b5"></a>
|
|
</p>
|
|
|
|
<p>
|
|
Now, from the extracted transfer functions from the disturbance force to the relative motion of the hexapod with respect to the granite (section <a href="#org122fad2">2</a>) and from the measured PSD of the relative motion (section <a href="#org19ee725">3</a>), we can compute the PSD of the disturbance force.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rz.psd_f = rz.pxsp_r<span class="org-type">./</span>abs<span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>freqresp<span class="org-rainbow-delimiters-depth-3">(</span>G<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-string">'Vm'</span>, <span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-4">)</span>, rz.f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">.^</span><span class="org-highlight-numbers-number">2</span>;
|
|
tyz.psd_f = tyz.pxz_ty_r<span class="org-type">./</span>abs<span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>freqresp<span class="org-rainbow-delimiters-depth-3">(</span>G<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-string">'Vm'</span>, <span class="org-string">'Fty'</span><span class="org-rainbow-delimiters-depth-4">)</span>, tyz.f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">.^</span><span class="org-highlight-numbers-number">2</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgafd3bc8" class="figure">
|
|
<p><img src="figs/dist_force_psd.png" alt="dist_force_psd.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Amplitude Spectral Density of the disturbance force (<a href="./figs/dist_force_psd.png">png</a>, <a href="./figs/dist_force_psd.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org40a18b0" class="outline-2">
|
|
<h2 id="org40a18b0"><span class="section-number-2">5</span> Noise Budget</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org7e2fa27"></a>
|
|
</p>
|
|
|
|
<p>
|
|
Now, from the compute spectral density of the disturbance sources, we can compute the resulting relative motion of the Hexapod with respect to the granite using the model.
|
|
We should verify that this is coherent with the measurements.
|
|
</p>
|
|
|
|
|
|
<div id="org3326f40" class="figure">
|
|
<p><img src="figs/psd_effect_dist_verif.png" alt="psd_effect_dist_verif.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Computed Effect of the disturbances on the relative displacement hexapod/granite (<a href="./figs/psd_effect_dist_verif.png">png</a>, <a href="./figs/psd_effect_dist_verif.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="orgead0a2c" class="figure">
|
|
<p><img src="figs/cas_computed_relative_displacement.png" alt="cas_computed_relative_displacement.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>CAS of the total Relative Displacement due to all considered sources of perturbation (<a href="./figs/cas_computed_relative_displacement.png">png</a>, <a href="./figs/cas_computed_relative_displacement.pdf">pdf</a>)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0562d36" class="outline-2">
|
|
<h2 id="org0562d36"><span class="section-number-2">6</span> Save</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
The PSD of the disturbance force are now saved for further noise budgeting when control is applied (the mat file is accessible <a href="mat/dist_psd.mat">here</a>).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">dist_f = struct<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
dist_f.f = gm.f; <span class="org-comment">% Frequency Vector [Hz]</span>
|
|
dist_f.psd_gm = gm.psd_gm; % Power Spectral Density of the Ground Motion [m<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>Hz]
|
|
dist_f.psd_ty = tyz.psd_f; % Power Spectral Density of the force induced by the Ty stage in the Z direction [N<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>Hz]
|
|
dist_f.psd_rz = rz.psd_f; % Power Spectral Density of the force induced by the Rz stage in the Z direction [N<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>Hz]
|
|
|
|
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2019-11-04 lun. 15:56</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|