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<h1 class="title">Noise Budgeting</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org6528281">1. Maximum Noise of the Relative Motion Sensors</a>
<ul>
<li><a href="#org5b3bcaa">1.1. Initialization</a></li>
<li><a href="#org9daa837">1.2. Control System</a></li>
<li><a href="#org538f5fe">1.3. Maximum induced vibration’s ASD</a></li>
<li><a href="#org626a300">1.4. Computation of the maximum relative motion sensor noise</a></li>
<li><a href="#org3e7c118">1.5. Verification of the induced motion error</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org6528281" class="outline-2">
<h2 id="org6528281"><span class="section-number-2">1</span> Maximum Noise of the Relative Motion Sensors</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org5b3bcaa" class="outline-3">
<h3 id="org5b3bcaa"><span class="section-number-3">1.1</span> Initialization</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<pre class="src src-matlab"> open(<span class="org-string">'nass_model.slx'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeSimscapeConfiguration();
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>);
initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
</pre>
</div>
<p>
We set the stiffness of the payload fixation:
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Kp = 1e8; <span class="org-comment">% [N/m]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> initializeNanoHexapod(<span class="org-string">'k'</span>, 1e5, <span class="org-string">'c'</span>, 2e2);
Ms = 50;
initializeSample(<span class="org-string">'mass'</span>, Ms, <span class="org-string">'freq'</span>, sqrt(Kp<span class="org-type">/</span>Ms)<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>ones(6,1));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, Ms);
</pre>
</div>
</div>
</div>
<div id="outline-container-org9daa837" class="outline-3">
<h3 id="org9daa837"><span class="section-number-3">1.2</span> Control System</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<pre class="src src-matlab"> Kdvf = 5e3<span class="org-type">*</span>s<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>1e3)<span class="org-type">*</span>eye(6);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> h = 2.0;
Kl = 2e7 <span class="org-type">*</span> eye(6) <span class="org-type">*</span> ...
1<span class="org-type">/</span>h<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>100<span class="org-type">/</span>h) <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>100<span class="org-type">*</span>h) <span class="org-type">+</span> 1) <span class="org-type">*</span> ...
1<span class="org-type">/</span>h<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>200<span class="org-type">/</span>h) <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>200<span class="org-type">*</span>h) <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>10 <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>10) <span class="org-type">*</span> ...
1<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>300);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
K = Kl<span class="org-type">*</span>nano_hexapod.kinematics.J<span class="org-type">*</span>diag([1, 1, 1, 1, 1, 0]);
</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 = linearize(mdl, io);
G.InputName = {<span class="org-string">'ndL1'</span>, <span class="org-string">'ndL2'</span>, <span class="org-string">'ndL3'</span>, <span class="org-string">'ndL4'</span>, <span class="org-string">'ndL5'</span>, <span class="org-string">'ndL6'</span>};
G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-org538f5fe" class="outline-3">
<h3 id="org538f5fe"><span class="section-number-3">1.3</span> Maximum induced vibration’s ASD</h3>
<div class="outline-text-3" id="text-1-3">
<p>
Required maximum induced ASD of the sample’s vibration due to the relative motion sensor noise.
\[ \bm{\Gamma}_x(\omega) = \begin{bmatrix} \Gamma_x(\omega) & \Gamma_y(\omega) & \Gamma_{R_x}(\omega) & \Gamma_{R_y}(\omega) \end{bmatrix} \]
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gamma_x = [(1e<span class="org-type">-</span>9)<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>100); <span class="org-comment">% Dx</span>
(1e<span class="org-type">-</span>9)<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>100); <span class="org-comment">% Dy</span>
(1e<span class="org-type">-</span>9)<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>100); <span class="org-comment">% Dz</span>
(2e<span class="org-type">-</span>8)<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>100); <span class="org-comment">% Rx</span>
(2e<span class="org-type">-</span>8)<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>100)]; <span class="org-comment">% Ry</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> freqs = logspace(0, 3, 1000);
</pre>
</div>
<p>
Corresponding RMS value in [nm rms, nrad rms]
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<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">Specifications</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Dx [nm]</td>
<td class="org-right">12.1</td>
</tr>
<tr>
<td class="org-left">Dy [nm]</td>
<td class="org-right">12.1</td>
</tr>
<tr>
<td class="org-left">Dz [nm]</td>
<td class="org-right">12.1</td>
</tr>
<tr>
<td class="org-left">Rx [nrad]</td>
<td class="org-right">241.8</td>
</tr>
<tr>
<td class="org-left">Ry [nrad]</td>
<td class="org-right">241.8</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org626a300" class="outline-3">
<h3 id="org626a300"><span class="section-number-3">1.4</span> Computation of the maximum relative motion sensor noise</h3>
<div class="outline-text-3" id="text-1-4">
<p>
Let’s note \(G\) the transfer function from the 6 sensor noise \(n\) to the 5dof pose error \(x\).
We have:
\[ x_i = \sum_{j=1}^6 G_{ij}(s) n_j, \quad i = 1 \dots 5 \]
In terms of ASD:
\[ \Gamma_{x_i}(\omega) = \sqrt{\sum_{j=1}^6 |G_{ij}(j\omega)|^2 \cdot {\Gamma_{n_j}}^2(\omega)}, \quad i = 1 \dots 5 \]
</p>
<p>
Let’s suppose that the ASD of all the sensor noise are equal:
\[ \Gamma_{n_j} = \Gamma_{n}, \quad j = 1 \dots 6 \]
</p>
<p>
We then have an upper bound of the sensor noise for each of the considered motion errors:
\[ \Gamma_{n_i, \text{max}}(\omega) = \frac{\Gamma_{x_i}(\omega)}{\sqrt{\sum_{j=1}^6 |G_{ij}(j\omega)|^2}}, \quad i = 1 \dots 5 \]
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gamma_ndL = zeros(5, length(freqs));
<span class="org-keyword">for</span> <span class="org-variable-name">in</span> = <span class="org-constant">1:5</span>
Gamma_ndL(in, <span class="org-type">:</span>) = abs(squeeze(freqresp(Gamma_x(in), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">./</span>sqrt(sum(abs(squeeze(freqresp(G(in, <span class="org-type">:</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2))<span class="org-type">'</span>;
<span class="org-keyword">end</span>
</pre>
</div>
<div id="org9e66f1d" class="figure">
<p><img src="figs/noise_budget_ndL_max_asd.png" alt="noise_budget_ndL_max_asd.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Maximum estimated ASD of the relative motion sensor noise</p>
</div>
<p>
If the noise ASD of the relative motion sensor is bellow the maximum specified ASD for all the considered motion:
\[ \Gamma_n < \Gamma_{n_i, \text{max}}, \quad i = 1 \dots 5 \]
Then, the motion error due to sensor noise should be bellow the one specified.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gamma_ndL_max = min(Gamma_ndL(1<span class="org-type">:</span>5, <span class="org-type">:</span>));
</pre>
</div>
<p>
Let’s take a sensor with a white noise up to 1kHz that is bellow the specified one:
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gamma_ndL_ex = abs(squeeze(freqresp(min(Gamma_ndL_max)<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>1e3), freqs, <span class="org-string">'Hz'</span>)));
</pre>
</div>
<div id="org21fc07c" class="figure">
<p><img src="figs/relative_motion_sensor_noise_ASD_example.png" alt="relative_motion_sensor_noise_ASD_example.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Requirement maximum ASD of the sensor noise + example of a sensor validating the requirements</p>
</div>
<p>
The corresponding RMS value of the sensor noise taken as an example is [nm RMS]:
</p>
<div class="org-src-container">
<pre class="src src-matlab"> 1e9<span class="org-type">*</span>sqrt(trapz(freqs, Gamma_ndL_max<span class="org-type">.^</span>2))
</pre>
</div>
<pre class="example">
519.29
</pre>
</div>
</div>
<div id="outline-container-org3e7c118" class="outline-3">
<h3 id="org3e7c118"><span class="section-number-3">1.5</span> Verification of the induced motion error</h3>
<div class="outline-text-3" id="text-1-5">
<p>
Verify that by taking the sensor noise, we have to wanted displacement error
From the sensor noise PSD \(\Gamma_n(\omega)\), we can estimate the obtained displacement PSD \(\Gamma_x(\omega)\):
\[ \Gamma_{x,i}(\omega) = \sqrt{ \sum_{j=1}^{6} |G_{ij}|^2(j\omega) \cdot \Gamma_{n,j}^2(\omega) }, \quad i = 1 \dots 5 \]
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gamma_xest = zeros(5, length(freqs));
<span class="org-keyword">for</span> <span class="org-variable-name">in</span> = <span class="org-constant">1:5</span>
Gamma_xest(in, <span class="org-type">:</span>) = sqrt(sum(abs(squeeze(freqresp(G(in, <span class="org-type">:</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>Gamma_ndL_max<span class="org-type">.^</span>2));
<span class="org-keyword">end</span>
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-right">Results</th>
<th scope="col" class="org-right">Specifications</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Dx [nm]</td>
<td class="org-right">8.9</td>
<td class="org-right">12.1</td>
</tr>
<tr>
<td class="org-left">Dy [nm]</td>
<td class="org-right">9.3</td>
<td class="org-right">12.1</td>
</tr>
<tr>
<td class="org-left">Dz [nm]</td>
<td class="org-right">10.2</td>
<td class="org-right">12.1</td>
</tr>
<tr>
<td class="org-left">Rx [nrad]</td>
<td class="org-right">110.2</td>
<td class="org-right">241.8</td>
</tr>
<tr>
<td class="org-left">Ry [nrad]</td>
<td class="org-right">107.8</td>
<td class="org-right">241.8</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-02-20 sam. 23:08</p>
</div>
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