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<title>Noise Budgeting</title>
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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="#orgc8b5888">1. Maximum Noise of the Relative Motion Sensors</a>
<li><a href="#org6528281">1. Maximum Noise of the Relative Motion Sensors</a>
<ul>
<li><a href="#org47d58ae">1.1. Initialization</a></li>
<li><a href="#org9b3405f">1.2. Control System</a></li>
<li><a href="#org4b1b358">1.3. Maximum induced vibration&rsquo;s ASD</a></li>
<li><a href="#org446dbf5">1.4. Computation of the maximum relative motion sensor noise</a></li>
<li><a href="#org65a9628">1.5. Verification of the induced motion error</a></li>
<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&rsquo;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-orgc8b5888" class="outline-2">
<h2 id="orgc8b5888"><span class="section-number-2">1</span> Maximum Noise of the Relative Motion Sensors</h2>
<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-org47d58ae" class="outline-3">
<h3 id="org47d58ae"><span class="section-number-3">1.1</span> Initialization</h3>
<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('nass_model.slx');
<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();
<pre class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
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('type', 'hac-dvf');
initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
</pre>
</div>
@@ -81,61 +86,61 @@ initializeController('type', 'hac-dvf');
We set the stiffness of the payload fixation:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kp = 1e8; % [N/m]
<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('k', 1e5, 'c', 2e2);
Ms = 50;
initializeSample('mass', Ms, 'freq', sqrt(Kp/Ms)/2/pi*ones(6,1));
<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('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms);
<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-org9b3405f" class="outline-3">
<h3 id="org9b3405f"><span class="section-number-3">1.2</span> Control System</h3>
<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*s/(1+s/2/pi/1e3)*eye(6);
<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 * eye(6) * ...
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
1/h*(s/(2*pi*200/h) + 1)/(s/(2*pi*200*h) + 1) * ...
(s/2/pi/10 + 1)/(s/2/pi/10) * ...
1/(1 + s/2/pi/300);
<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('mat/stages.mat', 'nano_hexapod');
K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
<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">%% Run the linearization
G = linearize(mdl, io);
G.InputName = {'ndL1', 'ndL2', 'ndL3', 'ndL4', 'ndL5', 'ndL6'};
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
<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-org4b1b358" class="outline-3">
<h3 id="org4b1b358"><span class="section-number-3">1.3</span> Maximum induced vibration&rsquo;s ASD</h3>
<div id="outline-container-org538f5fe" class="outline-3">
<h3 id="org538f5fe"><span class="section-number-3">1.3</span> Maximum induced vibration&rsquo;s ASD</h3>
<div class="outline-text-3" id="text-1-3">
<p>
Required maximum induced ASD of the sample&rsquo;s vibration due to the relative motion sensor noise.
@@ -143,16 +148,16 @@ Required maximum induced ASD of the sample&rsquo;s vibration due to the relative
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gamma_x = [(1e-9)/(1 + s/2/pi/100); % Dx
(1e-9)/(1 + s/2/pi/100); % Dy
(1e-9)/(1 + s/2/pi/100); % Dz
(2e-8)/(1 + s/2/pi/100); % Rx
(2e-8)/(1 + s/2/pi/100)]; % Ry
<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 class="src src-matlab"> freqs = logspace(0, 3, 1000);
</pre>
</div>
@@ -203,8 +208,8 @@ Corresponding RMS value in [nm rms, nrad rms]
</div>
</div>
<div id="outline-container-org446dbf5" class="outline-3">
<h3 id="org446dbf5"><span class="section-number-3">1.4</span> Computation of the maximum relative motion sensor noise</h3>
<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&rsquo;s note \(G\) the transfer function from the 6 sensor noise \(n\) to the 5dof pose error \(x\).
@@ -225,15 +230,15 @@ We then have an upper bound of the sensor noise for each of the considered motio
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gamma_ndL = zeros(5, length(freqs));
for in = 1:5
Gamma_ndL(in, :) = abs(squeeze(freqresp(Gamma_x(in), freqs, 'Hz')))./sqrt(sum(abs(squeeze(freqresp(G(in, :), freqs, 'Hz'))).^2))';
end
<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="orgf2f2139" class="figure">
<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>
@@ -246,7 +251,7 @@ 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(1:5, :));
<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>
@@ -254,12 +259,12 @@ Then, the motion error due to sensor noise should be bellow the one specified.
Let&rsquo;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)/(1 + s/2/pi/1e3), freqs, 'Hz')));
<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="org73ad463" class="figure">
<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>
@@ -269,7 +274,7 @@ Let&rsquo;s take a sensor with a white noise up to 1kHz that is bellow the speci
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*sqrt(trapz(freqs, Gamma_ndL_max.^2))
<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>
@@ -279,8 +284,8 @@ The corresponding RMS value of the sensor noise taken as an example is [nm RMS]:
</div>
</div>
<div id="outline-container-org65a9628" class="outline-3">
<h3 id="org65a9628"><span class="section-number-3">1.5</span> Verification of the induced motion error</h3>
<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
@@ -289,11 +294,11 @@ From the sensor noise PSD \(\Gamma_n(\omega)\), we can estimate the obtained dis
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gamma_xest = zeros(5, length(freqs));
<pre class="src src-matlab"> Gamma_xest = zeros(5, length(freqs));
for in = 1:5
Gamma_xest(in, :) = sqrt(sum(abs(squeeze(freqresp(G(in, :), freqs, 'Hz'))).^2.*Gamma_ndL_max.^2));
end
<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>
@@ -352,7 +357,7 @@ end
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-09-01 mar. 13:47</p>
<p class="date">Created: 2021-02-20 sam. 23:08</p>
</div>
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