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Add dynamic noise budgeting: sensor noise spec.

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Thomas Dehaeze 2020-07-31 18:00:29 +02:00
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<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-05-05 mar. 11:38 -->
<!-- 2020-07-31 ven. 18:00 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Simscape Model of the Nano-Active-Stabilization-System</title>
<meta name="generator" content="Org mode" />
@ -41,9 +40,10 @@
<li><a href="#orga323881">12. Effect of Experimental conditions on the plant dynamics (link)</a></li>
<li><a href="#orge7b9b41">13. Optimal Stiffness of the nano-hexapod to reduce plant uncertainty (link)</a></li>
<li><a href="#org5f73af9">14. Effect of flexible joints on the plant dynamics (link)</a></li>
<li><a href="#org14a10e8">15. Active Damping Techniques on the full Simscape Model (link)</a></li>
<li><a href="#orgd818a00">16. Control of the Nano-Active-Stabilization-System (link)</a></li>
<li><a href="#org361f405">17. Useful Matlab Functions (link)</a></li>
<li><a href="#org2852795">15. Dynamic Noise Budgeting (link)</a></li>
<li><a href="#org14a10e8">16. Active Damping Techniques on the full Simscape Model (link)</a></li>
<li><a href="#orgd818a00">17. Control of the Nano-Active-Stabilization-System (link)</a></li>
<li><a href="#org361f405">18. Useful Matlab Functions (link)</a></li>
</ul>
</div>
</div>
@ -216,18 +216,27 @@ Conclusion are drawn on the required stiffness properties of the flexible joints
</div>
</div>
<div id="outline-container-org14a10e8" class="outline-2">
<h2 id="org14a10e8"><span class="section-number-2">15</span> Active Damping Techniques on the full Simscape Model (<a href="control_active_damping.html">link</a>)</h2>
<div id="outline-container-org2852795" class="outline-2">
<h2 id="org2852795"><span class="section-number-2">15</span> Dynamic Noise Budgeting (<a href="noise_budgeting.html">link</a>)</h2>
<div class="outline-text-2" id="text-15">
<p>
The maximum allowed noise of the sensors in the system are estimated using a Dynamic Noise Budgeting.
</p>
</div>
</div>
<div id="outline-container-org14a10e8" class="outline-2">
<h2 id="org14a10e8"><span class="section-number-2">16</span> Active Damping Techniques on the full Simscape Model (<a href="control_active_damping.html">link</a>)</h2>
<div class="outline-text-2" id="text-16">
<p>
Active damping techniques are applied to the full Simscape model.
</p>
</div>
</div>
<div id="outline-container-orgd818a00" class="outline-2">
<h2 id="orgd818a00"><span class="section-number-2">16</span> Control of the Nano-Active-Stabilization-System (<a href="control.html">link</a>)</h2>
<div class="outline-text-2" id="text-16">
<h2 id="orgd818a00"><span class="section-number-2">17</span> Control of the Nano-Active-Stabilization-System (<a href="control.html">link</a>)</h2>
<div class="outline-text-2" id="text-17">
<p>
In this file are gathered all studies about the control the Nano-Active-Stabilization-System.
</p>
@ -235,8 +244,8 @@ In this file are gathered all studies about the control the Nano-Active-Stabiliz
</div>
<div id="outline-container-org361f405" class="outline-2">
<h2 id="org361f405"><span class="section-number-2">17</span> Useful Matlab Functions (<a href="./functions.html">link</a>)</h2>
<div class="outline-text-2" id="text-17">
<h2 id="org361f405"><span class="section-number-2">18</span> Useful Matlab Functions (<a href="./functions.html">link</a>)</h2>
<div class="outline-text-2" id="text-18">
<p>
Many matlab functions are shared among all the files of the projects.
</p>
@ -249,7 +258,7 @@ These functions are all defined <a href="./functions.html">here</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-05 mar. 11:38</p>
<p class="date">Created: 2020-07-31 ven. 18:00</p>
</div>
</body>
</html>

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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-07-31 ven. 17:58 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Noise Budgeting</title>
<meta name="generator" content="Org mode" />
<meta name="author" content="Dehaeze Thomas" />
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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<script src="./js/jquery.stickytableheaders.min.js"></script>
<script src="./js/readtheorg.js"></script>
<script>MathJax = {
tex: {
tags: 'ams',
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<body>
<div id="org-div-home-and-up">
<a accesskey="h" href="./index.html"> UP </a>
|
<a accesskey="H" href="./index.html"> HOME </a>
</div><div id="content">
<h1 class="title">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>
<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>
</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 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 class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<pre class="src src-matlab">open('nass_model.slx');
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'hac-dvf');
</pre>
</div>
<p>
We set the stiffness of the payload fixation:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kp = 1e8; % [N/m]
</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>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 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 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>
</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>
</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>
</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>
</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 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.
\[ \bm{\Gamma}_x(\omega) = \begin{bmatrix} \Gamma_x(\omega) & \Gamma_y(\omega) & \Gamma_{R_x}(\omega) & \Gamma_{R_y}(\omega) \end{bmatrix} \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gamma_x = [(1e-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>
</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>
<div class="org-src-container">
<pre class="src src-matlab">1e9*sqrt(trapz(freqs, (abs(squeeze(freqresp(Gamma_x, freqs, 'Hz')))').^2))
</pre>
</div>
<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">&#xa0;</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-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 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 6dof 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:
\[ \Gamma_{x_i}(\omega) = \sum_{j=1}^6 |G_{ij}(j\omega)|^2 \Gamma_{n_j}(\omega), \quad i = 1 \dots 5 \]
</p>
<p>
Let&rsquo;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:
\[ \Gamma_{n_i, \text{max}}(\omega) = \frac{\Gamma_{n_i}(\omega)}{\sum_{j=1}^6 |G_{ij}(j\omega)|^2}, \quad i = 1 \dots 5 \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gamma_ndL = zeros(5, length(freqs));
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>
</div>
<div id="orgf2f2139" 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(1:5, :));
</pre>
</div>
<p>
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>
</div>
<div id="org73ad463" 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*sqrt(trapz(freqs, Gamma_ndL_max.^2))
</pre>
</div>
<pre class="example">
519.29
</pre>
</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 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 \(\Gamma_x(\omega)\):
\[ \Gamma_{x,i}(\omega) = \sqrt{ \sum_{j=1}^{6} |G_{ij}|^2(j\omega) \Gamma_{n,j}^2(\omega) }, \quad i = 1 \dots 5 \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gamma_xest = zeros(5, length(freqs));
for in = 1:5
Gamma_xest(in, :) = sqrt(sum(abs(squeeze(freqresp(G(in, :), freqs, 'Hz'))).^2.*Gamma_ndL_max.^2));
end
</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">&#xa0;</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: 2020-07-31 ven. 17:58</p>
</div>
</body>
</html>

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org/index.org
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@ -72,6 +72,9 @@ Conclusion are drawn about what experimental conditions are critical on the vari
In this document is studied how the flexible joint stiffnesses (in flexion, torsion and compression) is affecting the plant dynamics.
Conclusion are drawn on the required stiffness properties of the flexible joints.
* Dynamic Noise Budgeting ([[file:noise_budgeting.org][link]])
The maximum allowed noise of the sensors in the system are estimated using a Dynamic Noise Budgeting.
* Active Damping Techniques on the full Simscape Model ([[file:control_active_damping.org][link]])
Active damping techniques are applied to the full Simscape model.