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auto/
*.tex
*.bbl
*.synctex.gz
.auctex-auto/
_minted*
# Windows default autosave extension
*.asv
*rtw/

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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-11-12 jeu. 10:16 -->
<!-- 2021-02-02 mar. 19:16 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Encoder - Test Bench</title>
<meta name="generator" content="Org mode" />
@ -22,19 +22,19 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org1c5bda2">1. Experimental Setup</a></li>
<li><a href="#orgdc41a88">2. Noise Spectral Density of the Encoder</a>
<li><a href="#org3c3af3a">1. Experimental Setup</a></li>
<li><a href="#orgdb3277a">2. Noise Spectral Density of the Encoder</a>
<ul>
<li><a href="#org9693b2a">2.1. Load Data</a></li>
<li><a href="#orgb24809d">2.2. Time Domain Results</a></li>
<li><a href="#org2228685">2.3. Frequency Domain Noise</a></li>
<li><a href="#org81a5e5f">2.1. Load Data</a></li>
<li><a href="#orgbed7f20">2.2. Time Domain Results</a></li>
<li><a href="#org319de75">2.3. Frequency Domain Noise</a></li>
</ul>
</li>
<li><a href="#orge121b74">3. Dynamics from Actuator to Encoder</a>
<li><a href="#orgb1ca2cf">3. Dynamics from Actuator to Encoder</a>
<ul>
<li><a href="#orgae3dfc0">3.1. Load Data</a></li>
<li><a href="#org83ba060">3.2. Excitation and Measured Signals</a></li>
<li><a href="#orge31f70d">3.3. Identification</a></li>
<li><a href="#orgfa505d1">3.1. Load Data</a></li>
<li><a href="#org3f21900">3.2. Excitation and Measured Signals</a></li>
<li><a href="#org0b79009">3.3. Identification</a></li>
</ul>
</li>
</ul>
@ -49,23 +49,23 @@ In this document, we wish to study the use of an encoder in parallel with an Amp
The document is divided into the following Sections:
</p>
<ul class="org-ul">
<li>Section <a href="#orgae74897">1</a>: the test-bench used is described</li>
<li>Section <a href="#org2f2ab76">2</a>: the noise spectral density of the encoder is estimated</li>
<li>Section <a href="#org3ffacc7">3</a>: the dynamics from the amplified piezoelectric actuator to the encoder measured displacement is identified</li>
<li>Section <a href="#org4c85aef">1</a>: the test-bench used is described</li>
<li>Section <a href="#org088f993">2</a>: the noise spectral density of the encoder is estimated</li>
<li>Section <a href="#org077ed39">3</a>: the dynamics from the amplified piezoelectric actuator to the encoder measured displacement is identified</li>
</ul>
<div id="outline-container-org1c5bda2" class="outline-2">
<h2 id="org1c5bda2"><span class="section-number-2">1</span> Experimental Setup</h2>
<div id="outline-container-org3c3af3a" class="outline-2">
<h2 id="org3c3af3a"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orgae74897"></a>
<a id="org4c85aef"></a>
</p>
<p>
The experimental Setup is schematically represented in Figure <a href="#orgb6cceaa">1</a>.
The experimental Setup is schematically represented in Figure <a href="#org87d981b">1</a>.
</p>
<div class="note" id="org72fff46">
<div class="note" id="org217bb34">
<p>
Here are the equipment used in the test bench:
</p>
@ -85,21 +85,21 @@ The displacement of the mass (relative to the mechanical frame) is measured both
</p>
<div id="orgb6cceaa" class="figure">
<div id="org87d981b" class="figure">
<p><img src="figs/exp_setup_schematic.png" alt="exp_setup_schematic.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the Experiment</p>
</div>
<div id="orge5d61dd" class="figure">
<div id="org4703eda" class="figure">
<p><img src="figs/IMG_20201023_153905.jpg" alt="IMG_20201023_153905.jpg" />
</p>
<p><span class="figure-number">Figure 2: </span>Side View of the encoder</p>
</div>
<div id="orgad29df1" class="figure">
<div id="orgd6a1cee" class="figure">
<p><img src="figs/IMG_20201023_153914.jpg" alt="IMG_20201023_153914.jpg" />
</p>
<p><span class="figure-number">Figure 3: </span>Front View of the encoder</p>
@ -107,11 +107,11 @@ The displacement of the mass (relative to the mechanical frame) is measured both
</div>
</div>
<div id="outline-container-orgdc41a88" class="outline-2">
<h2 id="orgdc41a88"><span class="section-number-2">2</span> Noise Spectral Density of the Encoder</h2>
<div id="outline-container-orgdb3277a" class="outline-2">
<h2 id="orgdb3277a"><span class="section-number-2">2</span> Noise Spectral Density of the Encoder</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org2f2ab76"></a>
<a id="org088f993"></a>
</p>
<p>
The goal in this section is the estimate the noise of both the encoder and the intereferometer.
@ -123,50 +123,50 @@ Ideally, a mechanical part would clamp the two together, we here suppose that th
</p>
</div>
<div id="outline-container-org9693b2a" class="outline-3">
<h3 id="org9693b2a"><span class="section-number-3">2.1</span> Load Data</h3>
<div id="outline-container-org81a5e5f" class="outline-3">
<h3 id="org81a5e5f"><span class="section-number-3">2.1</span> Load Data</h3>
<div class="outline-text-3" id="text-2-1">
<p>
The measurement data are loaded and the offset are removed using the <code>detrend</code> command.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'int_enc_huddle_test.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'t'</span>);
<pre class="src src-matlab"> load(<span class="org-string">'int_enc_huddle_test.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'t'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">interferometer = detrend(interferometer, 0);
encoder = detrend(encoder, 0);
<pre class="src src-matlab"> interferometer = detrend(interferometer, 0);
encoder = detrend(encoder, 0);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb24809d" class="outline-3">
<h3 id="orgb24809d"><span class="section-number-3">2.2</span> Time Domain Results</h3>
<div id="outline-container-orgbed7f20" class="outline-3">
<h3 id="orgbed7f20"><span class="section-number-3">2.2</span> Time Domain Results</h3>
<div class="outline-text-3" id="text-2-2">
<p>
The measurement of both the encoder and interferometer are shown in Figure <a href="#org481639f">4</a>.
The measurement of both the encoder and interferometer are shown in Figure <a href="#orgad4a9af">4</a>.
</p>
<div id="org481639f" class="figure">
<div id="orgad4a9af" class="figure">
<p><img src="figs/huddle_test_time_domain.png" alt="huddle_test_time_domain.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Huddle test - Time domain signals</p>
</div>
<p>
The raw signals are filtered with a Low Pass filter (defined below) such that we can see the low frequency motion (Figure <a href="#orgaea06bd">5</a>).
The raw signals are filtered with a Low Pass filter (defined below) such that we can see the low frequency motion (Figure <a href="#orgc981fe9">5</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_lpf = 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>10);
<pre class="src src-matlab"> G_lpf = 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>10);
</pre>
</div>
<div id="orgaea06bd" class="figure">
<div id="orgc981fe9" class="figure">
<p><img src="figs/huddle_test_time_domain_filtered.png" alt="huddle_test_time_domain_filtered.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Huddle test - Time domain signals filtered with a LPF at 10Hz</p>
@ -174,24 +174,24 @@ The raw signals are filtered with a Low Pass filter (defined below) such that we
</div>
</div>
<div id="outline-container-org2228685" class="outline-3">
<h3 id="org2228685"><span class="section-number-3">2.3</span> Frequency Domain Noise</h3>
<div id="outline-container-org319de75" class="outline-3">
<h3 id="org319de75"><span class="section-number-3">2.3</span> Frequency Domain Noise</h3>
<div class="outline-text-3" id="text-2-3">
<p>
The noise of the measurement (supposing there is no motion) is now translated in the frequency domain by computed the Amplitude Spectral Density.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
win = hann(ceil(10<span class="org-type">/</span>Ts));
<pre class="src src-matlab"> Ts = 1e<span class="org-type">-</span>4;
win = hann(ceil(10<span class="org-type">/</span>Ts));
[p_i, f] = pwelch(interferometer, win, [], [], 1<span class="org-type">/</span>Ts);
[p_e, <span class="org-type">~</span>] = pwelch(encoder, win, [], [], 1<span class="org-type">/</span>Ts);
[p_i, f] = pwelch(interferometer, win, [], [], 1<span class="org-type">/</span>Ts);
[p_e, <span class="org-type">~</span>] = pwelch(encoder, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<p>
The comparison of the ASD of the encoder and interferometer are shown in Figure <a href="#org38217d2">6</a>.
The comparison of the ASD of the encoder and interferometer are shown in Figure <a href="#orgeae7d8d">6</a>.
</p>
<p>
@ -199,7 +199,7 @@ It is clear that although the encoder exhibit higher frequency noise, is it more
</p>
<div id="org38217d2" class="figure">
<div id="orgeae7d8d" class="figure">
<p><img src="figs/huddle_test_asd.png" alt="huddle_test_asd.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Amplitude Spectral Density of the signals during the Huddle test</p>
@ -208,25 +208,25 @@ It is clear that although the encoder exhibit higher frequency noise, is it more
</div>
</div>
<div id="outline-container-orge121b74" class="outline-2">
<h2 id="orge121b74"><span class="section-number-2">3</span> Dynamics from Actuator to Encoder</h2>
<div id="outline-container-orgb1ca2cf" class="outline-2">
<h2 id="orgb1ca2cf"><span class="section-number-2">3</span> Dynamics from Actuator to Encoder</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org3ffacc7"></a>
<a id="org077ed39"></a>
</p>
<p>
Now the dynamics from the force actuator to the measurement by the encoder is identified.
</p>
</div>
<div id="outline-container-orgae3dfc0" class="outline-3">
<h3 id="orgae3dfc0"><span class="section-number-3">3.1</span> Load Data</h3>
<div id="outline-container-orgfa505d1" class="outline-3">
<h3 id="orgfa505d1"><span class="section-number-3">3.1</span> Load Data</h3>
<div class="outline-text-3" id="text-3-1">
<p>
As usual, the measurement data are loaded.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'int_enc_id_noise_bis.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>);
<pre class="src src-matlab"> load(<span class="org-string">'int_enc_id_noise_bis.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>);
</pre>
</div>
@ -234,10 +234,10 @@ As usual, the measurement data are loaded.
The first 0.1 seconds are removed as it corresponds to transient behavior.
</p>
<div class="org-src-container">
<pre class="src src-matlab">interferometer = interferometer(t<span class="org-type">&gt;</span>0.1);
encoder = encoder(t<span class="org-type">&gt;</span>0.1);
u = u(t<span class="org-type">&gt;</span>0.1);
t = t(t<span class="org-type">&gt;</span>0.1);
<pre class="src src-matlab"> interferometer = interferometer(t<span class="org-type">&gt;</span>0.1);
encoder = encoder(t<span class="org-type">&gt;</span>0.1);
u = u(t<span class="org-type">&gt;</span>0.1);
t = t(t<span class="org-type">&gt;</span>0.1);
</pre>
</div>
@ -245,79 +245,83 @@ t = t(t<span class="org-type">&gt;</span>0.1);
Finally the offset are removed using the <code>detrend</code> command.
</p>
<div class="org-src-container">
<pre class="src src-matlab">interferometer = detrend(interferometer, 0);
encoder = detrend(encoder, 0);
u = detrend(u, 0);
<pre class="src src-matlab"> interferometer = detrend(interferometer, 0);
encoder = detrend(encoder, 0);
u = detrend(u, 0);
</pre>
</div>
</div>
</div>
<div id="outline-container-org83ba060" class="outline-3">
<h3 id="org83ba060"><span class="section-number-3">3.2</span> Excitation and Measured Signals</h3>
<div id="outline-container-org3f21900" class="outline-3">
<h3 id="org3f21900"><span class="section-number-3">3.2</span> Excitation and Measured Signals</h3>
<div class="outline-text-3" id="text-3-2">
<p>
The excitation signal is a white noise filtered by a low pass filter to not excite too much the high frequency modes.
</p>
<p>
The excitation signal is shown in Figure <a href="#org93c938e">7</a>.
The excitation signal is shown in Figure <a href="#orgf417c0d">7</a>.
</p>
<div id="org93c938e" class="figure">
<div id="orgf417c0d" class="figure">
<p><img src="figs/encoder_identification_excitation_time.png" alt="encoder_identification_excitation_time.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Excitation Voltage</p>
</div>
<p>
The measured motion by the interferometer and encoder is shown in Figure
</p>
<div id="org85b6206" class="figure">
<div id="orgb870b1e" class="figure">
<p><img src="figs/encoder_identification_motion.png" alt="encoder_identification_motion.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Measured displacement by the encoder and interferometer</p>
</div>
</div>
</div>
<div id="outline-container-orge31f70d" class="outline-3">
<h3 id="orge31f70d"><span class="section-number-3">3.3</span> Identification</h3>
<div id="outline-container-org0b79009" class="outline-3">
<h3 id="org0b79009"><span class="section-number-3">3.3</span> Identification</h3>
<div class="outline-text-3" id="text-3-3">
<p>
Now the dynamics from the voltage sent to the voltage amplitude driving the APA95ML to the measured displacement by both the encoder and interferometer are computed.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
win = hann(ceil(10<span class="org-type">/</span>Ts));
<pre class="src src-matlab"> Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
win = hann(ceil(10<span class="org-type">/</span>Ts));
[tf_i_est, f] = tfestimate(u, interferometer, win, [], [], 1<span class="org-type">/</span>Ts);
[co_i_est, <span class="org-type">~</span>] = mscohere(u, interferometer, win, [], [], 1<span class="org-type">/</span>Ts);
[tf_i_est, f] = tfestimate(u, interferometer, win, [], [], 1<span class="org-type">/</span>Ts);
[co_i_est, <span class="org-type">~</span>] = mscohere(u, interferometer, win, [], [], 1<span class="org-type">/</span>Ts);
[tf_e_est, <span class="org-type">~</span>] = tfestimate(u, encoder, win, [], [], 1<span class="org-type">/</span>Ts);
[co_e_est, <span class="org-type">~</span>] = mscohere(u, encoder, win, [], [], 1<span class="org-type">/</span>Ts);
[tf_e_est, <span class="org-type">~</span>] = tfestimate(u, encoder, win, [], [], 1<span class="org-type">/</span>Ts);
[co_e_est, <span class="org-type">~</span>] = mscohere(u, encoder, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
<p>
The obtained coherence is shown in Figure <a href="#org646a3b0">9</a>.
The obtained coherence is shown in Figure <a href="#orgd2811d2">9</a>.
It is shown that the identification is good until 500Hz for the interferometer and until 1kHz for the encoder.
</p>
<div id="org646a3b0" class="figure">
<div id="orgd2811d2" class="figure">
<p><img src="figs/identification_dynamics_coherence.png" alt="identification_dynamics_coherence.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Obtained coherence for both the encoder and interferometer</p>
</div>
<p>
The compared dynamics as measured by the intereferometer and encoder are shown in Figure <a href="#orgbf0b43f">10</a>.
The compared dynamics as measured by the intereferometer and encoder are shown in Figure <a href="#org7032434">10</a>.
</p>
<div id="orgbf0b43f" class="figure">
<div id="org7032434" class="figure">
<p><img src="figs/identification_dynamics_bode.png" alt="identification_dynamics_bode.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Obtained dynamics from actuator voltage to displacement as measured by the interferometer and by the encoder</p>
</div>
@ -330,7 +334,7 @@ The second resonance at around 900Hz most likely corresponds to the resonance of
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-12 jeu. 10:16</p>
<p class="date">Created: 2021-02-02 mar. 19:16</p>
</div>
</body>
</html>

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@ -10,6 +10,14 @@
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#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+BIND: org-latex-image-default-width ""
#+LaTeX_CLASS: scrreprt
#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
#+EXPORT_FILE_NAME: test-bench-pd200.tex
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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@ -63,11 +71,13 @@ The displacement of the mass (relative to the mechanical frame) is measured both
#+name: fig:encoder_side_view
#+ATTR_ORG: :width 300
#+ATTR_LATEX: :width \linewidth
#+caption: Side View of the encoder
[[file:figs/IMG_20201023_153905.jpg]]
#+name: fig:encoder_front_view
#+caption: Front View of the encoder
#+ATTR_LATEX: :width \linewidth
[[file:figs/IMG_20201023_153914.jpg]]
* Noise Spectral Density of the Encoder
@ -239,7 +249,7 @@ The excitation signal is shown in Figure [[fig:encoder_identification_excitation
#+end_src
#+name: fig:encoder_identification_excitation_time
#+caption:
#+caption: Excitation Voltage
#+RESULTS:
[[file:figs/encoder_identification_excitation_time.png]]
@ -259,7 +269,7 @@ The measured motion by the interferometer and encoder is shown in Figure
#+end_src
#+name: fig:encoder_identification_motion
#+caption:
#+caption: Measured displacement by the encoder and interferometer
#+RESULTS:
[[file:figs/encoder_identification_motion.png]]
@ -297,7 +307,7 @@ It is shown that the identification is good until 500Hz for the interferometer a
#+end_src
#+name: fig:identification_dynamics_coherence
#+caption:
#+caption: Obtained coherence for both the encoder and interferometer
#+RESULTS:
[[file:figs/identification_dynamics_coherence.png]]
@ -336,7 +346,7 @@ The compared dynamics as measured by the intereferometer and encoder are shown i
#+end_src
#+name: fig:identification_dynamics_bode
#+caption:
#+caption: Obtained dynamics from actuator voltage to displacement as measured by the interferometer and by the encoder
#+RESULTS:
[[file:figs/identification_dynamics_bode.png]]

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