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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2021-02-10 mer. 15:14 --> <!-- 2021-02-11 jeu. 15:21 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Encoder Renishaw Vionic - Test Bench</title> <title>Encoder Renishaw Vionic - Test Bench</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -39,22 +39,21 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgee60877">1. Expected Performances</a></li> <li><a href="#orgacaf822">1. Expected Performances</a></li>
<li><a href="#org78808d1">2. Encoder Model</a></li> <li><a href="#orgd1b48b9">2. Encoder Model</a></li>
<li><a href="#org07e5c0c">3. Noise Measurement</a> <li><a href="#org9947f0d">3. Noise Measurement</a>
<ul> <ul>
<li><a href="#org1171cfb">3.1. Test Bench</a></li> <li><a href="#org7dd6ce0">3.1. Test Bench</a></li>
<li><a href="#org2d3c7ed">3.2. Thermal drifts</a></li> <li><a href="#orgd61ad80">3.2. Thermal drifts</a></li>
<li><a href="#org12c8422">3.3. Time Domain signals</a></li> <li><a href="#org8f23c76">3.3. Time Domain signals</a></li>
<li><a href="#orgcfb7422">3.4. Noise Spectral Density</a></li> <li><a href="#orgbd6cefe">3.4. Noise Spectral Density</a></li>
<li><a href="#orgf450d0e">3.5. Noise Model</a></li> <li><a href="#orgc14197f">3.5. Noise Model</a></li>
<li><a href="#org5d6e2aa">3.6. Validity of the noise model</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgbcdb22e">4. Linearity Measurement</a> <li><a href="#orgbc58807">4. Linearity Measurement</a>
<ul> <ul>
<li><a href="#org0508ec2">4.1. Test Bench</a></li> <li><a href="#org38d4317">4.1. Test Bench</a></li>
<li><a href="#org4e41106">4.2. Results</a></li> <li><a href="#org9a6927b">4.2. Results</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -64,7 +63,7 @@
<p>This report is also available as a <a href="./test-bench-vionic.pdf">pdf</a>.</p> <p>This report is also available as a <a href="./test-bench-vionic.pdf">pdf</a>.</p>
<hr> <hr>
<div class="note" id="orgf0dfbf1"> <div class="note" id="org34d0504">
<p> <p>
You can find below the documentation of: You can find below the documentation of:
</p> </p>
@ -89,25 +88,25 @@ In particular, we would like to measure:
This document is structured as follow: This document is structured as follow:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org5825e63">1</a>: the expected performance of the Vionic encoder system are described</li> <li>Section <a href="#orgafe2cb7">1</a>: the expected performance of the Vionic encoder system are described</li>
<li>Section <a href="#org886dc10">2</a>: a simple model of the encoder is developed</li> <li>Section <a href="#org1d1f36e">2</a>: a simple model of the encoder is developed</li>
<li>Section <a href="#orgce8febf">3</a>: the noise of the encoder is measured and a model of the noise is identified</li> <li>Section <a href="#orgf70a154">3</a>: the noise of the encoder is measured and a model of the noise is identified</li>
<li>Section <a href="#org0a6ada3">4</a>: the linearity of the sensor is estimated</li> <li>Section <a href="#org3767bd5">4</a>: the linearity of the sensor is estimated</li>
</ul> </ul>
<div id="outline-container-orgee60877" class="outline-2"> <div id="outline-container-orgacaf822" class="outline-2">
<h2 id="orgee60877"><span class="section-number-2">1</span> Expected Performances</h2> <h2 id="orgacaf822"><span class="section-number-2">1</span> Expected Performances</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="org5825e63"></a> <a id="orgafe2cb7"></a>
</p> </p>
<p> <p>
The Vionic encoder is shown in Figure <a href="#org8649a60">1</a>. The Vionic encoder is shown in Figure <a href="#org300cb52">1</a>.
</p> </p>
<div id="org8649a60" class="figure"> <div id="org300cb52" class="figure">
<p><img src="figs/encoder_vionic.png" alt="encoder_vionic.png" /> <p><img src="figs/encoder_vionic.png" alt="encoder_vionic.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Picture of the Vionic Encoder</p> <p><span class="figure-number">Figure 1: </span>Picture of the Vionic Encoder</p>
@ -135,21 +134,21 @@ Interpolation is within the readhead, with fine resolution versions being furthe
</blockquote> </blockquote>
<p> <p>
The expected interpolation errors (non-linearity) is shown in Figure <a href="#org35c5a3c">2</a>. The expected interpolation errors (non-linearity) is shown in Figure <a href="#org74b94f4">2</a>.
</p> </p>
<div id="org35c5a3c" class="figure"> <div id="org74b94f4" class="figure">
<p><img src="./figs/vionic_expected_noise.png" alt="vionic_expected_noise.png" /> <p><img src="./figs/vionic_expected_noise.png" alt="vionic_expected_noise.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Expected interpolation errors for the Vionic Encoder</p> <p><span class="figure-number">Figure 2: </span>Expected interpolation errors for the Vionic Encoder</p>
</div> </div>
<p> <p>
The characteristics as advertise in the manual as well as our specifications are shown in Table <a href="#org025a9b8">1</a>. The characteristics as advertise in the manual as well as our specifications are shown in Table <a href="#org12ad600">1</a>.
</p> </p>
<table id="org025a9b8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org12ad600" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Characteristics of the Vionic compared with the specifications</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Characteristics of the Vionic compared with the specifications</caption>
<colgroup> <colgroup>
@ -169,7 +168,7 @@ The characteristics as advertise in the manual as well as our specifications are
<tbody> <tbody>
<tr> <tr>
<td class="org-left">Time Delay</td> <td class="org-left">Time Delay</td>
<td class="org-center">&#xa0;</td> <td class="org-center">&lt; 10 ns</td>
<td class="org-center">&lt; 0.5 ms</td> <td class="org-center">&lt; 0.5 ms</td>
</tr> </tr>
@ -201,11 +200,11 @@ The characteristics as advertise in the manual as well as our specifications are
</div> </div>
</div> </div>
<div id="outline-container-org78808d1" class="outline-2"> <div id="outline-container-orgd1b48b9" class="outline-2">
<h2 id="org78808d1"><span class="section-number-2">2</span> Encoder Model</h2> <h2 id="orgd1b48b9"><span class="section-number-2">2</span> Encoder Model</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="org886dc10"></a> <a id="org1d1f36e"></a>
</p> </p>
<p> <p>
@ -218,38 +217,53 @@ It is also characterized by its measurement noise \(n\) that can be described by
</p> </p>
<p> <p>
The model of the encoder is shown in Figure <a href="#orgd01aa78">3</a>. The model of the encoder is shown in Figure <a href="#orge3dfe4a">3</a>.
</p> </p>
<div id="orgd01aa78" class="figure"> <div id="orge3dfe4a" class="figure">
<p><img src="figs/encoder-model-schematic.png" alt="encoder-model-schematic.png" /> <p><img src="figs/encoder-model-schematic.png" alt="encoder-model-schematic.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Model of the Encoder</p> <p><span class="figure-number">Figure 3: </span>Model of the Encoder</p>
</div> </div>
<p> <p>
We can also use a transfer function \(G_n(s)\) to shape a noise \(\tilde{n}\) with unity ASD as shown in Figure <a href="#org35c5a3c">2</a>. We can also use a transfer function \(G_n(s)\) to shape a noise \(\tilde{n}\) with unity ASD as shown in Figure <a href="#org74b94f4">2</a>.
</p> </p>
<div id="org0de813a" class="figure"> <div id="orgb259ef8" class="figure">
<p><img src="figs/encoder-model-schematic-with-asd.png" alt="encoder-model-schematic-with-asd.png" /> <p><img src="figs/encoder-model-schematic-with-asd.png" alt="encoder-model-schematic-with-asd.png" />
</p> </p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org07e5c0c" class="outline-2"> <div id="outline-container-org9947f0d" class="outline-2">
<h2 id="org07e5c0c"><span class="section-number-2">3</span> Noise Measurement</h2> <h2 id="org9947f0d"><span class="section-number-2">3</span> Noise Measurement</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="orgce8febf"></a> <a id="orgf70a154"></a>
</p> </p>
<p>
This part is structured as follow:
</p>
<ul class="org-ul">
<li>Section <a href="#org1bbddb3">3.1</a>: the measurement bench is described</li>
<li>Section <a href="#orge37ddeb">3.2</a>: long measurement is performed to estimate the low frequency drifts in the measurement</li>
<li>Section <a href="#orgbe1c0e1">3.3</a>: high frequency measurements are performed to estimate the high frequency noise</li>
<li>Section <a href="#orgfafa9fd">3.4</a>: the Spectral density of the measurement noise is estimated</li>
<li>Section <a href="#org2284feb">3.5</a>: finally, the measured noise is modeled</li>
</ul>
</div> </div>
<div id="outline-container-org1171cfb" class="outline-3">
<h3 id="org1171cfb"><span class="section-number-3">3.1</span> Test Bench</h3> <div id="outline-container-org7dd6ce0" class="outline-3">
<h3 id="org7dd6ce0"><span class="section-number-3">3.1</span> Test Bench</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p>
<a id="org1bbddb3"></a>
</p>
<p> <p>
To measure the noise \(n\) of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured. To measure the noise \(n\) of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured.
Then, the measured signal \(y_m\) corresponds to the noise \(n\). Then, the measured signal \(y_m\) corresponds to the noise \(n\).
@ -257,62 +271,138 @@ Then, the measured signal \(y_m\) corresponds to the noise \(n\).
</div> </div>
</div> </div>
<div id="outline-container-org2d3c7ed" class="outline-3"> <div id="outline-container-orgd61ad80" class="outline-3">
<h3 id="org2d3c7ed"><span class="section-number-3">3.2</span> Thermal drifts</h3> <h3 id="orgd61ad80"><span class="section-number-3">3.2</span> Thermal drifts</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<ul class="org-ul"> <p>
<li class="off"><code>[&#xa0;]</code> picture of the setup</li> <a id="orge37ddeb"></a>
<li class="off"><code>[&#xa0;]</code> long thermal drifts</li> Measured displacement were recording during approximately 40 hours with a sample frequency of 100Hz.
<li class="off"><code>[&#xa0;]</code> once stabilize, look at the noise</li> A first order low pass filter with a corner frequency of 1Hz
<li class="off"><code>[&#xa0;]</code> compute low frequency ASD (may still be thermal drifts of the mechanics and not noise)</li> </p>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">enc_l = load(<span class="org-string">'mat/noise_meas_40h_100Hz_1.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'x'</span>);
</pre>
</div>
<p>
The measured time domain data are shown in Figure <a href="#org55bfe2a">5</a>.
</p>
<div id="org55bfe2a" class="figure">
<p><img src="figs/vionic_drifts_time.png" alt="vionic_drifts_time.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Measured thermal drifts</p>
</div>
<p>
The measured data seems to experience a constant drift after approximately 20 hour.
Let&rsquo;s estimate this drift.
</p>
<pre class="example">
The mean drift is approximately 60.9 [nm/hour] or 1.0 [nm/min]
</pre>
<p>
Comparison between the data and the linear fit is shown in Figure <a href="#org1085735">6</a>.
</p>
<div id="org1085735" class="figure">
<p><img src="figs/vionic_drifts_linear_fit.png" alt="vionic_drifts_linear_fit.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Measured drift and linear fit</p>
</div>
<p>
Let&rsquo;s now estimate the Power Spectral Density of the measured displacement.
The obtained low frequency ASD is shown in Figure <a href="#orgf2675d7">7</a>.
</p>
<div id="orgf2675d7" class="figure">
<p><img src="figs/vionic_noise_asd_low_freq.png" alt="vionic_noise_asd_low_freq.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Amplitude Spectral density of the measured displacement</p>
</div>
</div> </div>
</div> </div>
<div id="outline-container-org12c8422" class="outline-3"> <div id="outline-container-org8f23c76" class="outline-3">
<h3 id="org12c8422"><span class="section-number-3">3.3</span> Time Domain signals</h3> <h3 id="org8f23c76"><span class="section-number-3">3.3</span> Time Domain signals</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
First we load the data. <a id="orgbe1c0e1"></a>
The raw measured data as well as the low pass filtered data (using a first order low pass filter with a cut-off at 10Hz) are shown in Figure <a href="#org0525912">5</a>.
</p> </p>
<div id="org0525912" class="figure"> <p>
Then, and for all the 7 encoders, we record the measured motion during 100s with a sampling frequency of 20kHz.
</p>
<p>
The raw measured data as well as the low pass filtered data (using a first order low pass filter with a cut-off at 10Hz) are shown in Figure <a href="#orgbd876dc">8</a>.
</p>
<div id="orgbd876dc" class="figure">
<p><img src="figs/vionic_noise_raw_lpf.png" alt="vionic_noise_raw_lpf.png" /> <p><img src="figs/vionic_noise_raw_lpf.png" alt="vionic_noise_raw_lpf.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Time domain measurement (raw data and low pass filtered data with first order 10Hz LPF)</p> <p><span class="figure-number">Figure 8: </span>Time domain measurement (raw data and low pass filtered data with first order 10Hz LPF)</p>
</div> </div>
<p> <p>
The time domain data for all the encoders are compared in Figure <a href="#org5c2c4fa">6</a>. The time domain data for all the encoders are compared in Figure <a href="#org63a82cb">9</a>.
</p> </p>
<div id="org5c2c4fa" class="figure"> <p>
We can see some drifts that are in the order of few nm to 20nm per minute.
As shown in Section <a href="#orge37ddeb">3.2</a>, these drifts should diminish over time down to 1nm/min.
</p>
<div id="org63a82cb" class="figure">
<p><img src="figs/vionic_noise_time.png" alt="vionic_noise_time.png" /> <p><img src="figs/vionic_noise_time.png" alt="vionic_noise_time.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Comparison of the time domain measurement</p> <p><span class="figure-number">Figure 9: </span>Comparison of the time domain measurement</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgcfb7422" class="outline-3"> <div id="outline-container-orgbd6cefe" class="outline-3">
<h3 id="orgcfb7422"><span class="section-number-3">3.4</span> Noise Spectral Density</h3> <h3 id="orgbd6cefe"><span class="section-number-3">3.4</span> Noise Spectral Density</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
The amplitude spectral density is computed and shown in Figure <a href="#orged52478">7</a>. <a id="orgfafa9fd"></a>
</p> </p>
<div id="orged52478" class="figure"> <p>
The amplitude spectral densities for all the encoder are computed and shown in Figure <a href="#org4b13cc6">10</a>.
</p>
<div id="org4b13cc6" class="figure">
<p><img src="figs/vionic_noise_asd.png" alt="vionic_noise_asd.png" /> <p><img src="figs/vionic_noise_asd.png" alt="vionic_noise_asd.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Amplitude Spectral Density of the measured signal</p> <p><span class="figure-number">Figure 10: </span>Amplitude Spectral Density of the measured signal</p>
</div>
<p>
We can combine these measurements with the low frequency noise computed in Section <a href="#orge37ddeb">3.2</a>.
The obtained ASD is shown in Figure <a href="#orgec960f3">11</a>.
</p>
<div id="orgec960f3" class="figure">
<p><img src="figs/vionic_noise_asd_combined.png" alt="vionic_noise_asd_combined.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Combined low frequency and high frequency noise measurements</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgf450d0e" class="outline-3"> <div id="outline-container-orgc14197f" class="outline-3">
<h3 id="orgf450d0e"><span class="section-number-3">3.5</span> Noise Model</h3> <h3 id="orgc14197f"><span class="section-number-3">3.5</span> Noise Model</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-3-5">
<p>
<a id="org2284feb"></a>
</p>
<p> <p>
Let&rsquo;s create a transfer function that approximate the measured noise of the encoder. Let&rsquo;s create a transfer function that approximate the measured noise of the encoder.
</p> </p>
@ -322,23 +412,18 @@ Let&rsquo;s create a transfer function that approximate the measured noise of th
</div> </div>
<p> <p>
The amplitude of the transfer function and the measured ASD are shown in Figure <a href="#orgd40fb21">8</a>. The amplitude of the transfer function and the measured ASD are shown in Figure <a href="#org904aecb">12</a>.
</p> </p>
<div id="orgd40fb21" class="figure"> <div id="org904aecb" class="figure">
<p><img src="figs/vionic_noise_asd_model.png" alt="vionic_noise_asd_model.png" /> <p><img src="figs/vionic_noise_asd_model.png" alt="vionic_noise_asd_model.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Measured ASD of the noise and modeled one</p> <p><span class="figure-number">Figure 12: </span>Measured ASD of the noise and modeled one</p>
</div>
</div>
</div> </div>
<div id="outline-container-org5d6e2aa" class="outline-3">
<h3 id="org5d6e2aa"><span class="section-number-3">3.6</span> Validity of the noise model</h3>
<div class="outline-text-3" id="text-3-6">
<p> <p>
The cumulative amplitude spectrum is now computed and shown in Figure <a href="#orgf87a6b7">9</a>. The cumulative amplitude spectrum is now computed and shown in Figure <a href="#orgff7d2cd">13</a>.
</p> </p>
<p> <p>
@ -346,24 +431,24 @@ We can see that the Root Mean Square value of the measurement noise is \(\approx
</p> </p>
<div id="orgf87a6b7" class="figure"> <div id="orgff7d2cd" class="figure">
<p><img src="figs/vionic_noise_cas_model.png" alt="vionic_noise_cas_model.png" /> <p><img src="figs/vionic_noise_cas_model.png" alt="vionic_noise_cas_model.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Meassured CAS of the noise and modeled one</p> <p><span class="figure-number">Figure 13: </span>Meassured CAS of the noise and modeled one</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgbcdb22e" class="outline-2"> <div id="outline-container-orgbc58807" class="outline-2">
<h2 id="orgbcdb22e"><span class="section-number-2">4</span> Linearity Measurement</h2> <h2 id="orgbc58807"><span class="section-number-2">4</span> Linearity Measurement</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="org0a6ada3"></a> <a id="org3767bd5"></a>
</p> </p>
</div> </div>
<div id="outline-container-org0508ec2" class="outline-3"> <div id="outline-container-org38d4317" class="outline-3">
<h3 id="org0508ec2"><span class="section-number-3">4.1</span> Test Bench</h3> <h3 id="org38d4317"><span class="section-number-3">4.1</span> Test Bench</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
In order to measure the linearity, we have to compare the measured displacement with a reference sensor with a known linearity. In order to measure the linearity, we have to compare the measured displacement with a reference sensor with a known linearity.
@ -372,7 +457,7 @@ An actuator should also be there so impose a displacement.
</p> </p>
<p> <p>
One idea is to use the test-bench shown in Figure <a href="#orge0a809b">10</a>. One idea is to use the test-bench shown in Figure <a href="#org5a7f983">14</a>.
</p> </p>
<p> <p>
@ -385,22 +470,22 @@ As the interferometer has a very large bandwidth, we should be able to estimate
</p> </p>
<div id="orge0a809b" class="figure"> <div id="org5a7f983" class="figure">
<p><img src="figs/test_bench_encoder_calibration.png" alt="test_bench_encoder_calibration.png" /> <p><img src="figs/test_bench_encoder_calibration.png" alt="test_bench_encoder_calibration.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Schematic of the test bench</p> <p><span class="figure-number">Figure 14: </span>Schematic of the test bench</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org4e41106" class="outline-3"> <div id="outline-container-org9a6927b" class="outline-3">
<h3 id="org4e41106"><span class="section-number-3">4.2</span> Results</h3> <h3 id="org9a6927b"><span class="section-number-3">4.2</span> Results</h3>
</div> </div>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-02-10 mer. 15:14</p> <p class="date">Created: 2021-02-11 jeu. 15:21</p>
</div> </div>
</body> </body>
</html> </html>

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test-bench-vionic.org
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@ -169,7 +169,18 @@ We can also use a transfer function $G_n(s)$ to shape a noise $\tilde{n}$ with u
* Noise Measurement * Noise Measurement
<<sec:noise_measurement>> <<sec:noise_measurement>>
** Introduction :ignore:
This part is structured as follow:
- Section [[sec:noise_bench]]: the measurement bench is described
- Section [[sec:thermal_drifts]]: long measurement is performed to estimate the low frequency drifts in the measurement
- Section [[sec:vionic_noise_time]]: high frequency measurements are performed to estimate the high frequency noise
- Section [[sec:noise_asd]]: the Spectral density of the measurement noise is estimated
- Section [[sec:vionic_noise_model]]: finally, the measured noise is modeled
** Test Bench ** Test Bench
<<sec:noise_bench>>
To measure the noise $n$ of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured. To measure the noise $n$ of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured.
Then, the measured signal $y_m$ corresponds to the noise $n$. Then, the measured signal $y_m$ corresponds to the noise $n$.
@ -192,16 +203,21 @@ addpath('./matlab/');
addpath('./mat/'); addpath('./mat/');
#+end_src #+end_src
** TODO Thermal drifts ** Thermal drifts
<<sec:thermal_drifts>>
- [ ] picture of the setup Measured displacement were recording during approximately 40 hours with a sample frequency of 100Hz.
- [ ] long thermal drifts A first order low pass filter with a corner frequency of 1Hz
- [ ] Identification of the drifts (exponential fit)
- [ ] once stabilize, look at the noise
- [ ] compute low frequency ASD (may still be thermal drifts of the mechanics and not noise)
#+begin_src matlab #+begin_src matlab
enc_l = load('mat/noise_meas_40h_200Hz_1.mat', 't', 'x'); enc_l = load('mat/noise_meas_40h_100Hz_1.mat', 't', 'x');
#+end_src
The measured time domain data are shown in Figure [[fig:vionic_drifts_time]].
#+begin_src matlab :exports none
enc_l.x = enc_l.x(enc_l.t > 5); % Remove first 5 seconds
enc_l.t = enc_l.t(enc_l.t > 5); % Remove first 5 seconds
enc_l.t = enc_l.t - enc_l.t(1); % Start at 0
enc_l.x = enc_l.x - mean(enc_l.x(enc_l.t < 1)); % Start at zero displacement enc_l.x = enc_l.x - mean(enc_l.x(enc_l.t < 1)); % Start at zero displacement
#+end_src #+end_src
@ -212,67 +228,94 @@ plot(enc_l.t/3600, 1e9*enc_l.x, '-');
hold off; hold off;
xlabel('Time [h]'); xlabel('Time [h]');
ylabel('Displacement [nm]'); ylabel('Displacement [nm]');
xlim([0, 40]);
#+end_src #+end_src
Exponential fit #+begin_src matlab :tangle no :exports results :results file replace
#+begin_src matlab exportFig('figs/vionic_drifts_time.pdf', 'width', 'wide', 'height', 'normal');
f = @(b,x) b(1)*(1 - exp(-x/b(2))); #+end_src
y_cur = enc_l.x; #+name: fig:vionic_drifts_time
t_cur = end_l.t; #+caption: Measured thermal drifts
#+RESULTS:
nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function [[file:figs/vionic_drifts_time.png]]
B0 = [400e-9, 2*60*60]; % Choose Appropriate Initial Estimates
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’ The measured data seems to experience a constant drift after approximately 20 hour.
Let's estimate this drift.
#+begin_src matlab :exports none
t0 = 20*3600; % Start time [s]
x_stab = enc_l.x(enc_l.t > t0);
x_stab = x_stab - x_stab(1);
t_stab = enc_l.t(enc_l.t > t0);
t_stab = t_stab - t_stab(1);
#+end_src #+end_src
The corresponding time constant is (in [h]):
#+begin_src matlab :results value replace :exports results #+begin_src matlab :results value replace :exports results
B(2)/60/60 sprintf('The mean drift is approximately %.1f [nm/hour] or %.1f [nm/min]', 3600*1e9*(t_stab\x_stab), 60*1e9*(t_stab\x_stab))
#+end_src #+end_src
Comparison of the data and exponential fit #+RESULTS:
: The mean drift is approximately 60.9 [nm/hour] or 1.0 [nm/min]
Comparison between the data and the linear fit is shown in Figure [[fig:vionic_drifts_linear_fit]].
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
hold on; hold on;
plot(enc_l.t/60/60, 1e9*enc_l.x); plot(t_stab/3600, 1e9*x_stab, '-');
plot(enc_l.t/60/60, 1e9*f(B, enc_l.t)); plot(t_stab/3600, 1e9*t_stab*(t_stab\x_stab), 'k--');
hold off; hold off;
xlim([0, 17.5]) xlabel('Time [h]');
xlabel('Time [h]'); ylabel('Displacement [nm]'); ylabel('Displacement [nm]');
xlim([0, 20]);
#+end_src #+end_src
Let's get only the data once it is stabilized #+begin_src matlab :tangle no :exports results :results file replace
#+begin_src matlab exportFig('figs/vionic_drifts_linear_fit.pdf', 'width', 'wide', 'height', 'normal');
x_stab = enc_l.x(enc_l.t > 20*3600);
x_stab = x_stab - mean(x_stab);
t_stab = enc_l.t(enc_l.t > 20*3600);
x_stab = x_stab - x_stab(1);
#+end_src #+end_src
#+name: fig:vionic_drifts_linear_fit
#+caption: Measured drift and linear fit
#+RESULTS:
[[file:figs/vionic_drifts_linear_fit.png]]
Let's now estimate the Power Spectral Density of the measured displacement.
The obtained low frequency ASD is shown in Figure [[fig:vionic_noise_asd_low_freq]].
#+begin_src matlab :exports none #+begin_src matlab :exports none
% Compute sampling Frequency % Compute sampling Frequency
Ts = (enc{1}.t(end) - enc{1}.t(1))/(length(enc{1}.t)-1); Ts = (enc_l.t(end) - enc_l.t(1))/(length(enc_l.t)-1);
Fs = 1/Ts; Fs = 1/Ts;
% Hannning Windows % Hannning Windows
win = hanning(ceil(60/Ts)); win = hanning(ceil(60*10/Ts));
[pxx, f] = pwelch(x_stab, win, [], [], Fs); [pxx_l, f_l] = pwelch(x_stab, win, [], [], Fs);
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
hold on; hold on;
plot(f, sqrt(pxx)) plot(f_l, sqrt(pxx_l))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
% xlim([10, Fs/2]); xlim([1e-2, 1e0]);
% ylim([1e-11, 1e-10]); ylim([1e-11, 1e-8]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/vionic_noise_asd_low_freq.pdf', 'width', 'side', 'height', 'normal');
#+end_src
#+name: fig:vionic_noise_asd_low_freq
#+caption: Amplitude Spectral density of the measured displacement
#+RESULTS:
[[file:figs/vionic_noise_asd_low_freq.png]]
** Time Domain signals ** Time Domain signals
First we load the data. <<sec:vionic_noise_time>>
Then, and for all the 7 encoders, we record the measured motion during 100s with a sampling frequency of 20kHz.
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Load all the measurements %% Load all the measurements
enc = {}; enc = {};
@ -310,6 +353,9 @@ exportFig('figs/vionic_noise_raw_lpf.pdf', 'width', 'wide', 'height', 'normal');
[[file:figs/vionic_noise_raw_lpf.png]] [[file:figs/vionic_noise_raw_lpf.png]]
The time domain data for all the encoders are compared in Figure [[fig:vionic_noise_time]]. The time domain data for all the encoders are compared in Figure [[fig:vionic_noise_time]].
We can see some drifts that are in the order of few nm to 20nm per minute.
As shown in Section [[sec:thermal_drifts]], these drifts should diminish over time down to 1nm/min.
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
hold on; hold on;
@ -333,7 +379,9 @@ exportFig('figs/vionic_noise_time.pdf', 'width', 'wide', 'height', 'normal');
[[file:figs/vionic_noise_time.png]] [[file:figs/vionic_noise_time.png]]
** Noise Spectral Density ** Noise Spectral Density
The amplitude spectral density is computed and shown in Figure [[fig:vionic_noise_asd]]. <<sec:noise_asd>>
The amplitude spectral densities for all the encoder are computed and shown in Figure [[fig:vionic_noise_asd]].
#+begin_src matlab :exports none #+begin_src matlab :exports none
% Compute sampling Frequency % Compute sampling Frequency
Ts = (enc{1}.t(end) - enc{1}.t(1))/(length(enc{1}.t)-1); Ts = (enc{1}.t(end) - enc{1}.t(1))/(length(enc{1}.t)-1);
@ -361,7 +409,7 @@ end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
xlim([10, Fs/2]); xlim([10, Fs/2]);
ylim([1e-11, 1e-10]); ylim([1e-11, 1e-9]);
legend('location', 'northeast'); legend('location', 'northeast');
#+end_src #+end_src
@ -374,7 +422,33 @@ exportFig('figs/vionic_noise_asd.pdf', 'width', 'wide', 'height', 'normal');
#+RESULTS: #+RESULTS:
[[file:figs/vionic_noise_asd.png]] [[file:figs/vionic_noise_asd.png]]
We can combine these measurements with the low frequency noise computed in Section [[sec:thermal_drifts]].
The obtained ASD is shown in Figure [[fig:vionic_noise_asd_combined]].
#+begin_src matlab :exports none
[pxx_h, f_h] = pwelch(enc{2}.x, hanning(ceil(10/Ts)), [], [], Fs);
figure;
hold on;
plot(f_h(f_h>0.6), sqrt(pxx_h(f_h>0.6)), 'k-');
plot(f_l(f_l<1), sqrt(pxx_l(f_l<1)), 'k-')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
xlim([1e-2, Fs/2]);
ylim([1e-12, 1e-8]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/vionic_noise_asd_combined.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:vionic_noise_asd_combined
#+caption: Combined low frequency and high frequency noise measurements
#+RESULTS:
[[file:figs/vionic_noise_asd_combined.png]]
** Noise Model ** Noise Model
<<sec:vionic_noise_model>>
Let's create a transfer function that approximate the measured noise of the encoder. Let's create a transfer function that approximate the measured noise of the encoder.
#+begin_src matlab #+begin_src matlab
Gn_e = 1.8e-11/(1 + s/2/pi/1e4); Gn_e = 1.8e-11/(1 + s/2/pi/1e4);
@ -385,7 +459,7 @@ The amplitude of the transfer function and the measured ASD are shown in Figure
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
hold on; hold on;
plot(f, sqrt(p1), 'color', [0, 0, 0, 0.5], 'DisplayName', '$\Gamma_n(\omega)$'); plot(f, sqrt(enc{1}.pxx), 'color', [0, 0, 0, 0.5], 'DisplayName', '$\Gamma_n(\omega)$');
for i=2:7 for i=2:7
plot(f, sqrt(enc{i}.pxx), 'color', [0, 0, 0, 0.5], ... plot(f, sqrt(enc{i}.pxx), 'color', [0, 0, 0, 0.5], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
@ -408,7 +482,6 @@ exportFig('figs/vionic_noise_asd_model.pdf', 'width', 'wide', 'height', 'normal'
#+RESULTS: #+RESULTS:
[[file:figs/vionic_noise_asd_model.png]] [[file:figs/vionic_noise_asd_model.png]]
** Validity of the noise model
The cumulative amplitude spectrum is now computed and shown in Figure [[fig:vionic_noise_cas_model]]. The cumulative amplitude spectrum is now computed and shown in Figure [[fig:vionic_noise_cas_model]].
We can see that the Root Mean Square value of the measurement noise is $\approx 1.6 \, nm$ as advertise in the datasheet. We can see that the Root Mean Square value of the measurement noise is $\approx 1.6 \, nm$ as advertise in the datasheet.

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