Analyse long measurement
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@ -3,7 +3,7 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2021-02-10 mer. 15:14 -->
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<!-- 2021-02-11 jeu. 15:21 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Encoder Renishaw Vionic - Test Bench</title>
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<meta name="generator" content="Org mode" />
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@ -39,22 +39,21 @@
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgee60877">1. Expected Performances</a></li>
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<li><a href="#org78808d1">2. Encoder Model</a></li>
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<li><a href="#org07e5c0c">3. Noise Measurement</a>
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<li><a href="#orgacaf822">1. Expected Performances</a></li>
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<li><a href="#orgd1b48b9">2. Encoder Model</a></li>
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<li><a href="#org9947f0d">3. Noise Measurement</a>
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<ul>
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<li><a href="#org1171cfb">3.1. Test Bench</a></li>
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<li><a href="#org2d3c7ed">3.2. Thermal drifts</a></li>
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<li><a href="#org12c8422">3.3. Time Domain signals</a></li>
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<li><a href="#orgcfb7422">3.4. Noise Spectral Density</a></li>
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<li><a href="#orgf450d0e">3.5. Noise Model</a></li>
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<li><a href="#org5d6e2aa">3.6. Validity of the noise model</a></li>
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<li><a href="#org7dd6ce0">3.1. Test Bench</a></li>
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<li><a href="#orgd61ad80">3.2. Thermal drifts</a></li>
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<li><a href="#org8f23c76">3.3. Time Domain signals</a></li>
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<li><a href="#orgbd6cefe">3.4. Noise Spectral Density</a></li>
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<li><a href="#orgc14197f">3.5. Noise Model</a></li>
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</ul>
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</li>
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<li><a href="#orgbcdb22e">4. Linearity Measurement</a>
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<li><a href="#orgbc58807">4. Linearity Measurement</a>
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<ul>
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<li><a href="#org0508ec2">4.1. Test Bench</a></li>
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<li><a href="#org4e41106">4.2. Results</a></li>
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<li><a href="#org38d4317">4.1. Test Bench</a></li>
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<li><a href="#org9a6927b">4.2. Results</a></li>
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</ul>
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</li>
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</ul>
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@ -64,7 +63,7 @@
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<p>This report is also available as a <a href="./test-bench-vionic.pdf">pdf</a>.</p>
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<hr>
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<div class="note" id="orgf0dfbf1">
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<div class="note" id="org34d0504">
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<p>
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You can find below the documentation of:
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</p>
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@ -89,25 +88,25 @@ In particular, we would like to measure:
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This document is structured as follow:
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</p>
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<ul class="org-ul">
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<li>Section <a href="#org5825e63">1</a>: the expected performance of the Vionic encoder system are described</li>
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<li>Section <a href="#org886dc10">2</a>: a simple model of the encoder is developed</li>
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<li>Section <a href="#orgce8febf">3</a>: the noise of the encoder is measured and a model of the noise is identified</li>
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<li>Section <a href="#org0a6ada3">4</a>: the linearity of the sensor is estimated</li>
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<li>Section <a href="#orgafe2cb7">1</a>: the expected performance of the Vionic encoder system are described</li>
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<li>Section <a href="#org1d1f36e">2</a>: a simple model of the encoder is developed</li>
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<li>Section <a href="#orgf70a154">3</a>: the noise of the encoder is measured and a model of the noise is identified</li>
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<li>Section <a href="#org3767bd5">4</a>: the linearity of the sensor is estimated</li>
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</ul>
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<div id="outline-container-orgee60877" class="outline-2">
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<h2 id="orgee60877"><span class="section-number-2">1</span> Expected Performances</h2>
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<div id="outline-container-orgacaf822" class="outline-2">
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<h2 id="orgacaf822"><span class="section-number-2">1</span> Expected Performances</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org5825e63"></a>
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<a id="orgafe2cb7"></a>
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</p>
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<p>
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The Vionic encoder is shown in Figure <a href="#org8649a60">1</a>.
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The Vionic encoder is shown in Figure <a href="#org300cb52">1</a>.
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</p>
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<div id="org8649a60" class="figure">
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<div id="org300cb52" class="figure">
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<p><img src="figs/encoder_vionic.png" alt="encoder_vionic.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Picture of the Vionic Encoder</p>
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@ -135,21 +134,21 @@ Interpolation is within the readhead, with fine resolution versions being furthe
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</blockquote>
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<p>
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The expected interpolation errors (non-linearity) is shown in Figure <a href="#org35c5a3c">2</a>.
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The expected interpolation errors (non-linearity) is shown in Figure <a href="#org74b94f4">2</a>.
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</p>
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<div id="org35c5a3c" class="figure">
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<div id="org74b94f4" class="figure">
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<p><img src="./figs/vionic_expected_noise.png" alt="vionic_expected_noise.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Expected interpolation errors for the Vionic Encoder</p>
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</div>
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<p>
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The characteristics as advertise in the manual as well as our specifications are shown in Table <a href="#org025a9b8">1</a>.
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The characteristics as advertise in the manual as well as our specifications are shown in Table <a href="#org12ad600">1</a>.
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</p>
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<table id="org025a9b8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org12ad600" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Characteristics of the Vionic compared with the specifications</caption>
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<colgroup>
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@ -169,7 +168,7 @@ The characteristics as advertise in the manual as well as our specifications are
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<tbody>
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<tr>
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<td class="org-left">Time Delay</td>
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<td class="org-center"> </td>
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<td class="org-center">< 10 ns</td>
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<td class="org-center">< 0.5 ms</td>
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</tr>
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@ -201,11 +200,11 @@ The characteristics as advertise in the manual as well as our specifications are
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</div>
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</div>
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<div id="outline-container-org78808d1" class="outline-2">
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<h2 id="org78808d1"><span class="section-number-2">2</span> Encoder Model</h2>
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<div id="outline-container-orgd1b48b9" class="outline-2">
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<h2 id="orgd1b48b9"><span class="section-number-2">2</span> Encoder Model</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org886dc10"></a>
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<a id="org1d1f36e"></a>
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</p>
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<p>
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@ -218,38 +217,53 @@ It is also characterized by its measurement noise \(n\) that can be described by
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</p>
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<p>
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The model of the encoder is shown in Figure <a href="#orgd01aa78">3</a>.
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The model of the encoder is shown in Figure <a href="#orge3dfe4a">3</a>.
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</p>
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<div id="orgd01aa78" class="figure">
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<div id="orge3dfe4a" class="figure">
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<p><img src="figs/encoder-model-schematic.png" alt="encoder-model-schematic.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Model of the Encoder</p>
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</div>
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<p>
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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>.
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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>.
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</p>
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<div id="org0de813a" class="figure">
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<div id="orgb259ef8" class="figure">
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||||
<p><img src="figs/encoder-model-schematic-with-asd.png" alt="encoder-model-schematic-with-asd.png" />
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||||
</p>
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||||
</div>
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||||
</div>
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||||
</div>
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<div id="outline-container-org07e5c0c" class="outline-2">
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||||
<h2 id="org07e5c0c"><span class="section-number-2">3</span> Noise Measurement</h2>
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<div id="outline-container-org9947f0d" class="outline-2">
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||||
<h2 id="org9947f0d"><span class="section-number-2">3</span> Noise Measurement</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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<a id="orgce8febf"></a>
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||||
<a id="orgf70a154"></a>
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||||
</p>
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||||
<p>
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||||
This part is structured as follow:
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</p>
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||||
<ul class="org-ul">
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||||
<li>Section <a href="#org1bbddb3">3.1</a>: the measurement bench is described</li>
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<li>Section <a href="#orge37ddeb">3.2</a>: long measurement is performed to estimate the low frequency drifts in the measurement</li>
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<li>Section <a href="#orgbe1c0e1">3.3</a>: high frequency measurements are performed to estimate the high frequency noise</li>
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<li>Section <a href="#orgfafa9fd">3.4</a>: the Spectral density of the measurement noise is estimated</li>
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<li>Section <a href="#org2284feb">3.5</a>: finally, the measured noise is modeled</li>
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</ul>
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</div>
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<div id="outline-container-org1171cfb" class="outline-3">
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<h3 id="org1171cfb"><span class="section-number-3">3.1</span> Test Bench</h3>
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<div id="outline-container-org7dd6ce0" class="outline-3">
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<h3 id="org7dd6ce0"><span class="section-number-3">3.1</span> Test Bench</h3>
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<div class="outline-text-3" id="text-3-1">
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<p>
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<a id="org1bbddb3"></a>
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</p>
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<p>
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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.
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Then, the measured signal \(y_m\) corresponds to the noise \(n\).
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@ -257,62 +271,138 @@ Then, the measured signal \(y_m\) corresponds to the noise \(n\).
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</div>
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</div>
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<div id="outline-container-org2d3c7ed" class="outline-3">
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<h3 id="org2d3c7ed"><span class="section-number-3">3.2</span> Thermal drifts</h3>
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<div id="outline-container-orgd61ad80" class="outline-3">
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<h3 id="orgd61ad80"><span class="section-number-3">3.2</span> Thermal drifts</h3>
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<div class="outline-text-3" id="text-3-2">
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<ul class="org-ul">
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<li class="off"><code>[ ]</code> picture of the setup</li>
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<li class="off"><code>[ ]</code> long thermal drifts</li>
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<li class="off"><code>[ ]</code> once stabilize, look at the noise</li>
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<li class="off"><code>[ ]</code> compute low frequency ASD (may still be thermal drifts of the mechanics and not noise)</li>
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</ul>
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<p>
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||||
<a id="orge37ddeb"></a>
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Measured displacement were recording during approximately 40 hours with a sample frequency of 100Hz.
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A first order low pass filter with a corner frequency of 1Hz
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</p>
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<div class="org-src-container">
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<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>);
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</pre>
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</div>
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<p>
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The measured time domain data are shown in Figure <a href="#org55bfe2a">5</a>.
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</p>
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<div id="org55bfe2a" class="figure">
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<p><img src="figs/vionic_drifts_time.png" alt="vionic_drifts_time.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Measured thermal drifts</p>
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</div>
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<p>
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The measured data seems to experience a constant drift after approximately 20 hour.
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Let’s estimate this drift.
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</p>
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<pre class="example">
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The mean drift is approximately 60.9 [nm/hour] or 1.0 [nm/min]
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</pre>
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<p>
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Comparison between the data and the linear fit is shown in Figure <a href="#org1085735">6</a>.
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</p>
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<div id="org1085735" class="figure">
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<p><img src="figs/vionic_drifts_linear_fit.png" alt="vionic_drifts_linear_fit.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Measured drift and linear fit</p>
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</div>
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<p>
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Let’s now estimate the Power Spectral Density of the measured displacement.
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The obtained low frequency ASD is shown in Figure <a href="#orgf2675d7">7</a>.
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</p>
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<div id="orgf2675d7" class="figure">
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<p><img src="figs/vionic_noise_asd_low_freq.png" alt="vionic_noise_asd_low_freq.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Amplitude Spectral density of the measured displacement</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org12c8422" class="outline-3">
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||||
<h3 id="org12c8422"><span class="section-number-3">3.3</span> Time Domain signals</h3>
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<div id="outline-container-org8f23c76" class="outline-3">
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||||
<h3 id="org8f23c76"><span class="section-number-3">3.3</span> Time Domain signals</h3>
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<div class="outline-text-3" id="text-3-3">
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<p>
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First we load the data.
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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>.
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<a id="orgbe1c0e1"></a>
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</p>
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<div id="org0525912" class="figure">
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<p>
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Then, and for all the 7 encoders, we record the measured motion during 100s with a sampling frequency of 20kHz.
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</p>
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<p>
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||||
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>.
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||||
</p>
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||||
|
||||
<div id="orgbd876dc" class="figure">
|
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<p><img src="figs/vionic_noise_raw_lpf.png" alt="vionic_noise_raw_lpf.png" />
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||||
</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>
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||||
<p><span class="figure-number">Figure 8: </span>Time domain measurement (raw data and low pass filtered data with first order 10Hz LPF)</p>
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</div>
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<p>
|
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The time domain data for all the encoders are compared in Figure <a href="#org5c2c4fa">6</a>.
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The time domain data for all the encoders are compared in Figure <a href="#org63a82cb">9</a>.
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</p>
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<div id="org5c2c4fa" class="figure">
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<p>
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We can see some drifts that are in the order of few nm to 20nm per minute.
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As shown in Section <a href="#orge37ddeb">3.2</a>, these drifts should diminish over time down to 1nm/min.
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</p>
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<div id="org63a82cb" class="figure">
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<p><img src="figs/vionic_noise_time.png" alt="vionic_noise_time.png" />
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||||
</p>
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||||
<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>
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||||
</div>
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||||
</div>
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||||
</div>
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||||
|
||||
<div id="outline-container-orgcfb7422" class="outline-3">
|
||||
<h3 id="orgcfb7422"><span class="section-number-3">3.4</span> Noise Spectral Density</h3>
|
||||
<div id="outline-container-orgbd6cefe" class="outline-3">
|
||||
<h3 id="orgbd6cefe"><span class="section-number-3">3.4</span> Noise Spectral Density</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
The amplitude spectral density is computed and shown in Figure <a href="#orged52478">7</a>.
|
||||
<a id="orgfafa9fd"></a>
|
||||
</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>
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||||
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||||
<div id="org4b13cc6" class="figure">
|
||||
<p><img src="figs/vionic_noise_asd.png" alt="vionic_noise_asd.png" />
|
||||
</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 id="outline-container-orgf450d0e" class="outline-3">
|
||||
<h3 id="orgf450d0e"><span class="section-number-3">3.5</span> Noise Model</h3>
|
||||
<div id="outline-container-orgc14197f" class="outline-3">
|
||||
<h3 id="orgc14197f"><span class="section-number-3">3.5</span> Noise Model</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
<a id="org2284feb"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Let’s create a transfer function that approximate the measured noise of the encoder.
|
||||
</p>
|
||||
@ -322,23 +412,18 @@ Let’s create a transfer function that approximate the measured noise of th
|
||||
</div>
|
||||
|
||||
<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>
|
||||
|
||||
|
||||
<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>
|
||||
<p><span class="figure-number">Figure 8: </span>Measured ASD of the noise and modeled one</p>
|
||||
</div>
|
||||
</div>
|
||||
<p><span class="figure-number">Figure 12: </span>Measured ASD of the noise and modeled one</p>
|
||||
</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>
|
||||
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>
|
||||
@ -346,24 +431,24 @@ We can see that the Root Mean Square value of the measurement noise is \(\approx
|
||||
</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>
|
||||
<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 id="outline-container-orgbcdb22e" class="outline-2">
|
||||
<h2 id="orgbcdb22e"><span class="section-number-2">4</span> Linearity Measurement</h2>
|
||||
<div id="outline-container-orgbc58807" class="outline-2">
|
||||
<h2 id="orgbc58807"><span class="section-number-2">4</span> Linearity Measurement</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org0a6ada3"></a>
|
||||
<a id="org3767bd5"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org0508ec2" class="outline-3">
|
||||
<h3 id="org0508ec2"><span class="section-number-3">4.1</span> Test Bench</h3>
|
||||
<div id="outline-container-org38d4317" class="outline-3">
|
||||
<h3 id="org38d4317"><span class="section-number-3">4.1</span> Test Bench</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
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>
|
||||
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>
|
||||
@ -385,22 +470,22 @@ As the interferometer has a very large bandwidth, we should be able to estimate
|
||||
</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>
|
||||
<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 id="outline-container-org4e41106" class="outline-3">
|
||||
<h3 id="org4e41106"><span class="section-number-3">4.2</span> Results</h3>
|
||||
<div id="outline-container-org9a6927b" class="outline-3">
|
||||
<h3 id="org9a6927b"><span class="section-number-3">4.2</span> Results</h3>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<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>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -169,7 +169,18 @@ We can also use a transfer function $G_n(s)$ to shape a noise $\tilde{n}$ with u
|
||||
|
||||
* 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
|
||||
<<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.
|
||||
Then, the measured signal $y_m$ corresponds to the noise $n$.
|
||||
@ -192,16 +203,21 @@ addpath('./matlab/');
|
||||
addpath('./mat/');
|
||||
#+end_src
|
||||
|
||||
** TODO Thermal drifts
|
||||
|
||||
- [ ] picture of the setup
|
||||
- [ ] long thermal drifts
|
||||
- [ ] 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)
|
||||
** Thermal drifts
|
||||
<<sec:thermal_drifts>>
|
||||
Measured displacement were recording during approximately 40 hours with a sample frequency of 100Hz.
|
||||
A first order low pass filter with a corner frequency of 1Hz
|
||||
|
||||
#+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
|
||||
#+end_src
|
||||
|
||||
@ -212,67 +228,94 @@ plot(enc_l.t/3600, 1e9*enc_l.x, '-');
|
||||
hold off;
|
||||
xlabel('Time [h]');
|
||||
ylabel('Displacement [nm]');
|
||||
xlim([0, 40]);
|
||||
#+end_src
|
||||
|
||||
Exponential fit
|
||||
#+begin_src matlab
|
||||
f = @(b,x) b(1)*(1 - exp(-x/b(2)));
|
||||
|
||||
y_cur = enc_l.x;
|
||||
t_cur = end_l.t;
|
||||
|
||||
nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
|
||||
B0 = [400e-9, 2*60*60]; % Choose Appropriate Initial Estimates
|
||||
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/vionic_drifts_time.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:vionic_drifts_time
|
||||
#+caption: Measured thermal drifts
|
||||
#+RESULTS:
|
||||
[[file:figs/vionic_drifts_time.png]]
|
||||
|
||||
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
|
||||
|
||||
The corresponding time constant is (in [h]):
|
||||
#+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
|
||||
|
||||
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
|
||||
figure;
|
||||
hold on;
|
||||
plot(enc_l.t/60/60, 1e9*enc_l.x);
|
||||
plot(enc_l.t/60/60, 1e9*f(B, enc_l.t));
|
||||
plot(t_stab/3600, 1e9*x_stab, '-');
|
||||
plot(t_stab/3600, 1e9*t_stab*(t_stab\x_stab), 'k--');
|
||||
hold off;
|
||||
xlim([0, 17.5])
|
||||
xlabel('Time [h]'); ylabel('Displacement [nm]');
|
||||
xlabel('Time [h]');
|
||||
ylabel('Displacement [nm]');
|
||||
xlim([0, 20]);
|
||||
#+end_src
|
||||
|
||||
Let's get only the data once it is stabilized
|
||||
#+begin_src matlab
|
||||
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);
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/vionic_drifts_linear_fit.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+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
|
||||
% 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;
|
||||
|
||||
% 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
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, sqrt(pxx))
|
||||
plot(f_l, sqrt(pxx_l))
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
|
||||
% xlim([10, Fs/2]);
|
||||
% ylim([1e-11, 1e-10]);
|
||||
xlim([1e-2, 1e0]);
|
||||
ylim([1e-11, 1e-8]);
|
||||
#+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
|
||||
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
|
||||
%% Load all the measurements
|
||||
enc = {};
|
||||
@ -310,6 +353,9 @@ exportFig('figs/vionic_noise_raw_lpf.pdf', 'width', 'wide', 'height', 'normal');
|
||||
[[file:figs/vionic_noise_raw_lpf.png]]
|
||||
|
||||
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
|
||||
figure;
|
||||
hold on;
|
||||
@ -333,7 +379,9 @@ exportFig('figs/vionic_noise_time.pdf', 'width', 'wide', 'height', 'normal');
|
||||
[[file:figs/vionic_noise_time.png]]
|
||||
|
||||
** 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
|
||||
% Compute sampling Frequency
|
||||
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');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
|
||||
xlim([10, Fs/2]);
|
||||
ylim([1e-11, 1e-10]);
|
||||
ylim([1e-11, 1e-9]);
|
||||
legend('location', 'northeast');
|
||||
#+end_src
|
||||
|
||||
@ -374,7 +422,33 @@ exportFig('figs/vionic_noise_asd.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+RESULTS:
|
||||
[[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
|
||||
<<sec:vionic_noise_model>>
|
||||
|
||||
Let's create a transfer function that approximate the measured noise of the encoder.
|
||||
#+begin_src matlab
|
||||
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
|
||||
figure;
|
||||
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
|
||||
plot(f, sqrt(enc{i}.pxx), 'color', [0, 0, 0, 0.5], ...
|
||||
'HandleVisibility', 'off');
|
||||
@ -408,7 +482,6 @@ exportFig('figs/vionic_noise_asd_model.pdf', 'width', 'wide', 'height', 'normal'
|
||||
#+RESULTS:
|
||||
[[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]].
|
||||
|
||||
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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