522 lines
21 KiB
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522 lines
21 KiB
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<!-- 2021-05-12 mer. 13:32 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Attocube - Test Bench</title>
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<meta name="author" content="Dehaeze Thomas" />
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Attocube - Test Bench</h1>
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<div id="table-of-contents">
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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="#orgad0ac54">1. Estimation of the Spectral Density of the Attocube Noise</a>
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<ul>
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<li><a href="#org757fb16">1.1. Long and Slow measurement</a></li>
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<li><a href="#orgb6208e7">1.2. Short and Fast measurement</a></li>
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<li><a href="#org3bd5ef2">1.3. Obtained Amplitude Spectral Density of the measured displacement</a></li>
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</ul>
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</li>
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<li><a href="#org4a27f63">2. Effect of the “bubble sheet” and “Aluminium tube”</a></li>
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<li><a href="#org1df7d3a">3. Measurement of the Attocube’s non-linearity</a>
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<ul>
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<li><a href="#orgcff7a2a">3.1. Load Data</a></li>
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<li><a href="#orgf79eda1">3.2. Time Domain Results</a></li>
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<li><a href="#org46f4beb">3.3. Difference between Encoder and Interferometer as a function of time</a></li>
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<li><a href="#org044dbe7">3.4. Difference between Encoder and Interferometer as a function of position</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<hr>
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<p>This report is also available as a <a href="./test-bench-attocube.pdf">pdf</a>.</p>
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<hr>
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<p>
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In this document, few caracteristics of the Attocube Displacement Measuring Interferometer IDS3010 (<a href="https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer">link</a>) are studied:
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</p>
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<ul class="org-ul">
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<li>Section <a href="#org8ac57f2">1</a>: the ASD noise of the measured displacement is estimated</li>
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<li>Section <a href="#org5a54545">2</a>: the effect of two air protections on the stability of the measurement is studied</li>
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<li>Section <a href="#org477e3cc">3</a>: the cyclic non-linearity of the attocube is estimated using a encoder</li>
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</ul>
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<div id="outline-container-orgad0ac54" class="outline-2">
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<h2 id="orgad0ac54"><span class="section-number-2">1</span> Estimation of the Spectral Density of the Attocube Noise</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org8ac57f2"></a>
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</p>
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<div id="org68c205b" class="figure">
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<p><img src="figs/test-bench-schematic.png" alt="test-bench-schematic.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Test Bench Schematic</p>
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</div>
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<div id="org4669be2" class="figure">
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<p><img src="figs/IMG-7865.JPG" alt="IMG-7865.JPG" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Picture of the test bench. The Attocube and mirror are covered by a “bubble sheet”</p>
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</div>
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</div>
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<div id="outline-container-org757fb16" class="outline-3">
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<h3 id="org757fb16"><span class="section-number-3">1.1</span> Long and Slow measurement</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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The first measurement was made during ~17 hours with a sampling time of \(T_s = 0.1\,s\).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> load(<span class="org-string">'long_test_plastic.mat'</span>, <span class="org-string">'x'</span>, <span class="org-string">'t'</span>)
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Ts = 0.1; <span class="org-comment">% [s]</span>
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</pre>
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</div>
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<div id="orgc4044aa" class="figure">
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<p><img src="figs/long_meas_time_domain_full.png" alt="long_meas_time_domain_full.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Long measurement time domain data</p>
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</div>
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<p>
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Let’s fit the data with a step response to a first order low pass filter (Figure <a href="#org6d464f2">4</a>).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> f = @(b,x) b(1)<span class="org-type">*</span>(1 <span class="org-type">-</span> exp(<span class="org-type">-</span>x<span class="org-type">/</span>b(2)));
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y_cur = x(t <span class="org-type"><</span> 17.5<span class="org-type">*</span>60<span class="org-type">*</span>60);
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t_cur = t(t <span class="org-type"><</span> 17.5<span class="org-type">*</span>60<span class="org-type">*</span>60);
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nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
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B0 = [400e<span class="org-type">-</span>9, 2<span class="org-type">*</span>60<span class="org-type">*</span>60]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
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[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters ‘B’</span>
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</pre>
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</div>
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<p>
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The corresponding time constant is (in [h]):
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</p>
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<pre class="example">
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2.0658
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</pre>
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<div id="org6d464f2" class="figure">
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<p><img src="figs/long_meas_time_domain_fit.png" alt="long_meas_time_domain_fit.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Fit of the measurement data with a step response of a first order low pass filter</p>
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</div>
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<p>
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We can see in Figure <a href="#orgc4044aa">3</a> that there is a transient period where the measured displacement experiences some drifts.
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This is probably due to thermal effects.
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We only select the data between <code>t1</code> and <code>t2</code>.
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The obtained displacement is shown in Figure <a href="#org551a453">5</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> t1 = 10.5; t2 = 17.5; <span class="org-comment">% [h]</span>
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x = x(t <span class="org-type">></span> t1<span class="org-type">*</span>60<span class="org-type">*</span>60 <span class="org-type">&</span> t <span class="org-type"><</span> t2<span class="org-type">*</span>60<span class="org-type">*</span>60);
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x = x <span class="org-type">-</span> mean(x);
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t = t(t <span class="org-type">></span> t1<span class="org-type">*</span>60<span class="org-type">*</span>60 <span class="org-type">&</span> t <span class="org-type"><</span> t2<span class="org-type">*</span>60<span class="org-type">*</span>60);
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t = t <span class="org-type">-</span> t(1);
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</pre>
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</div>
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<div id="org551a453" class="figure">
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<p><img src="figs/long_meas_time_domain_zoom.png" alt="long_meas_time_domain_zoom.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Kept data (removed slow drifts during the first hours)</p>
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</div>
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<p>
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The Power Spectral Density of the measured displacement is computed
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> win = hann(ceil(length(x)<span class="org-type">/</span>20));
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[p_1, f_1] = pwelch(x, win, [], [], 1<span class="org-type">/</span>Ts);
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</pre>
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</div>
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<p>
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As a low pass filter was used in the measurement process, we multiply the PSD by the square of the inverse of the filter’s norm.
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</p>
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<div class="org-src-container">
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<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>);
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p_1 = p_1<span class="org-type">./</span>abs(squeeze(freqresp(G_lpf, f_1, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2;
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</pre>
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</div>
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<p>
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Only frequencies below 2Hz are taken into account (high frequency noise will be measured afterwards).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> p_1 = p_1(f_1 <span class="org-type"><</span> 2);
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f_1 = f_1(f_1 <span class="org-type"><</span> 2);
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orgb6208e7" class="outline-3">
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<h3 id="orgb6208e7"><span class="section-number-3">1.2</span> Short and Fast measurement</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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An second measurement is done in order to estimate the high frequency noise of the interferometer.
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The measurement is done with a sampling time of \(T_s = 0.1\,ms\) and a duration of ~100s.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> load(<span class="org-string">'short_test_plastic.mat'</span>)
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"> x = detrend(x, 0);
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</pre>
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</div>
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<p>
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The time domain measurement is shown in Figure <a href="#orgc1ba734">6</a>.
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</p>
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<div id="orgc1ba734" class="figure">
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<p><img src="figs/short_meas_time_domain.png" alt="short_meas_time_domain.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Time domain measurement with the high sampling rate</p>
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</div>
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<p>
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The Power Spectral Density of the measured displacement is computed
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> win = hann(ceil(length(x)<span class="org-type">/</span>20));
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[p_2, f_2] = pwelch(x, win, [], [], 1<span class="org-type">/</span>Ts);
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org3bd5ef2" class="outline-3">
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<h3 id="org3bd5ef2"><span class="section-number-3">1.3</span> Obtained Amplitude Spectral Density of the measured displacement</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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The computed ASD of the two measurements are combined in Figure <a href="#org7f818d7">7</a>.
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</p>
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<div id="org7f818d7" class="figure">
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<p><img src="figs/psd_combined.png" alt="psd_combined.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Obtained 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>
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<div id="outline-container-org4a27f63" class="outline-2">
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<h2 id="org4a27f63"><span class="section-number-2">2</span> Effect of the “bubble sheet” and “Aluminium tube”</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org5a54545"></a>
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</p>
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<div id="org95d7372" class="figure">
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<p><img src="figs/IMG-7864.JPG" alt="IMG-7864.JPG" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Aluminium tube used to protect the beam path from disturbances</p>
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</div>
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<p>
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Measurements corresponding to the used of both the aluminium tube and the “bubble sheet ”are loaded and PSD is computed.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> load(<span class="org-string">'short_test_plastic.mat'</span>);
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
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t_1 = t;
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x_1 = detrend(x, 0);
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[p_1, f_1] = pwelch(x_1, win, [], [], 1<span class="org-type">/</span>Ts);
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</pre>
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</div>
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<p>
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Then, the measurement corresponding to the use of the aluminium tube only are loaded and the PSD is computed.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> load(<span class="org-string">'short_test_alu_tube.mat'</span>);
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
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t_2 = t;
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x_2 = detrend(x, 0);
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[p_2, f_2] = pwelch(x_2, win, [], [], 1<span class="org-type">/</span>Ts);
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</pre>
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</div>
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<p>
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Finally, the measurements when neither using the aluminium tube nor the “bubble sheet” are used.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> load(<span class="org-string">'short_test_without_material.mat'</span>);
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
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t_3 = t;
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x_3 = detrend(x, 0);
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[p_3, f_3] = pwelch(x_3, win, [], [], 1<span class="org-type">/</span>Ts);
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</pre>
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</div>
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<p>
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The time domain signals are compared in Figure <a href="#org58f5584">9</a> and the power spectral densities are compared in Figure <a href="#orgcc1019d">10</a>.
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</p>
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<div id="org58f5584" class="figure">
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<p><img src="figs/time_domain_comp_bubble_aluminium.png" alt="time_domain_comp_bubble_aluminium.png" />
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</p>
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<p><span class="figure-number">Figure 9: </span>Time domain signals</p>
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</div>
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<div id="orgcc1019d" class="figure">
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<p><img src="figs/asd_noise_comp_bubble_aluminium.png" alt="asd_noise_comp_bubble_aluminium.png" />
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</p>
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<p><span class="figure-number">Figure 10: </span>Comparison of the noise ASD with and without bubble sheet</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org1df7d3a" class="outline-2">
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<h2 id="org1df7d3a"><span class="section-number-2">3</span> Measurement of the Attocube’s non-linearity</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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<a id="org477e3cc"></a>
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</p>
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<p>
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The measurement setup is shown in Figure <a href="#orgce55d14">11</a>.
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</p>
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<div class="note" id="org6167c44">
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<p>
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Here are the equipment used in the test bench:
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</p>
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<ul class="org-ul">
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<li>Renishaw Resolution Encoder with 1nm resolution (<a href="doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf">doc</a>)</li>
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<li>Attocube interferometer (<a href="doc/IDS3010.pdf">doc</a>)</li>
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<li>Cedrat Amplified Piezoelectric Actuator APA95ML (<a href="doc/APA95ML.pdf">doc</a>)</li>
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<li>Voltage Amplifier LA75B (<a href="doc/LA75B.pdf">doc</a>)</li>
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<li>Speedgoat IO131 with 16bits ADC and DAC (<a href="doc/IO130 IO131 OEM Datasheet.pdf">doc</a>)</li>
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</ul>
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</div>
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<div id="orgce55d14" class="figure">
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<p><img src="figs/exp_setup_schematic.png" alt="exp_setup_schematic.png" />
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</p>
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<p><span class="figure-number">Figure 11: </span>Schematic of the Experiment</p>
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</div>
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<p>
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A DAC and voltage amplified are used to move the mass with the Amplified Piezoelectric Actuator (APA95ML).
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The encoder and the attocube are measure ring the same motion.
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</p>
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<p>
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As will be shown shortly, this measurement permitted to measure the period non-linearity of the Attocube.
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</p>
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</div>
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<div id="outline-container-orgcff7a2a" class="outline-3">
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<h3 id="orgcff7a2a"><span class="section-number-3">3.1</span> Load Data</h3>
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<div class="outline-text-3" id="text-3-1">
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<p>
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The measurement data are loaded and the offset are removed using the <code>detrend</code> command.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"> load(<span class="org-string">'int_enc_comp.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>);
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
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</pre>
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</div>
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|
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<div class="org-src-container">
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<pre class="src src-matlab"> interferometer = detrend(interferometer, 0);
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encoder = detrend(encoder, 0);
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u = detrend(u, 0);
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</pre>
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</div>
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</div>
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</div>
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|
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<div id="outline-container-orgf79eda1" class="outline-3">
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<h3 id="orgf79eda1"><span class="section-number-3">3.2</span> Time Domain Results</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
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|
One period of the displacement of the mass as measured by the encoder and interferometer are shown in Figure <a href="#org0f63236">12</a>.
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It consist of the sinusoidal motion at 0.5Hz with an amplitude of approximately \(70\mu m\).
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</p>
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|
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<p>
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|
The frequency of the motion is chosen such that no resonance in the system is excited.
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This should improve the coherence between the measurements made by the encoder and interferometer.
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|
</p>
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|
|
|
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|
<div id="org0f63236" class="figure">
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|
<p><img src="figs/int_enc_one_cycle.png" alt="int_enc_one_cycle.png" />
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|
</p>
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|
<p><span class="figure-number">Figure 12: </span>One cycle measurement</p>
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|
</div>
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|
|
|
<p>
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|
The difference between the two measurements during the same period is shown in Figure <a href="#org62cb585">13</a>.
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|
</p>
|
|
|
|
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|
<div id="org62cb585" class="figure">
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|
<p><img src="figs/int_enc_one_cycle_error.png" alt="int_enc_one_cycle_error.png" />
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|
</p>
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|
<p><span class="figure-number">Figure 13: </span>Difference between the Encoder and the interferometer during one cycle</p>
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|
</div>
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|
</div>
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|
</div>
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|
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|
<div id="outline-container-org46f4beb" class="outline-3">
|
|
<h3 id="org46f4beb"><span class="section-number-3">3.3</span> Difference between Encoder and Interferometer as a function of time</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
The data is filtered using a second order low pass filter with a cut-off frequency \(\omega_0\) as defined below.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5; <span class="org-comment">% [rad/s]</span>
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|
xi = 0.7;
|
|
|
|
G_lpf = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">/</span>w0<span class="org-type">*</span>s <span class="org-type">+</span> s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
After filtering, the data is “re-shaped” such that we can superimpose all the measured periods as shown in Figure <a href="#org3caa2df">14</a>.
|
|
This gives an idea of the measurement error as given by the Attocube during a \(70 \mu m\) motion.
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|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> d_err_mean = reshape(lsim(G_lpf, encoder <span class="org-type">-</span> interferometer, t), [2<span class="org-type">/</span>Ts floor(Ts<span class="org-type">/</span>2<span class="org-type">*</span>length(encoder))]);
|
|
d_err_mean = d_err_mean <span class="org-type">-</span> mean(d_err_mean);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org3caa2df" class="figure">
|
|
<p><img src="figs/int_enc_error_mean_time.png" alt="int_enc_error_mean_time.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Difference between the two measurement in the time domain, averaged for all the cycles</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org044dbe7" class="outline-3">
|
|
<h3 id="org044dbe7"><span class="section-number-3">3.4</span> Difference between Encoder and Interferometer as a function of position</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<p>
|
|
Figure <a href="#org3caa2df">14</a> gives the measurement error as a function of time.
|
|
We here wish the compute this measurement error as a function of the position (as measured by the encoer).
|
|
</p>
|
|
|
|
<p>
|
|
To do so, all the attocube measurements corresponding to each position measured by the Encoder (resolution of \(1nm\)) are averaged.
|
|
Figure <a href="#org706801f">15</a> is obtained where we clearly see an error with a period comparable to the motion range and a much smaller period corresponding to the non-linear period errors that we wish the estimate.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> [e_sorted, <span class="org-type">~</span>, e_ind] = unique(encoder);
|
|
|
|
i_mean = zeros(length(e_sorted), 1);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(e_sorted)</span>
|
|
i_mean(<span class="org-constant">i</span>) = mean(interferometer(e_ind <span class="org-type">==</span> <span class="org-constant">i</span>));
|
|
<span class="org-keyword">end</span>
|
|
|
|
i_mean_error = (i_mean <span class="org-type">-</span> e_sorted);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org706801f" class="figure">
|
|
<p><img src="figs/int_enc_error_mean_position.png" alt="int_enc_error_mean_position.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Difference between the two measurement as a function of the measured position by the encoder, averaged for all the cycles</p>
|
|
</div>
|
|
|
|
<p>
|
|
The period of the non-linearity seems to be equal to \(765 nm\) which corresponds to half the wavelength of the Laser (\(1.53 \mu m\)).
|
|
For the motion range done here, the non-linearity is measured over ~18 periods which permits to do some averaging.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"> win_length = 1530<span class="org-type">/</span>2; <span class="org-comment">% length of the windows (corresponds to 765 nm)</span>
|
|
num_avg = floor(length(e_sorted)<span class="org-type">/</span>win_length); <span class="org-comment">% number of averaging</span>
|
|
|
|
i_init = ceil((length(e_sorted) <span class="org-type">-</span> win_length<span class="org-type">*</span>num_avg)<span class="org-type">/</span>2); <span class="org-comment">% does not start at the extremity</span>
|
|
|
|
e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init<span class="org-type">:</span>i_init<span class="org-type">+</span>win_length<span class="org-type">*</span>num_avg<span class="org-type">-</span>1), [win_length num_avg]), 2);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained periodic non-linearity is shown in Figure <a href="#orgfc3683a">16</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgfc3683a" class="figure">
|
|
<p><img src="figs/int_non_linearity_period_wavelength.png" alt="int_non_linearity_period_wavelength.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Non-Linearity of the Interferometer over the period of the wavelength</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2021-05-12 mer. 13:32</p>
|
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</div>
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</body>
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</html>
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