339 lines
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339 lines
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<!-- 2021-06-08 mar. 22:38 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Nano-Hexapod - Test Bench</title>
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<meta name="author" content="Dehaeze Thomas" />
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<div id="org-div-home-and-up">
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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">Nano-Hexapod - 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="#orgd7e7f5e">1. Encoders fixed to the Struts</a>
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<ul>
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<li><a href="#org00dcf35">1.1. Introduction</a></li>
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<li><a href="#orgb763144">1.2. Load Data</a></li>
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<li><a href="#orgb853f20">1.3. Spectral Analysis - Setup</a></li>
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<li><a href="#orge1489cc">1.4. DVF Plant</a></li>
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<li><a href="#org0c1cf8a">1.5. IFF Plant</a></li>
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<li><a href="#orgc6ecc36">1.6. Jacobian</a>
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<ul>
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<li><a href="#org1c3941e">1.6.1. DVF Plant</a></li>
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<li><a href="#org31caf05">1.6.2. IFF Plant</a></li>
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</ul>
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</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-nano-hexapod.pdf">pdf</a>.</p>
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<hr>
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<div class="note" id="orgf7b18a3">
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<p>
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Here are the documentation of the equipment used for this test bench:
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</p>
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<ul class="org-ul">
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<li>Voltage Amplifier: PiezoDrive <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
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<li>Amplified Piezoelectric Actuator: Cedrat <a href="doc/APA300ML.pdf">APA300ML</a></li>
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<li>DAC/ADC: Speedgoat <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
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<li>Encoder: Renishaw <a href="doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf">Vionic</a> and used <a href="doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf">Ruler</a></li>
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<li>Interferometers: Attocube</li>
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</ul>
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</div>
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<div id="org2a6b667" class="figure">
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<p><img src="figs/IMG_20210608_152917.jpg" alt="IMG_20210608_152917.jpg" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Nano-Hexapod</p>
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</div>
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<div id="org7b600dc" class="figure">
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<p><img src="figs/IMG_20210608_154722.jpg" alt="IMG_20210608_154722.jpg" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Nano-Hexapod and the control electronics</p>
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</div>
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<div id="outline-container-orgd7e7f5e" class="outline-2">
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<h2 id="orgd7e7f5e"><span class="section-number-2">1</span> Encoders fixed to the Struts</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org00dcf35" class="outline-3">
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<h3 id="org00dcf35"><span class="section-number-3">1.1</span> Introduction</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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In this section, the encoders are fixed to the struts.
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</p>
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</div>
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</div>
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<div id="outline-container-orgb763144" class="outline-3">
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<h3 id="orgb763144"><span class="section-number-3">1.2</span> Load Data</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab">meas_data_lf = {};
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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meas_data_lf(<span class="org-constant">i</span>) = {load(sprintf(<span class="org-string">'mat/frf_data_exc_strut_%i_noise_lf.mat'</span>, <span class="org-constant">i</span>), <span class="org-string">'t'</span>, <span class="org-string">'Va'</span>, <span class="org-string">'Vs'</span>, <span class="org-string">'de'</span>)};
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meas_data_hf(<span class="org-constant">i</span>) = {load(sprintf(<span class="org-string">'mat/frf_data_exc_strut_%i_noise_hf.mat'</span>, <span class="org-constant">i</span>), <span class="org-string">'t'</span>, <span class="org-string">'Va'</span>, <span class="org-string">'Vs'</span>, <span class="org-string">'de'</span>)};
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<span class="org-keyword">end</span>
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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-orgb853f20" class="outline-3">
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<h3 id="orgb853f20"><span class="section-number-3">1.3</span> Spectral Analysis - Setup</h3>
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<div class="outline-text-3" id="text-1-3">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-comment">% Sampling Time [s]</span>
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Ts = (meas_data_lf{1}.t(end) <span class="org-type">-</span> (meas_data_lf{1}.t(1)))<span class="org-type">/</span>(length(meas_data_lf{1}.t)<span class="org-type">-</span>1);
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<span class="org-comment">% Sampling Frequency [Hz]</span>
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Fs = 1<span class="org-type">/</span>Ts;
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<span class="org-comment">% Hannning Windows</span>
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win = hanning(ceil(1<span class="org-type">*</span>Fs));
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</pre>
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</div>
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<p>
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And we get the frequency vector.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">[<span class="org-type">~</span>, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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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">i_lf = f <span class="org-type"><</span> 250; <span class="org-comment">% Points for low frequency excitation</span>
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i_hf = f <span class="org-type">></span> 250; <span class="org-comment">% Points for high frequency excitation</span>
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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-orge1489cc" class="outline-3">
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<h3 id="orge1489cc"><span class="section-number-3">1.4</span> DVF Plant</h3>
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<div class="outline-text-3" id="text-1-4">
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<p>
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First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#orgdf189a0">3</a>).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Coherence</span></span>
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coh_dvf_lf = zeros(length(f), 6, 6);
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coh_dvf_hf = zeros(length(f), 6, 6);
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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coh_dvf_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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coh_dvf_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="orgdf189a0" class="figure">
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<p><img src="figs/enc_struts_dvf_coh.png" alt="enc_struts_dvf_coh.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Obtained coherence for the DVF plant</p>
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</div>
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<p>
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Then the 6x6 transfer function matrix is estimated (Figure <a href="#orgce0ab32">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"><span class="org-matlab-cellbreak"><span class="org-comment">%% DVF Plant</span></span>
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G_dvf_lf = zeros(length(f), 6, 6);
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G_dvf_hf = zeros(length(f), 6, 6);
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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G_dvf_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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G_dvf_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.de, win, [], [], 1<span class="org-type">/</span>Ts);
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="orgce0ab32" class="figure">
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<p><img src="figs/enc_struts_dvf_frf.png" alt="enc_struts_dvf_frf.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Measured FRF for the DVF plant</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org0c1cf8a" class="outline-3">
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<h3 id="org0c1cf8a"><span class="section-number-3">1.5</span> IFF Plant</h3>
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<div class="outline-text-3" id="text-1-5">
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<p>
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First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#org1ba438b">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"><span class="org-matlab-cellbreak"><span class="org-comment">%% Coherence</span></span>
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coh_iff_lf = zeros(length(f), 6, 6);
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coh_iff_hf = zeros(length(f), 6, 6);
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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coh_iff_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
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coh_iff_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = mscohere(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="org1ba438b" class="figure">
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<p><img src="figs/enc_struts_iff_coh.png" alt="enc_struts_iff_coh.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Obtained coherence for the IFF plant</p>
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</div>
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<p>
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Then the 6x6 transfer function matrix is estimated (Figure <a href="#orge2cbf29">6</a>).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF Plant</span></span>
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G_iff_lf = zeros(length(f), 6, 6);
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G_iff_hf = zeros(length(f), 6, 6);
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
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G_iff_lf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_lf{<span class="org-constant">i</span>}.Va, meas_data_lf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
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G_iff_hf(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = tfestimate(meas_data_hf{<span class="org-constant">i</span>}.Va, meas_data_hf{<span class="org-constant">i</span>}.Vs, win, [], [], 1<span class="org-type">/</span>Ts);
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="orge2cbf29" class="figure">
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<p><img src="figs/enc_struts_iff_frf.png" alt="enc_struts_iff_frf.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Measured FRF for the IFF plant</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgc6ecc36" class="outline-3">
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<h3 id="orgc6ecc36"><span class="section-number-3">1.6</span> Jacobian</h3>
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<div class="outline-text-3" id="text-1-6">
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<p>
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The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements.
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</p>
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<p>
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Consider the plant shown in Figure <a href="#org573cce0">7</a> with:
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</p>
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<ul class="org-ul">
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<li>\(\tau\) the 6 input voltages (going to the PD200 amplifier and then to the APA)</li>
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<li>\(d\mathcal{L}\) the relative motion sensor outputs (encoders)</li>
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<li>\(\bm{\tau}_m\) the generated voltage of the force sensor stacks</li>
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<li>\(J_a\) and \(J_s\) the Jacobians for the actuators and sensors</li>
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</ul>
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<div id="org573cce0" class="figure">
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<p><img src="figs/schematic_jacobian_in_out.png" alt="schematic_jacobian_in_out.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Plant in the cartesian Frame</p>
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</div>
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<p>
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First, we load the Jacobian matrix (same for the actuators and sensors).
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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">'jacobian.mat'</span>, <span class="org-string">'J'</span>);
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</pre>
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</div>
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</div>
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<div id="outline-container-org1c3941e" class="outline-4">
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<h4 id="org1c3941e"><span class="section-number-4">1.6.1</span> DVF Plant</h4>
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<div class="outline-text-4" id="text-1-6-1">
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<p>
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The transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\) is computed and shown in Figure <a href="#org92f038e">8</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">G_dvf_J_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_lf, [2 3 1]), inv(J<span class="org-type">'</span>))), [3 1 2]);
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G_dvf_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_hf, [2 3 1]), inv(J<span class="org-type">'</span>))), [3 1 2]);
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</pre>
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</div>
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<div id="org92f038e" class="figure">
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<p><img src="figs/enc_struts_dvf_cart_frf.png" alt="enc_struts_dvf_cart_frf.png" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Measured FRF for the DVF plant in the cartesian frame</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org31caf05" class="outline-4">
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<h4 id="org31caf05"><span class="section-number-4">1.6.2</span> IFF Plant</h4>
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<div class="outline-text-4" id="text-1-6-2">
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<p>
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The transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{F}}_m\) is computed and shown in Figure <a href="#orge1b3404">9</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">G_iff_J_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_lf, [2 3 1]), inv(J<span class="org-type">'</span>))), [3 1 2]);
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G_iff_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_hf, [2 3 1]), inv(J<span class="org-type">'</span>))), [3 1 2]);
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</pre>
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</div>
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<div id="orge1b3404" class="figure">
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<p><img src="figs/enc_struts_iff_cart_frf.png" alt="enc_struts_iff_cart_frf.png" />
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</p>
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<p><span class="figure-number">Figure 9: </span>Measured FRF for the IFF plant in the cartesian frame</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>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2021-06-08 mar. 22:38</p>
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</div>
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</body>
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</html>
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