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IFF analysis with Simscape model

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Thomas Dehaeze 2021-06-09 18:14:45 +02:00
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2021-06-08 mar. 22:38 --> <!-- 2021-06-09 mer. 18:13 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Nano-Hexapod - Test Bench</title> <title>Nano-Hexapod - Test Bench</title>
<meta name="author" content="Dehaeze Thomas" /> <meta name="author" content="Dehaeze Thomas" />
@ -39,21 +39,33 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgd7e7f5e">1. Encoders fixed to the Struts</a> <li><a href="#orgd5a3ff2">1. Encoders fixed to the Struts</a>
<ul> <ul>
<li><a href="#org00dcf35">1.1. Introduction</a></li> <li><a href="#orgaaf36d1">1.1. Introduction</a></li>
<li><a href="#orgb763144">1.2. Load Data</a></li> <li><a href="#org4eac0e4">1.2. Identification of the dynamics</a>
<li><a href="#orgb853f20">1.3. Spectral Analysis - Setup</a></li>
<li><a href="#orge1489cc">1.4. DVF Plant</a></li>
<li><a href="#org0c1cf8a">1.5. IFF Plant</a></li>
<li><a href="#orgc6ecc36">1.6. Jacobian</a>
<ul> <ul>
<li><a href="#org1c3941e">1.6.1. DVF Plant</a></li> <li><a href="#orge7631cb">1.2.1. Load Data</a></li>
<li><a href="#org31caf05">1.6.2. IFF Plant</a></li> <li><a href="#org3d8f0db">1.2.2. Spectral Analysis - Setup</a></li>
<li><a href="#orgfe475e0">1.2.3. DVF Plant</a></li>
<li><a href="#org9c55cb0">1.2.4. IFF Plant</a></li>
</ul>
</li>
<li><a href="#orgb32a800">1.3. Comparison with the Simscape Model</a>
<ul>
<li><a href="#org49d6b51">1.3.1. Dynamics from Actuator to Force Sensors</a></li>
<li><a href="#org68f8e6c">1.3.2. Dynamics from Actuator to Encoder</a></li>
</ul>
</li>
<li><a href="#orge6221eb">1.4. Integral Force Feedback</a>
<ul>
<li><a href="#org1ccd985">1.4.1. Root Locus and Decentralized Loop gain</a></li>
<li><a href="#orgd6bc33c">1.4.2. Multiple Gains - Simulation</a></li>
<li><a href="#orgcbdb9eb">1.4.3. Experimental Results</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org16300e1">2. Encoders fixed to the plates</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -61,7 +73,11 @@
<p>This report is also available as a <a href="./test-bench-nano-hexapod.pdf">pdf</a>.</p> <p>This report is also available as a <a href="./test-bench-nano-hexapod.pdf">pdf</a>.</p>
<hr> <hr>
<div class="note" id="orgf7b18a3"> <p>
In this document, the dynamics of the nano-hexapod shown in Figure <a href="#orgcaac3cd">1</a> is identified.
</p>
<div class="note" id="org64d5e50">
<p> <p>
Here are the documentation of the equipment used for this test bench: Here are the documentation of the equipment used for this test bench:
</p> </p>
@ -76,25 +92,149 @@ Here are the documentation of the equipment used for this test bench:
</div> </div>
<div id="org2a6b667" class="figure"> <div id="orgcaac3cd" class="figure">
<p><img src="figs/IMG_20210608_152917.jpg" alt="IMG_20210608_152917.jpg" /> <p><img src="figs/IMG_20210608_152917.jpg" alt="IMG_20210608_152917.jpg" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Nano-Hexapod</p> <p><span class="figure-number">Figure 1: </span>Nano-Hexapod</p>
</div> </div>
<div id="org7b600dc" class="figure"> <div id="org6004b44" class="figure">
<p><img src="figs/IMG_20210608_154722.jpg" alt="IMG_20210608_154722.jpg" /> <p><img src="figs/IMG_20210608_154722.jpg" alt="IMG_20210608_154722.jpg" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Nano-Hexapod and the control electronics</p> <p><span class="figure-number">Figure 2: </span>Nano-Hexapod and the control electronics</p>
</div> </div>
<div id="outline-container-orgd7e7f5e" class="outline-2">
<h2 id="orgd7e7f5e"><span class="section-number-2">1</span> Encoders fixed to the Struts</h2> <div id="orgc32dab5" class="figure">
<p><img src="figs/nano_hexapod_signals.png" alt="nano_hexapod_signals.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Block diagram of the system with named signals</p>
</div>
<table id="orgcb52f65" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> List of signals</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left"><b>Unit</b></th>
<th scope="col" class="org-left"><b>Matlab</b></th>
<th scope="col" class="org-left"><b>Vector</b></th>
<th scope="col" class="org-left"><b>Elements</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Control Input (wanted DAC voltage)</td>
<td class="org-left"><code>[V]</code></td>
<td class="org-left"><code>u</code></td>
<td class="org-left">\(\bm{u}\)</td>
<td class="org-left">\(u_i\)</td>
</tr>
<tr>
<td class="org-left">DAC Output Voltage</td>
<td class="org-left"><code>[V]</code></td>
<td class="org-left"><code>u</code></td>
<td class="org-left">\(\tilde{\bm{u}}\)</td>
<td class="org-left">\(\tilde{u}_i\)</td>
</tr>
<tr>
<td class="org-left">PD200 Output Voltage</td>
<td class="org-left"><code>[V]</code></td>
<td class="org-left"><code>ua</code></td>
<td class="org-left">\(\bm{u}_a\)</td>
<td class="org-left">\(u_{a,i}\)</td>
</tr>
<tr>
<td class="org-left">Actuator applied force</td>
<td class="org-left"><code>[N]</code></td>
<td class="org-left"><code>tau</code></td>
<td class="org-left">\(\bm{\tau}\)</td>
<td class="org-left">\(\tau_i\)</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left">Strut motion</td>
<td class="org-left"><code>[m]</code></td>
<td class="org-left"><code>dL</code></td>
<td class="org-left">\(d\bm{\mathcal{L}}\)</td>
<td class="org-left">\(d\mathcal{L}_i\)</td>
</tr>
<tr>
<td class="org-left">Encoder measured displacement</td>
<td class="org-left"><code>[m]</code></td>
<td class="org-left"><code>dLm</code></td>
<td class="org-left">\(d\bm{\mathcal{L}}_m\)</td>
<td class="org-left">\(d\mathcal{L}_{m,i}\)</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left">Force Sensor strain</td>
<td class="org-left"><code>[m]</code></td>
<td class="org-left"><code>epsilon</code></td>
<td class="org-left">\(\bm{\epsilon}\)</td>
<td class="org-left">\(\epsilon_i\)</td>
</tr>
<tr>
<td class="org-left">Force Sensor Generated Voltage</td>
<td class="org-left"><code>[V]</code></td>
<td class="org-left"><code>taum</code></td>
<td class="org-left">\(\tilde{\bm{\tau}}_m\)</td>
<td class="org-left">\(\tilde{\tau}_{m,i}\)</td>
</tr>
<tr>
<td class="org-left">Measured Generated Voltage</td>
<td class="org-left"><code>[V]</code></td>
<td class="org-left"><code>taum</code></td>
<td class="org-left">\(\bm{\tau}_m\)</td>
<td class="org-left">\(\tau_{m,i}\)</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left">Motion of the top platform</td>
<td class="org-left"><code>[m,rad]</code></td>
<td class="org-left"><code>dX</code></td>
<td class="org-left">\(d\bm{\mathcal{X}}\)</td>
<td class="org-left">\(d\mathcal{X}_i\)</td>
</tr>
<tr>
<td class="org-left">Metrology measured displacement</td>
<td class="org-left"><code>[m,rad]</code></td>
<td class="org-left"><code>dXm</code></td>
<td class="org-left">\(d\bm{\mathcal{X}}_m\)</td>
<td class="org-left">\(d\mathcal{X}_{m,i}\)</td>
</tr>
</tbody>
</table>
<div id="outline-container-orgd5a3ff2" class="outline-2">
<h2 id="orgd5a3ff2"><span class="section-number-2">1</span> Encoders fixed to the Struts</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org00dcf35" class="outline-3"> <div id="outline-container-orgaaf36d1" class="outline-3">
<h3 id="org00dcf35"><span class="section-number-3">1.1</span> Introduction</h3> <h3 id="orgaaf36d1"><span class="section-number-3">1.1</span> Introduction</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
In this section, the encoders are fixed to the struts. In this section, the encoders are fixed to the struts.
@ -102,11 +242,16 @@ In this section, the encoders are fixed to the struts.
</div> </div>
</div> </div>
<div id="outline-container-orgb763144" class="outline-3"> <div id="outline-container-org4eac0e4" class="outline-3">
<h3 id="orgb763144"><span class="section-number-3">1.2</span> Load Data</h3> <h3 id="org4eac0e4"><span class="section-number-3">1.2</span> Identification of the dynamics</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
</div>
<div id="outline-container-orge7631cb" class="outline-4">
<h4 id="orge7631cb"><span class="section-number-4">1.2.1</span> Load Data</h4>
<div class="outline-text-4" id="text-1-2-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">meas_data_lf = {}; <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load Identification Data</span></span>
meas_data_lf = {};
<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> <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>
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>)}; 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>)};
@ -117,11 +262,12 @@ In this section, the encoders are fixed to the struts.
</div> </div>
</div> </div>
<div id="outline-container-orgb853f20" class="outline-3"> <div id="outline-container-org3d8f0db" class="outline-4">
<h3 id="orgb853f20"><span class="section-number-3">1.3</span> Spectral Analysis - Setup</h3> <h4 id="org3d8f0db"><span class="section-number-4">1.2.2</span> Spectral Analysis - Setup</h4>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-4" id="text-1-2-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% Sampling Time [s]</span> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Setup useful variables</span></span>
<span class="org-comment">% Sampling Time [s]</span>
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); 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);
<span class="org-comment">% Sampling Frequency [Hz]</span> <span class="org-comment">% Sampling Frequency [Hz]</span>
@ -129,30 +275,22 @@ Fs = 1<span class="org-type">/</span>Ts;
<span class="org-comment">% Hannning Windows</span> <span class="org-comment">% Hannning Windows</span>
win = hanning(ceil(1<span class="org-type">*</span>Fs)); win = hanning(ceil(1<span class="org-type">*</span>Fs));
</pre>
</div>
<p> <span class="org-comment">% And we get the frequency vector</span>
And we get the frequency vector. [<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);
</p>
<div class="org-src-container">
<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);
</pre>
</div>
<div class="org-src-container"> i_lf = f <span class="org-type">&lt;</span> 250; <span class="org-comment">% Points for low frequency excitation</span>
<pre class="src src-matlab">i_lf = f <span class="org-type">&lt;</span> 250; <span class="org-comment">% Points for low frequency excitation</span>
i_hf = f <span class="org-type">&gt;</span> 250; <span class="org-comment">% Points for high frequency excitation</span> i_hf = f <span class="org-type">&gt;</span> 250; <span class="org-comment">% Points for high frequency excitation</span>
</pre> </pre>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orge1489cc" class="outline-3"> <div id="outline-container-orgfe475e0" class="outline-4">
<h3 id="orge1489cc"><span class="section-number-3">1.4</span> DVF Plant</h3> <h4 id="orgfe475e0"><span class="section-number-4">1.2.3</span> DVF Plant</h4>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-4" id="text-1-2-3">
<p> <p>
First, let&rsquo;s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#orgdf189a0">3</a>). First, let&rsquo;s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#org7027095">4</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -164,22 +302,21 @@ coh_dvf_hf = zeros(length(f), 6, 6);
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); 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);
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); 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);
<span class="org-keyword">end</span> <span class="org-keyword">end</span>
</pre> </pre>
</div> </div>
<div id="orgdf189a0" class="figure"> <div id="org7027095" class="figure">
<p><img src="figs/enc_struts_dvf_coh.png" alt="enc_struts_dvf_coh.png" /> <p><img src="figs/enc_struts_dvf_coh.png" alt="enc_struts_dvf_coh.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Obtained coherence for the DVF plant</p> <p><span class="figure-number">Figure 4: </span>Obtained coherence for the DVF plant</p>
</div> </div>
<p> <p>
Then the 6x6 transfer function matrix is estimated (Figure <a href="#orgce0ab32">4</a>). Then the 6x6 transfer function matrix is estimated (Figure <a href="#orgeda62ff">5</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% DVF Plant</span></span> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% DVF Plant (transfer function from u to dLm)</span></span>
G_dvf_lf = zeros(length(f), 6, 6); G_dvf_lf = zeros(length(f), 6, 6);
G_dvf_hf = zeros(length(f), 6, 6); G_dvf_hf = zeros(length(f), 6, 6);
@ -191,24 +328,24 @@ G_dvf_hf = zeros(length(f), 6, 6);
</div> </div>
<div id="orgce0ab32" class="figure"> <div id="orgeda62ff" class="figure">
<p><img src="figs/enc_struts_dvf_frf.png" alt="enc_struts_dvf_frf.png" /> <p><img src="figs/enc_struts_dvf_frf.png" alt="enc_struts_dvf_frf.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Measured FRF for the DVF plant</p> <p><span class="figure-number">Figure 5: </span>Measured FRF for the DVF plant</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org0c1cf8a" class="outline-3"> <div id="outline-container-org9c55cb0" class="outline-4">
<h3 id="org0c1cf8a"><span class="section-number-3">1.5</span> IFF Plant</h3> <h4 id="org9c55cb0"><span class="section-number-4">1.2.4</span> IFF Plant</h4>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-4" id="text-1-2-4">
<p> <p>
First, let&rsquo;s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#org1ba438b">5</a>). First, let&rsquo;s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure <a href="#orga958a00">6</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Coherence</span></span> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Coherence for the IFF plant</span></span>
coh_iff_lf = zeros(length(f), 6, 6); coh_iff_lf = zeros(length(f), 6, 6);
coh_iff_hf = zeros(length(f), 6, 6); coh_iff_hf = zeros(length(f), 6, 6);
@ -221,14 +358,14 @@ coh_iff_hf = zeros(length(f), 6, 6);
</div> </div>
<div id="org1ba438b" class="figure"> <div id="orga958a00" class="figure">
<p><img src="figs/enc_struts_iff_coh.png" alt="enc_struts_iff_coh.png" /> <p><img src="figs/enc_struts_iff_coh.png" alt="enc_struts_iff_coh.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Obtained coherence for the IFF plant</p> <p><span class="figure-number">Figure 6: </span>Obtained coherence for the IFF plant</p>
</div> </div>
<p> <p>
Then the 6x6 transfer function matrix is estimated (Figure <a href="#orge2cbf29">6</a>). Then the 6x6 transfer function matrix is estimated (Figure <a href="#orgaa3ad1c">7</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF Plant</span></span> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF Plant</span></span>
@ -243,96 +380,199 @@ G_iff_hf = zeros(length(f), 6, 6);
</div> </div>
<div id="orge2cbf29" class="figure"> <div id="orgaa3ad1c" class="figure">
<p><img src="figs/enc_struts_iff_frf.png" alt="enc_struts_iff_frf.png" /> <p><img src="figs/enc_struts_iff_frf.png" alt="enc_struts_iff_frf.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Measured FRF for the IFF plant</p> <p><span class="figure-number">Figure 7: </span>Measured FRF for the IFF plant</p>
</div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgc6ecc36" class="outline-3"> <div id="outline-container-orgb32a800" class="outline-3">
<h3 id="orgc6ecc36"><span class="section-number-3">1.6</span> Jacobian</h3> <h3 id="orgb32a800"><span class="section-number-3">1.3</span> Comparison with the Simscape Model</h3>
<div class="outline-text-3" id="text-1-6"> <div class="outline-text-3" id="text-1-3">
<p> <p>
The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements. In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.
</p> </p>
<p>
Consider the plant shown in Figure <a href="#org573cce0">7</a> with:
</p>
<ul class="org-ul">
<li>\(\tau\) the 6 input voltages (going to the PD200 amplifier and then to the APA)</li>
<li>\(d\mathcal{L}\) the relative motion sensor outputs (encoders)</li>
<li>\(\bm{\tau}_m\) the generated voltage of the force sensor stacks</li>
<li>\(J_a\) and \(J_s\) the Jacobians for the actuators and sensors</li>
</ul>
<div id="org573cce0" class="figure">
<p><img src="figs/schematic_jacobian_in_out.png" alt="schematic_jacobian_in_out.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Plant in the cartesian Frame</p>
</div> </div>
<p> <div id="outline-container-org49d6b51" class="outline-4">
First, we load the Jacobian matrix (same for the actuators and sensors). <h4 id="org49d6b51"><span class="section-number-4">1.3.1</span> Dynamics from Actuator to Force Sensors</h4>
</p> <div class="outline-text-4" id="text-1-3-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'jacobian.mat'</span>, <span class="org-string">'J'</span>); <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Initialize Nano-Hexapod</span></span>
n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ...
<span class="org-string">'flex_top_type'</span>, <span class="org-string">'4dof'</span>, ...
<span class="org-string">'motion_sensor_type'</span>, <span class="org-string">'struts'</span>, ...
<span class="org-string">'actuator_type'</span>, <span class="org-string">'2dof'</span>);
</pre> </pre>
</div> </div>
</div>
<div id="outline-container-org1c3941e" class="outline-4">
<h4 id="org1c3941e"><span class="section-number-4">1.6.1</span> DVF Plant</h4>
<div class="outline-text-4" id="text-1-6-1">
<p>
The transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\) is computed and shown in Figure <a href="#org92f038e">8</a>.
</p>
<div class="org-src-container"> <div class="org-src-container">
<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]); <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Identify the IFF Plant (transfer function from u to taum)</span></span>
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]); clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/Fm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Force Sensors</span>
Giff = exp(<span class="org-type">-</span>s<span class="org-type">*</span>Ts)<span class="org-type">*</span>linearize(mdl, io, 0.0, options);
</pre> </pre>
</div> </div>
<div id="org92f038e" class="figure"> <div id="orgb002d1f" class="figure">
<p><img src="figs/enc_struts_dvf_cart_frf.png" alt="enc_struts_dvf_cart_frf.png" /> <p><img src="figs/enc_struts_iff_comp_simscape.png" alt="enc_struts_iff_comp_simscape.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Measured FRF for the DVF plant in the cartesian frame</p> <p><span class="figure-number">Figure 8: </span>Diagonal elements of the IFF Plant</p>
</div>
<div id="orgef9afdd" class="figure">
<p><img src="figs/enc_struts_iff_comp_offdiag_simscape.png" alt="enc_struts_iff_comp_offdiag_simscape.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Off diagonal elements of the IFF Plant</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org31caf05" class="outline-4"> <div id="outline-container-org68f8e6c" class="outline-4">
<h4 id="org31caf05"><span class="section-number-4">1.6.2</span> IFF Plant</h4> <h4 id="org68f8e6c"><span class="section-number-4">1.3.2</span> Dynamics from Actuator to Encoder</h4>
<div class="outline-text-4" id="text-1-6-2"> <div class="outline-text-4" id="text-1-3-2">
<p> <div class="org-src-container">
The transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{F}}_m\) is computed and shown in Figure <a href="#orge1b3404">9</a>. <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Initialization of the Nano-Hexapod</span></span>
</p> n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ...
<span class="org-string">'flex_top_type'</span>, <span class="org-string">'4dof'</span>, ...
<span class="org-string">'motion_sensor_type'</span>, <span class="org-string">'struts'</span>, ...
<span class="org-string">'actuator_type'</span>, <span class="org-string">'2dof'</span>);
</pre>
</div>
<div class="org-src-container"> <div class="org-src-container">
<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]); <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Identify the DVF Plant (transfer function from u to dLm)</span></span>
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]); clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/D'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Encoders</span>
Gdvf = exp(<span class="org-type">-</span>s<span class="org-type">*</span>Ts)<span class="org-type">*</span>linearize(mdl, io, 0.0, options);
</pre> </pre>
</div> </div>
<div id="orge1b3404" class="figure"> <div id="org8001ef8" class="figure">
<p><img src="figs/enc_struts_iff_cart_frf.png" alt="enc_struts_iff_cart_frf.png" /> <p><img src="figs/enc_struts_dvf_comp_simscape.png" alt="enc_struts_dvf_comp_simscape.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Measured FRF for the IFF plant in the cartesian frame</p> <p><span class="figure-number">Figure 10: </span>Diagonal elements of the DVF Plant</p>
</div>
<div id="org8a8dc6a" class="figure">
<p><img src="figs/enc_struts_dvf_comp_offdiag_simscape.png" alt="enc_struts_dvf_comp_offdiag_simscape.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Off diagonal elements of the DVF Plant</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orge6221eb" class="outline-3">
<h3 id="orge6221eb"><span class="section-number-3">1.4</span> Integral Force Feedback</h3>
<div class="outline-text-3" id="text-1-4">
</div>
<div id="outline-container-org1ccd985" class="outline-4">
<h4 id="org1ccd985"><span class="section-number-4">1.4.1</span> Root Locus and Decentralized Loop gain</h4>
<div class="outline-text-4" id="text-1-4-1">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF Controller</span></span>
Kiff_g1 = (1<span class="org-type">/</span>(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>40))<span class="org-type">*</span>...<span class="org-comment"> % Low pass filter (provides integral action above 40Hz)</span>
(s<span class="org-type">/</span>(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>30))<span class="org-type">*</span>...<span class="org-comment"> % High pass filter to limit low frequency gain</span>
(1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>500))<span class="org-type">*</span>...<span class="org-comment"> % Low pass filter to be more robust to high frequency resonances</span>
eye(6); <span class="org-comment">% Diagonal 6x6 controller</span>
</pre>
</div>
<div id="org9d7fb85" class="figure">
<p><img src="figs/enc_struts_iff_root_locus.png" alt="enc_struts_iff_root_locus.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Root Locus for the IFF control strategy</p>
</div>
<p>
Then the &ldquo;optimal&rdquo; IFF controller is:
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF controller with Optimal gain</span></span>
Kiff = g<span class="org-type">*</span>Kiff_g1;
</pre>
</div>
<div id="org879ceab" class="figure">
<p><img src="figs/enc_struts_iff_opt_loop_gain.png" alt="enc_struts_iff_opt_loop_gain.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Bode plot of the &ldquo;decentralized loop gain&rdquo; \(G_\text{iff}(i,i) \times K_\text{iff}(i,i)\)</p>
</div>
</div>
</div>
<div id="outline-container-orgd6bc33c" class="outline-4">
<h4 id="orgd6bc33c"><span class="section-number-4">1.4.2</span> Multiple Gains - Simulation</h4>
<div class="outline-text-4" id="text-1-4-2">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Tested IFF gains</span></span>
iff_gains = [4, 10, 20, 40, 100, 200, 400, 1000];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Initialize the Simscape model in closed loop</span></span>
n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ...
<span class="org-string">'flex_top_type'</span>, <span class="org-string">'4dof'</span>, ...
<span class="org-string">'motion_sensor_type'</span>, <span class="org-string">'struts'</span>, ...
<span class="org-string">'actuator_type'</span>, <span class="org-string">'2dof'</span>, ...
<span class="org-string">'controller_type'</span>, <span class="org-string">'iff'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain</span></span>
Gd_iff = {zeros(1, length(iff_gains))};
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/D'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Strut Displacement (encoder)</span>
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(iff_gains)</span>
Kiff = iff_gains(<span class="org-constant">i</span>)<span class="org-type">*</span>Kiff_g1<span class="org-type">*</span>eye(6); <span class="org-comment">% IFF Controller</span>
Gd_iff(<span class="org-constant">i</span>) = {exp(<span class="org-type">-</span>s<span class="org-type">*</span>Ts)<span class="org-type">*</span>linearize(mdl, io, 0.0, options)};
isstable(Gd_iff{<span class="org-constant">i</span>})
<span class="org-keyword">end</span>
</pre>
</div>
<div id="orgb5b5f55" class="figure">
<p><img src="figs/enc_struts_iff_gains_effect_dvf_plant.png" alt="enc_struts_iff_gains_effect_dvf_plant.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Effect of the IFF gain \(g\) on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)</p>
</div>
</div>
</div>
<div id="outline-container-orgcbdb9eb" class="outline-4">
<h4 id="orgcbdb9eb"><span class="section-number-4">1.4.3</span> Experimental Results</h4>
</div>
</div>
</div>
<div id="outline-container-org16300e1" class="outline-2">
<h2 id="org16300e1"><span class="section-number-2">2</span> Encoders fixed to the plates</h2>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-06-08 mar. 22:38</p> <p class="date">Created: 2021-06-09 mer. 18:13</p>
</div> </div>
</body> </body>
</html> </html>

347
test-bench-nano-hexapod.org
View File

@ -46,7 +46,7 @@
<hr> <hr>
#+end_export #+end_export
\clearpage #+latex: \clearpage
* Introduction :ignore: * Introduction :ignore:
In this document, the dynamics of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]] is identified. In this document, the dynamics of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]] is identified.
@ -136,7 +136,6 @@ Here are the documentation of the equipment used for this test bench:
| Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ | | Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ |
| Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ | | Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ |
*
* Encoders fixed to the Struts * Encoders fixed to the Struts
** Introduction ** Introduction
In this section, the encoders are fixed to the struts. In this section, the encoders are fixed to the struts.
@ -164,6 +163,7 @@ addpath('./src/');
** Identification of the dynamics ** Identification of the dynamics
*** Load Data *** Load Data
#+begin_src matlab #+begin_src matlab
%% Load Identification Data
meas_data_lf = {}; meas_data_lf = {};
for i = 1:6 for i = 1:6
@ -174,6 +174,7 @@ end
*** Spectral Analysis - Setup *** Spectral Analysis - Setup
#+begin_src matlab #+begin_src matlab
%% Setup useful variables
% Sampling Time [s] % Sampling Time [s]
Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1); Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1);
@ -182,14 +183,10 @@ Fs = 1/Ts;
% Hannning Windows % Hannning Windows
win = hanning(ceil(1*Fs)); win = hanning(ceil(1*Fs));
#+end_src
And we get the frequency vector. % And we get the frequency vector
#+begin_src matlab
[~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts); [~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab
i_lf = f < 250; % Points for low frequency excitation i_lf = f < 250; % Points for low frequency excitation
i_hf = f > 250; % Points for high frequency excitation i_hf = f > 250; % Points for high frequency excitation
#+end_src #+end_src
@ -206,10 +203,10 @@ for i = 1:6
coh_dvf_lf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); coh_dvf_lf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
coh_dvf_hf(:, :, i) = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts); coh_dvf_hf(:, :, i) = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts);
end end
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Coherence for the transfer function from u to dLm
figure; figure;
hold on; hold on;
for i = 1:5 for i = 1:5
@ -248,7 +245,7 @@ exportFig('figs/enc_struts_dvf_coh.pdf', 'width', 'wide', 'height', 'normal');
Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_struts_dvf_frf]]). Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_struts_dvf_frf]]).
#+begin_src matlab #+begin_src matlab
%% DVF Plant %% DVF Plant (transfer function from u to dLm)
G_dvf_lf = zeros(length(f), 6, 6); G_dvf_lf = zeros(length(f), 6, 6);
G_dvf_hf = zeros(length(f), 6, 6); G_dvf_hf = zeros(length(f), 6, 6);
@ -259,6 +256,7 @@ end
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
@ -320,7 +318,7 @@ exportFig('figs/enc_struts_dvf_frf.pdf', 'width', 'wide', 'height', 'tall');
First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_iff_coh]]). First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_iff_coh]]).
#+begin_src matlab #+begin_src matlab
%% Coherence %% Coherence for the IFF plant
coh_iff_lf = zeros(length(f), 6, 6); coh_iff_lf = zeros(length(f), 6, 6);
coh_iff_hf = zeros(length(f), 6, 6); coh_iff_hf = zeros(length(f), 6, 6);
@ -332,6 +330,7 @@ end
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Coherence of the IFF Plant (transfer function from u to taum)
figure; figure;
hold on; hold on;
for i = 1:5 for i = 1:5
@ -381,6 +380,7 @@ end
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Bode plot of the IFF Plant (transfer function from u to taum)
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
@ -619,6 +619,7 @@ In this section, the measured dynamics is compared with the dynamics estimated f
*** Initialize :noexport: *** Initialize :noexport:
#+begin_src matlab :tangle no #+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/') addpath('matlab/')
addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
@ -628,6 +629,7 @@ addpath('matlab/nass-simscape/mat/')
#+end_src #+end_src
#+begin_src matlab :eval no #+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/STEPS/png/')
@ -636,6 +638,7 @@ addpath('nass-simscape/mat/')
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape'; mdl = 'nano_hexapod_simscape';
options = linearizeOptions; options = linearizeOptions;
@ -646,22 +649,24 @@ open(mdl)
*** Dynamics from Actuator to Force Sensors *** Dynamics from Actuator to Force Sensors
#+begin_src matlab #+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... %% Initialize Nano-Hexapod
'flex_top_type', '2dof', ... n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
'flex_top_type', '4dof', ...
'motion_sensor_type', 'struts', ... 'motion_sensor_type', 'struts', ...
'actuator_type', '2dof'); 'actuator_type', '2dof');
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
%% Input/Output definition %% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1; clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
Giff = 20*exp(-s*Ts)*linearize(mdl, io, 0.0, options); Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data
freqs = 2*logspace(1, 3, 1000); freqs = 2*logspace(1, 3, 1000);
figure; figure;
@ -669,21 +674,22 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]); ax1 = nexttile([2,1]);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... plot(f(i_lf), abs(G_iff_lf(i_lf,1, 1)), 'color', [0,0,0,0.2], ...
'DisplayName', '$\tau_{m,i}/u_i$ - Model')
plot(f(i_lf), abs(G_iff_lf(i_lf,1, 1)), ...
'DisplayName', '$\tau_{m,i}/u_i$ - FRF') 'DisplayName', '$\tau_{m,i}/u_i$ - FRF')
for i = 2:6 for i = 2:6
set(gca,'ColorOrderIndex',1); set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... plot(f(i_lf), abs(G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot(f(i_hf), abs(G_iff_hf(i_hf,i, i)), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ...
'DisplayName', '$\tau_{m,i}/u_i$ - Model')
for i = 2:6 for i = 2:6
set(gca,'ColorOrderIndex',2) set(gca,'ColorOrderIndex',2);
plot(f(i_lf), abs(G_iff_lf(i_lf,i, i)), ... plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ...
'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot(f(i_hf), abs(G_iff_hf(i_hf,i, i)), ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
hold off; hold off;
@ -694,14 +700,12 @@ legend('location', 'southeast');
ax2 = nexttile; ax2 = nexttile;
hold on; hold on;
for i = 1:6 for i = 1:6
set(gca,'ColorOrderIndex',1); plot(f(i_lf), 180/pi*angle(G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2]);
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-'); plot(f(i_hf), 180/pi*angle(G_iff_hf(i_hf,i, i)), 'color', [0,0,0,0.2]);
end end
for i = 1:6 for i = 1:6
set(gca,'ColorOrderIndex',2) set(gca,'ColorOrderIndex',2);
plot(f(i_lf), 180/pi*angle(G_iff_lf(i_lf,i, i))); plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',2)
plot(f(i_hf), 180/pi*angle(G_iff_hf(i_hf,i, i)));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
@ -723,31 +727,29 @@ exportFig('figs/enc_struts_iff_comp_simscape.pdf', 'width', 'wide', 'height', 't
[[file:figs/enc_struts_iff_comp_simscape.png]] [[file:figs/enc_struts_iff_comp_simscape.png]]
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements)
freqs = 2*logspace(1, 3, 1000); freqs = 2*logspace(1, 3, 1000);
figure; figure;
hold on; hold on;
% Off diagonal terms % Off diagonal terms
set(gca,'ColorOrderIndex',1); plot(f(i_lf), abs(G_iff_lf(i_lf, 1, 2)), 'color', [0,0,0,0.2], ...
plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), ... 'DisplayName', '$\tau_{m,i}/u_j$ - FRF')
'DisplayName', '$\tau_{m,i}/u_j$ - Model')
for i = 1:5 for i = 1:5
for j = i+1:6 for j = i+1:6
set(gca,'ColorOrderIndex',1); plot(f(i_lf), abs(G_iff_lf(i_lf, i, j)), 'color', [0,0,0,0.2], ...
plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), ... 'HandleVisibility', 'off');
plot(f(i_hf), abs(G_iff_hf(i_hf, i, j)), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
set(gca,'ColorOrderIndex',2); set(gca,'ColorOrderIndex',2);
plot(f(i_lf), abs(G_iff_lf(i_lf, 1, 2)), ... plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), ...
'DisplayName', '$\tau_{m,i}/u_j$ - FRF') 'DisplayName', '$\tau_{m,i}/u_j$ - Model')
for i = 1:5 for i = 1:5
for j = i+1:6 for j = i+1:6
set(gca,'ColorOrderIndex',2); set(gca,'ColorOrderIndex',2);
plot(f(i_lf), abs(G_iff_lf(i_lf, i, j)), ... plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), ...
'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2);
plot(f(i_hf), abs(G_iff_hf(i_hf, i, j)), ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
@ -769,22 +771,24 @@ exportFig('figs/enc_struts_iff_comp_offdiag_simscape.pdf', 'width', 'wide', 'hei
*** Dynamics from Actuator to Encoder *** Dynamics from Actuator to Encoder
#+begin_src matlab #+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... %% Initialization of the Nano-Hexapod
'flex_top_type', '2dof', ... n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
'flex_top_type', '4dof', ...
'motion_sensor_type', 'struts', ... 'motion_sensor_type', 'struts', ...
'actuator_type', '2dof'); 'actuator_type', '2dof');
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
%% Input/Output definition %% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1; clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders
Gdvf = 20*exp(-s*Ts)*linearize(mdl, io, 0.0, options); Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Diagonal elements of the DVF plant
freqs = 2*logspace(1, 3, 1000); freqs = 2*logspace(1, 3, 1000);
figure; figure;
@ -792,40 +796,39 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]); ax1 = nexttile([2,1]);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... plot(f(i_lf), abs(G_dvf_lf(i_lf,1, 1)), 'color', [0,0,0,0.2], ...
'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
plot(f(i_lf), abs(G_dvf_lf(i_lf,1, 1)), ...
'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF')
for i = 2:6 for i = 2:6
set(gca,'ColorOrderIndex',1); set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... plot(f(i_lf), abs(G_dvf_lf(i_lf,i, i)), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot(f(i_hf), abs(G_dvf_hf(i_hf,i, i)), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ...
'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
for i = 2:6 for i = 2:6
set(gca,'ColorOrderIndex',2) set(gca,'ColorOrderIndex',2);
plot(f(i_lf), abs(G_dvf_lf(i_lf,i, i)), ... plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ...
'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
plot(f(i_hf), abs(G_dvf_hf(i_hf,i, i)), ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ylim([1e-8, 1e-3]); ylim([1e-8, 1e-3]);
legend('location', 'northeast');
ax2 = nexttile; ax2 = nexttile;
hold on; hold on;
for i = 1:6 for i = 1:6
set(gca,'ColorOrderIndex',1); plot(f(i_lf), 180/pi*angle(G_dvf_lf(i_lf,i, i)), 'color', [0,0,0,0.2]);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-'); plot(f(i_hf), 180/pi*angle(G_dvf_hf(i_hf,i, i)), 'color', [0,0,0,0.2]);
end end
for i = 1:6 for i = 1:6
set(gca,'ColorOrderIndex',2) set(gca,'ColorOrderIndex',2);
plot(f(i_lf), 180/pi*angle(G_dvf_lf(i_lf,i, i))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',2)
plot(f(i_hf), 180/pi*angle(G_dvf_hf(i_hf,i, i)));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
@ -847,31 +850,29 @@ exportFig('figs/enc_struts_dvf_comp_simscape.pdf', 'width', 'wide', 'height', 't
[[file:figs/enc_struts_dvf_comp_simscape.png]] [[file:figs/enc_struts_dvf_comp_simscape.png]]
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Off-diagonal elements of the DVF plant
freqs = 2*logspace(1, 3, 1000); freqs = 2*logspace(1, 3, 1000);
figure; figure;
hold on; hold on;
% Off diagonal terms % Off diagonal terms
set(gca,'ColorOrderIndex',1); plot(f(i_lf), abs(G_dvf_lf(i_lf, 1, 2)), 'color', [0,0,0,0.2], ...
plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF')
'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model')
for i = 1:5 for i = 1:5
for j = i+1:6 for j = i+1:6
set(gca,'ColorOrderIndex',1); plot(f(i_lf), abs(G_dvf_lf(i_lf, i, j)), 'color', [0,0,0,0.2], ...
plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), ... 'HandleVisibility', 'off');
plot(f(i_hf), abs(G_dvf_hf(i_hf, i, j)), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
set(gca,'ColorOrderIndex',2); set(gca,'ColorOrderIndex',2);
plot(f(i_lf), abs(G_dvf_lf(i_lf, 1, 2)), ... plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), ...
'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model')
for i = 1:5 for i = 1:5
for j = i+1:6 for j = i+1:6
set(gca,'ColorOrderIndex',2); set(gca,'ColorOrderIndex',2);
plot(f(i_lf), abs(G_dvf_lf(i_lf, i, j)), ... plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), ...
'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2);
plot(f(i_hf), abs(G_dvf_hf(i_hf, i, j)), ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
end end
@ -891,58 +892,196 @@ exportFig('figs/enc_struts_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'hei
#+RESULTS: #+RESULTS:
[[file:figs/enc_struts_dvf_comp_offdiag_simscape.png]] [[file:figs/enc_struts_dvf_comp_offdiag_simscape.png]]
** TODO Integral Force Feedback ** Integral Force Feedback
*** Plant *** Root Locus and Decentralized Loop gain
#+begin_src matlab
%% IFF Controller
Kiff_g1 = (1/(s + 2*pi*40))*... % Low pass filter (provides integral action above 40Hz)
(s/(s + 2*pi*30))*... % High pass filter to limit low frequency gain
(1/(1 + s/2/pi/500))*... % Low pass filter to be more robust to high frequency resonances
eye(6); % Diagonal 6x6 controller
#+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
%% Root Locus for IFF
gains = logspace(1, 4, 100);
figure;
hold on;
% Pure Integrator
set(gca,'ColorOrderIndex',1);
plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', '$g = 0$');
set(gca,'ColorOrderIndex',1);
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(Giff, g*Kiff_g1*eye(6)));
set(gca,'ColorOrderIndex',1);
plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off');
end
g = 4e2;
clpoles = pole(feedback(Giff, g*Kiff_g1*eye(6)));
set(gca,'ColorOrderIndex',2);
plot(real(clpoles), imag(clpoles), 'x', 'DisplayName', sprintf('$g=%.0f$', g));
hold off;
axis square;
xlim([-1250, 0]); ylim([0, 1250]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/enc_struts_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:enc_struts_iff_root_locus
#+caption: Root Locus for the IFF control strategy
#+RESULTS:
[[file:figs/enc_struts_iff_root_locus.png]]
Then the "optimal" IFF controller is:
#+begin_src matlab
%% IFF controller with Optimal gain
Kiff = g*Kiff_g1;
#+end_src
#+begin_src matlab :exports none
%% Bode plot of the "decentralized loop gain"
freqs = 2*logspace(1, 3, 1000);
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]); ax1 = nexttile([2,1]);
hold on; hold on;
for i = 1:5 plot(f(i_lf), abs(squeeze(freqresp(Kiff(1,1), f(i_lf), 'Hz')).*G_iff_lf(i_lf,1, 1)), 'color', [0,0,0,0.2], ...
for j = i+1:6 'DisplayName', '$\tau_{m,i}/u_i \cdot K_{iff}$ - FRF')
plot(f(i_lf), abs(G_iff_lf(i_lf, i, j)), 'color', [0, 0, 0, 0.2], ... for i = 2:6
set(gca,'ColorOrderIndex',2)
plot(f(i_lf), abs(squeeze(freqresp(Kiff(1,1), f(i_lf), 'Hz')).*G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
plot(f(i_hf), abs(G_iff_hf(i_hf, i, j)), 'color', [0, 0, 0, 0.2], ... set(gca,'ColorOrderIndex',2)
'HandleVisibility', 'off'); plot(f(i_hf), abs(squeeze(freqresp(Kiff(1,1), f(i_hf), 'Hz')).*G_iff_hf(i_hf,i, i)), 'color', [0,0,0,0.2], ...
end 'HandleVisibility', 'off');
end end
for i =1:6 set(gca,'ColorOrderIndex',2);
set(gca,'ColorOrderIndex',i) plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*Giff(1,1), freqs, 'Hz'))), '-', ...
plot(f(i_lf), abs(G_iff_lf(i_lf,i, i)), ... 'DisplayName', '$\tau_{m,i}/u_i \cdot K_{iff}$ - Model')
'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); for i = 2:6
set(gca,'ColorOrderIndex',i) set(gca,'ColorOrderIndex',2);
plot(f(i_hf), abs(G_iff_hf(i_hf,i, i)), ... plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*Giff(i,i), freqs, 'Hz'))), '-', ...
'HandleVisibility', 'off'); 'HandleVisibility', 'off');
end end
plot(f(i_lf), abs(G_iff_lf(i_lf, 1, 2)), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$G_{iff}(i,j)$');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); legend('location', 'northeast');
ylim([1e-3, 1e2]);
ax2 = nexttile; ax2 = nexttile;
hold on; hold on;
for i =1:6 for i = 1:6
set(gca,'ColorOrderIndex',i) plot(f(i_lf), 180/pi*angle(squeeze(freqresp(Kiff(1,1), f(i_lf), 'Hz')).*G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2]);
plot(f(i_lf), 180/pi*angle(G_iff_lf(i_lf,i, i))); plot(f(i_hf), 180/pi*angle(squeeze(freqresp(Kiff(1,1), f(i_hf), 'Hz')).*G_iff_hf(i_hf,i, i)), 'color', [0,0,0,0.2]);
set(gca,'ColorOrderIndex',i) end
plot(f(i_hf), 180/pi*angle(G_iff_hf(i_hf,i, i))); for i = 1:6
set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*angle(squeeze(freqresp(Kiff(1,1)*Giff(i,i), freqs, 'Hz'))), '-');
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
hold off; ylim([-180, 180]);
yticks(-360:90:360); yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x'); linkaxes([ax1,ax2],'x');
xlim([20, 2e3]); xlim([freqs(1), freqs(end)]);
#+end_src #+end_src
*** Root Locus #+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/enc_struts_iff_opt_loop_gain.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
*** Gains #+name: fig:enc_struts_iff_opt_loop_gain
#+caption: Bode plot of the "decentralized loop gain" $G_\text{iff}(i,i) \times K_\text{iff}(i,i)$
#+RESULTS:
[[file:figs/enc_struts_iff_opt_loop_gain.png]]
*** Multiple Gains - Simulation
#+begin_src matlab
%% Tested IFF gains
iff_gains = [4, 10, 20, 40, 100, 200, 400, 1000];
#+end_src
#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
'flex_top_type', '4dof', ...
'motion_sensor_type', 'struts', ...
'actuator_type', '2dof', ...
'controller_type', 'iff');
#+end_src
#+begin_src matlab
%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain
Gd_iff = {zeros(1, length(iff_gains))};
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Strut Displacement (encoder)
for i = 1:length(iff_gains)
Kiff = iff_gains(i)*Kiff_g1*eye(6); % IFF Controller
Gd_iff(i) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)};
isstable(Gd_iff{i})
end
#+end_src
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain
freqs = 2*logspace(1, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(iff_gains)
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-', ...
'DisplayName', sprintf('$g = %.0f$', iff_gains(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ax2 = nexttile;
hold on;
for i = 1:length(iff_gains)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/enc_struts_iff_gains_effect_dvf_plant.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:enc_struts_iff_gains_effect_dvf_plant
#+caption: Effect of the IFF gain $g$ on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$
#+RESULTS:
[[file:figs/enc_struts_iff_gains_effect_dvf_plant.png]]
*** Experimental Results *** Experimental Results
* Encoders fixed to the plates
** Introduction :ignore:

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test-bench-nano-hexapod.pdf
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