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<title>Nano-Hexapod - Test Bench</title>
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<h1 class="title">Nano-Hexapod - Test Bench</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgd5a3ff2">1. Encoders fixed to the Struts</a>
<ul>
<li><a href="#orgaaf36d1">1.1. Introduction</a></li>
<li><a href="#org4eac0e4">1.2. Identification of the dynamics</a>
<ul>
<li><a href="#orge7631cb">1.2.1. Load Data</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>
</li>
</ul>
</li>
<li><a href="#org16300e1">2. Encoders fixed to the plates</a></li>
</ul>
</div>
</div>
<hr>
<p>This report is also available as a <a href="./test-bench-nano-hexapod.pdf">pdf</a>.</p>
<hr>
<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>
Here are the documentation of the equipment used for this test bench:
</p>
<ul class="org-ul">
<li>Voltage Amplifier: PiezoDrive <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
<li>Amplified Piezoelectric Actuator: Cedrat <a href="doc/APA300ML.pdf">APA300ML</a></li>
<li>DAC/ADC: Speedgoat <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
<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>
<li>Interferometers: Attocube</li>
</ul>
</div>
<div id="orgcaac3cd" class="figure">
<p><img src="figs/IMG_20210608_152917.jpg" alt="IMG_20210608_152917.jpg" />
</p>
<p><span class="figure-number">Figure 1: </span>Nano-Hexapod</p>
</div>
<div id="org6004b44" class="figure">
<p><img src="figs/IMG_20210608_154722.jpg" alt="IMG_20210608_154722.jpg" />
</p>
<p><span class="figure-number">Figure 2: </span>Nano-Hexapod and the control electronics</p>
</div>
<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>
<div id="outline-container-orgaaf36d1" class="outline-3">
<h3 id="orgaaf36d1"><span class="section-number-3">1.1</span> Introduction</h3>
<div class="outline-text-3" id="text-1-1">
<p>
In this section, the encoders are fixed to the struts.
</p>
</div>
</div>
<div id="outline-container-org4eac0e4" class="outline-3">
<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>
<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">
<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>
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_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>)};
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org3d8f0db" class="outline-4">
<h4 id="org3d8f0db"><span class="section-number-4">1.2.2</span> Spectral Analysis - Setup</h4>
<div class="outline-text-4" id="text-1-2-2">
<div class="org-src-container">
<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);
<span class="org-comment">% Sampling Frequency [Hz]</span>
Fs = 1<span class="org-type">/</span>Ts;
<span class="org-comment">% Hannning Windows</span>
win = hanning(ceil(1<span class="org-type">*</span>Fs));
<span class="org-comment">% And we get the frequency vector</span>
[<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);
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>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgfe475e0" class="outline-4">
<h4 id="orgfe475e0"><span class="section-number-4">1.2.3</span> DVF Plant</h4>
<div class="outline-text-4" id="text-1-2-3">
<p>
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>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Coherence</span></span>
coh_dvf_lf = zeros(length(f), 6, 6);
coh_dvf_hf = zeros(length(f), 6, 6);
<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>
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);
<span class="org-keyword">end</span>
</pre>
</div>
<div id="org7027095" class="figure">
<p><img src="figs/enc_struts_dvf_coh.png" alt="enc_struts_dvf_coh.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Obtained coherence for the DVF plant</p>
</div>
<p>
Then the 6x6 transfer function matrix is estimated (Figure <a href="#orgeda62ff">5</a>).
</p>
<div class="org-src-container">
<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_hf = zeros(length(f), 6, 6);
<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>
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);
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);
<span class="org-keyword">end</span>
</pre>
</div>
<div id="orgeda62ff" class="figure">
<p><img src="figs/enc_struts_dvf_frf.png" alt="enc_struts_dvf_frf.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Measured FRF for the DVF plant</p>
</div>
</div>
</div>
<div id="outline-container-org9c55cb0" class="outline-4">
<h4 id="org9c55cb0"><span class="section-number-4">1.2.4</span> IFF Plant</h4>
<div class="outline-text-4" id="text-1-2-4">
<p>
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>
<div class="org-src-container">
<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_hf = zeros(length(f), 6, 6);
<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>
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);
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);
<span class="org-keyword">end</span>
</pre>
</div>
<div id="orga958a00" class="figure">
<p><img src="figs/enc_struts_iff_coh.png" alt="enc_struts_iff_coh.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Obtained coherence for the IFF plant</p>
</div>
<p>
Then the 6x6 transfer function matrix is estimated (Figure <a href="#orgaa3ad1c">7</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% IFF Plant</span></span>
G_iff_lf = zeros(length(f), 6, 6);
G_iff_hf = zeros(length(f), 6, 6);
<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>
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);
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);
<span class="org-keyword">end</span>
</pre>
</div>
<div id="orgaa3ad1c" class="figure">
<p><img src="figs/enc_struts_iff_frf.png" alt="enc_struts_iff_frf.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Measured FRF for the IFF plant</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb32a800" class="outline-3">
<h3 id="orgb32a800"><span class="section-number-3">1.3</span> Comparison with the Simscape Model</h3>
<div class="outline-text-3" id="text-1-3">
<p>
In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.
</p>
</div>
<div id="outline-container-org49d6b51" class="outline-4">
<h4 id="org49d6b51"><span class="section-number-4">1.3.1</span> Dynamics from Actuator to Force Sensors</h4>
<div class="outline-text-4" id="text-1-3-1">
<div class="org-src-container">
<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>
</div>
<div class="org-src-container">
<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>
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>
</div>
<div id="orgb002d1f" class="figure">
<p><img src="figs/enc_struts_iff_comp_simscape.png" alt="enc_struts_iff_comp_simscape.png" />
</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 id="outline-container-org68f8e6c" class="outline-4">
<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-1-3-2">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Initialization of the 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>
</div>
<div class="org-src-container">
<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>
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>
</div>
<div id="org8001ef8" class="figure">
<p><img src="figs/enc_struts_dvf_comp_simscape.png" alt="enc_struts_dvf_comp_simscape.png" />
</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 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 id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-06-09 mer. 18:13</p>
</div>
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