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<h1 class="title">Decentralize control to add virtual mass</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org982b263">1. Initialization</a></li>
<li><a href="#org35a3822">2. Identification</a></li>
<li><a href="#orgd6fc719">3. Adding Virtual Mass in the Leg’s Space</a>
<ul>
<li><a href="#orga27c9a0">3.1. Plant</a></li>
<li><a href="#orgcbce41a">3.2. Controller Design</a></li>
<li><a href="#orgca1f525">3.3. Identification of the Primary Plant</a></li>
</ul>
</li>
<li><a href="#orgc9131d0">4. Adding Virtual Mass in the Task Space</a>
<ul>
<li><a href="#orgdbe6a25">4.1. Plant</a></li>
<li><a href="#org571922f">4.2. Controller Design</a></li>
<li><a href="#org4960701">4.3. Identification of the Primary Plant</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org982b263" class="outline-2">
<h2 id="org982b263"><span class="section-number-2">1</span> Initialization</h2>
<div class="outline-text-2" id="text-1">
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'hac-dvf');
</pre>
</div>
<p>
The nano-hexapod has the following leg’s stiffness and damping.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeNanoHexapod('k', 1e5, 'c', 2e2);
</pre>
</div>
<p>
We set the stiffness of the payload fixation:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kp = 1e8; % [N/m]
</pre>
</div>
</div>
</div>
<div id="outline-container-org35a3822" class="outline-2">
<h2 id="org35a3822"><span class="section-number-2">2</span> Identification</h2>
<div class="outline-text-2" id="text-2">
<p>
We identify the system for the following payload masses:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ms = [1, 10, 50];
</pre>
</div>
<p>
Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\).
Identification of the Primary plant without virtual add of mass
</p>
</div>
</div>
<div id="outline-container-orgd6fc719" class="outline-2">
<h2 id="orgd6fc719"><span class="section-number-2">3</span> Adding Virtual Mass in the Leg’s Space</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orga27c9a0" class="outline-3">
<h3 id="orga27c9a0"><span class="section-number-3">3.1</span> Plant</h3>
<div class="outline-text-3" id="text-3-1">
<div id="org98e7ba8" class="figure">
<p><img src="figs/virtual_mass_plant_L.png" alt="virtual_mass_plant_L.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Transfer function from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses</p>
</div>
</div>
</div>
<div id="outline-container-orgcbce41a" class="outline-3">
<h3 id="orgcbce41a"><span class="section-number-3">3.2</span> Controller Design</h3>
<div class="outline-text-3" id="text-3-2">
<div class="org-src-container">
<pre class="src src-matlab">Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6);
</pre>
</div>
<div id="orgccb3b9e" class="figure">
<p><img src="figs/virtual_mass_loop_gain_L.png" alt="virtual_mass_loop_gain_L.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Loop Gain for the addition of virtual mass in the leg’s space</p>
</div>
</div>
</div>
<div id="outline-container-orgca1f525" class="outline-3">
<h3 id="orgca1f525"><span class="section-number-3">3.3</span> Identification of the Primary Plant</h3>
<div class="outline-text-3" id="text-3-3">
<div id="orgd49505e" class="figure">
<p><img src="figs/virtual_mass_L_primary_plant_X.png" alt="virtual_mass_L_primary_plant_X.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the leg’s space</p>
</div>
<div id="org2281744" class="figure">
<p><img src="figs/virtual_mass_L_primary_plant_L.png" alt="virtual_mass_L_primary_plant_L.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the leg’s space</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgc9131d0" class="outline-2">
<h2 id="orgc9131d0"><span class="section-number-2">4</span> Adding Virtual Mass in the Task Space</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-orgdbe6a25" class="outline-3">
<h3 id="orgdbe6a25"><span class="section-number-3">4.1</span> Plant</h3>
<div class="outline-text-3" id="text-4-1">
<p>
Let’s look at the transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\):
\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \]
</p>
<div id="org6488b4c" class="figure">
<p><img src="figs/virtual_mass_plant_X.png" alt="virtual_mass_plant_X.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Dynamics from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) used for virtual mass addition in the task space</p>
</div>
</div>
</div>
<div id="outline-container-org571922f" class="outline-3">
<h3 id="org571922f"><span class="section-number-3">4.2</span> Controller Design</h3>
<div class="outline-text-3" id="text-4-2">
<div class="org-src-container">
<pre class="src src-matlab">KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0]));
</pre>
</div>
<div id="orgf411330" class="figure">
<p><img src="figs/virtual_mass_loop_gain_X.png" alt="virtual_mass_loop_gain_X.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Loop gain for virtual mass addition in the task space</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Kdvf = inv(nano_hexapod.kinematics.J')*KmX*inv(nano_hexapod.kinematics.J);
</pre>
</div>
</div>
</div>
<div id="outline-container-org4960701" class="outline-3">
<h3 id="org4960701"><span class="section-number-3">4.3</span> Identification of the Primary Plant</h3>
<div class="outline-text-3" id="text-4-3">
<div id="orge1df87b" class="figure">
<p><img src="figs/virtual_mass_X_primary_plant_X.png" alt="virtual_mass_X_primary_plant_X.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the task space</p>
</div>
<div id="org647b748" class="figure">
<p><img src="figs/virtual_mass_X_primary_plant_L.png" alt="virtual_mass_X_primary_plant_L.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the task space</p>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-05 mar. 10:34</p>
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