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Add analysis about virtual mass addition

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Thomas Dehaeze
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<title>Decentralize control to add virtual mass</title>
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</div><div id="content">
<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>
<ul>
<li><a href="#org33f35d2">2.1. Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\)</a></li>
<li><a href="#org6663ed2">2.2. Identification of the Primary plant without virtual add of mass</a></li>
</ul>
</li>
<li><a href="#orgd6fc719">3. Adding Virtual Mass in the Leg&rsquo;s Space</a>
<ul>
<li><a href="#orgc37faa7">3.1. Plant</a></li>
<li><a href="#org4ae3263">3.2. Controller Design</a></li>
<li><a href="#orgb270293">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="#org9ed2d4c">4.1. Plant</a></li>
<li><a href="#org4f03a34">4.2. Controller Design</a></li>
<li><a href="#org2fe0ce0">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(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
</pre>
</div>
<p>
We set the stiffness of the payload fixation:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kp = 1e8; <span class="org-comment">% [N/m]</span>
</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">
</div>
<div id="outline-container-org33f35d2" class="outline-3">
<h3 id="org33f35d2"><span class="section-number-3">2.1</span> Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\)</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab">K = tf(zeros(6));
Kdvf = tf(zeros(6));
</pre>
</div>
<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>
The nano-hexapod has the following leg&rsquo;s stiffness and damping.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'k'</span>, 1e5, <span class="org-string">'c'</span>, 2e2);
</pre>
</div>
</div>
</div>
<div id="outline-container-org6663ed2" class="outline-3">
<h3 id="org6663ed2"><span class="section-number-3">2.2</span> Identification of the Primary plant without virtual add of mass</h3>
</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&rsquo;s Space</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orgc37faa7" class="outline-3">
<h3 id="orgc37faa7"><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-org4ae3263" class="outline-3">
<h3 id="org4ae3263"><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<span class="org-type">*</span>s<span class="org-type">^</span>2<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>2<span class="org-type">*</span>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&rsquo;s space</p>
</div>
</div>
</div>
<div id="outline-container-orgb270293" class="outline-3">
<h3 id="orgb270293"><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&rsquo;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&rsquo;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-org9ed2d4c" class="outline-3">
<h3 id="org9ed2d4c"><span class="section-number-3">4.1</span> Plant</h3>
<div class="outline-text-3" id="text-4-1">
<p>
Let&rsquo;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-org4f03a34" class="outline-3">
<h3 id="org4f03a34"><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<span class="org-type">^</span>2<span class="org-type">*</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>2<span class="org-type">*</span>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.J<span class="org-type">'</span>)<span class="org-type">*</span>KmX<span class="org-type">*</span>inv(nano_hexapod.J);
</pre>
</div>
</div>
</div>
<div id="outline-container-org2fe0ce0" class="outline-3">
<h3 id="org2fe0ce0"><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: 2020-04-17 ven. 14:10</p>
</div>
</body>
</html>

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@@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-04-17 ven. 10:25 -->
<!-- 2020-04-17 ven. 14:10 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Control of the NASS with optimal stiffness</title>
<meta name="generator" content="Org mode" />
@@ -42,7 +42,7 @@
<li><a href="#orgfef1a3f">1.3. Controller Design</a></li>
<li><a href="#org3c73014">1.4. Effect of the Low Authority Control on the Primary Plant</a></li>
<li><a href="#orgee5dbee">1.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
<li><a href="#org27f255e">1.6. Conclusion</a></li>
<li><a href="#org882e1ac">1.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#org81dc0a8">2. Primary Control in the leg space</a>
@@ -52,7 +52,7 @@
<li><a href="#org16d192f">2.3. Sensibility to Disturbances and Noise Budget</a></li>
<li><a href="#org84f68cc">2.4. Simulations</a></li>
<li><a href="#orgbeadec8">2.5. Results</a></li>
<li><a href="#org87fa1ac">2.6. Conclusion</a></li>
<li><a href="#orgd61852c">2.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#org9bd2bf8">3. Primary Control in the task space</a>
@@ -64,7 +64,7 @@
</ul>
</li>
<li><a href="#org57e2cfd">3.3. Simulation</a></li>
<li><a href="#org882e1ac">3.4. Conclusion</a></li>
<li><a href="#org8c0882d">3.4. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -207,6 +207,12 @@ Then, we compute the transfer function from forces applied by the actuators \(\b
The obtained dynamics is shown in Figure <a href="#org45c1265">5</a>.
</p>
<div class="important">
<p>
A zero with a positive real part is introduced in the transfer function from \(\mathcal{F}_y\) to \(\mathcal{X}_y\) after Decentralized Direct Velocity Feedback is applied.
</p>
</div>
<p>
And we compute the transfer function from actuator forces \(\bm{\tau}\) to position error of each leg \(\bm{\epsilon}_\mathcal{L}\):
@@ -237,9 +243,15 @@ The coupling does not change a lot with DVF.
<p>
The coupling in the space of the legs \(\bm{G}_\mathcal{L}\) are shown in Figure <a href="#orgc43d759">8</a>.
The magnitude of the coupling around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
</p>
<div class="important">
<p>
The magnitude of the coupling between \(\tau_i\) and \(d\mathcal{L}_j\) (Figure <a href="#orgc43d759">8</a>) around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
</p>
</div>
<div id="orgbb4e497" class="figure">
<p><img src="figs/opt_stiff_primary_plant_damped_coupling_X.png" alt="opt_stiff_primary_plant_damped_coupling_X.png" />
@@ -286,12 +298,18 @@ The norm of these transfer functions are shown in Figure <a href="#org199898b">9
<p><img src="figs/opt_stiff_sensibility_dist_dvf.png" alt="opt_stiff_sensibility_dist_dvf.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied</p>
</div>
<div class="important">
<p>
Decentralized Direct Velocity Feedback is shown to increase the effect of stages vibrations at high frequency and to reduce the effect of ground motion and direct forces at low frequency.
</p>
</div>
</div>
</div>
<div id="outline-container-org27f255e" class="outline-3">
<h3 id="org27f255e"><span class="section-number-3">1.6</span> Conclusion</h3>
<div id="outline-container-org882e1ac" class="outline-3">
<h3 id="org882e1ac"><span class="section-number-3">1.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-6">
<div class="important">
<p>
@@ -332,7 +350,7 @@ The controller for decentralized direct velocity feedback is the one designed in
<h3 id="org1e7a412"><span class="section-number-3">2.1</span> Plant in the leg space</h3>
<div class="outline-text-3" id="text-2-1">
<p>
We now loop at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\).
We now look at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\).
</p>
<p>
@@ -400,17 +418,17 @@ Kl = 2e7 <span class="org-type">*</span> eye(6) <span class="org-type">*</span>
<p><span class="figure-number">Figure 12: </span>Loop gain for the primary plant</p>
</div>
<p>
Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
K = Kl<span class="org-type">*</span>nano_hexapod.J<span class="org-type">*</span>diag([1, 1, 1, 1, 1, 0]);
</pre>
</div>
</div>
</div>
<p>
Check the MIMO stability
</p>
</div>
</div>
<div id="outline-container-org16d192f" class="outline-3">
<h3 id="org16d192f"><span class="section-number-3">2.3</span> Sensibility to Disturbances and Noise Budget</h3>
<div class="outline-text-3" id="text-2-3">
@@ -519,8 +537,8 @@ Finally, the time domain position error signals are shown in Figure <a href="#or
</div>
</div>
<div id="outline-container-org87fa1ac" class="outline-3">
<h3 id="org87fa1ac"><span class="section-number-3">2.6</span> Conclusion</h3>
<div id="outline-container-orgd61852c" class="outline-3">
<h3 id="orgd61852c"><span class="section-number-3">2.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-6">
<div class="important">
<p>
@@ -614,8 +632,8 @@ Kx<span class="org-type">(6,6) </span>= 5e4 <span class="org-type">*</span> ...
<div id="outline-container-org57e2cfd" class="outline-3">
<h3 id="org57e2cfd"><span class="section-number-3">3.3</span> Simulation</h3>
</div>
<div id="outline-container-org882e1ac" class="outline-3">
<h3 id="org882e1ac"><span class="section-number-3">3.4</span> Conclusion</h3>
<div id="outline-container-org8c0882d" class="outline-3">
<h3 id="org8c0882d"><span class="section-number-3">3.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-4">
<div class="important">
<p>
@@ -629,7 +647,7 @@ Kx<span class="org-type">(6,6) </span>= 5e4 <span class="org-type">*</span> ...
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-17 ven. 10:25</p>
<p class="date">Created: 2020-04-17 ven. 14:10</p>
</div>
</body>
</html>