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Add analysis about virtual mass addition

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Thomas Dehaeze
2020-04-17 14:11:34 +02:00
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25 changed files with 8708 additions and 22 deletions
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docs/optimal_stiffness_control.html
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<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>
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