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2020-05-25 11:13:54 +02:00
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@@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-05-20 mer. 16:56 -->
<!-- 2020-05-25 lun. 11:13 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Amplified Piezoelectric Stack Actuator</title>
<meta name="generator" content="Org mode" />
@@ -36,26 +36,30 @@
<ul>
<li><a href="#org996fd7c">1. Simplified Model</a>
<ul>
<li><a href="#org1e2b26f">1.1. Parameters</a></li>
<li><a href="#orgac18acf">1.2. Identification</a></li>
<li><a href="#orgd6c324c">1.3. Root Locus</a></li>
<li><a href="#orgd4866c5">1.1. Parameters</a></li>
<li><a href="#orgf0cb0e7">1.2. Identification</a></li>
<li><a href="#org8d3f9bd">1.3. Root Locus</a></li>
<li><a href="#orged9310d">1.4. Analytical Model</a></li>
<li><a href="#org2f351a4">1.5. Analytical Analysis</a></li>
</ul>
</li>
<li><a href="#orgf1a765f">2. Rotating X-Y platform</a>
<ul>
<li><a href="#orgd4866c5">2.1. Parameters</a></li>
<li><a href="#orgcfc57a7">2.2. Identification</a></li>
<li><a href="#org8d3f9bd">2.3. Root Locus</a></li>
<li><a href="#org6594475">2.1. Parameters</a></li>
<li><a href="#orgf86cabd">2.2. Identification</a></li>
<li><a href="#org5c898f6">2.3. Root Locus</a></li>
<li><a href="#org069f401">2.4. Analysis</a></li>
</ul>
</li>
<li><a href="#org3c74f7f">3. Stewart Platform with Amplified Actuators</a>
<ul>
<li><a href="#org5a7c6dc">3.1. Initialization</a></li>
<li><a href="#orgf0cb0e7">3.2. Identification</a></li>
<li><a href="#org206d2b9">3.2. Identification</a></li>
<li><a href="#org14c7063">3.3. Controller Design</a></li>
<li><a href="#org043ce40">3.4. Effect of the Low Authority Control on the Primary Plant</a></li>
<li><a href="#orgbc2f246">3.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
<li><a href="#org297c2ad">3.6. Optimal Stiffnesses</a></li>
<li><a href="#org1e2f810">3.7. Direct Velocity Feedback with Amplified Actuators</a></li>
</ul>
</li>
</ul>
@@ -132,8 +136,8 @@ The parameters are shown in the table below.
<h2 id="org996fd7c"><span class="section-number-2">1</span> Simplified Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org1e2b26f" class="outline-3">
<h3 id="org1e2b26f"><span class="section-number-3">1.1</span> Parameters</h3>
<div id="outline-container-orgd4866c5" class="outline-3">
<h3 id="orgd4866c5"><span class="section-number-3">1.1</span> Parameters</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<pre class="src src-matlab">m = 1; % [kg]
@@ -163,8 +167,8 @@ IFF Controller:
</div>
</div>
<div id="outline-container-orgac18acf" class="outline-3">
<h3 id="orgac18acf"><span class="section-number-3">1.2</span> Identification</h3>
<div id="outline-container-orgf0cb0e7" class="outline-3">
<h3 id="orgf0cb0e7"><span class="section-number-3">1.2</span> Identification</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Identification in open-loop.
@@ -219,8 +223,8 @@ Giff.OutputName = {'Fs', 'x1'};
</div>
</div>
<div id="outline-container-orgd6c324c" class="outline-3">
<h3 id="orgd6c324c"><span class="section-number-3">1.3</span> Root Locus</h3>
<div id="outline-container-org8d3f9bd" class="outline-3">
<h3 id="org8d3f9bd"><span class="section-number-3">1.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-1-3">
<div id="org85cd6e5" class="figure">
@@ -230,14 +234,82 @@ Giff.OutputName = {'Fs', 'x1'};
</div>
</div>
</div>
<div id="outline-container-orged9310d" class="outline-3">
<h3 id="orged9310d"><span class="section-number-3">1.4</span> Analytical Model</h3>
<div class="outline-text-3" id="text-1-4">
<p>
If we apply the Newton&rsquo;s second law of motion on the top mass, we obtain:
\[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \]
</p>
<p>
Then, we can write that the measured force \(F_s\) is equal to:
\[ F_s = k_a(w - x_e) + f = -k_e (x_1 - x_e) \]
which gives:
\[ x_e = \frac{k_a}{k_e + k_a} w + \frac{1}{k_e + k_a} f + \frac{k_e}{k_e + k_a} x_1 \]
</p>
<p>
Re-injecting that into the previous equations gives:
\[ x_1 = F \frac{1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + f \frac{\frac{k_e}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \]
\[ F_s = - F \frac{\frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_e k_a}{k_e + k_a} \Big( \frac{ms^2}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) - f \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ga = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * ...
[ 1 , k1 + ke*ka/(ke + ka) , ke/(ke + ka) ;
-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke+ka)*(m*s^2 + k1)];
Ga.InputName = {'F', 'w', 'f'};
Ga.OutputName = {'x1', 'Fs'};
</pre>
</div>
<div id="org7970b34" class="figure">
<p><img src="figs/comp_simscape_analytical.png" alt="comp_simscape_analytical.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed)</p>
</div>
</div>
</div>
<div id="outline-container-org2f351a4" class="outline-3">
<h3 id="org2f351a4"><span class="section-number-3">1.5</span> Analytical Analysis</h3>
<div class="outline-text-3" id="text-1-5">
<p>
For Integral Force Feedback Control, the plant is:
\[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
</p>
<p>
As high frequency, this converge to:
\[ \frac{F_s}{f} \underset{\omega\to\infty}{\longrightarrow} \frac{k_e}{k_e + k_a} \]
And at low frequency:
\[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
</p>
<p>
It has two complex conjugate zeros at:
\[ z = \pm j \sqrt{\frac{k_1}{m}} \]
And two complex conjugate poles at:
\[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \]
</p>
<p>
If maximal damping is to be attained with IFF, the distance between the zero and the pole is to be maximized.
Thus, we wish to maximize \(p/z\), which is equivalent as to minimize \(k_1\) and have \(k_e \approx k_a\) (supposing \(k_e + k_a \approx \text{cst}\)).
</p>
</div>
</div>
</div>
<div id="outline-container-orgf1a765f" class="outline-2">
<h2 id="orgf1a765f"><span class="section-number-2">2</span> Rotating X-Y platform</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-orgd4866c5" class="outline-3">
<h3 id="orgd4866c5"><span class="section-number-3">2.1</span> Parameters</h3>
<div id="outline-container-org6594475" class="outline-3">
<h3 id="org6594475"><span class="section-number-3">2.1</span> Parameters</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab">m = 1; % [kg]
@@ -264,8 +336,8 @@ h = 0.2; % [m]
</div>
</div>
<div id="outline-container-orgcfc57a7" class="outline-3">
<h3 id="orgcfc57a7"><span class="section-number-3">2.2</span> Identification</h3>
<div id="outline-container-orgf86cabd" class="outline-3">
<h3 id="orgf86cabd"><span class="section-number-3">2.2</span> Identification</h3>
<div class="outline-text-3" id="text-2-2">
<p>
Rotating speed in rad/s:
@@ -309,19 +381,19 @@ end
<div id="orga4fc975" class="figure">
<p><img src="figs/amplitifed_piezo_xy_rotation_plant_iff.png" alt="amplitifed_piezo_xy_rotation_plant_iff.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Transfer function matrix from forces to force sensors for multiple rotation speed</p>
<p><span class="figure-number">Figure 5: </span>Transfer function matrix from forces to force sensors for multiple rotation speed</p>
</div>
</div>
</div>
<div id="outline-container-org8d3f9bd" class="outline-3">
<h3 id="org8d3f9bd"><span class="section-number-3">2.3</span> Root Locus</h3>
<div id="outline-container-org5c898f6" class="outline-3">
<h3 id="org5c898f6"><span class="section-number-3">2.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-2-3">
<div id="orgccd3396" class="figure">
<p><img src="figs/amplified_piezo_xy_rotation_root_locus.png" alt="amplified_piezo_xy_rotation_root_locus.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Root locus for 3 rotating speed</p>
<p><span class="figure-number">Figure 6: </span>Root locus for 3 rotating speed</p>
</div>
</div>
</div>
@@ -372,7 +444,7 @@ end
<div id="orgca23612" class="figure">
<p><img src="figs/amplified_piezo_xy_rotating_unstable_root_locus.png" alt="amplified_piezo_xy_rotating_unstable_root_locus.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Root Locus for the two considered rotation speed. For the red curve, the system is unstable.</p>
<p><span class="figure-number">Figure 7: </span>Root Locus for the two considered rotation speed. For the red curve, the system is unstable.</p>
</div>
</div>
</div>
@@ -413,8 +485,8 @@ We set the stiffness of the payload fixation:
</div>
</div>
<div id="outline-container-orgf0cb0e7" class="outline-3">
<h3 id="orgf0cb0e7"><span class="section-number-3">3.2</span> Identification</h3>
<div id="outline-container-org206d2b9" class="outline-3">
<h3 id="org206d2b9"><span class="section-number-3">3.2</span> Identification</h3>
<div class="outline-text-3" id="text-3-2">
<div class="org-src-container">
<pre class="src src-matlab">K = tf(zeros(6));
@@ -447,7 +519,7 @@ The nano-hexapod has the following leg&rsquo;s stiffness and damping.
<div id="org0e2911a" class="figure">
<p><img src="figs/amplified_piezo_iff_loop_gain.png" alt="amplified_piezo_iff_loop_gain.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Dynamics for the Integral Force Feedback for three payload masses</p>
<p><span class="figure-number">Figure 8: </span>Dynamics for the Integral Force Feedback for three payload masses</p>
</div>
@@ -455,7 +527,7 @@ The nano-hexapod has the following leg&rsquo;s stiffness and damping.
<div id="org5d7f6d3" class="figure">
<p><img src="figs/amplified_piezo_iff_root_locus.png" alt="amplified_piezo_iff_root_locus.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Root Locus for the IFF control for three payload masses</p>
<p><span class="figure-number">Figure 9: </span>Root Locus for the IFF control for three payload masses</p>
</div>
<p>
@@ -465,7 +537,7 @@ Damping as function of the gain
<div id="org4743c83" class="figure">
<p><img src="figs/amplified_piezo_iff_damping_gain.png" alt="amplified_piezo_iff_damping_gain.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Damping ratio of the poles as a function of the IFF gain</p>
<p><span class="figure-number">Figure 10: </span>Damping ratio of the poles as a function of the IFF gain</p>
</div>
<p>
@@ -485,7 +557,7 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
<div id="org904efc3" class="figure">
<p><img src="figs/amplified_piezo_iff_plant_damped_X.png" alt="amplified_piezo_iff_plant_damped_X.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Primary plant in the task space with (dashed) and without (solid) IFF</p>
<p><span class="figure-number">Figure 11: </span>Primary plant in the task space with (dashed) and without (solid) IFF</p>
</div>
@@ -493,13 +565,13 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
<div id="orgddf3013" class="figure">
<p><img src="figs/amplified_piezo_iff_damped_plant_L.png" alt="amplified_piezo_iff_damped_plant_L.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
<p><span class="figure-number">Figure 12: </span>Primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
</div>
<div id="orgd940ce9" class="figure">
<p><img src="figs/amplified_piezo_iff_damped_coupling_X.png" alt="amplified_piezo_iff_damped_coupling_X.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Coupling in the primary plant in the task with (dashed) and without (solid) IFF</p>
<p><span class="figure-number">Figure 13: </span>Coupling in the primary plant in the task with (dashed) and without (solid) IFF</p>
</div>
@@ -507,7 +579,7 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
<div id="org4278690" class="figure">
<p><img src="figs/amplified_piezo_iff_damped_coupling_L.png" alt="amplified_piezo_iff_damped_coupling_L.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
<p><span class="figure-number">Figure 14: </span>Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
</div>
</div>
</div>
@@ -519,18 +591,26 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
<div id="org56179cd" class="figure">
<p><img src="figs/amplified_piezo_iff_disturbances.png" alt="amplified_piezo_iff_disturbances.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied</p>
<p><span class="figure-number">Figure 15: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied</p>
</div>
<div class="important">
</div>
</div>
</div>
<div id="outline-container-org297c2ad" class="outline-3">
<h3 id="org297c2ad"><span class="section-number-3">3.6</span> Optimal Stiffnesses</h3>
</div>
<div id="outline-container-org1e2f810" class="outline-3">
<h3 id="org1e2f810"><span class="section-number-3">3.7</span> Direct Velocity Feedback with Amplified Actuators</h3>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-20 mer. 16:56</p>
<p class="date">Created: 2020-05-25 lun. 11:13</p>
</div>
</body>
</html>