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<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>
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<h1 class="title">Amplified Piezoelectric Stack Actuator</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orga1734d6">1. Simplified Model</a>
<ul>
<li><a href="#org9543c57">1.1. Parameters</a></li>
<li><a href="#org20bf7c7">1.2. Identification</a></li>
<li><a href="#org1f0fec3">1.3. Root Locus</a></li>
<li><a href="#orgc967e9c">1.4. Analytical Model</a></li>
<li><a href="#orgb2d3e3a">1.5. Analytical Analysis</a></li>
</ul>
</li>
<li><a href="#org51b7142">2. Rotating X-Y platform</a>
<ul>
<li><a href="#orgf847a9d">2.1. Parameters</a></li>
<li><a href="#orga0c7065">2.2. Identification</a></li>
<li><a href="#org0815f8b">2.3. Root Locus</a></li>
<li><a href="#org6964694">2.4. Analysis</a></li>
</ul>
</li>
<li><a href="#org630b8fc">3. Stewart Platform with Amplified Actuators</a>
<ul>
<li><a href="#org0b7c221">3.1. Initialization</a></li>
<li><a href="#orgff8fe24">3.2. Identification</a></li>
<li><a href="#org58ae516">3.3. Controller Design</a></li>
<li><a href="#org1c125d1">3.4. Effect of the Low Authority Control on the Primary Plant</a></li>
<li><a href="#org3afc321">3.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
<li><a href="#orgface252">3.6. Optimal Stiffnesses</a></li>
</ul>
</li>
</ul>
</div>
</div>
<p>
The presented model is based on <a class='org-ref-reference' href="#souleille18_concep_activ_mount_space_applic">souleille18_concep_activ_mount_space_applic</a>.
</p>
<p>
The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure <a href="#org9bfac50">1</a>).
The parameters are shown in the table below.
</p>
<div id="org9bfac50" class="figure">
<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Parameters used for the model of the APA 100M</caption>
<colgroup>
<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">Value</th>
<th scope="col" class="org-left">Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(m\)</td>
<td class="org-left">\(1\,[kg]\)</td>
<td class="org-left">Payload mass</td>
</tr>
<tr>
<td class="org-left">\(k_e\)</td>
<td class="org-left">\(4.8\,[N/\mu m]\)</td>
<td class="org-left">Stiffness used to adjust the pole of the isolator</td>
</tr>
<tr>
<td class="org-left">\(k_1\)</td>
<td class="org-left">\(0.96\,[N/\mu m]\)</td>
<td class="org-left">Stiffness of the metallic suspension when the stack is removed</td>
</tr>
<tr>
<td class="org-left">\(k_a\)</td>
<td class="org-left">\(65\,[N/\mu m]\)</td>
<td class="org-left">Stiffness of the actuator</td>
</tr>
<tr>
<td class="org-left">\(c_1\)</td>
<td class="org-left">\(10\,[N/(m/s)]\)</td>
<td class="org-left">Added viscous damping</td>
</tr>
</tbody>
</table>
<div id="outline-container-orga1734d6" class="outline-2">
<h2 id="orga1734d6"><span class="section-number-2">1</span> Simplified Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org9543c57" class="outline-3">
<h3 id="org9543c57"><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]
ke = 4.8e6; % [N/m]
ce = 5; % [N/(m/s)]
me = 0.001; % [kg]
k1 = 0.96e6; % [N/m]
c1 = 10; % [N/(m/s)]
ka = 65e6; % [N/m]
ca = 5; % [N/(m/s)]
ma = 0.001; % [kg]
h = 0.2; % [m]
</pre>
</div>
<p>
IFF Controller:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kiff = -8000/s;
</pre>
</div>
</div>
</div>
<div id="outline-container-org20bf7c7" class="outline-3">
<h3 id="org20bf7c7"><span class="section-number-3">1.2</span> Identification</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Identification in open-loop.
</p>
<div class="org-src-container">
<pre class="src src-matlab">%% Name of the Simulink File
mdl = 'amplified_piezo_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion
io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
G = linearize(mdl, io, 0);
G.InputName = {'w', 'f', 'F'};
G.OutputName = {'Fs', 'x1'};
</pre>
</div>
<p>
Identification in closed-loop.
</p>
<div class="org-src-container">
<pre class="src src-matlab">%% Name of the Simulink File
mdl = 'amplified_piezo_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion
io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
Giff = linearize(mdl, io, 0);
Giff.InputName = {'w', 'f', 'F'};
Giff.OutputName = {'Fs', 'x1'};
</pre>
</div>
<div id="org3c557c6" class="figure">
<p><img src="figs/amplified_piezo_tf_ol_and_cl.png" alt="amplified_piezo_tf_ol_and_cl.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)</p>
</div>
</div>
</div>
<div id="outline-container-org1f0fec3" class="outline-3">
<h3 id="org1f0fec3"><span class="section-number-3">1.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-1-3">
<div id="org6370599" class="figure">
<p><img src="figs/amplified_piezo_root_locus.png" alt="amplified_piezo_root_locus.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Root Locus</p>
</div>
</div>
</div>
<div id="outline-container-orgc967e9c" class="outline-3">
<h3 id="orgc967e9c"><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="orgf78178e" 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-orgb2d3e3a" class="outline-3">
<h3 id="orgb2d3e3a"><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-org51b7142" class="outline-2">
<h2 id="org51b7142"><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-orgf847a9d" class="outline-3">
<h3 id="orgf847a9d"><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]
ke = 4.8e6; % [N/m]
ce = 5; % [N/(m/s)]
me = 0.001; % [kg]
k1 = 0.96e6; % [N/m]
c1 = 10; % [N/(m/s)]
ka = 65e6; % [N/m]
ca = 5; % [N/(m/s)]
ma = 0.001; % [kg]
h = 0.2; % [m]
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Kiff = tf(0);
</pre>
</div>
</div>
</div>
<div id="outline-container-orga0c7065" class="outline-3">
<h3 id="orga0c7065"><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:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ws = 2*pi*[0, 1, 10, 100];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Gs = {zeros(length(Ws), 1)};
</pre>
</div>
<p>
Identification in open-loop.
</p>
<div class="org-src-container">
<pre class="src src-matlab">%% Name of the Simulink File
mdl = 'amplified_piezo_xy_rotating_stage';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
for i = 1:length(Ws)
ws = Ws(i);
G = linearize(mdl, io, 0);
G.InputName = {'fx', 'fy'};
G.OutputName = {'Fsx', 'Fsy'};
Gs(i) = {G};
end
</pre>
</div>
<div id="org49fe0d0" 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 5: </span>Transfer function matrix from forces to force sensors for multiple rotation speed</p>
</div>
</div>
</div>
<div id="outline-container-org0815f8b" class="outline-3">
<h3 id="org0815f8b"><span class="section-number-3">2.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-2-3">
<div id="orgbf543c5" 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 6: </span>Root locus for 3 rotating speed</p>
</div>
</div>
</div>
<div id="outline-container-org6964694" class="outline-3">
<h3 id="org6964694"><span class="section-number-3">2.4</span> Analysis</h3>
<div class="outline-text-3" id="text-2-4">
<p>
The negative stiffness induced by the rotation is equal to \(m \omega_0^2\).
Thus, the maximum rotation speed where IFF can be applied is:
\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
</p>
<p>
Let&rsquo;s verify that.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ws = 2*pi*[140, 160];
</pre>
</div>
<p>
Identification
</p>
<div class="org-src-container">
<pre class="src src-matlab">%% Name of the Simulink File
mdl = 'amplified_piezo_xy_rotating_stage';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
for i = 1:length(Ws)
ws = Ws(i);
G = linearize(mdl, io, 0);
G.InputName = {'fx', 'fy'};
G.OutputName = {'Fsx', 'Fsy'};
Gs(i) = {G};
end
</pre>
</div>
<div id="org1b0c04f" 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 7: </span>Root Locus for the two considered rotation speed. For the red curve, the system is unstable.</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org630b8fc" class="outline-2">
<h2 id="org630b8fc"><span class="section-number-2">3</span> Stewart Platform with Amplified Actuators</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-org0b7c221" class="outline-3">
<h3 id="org0b7c221"><span class="section-number-3">3.1</span> Initialization</h3>
<div class="outline-text-3" id="text-3-1">
<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-iff');
</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-orgff8fe24" class="outline-3">
<h3 id="orgff8fe24"><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));
Kiff = 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('actuator', 'amplified');
</pre>
</div>
</div>
</div>
<div id="outline-container-org58ae516" class="outline-3">
<h3 id="org58ae516"><span class="section-number-3">3.3</span> Controller Design</h3>
<div class="outline-text-3" id="text-3-3">
<div id="orgd36ec15" 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 8: </span>Dynamics for the Integral Force Feedback for three payload masses</p>
</div>
<div id="org7816aa3" 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 9: </span>Root Locus for the IFF control for three payload masses</p>
</div>
<p>
Damping as function of the gain
</p>
<div id="orgfe47a62" 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 10: </span>Damping ratio of the poles as a function of the IFF gain</p>
</div>
<p>
Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kiff = -1e4/s*eye(6);
</pre>
</div>
</div>
</div>
<div id="outline-container-org1c125d1" class="outline-3">
<h3 id="org1c125d1"><span class="section-number-3">3.4</span> Effect of the Low Authority Control on the Primary Plant</h3>
<div class="outline-text-3" id="text-3-4">
<div id="orgd199337" 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 11: </span>Primary plant in the task space with (dashed) and without (solid) IFF</p>
</div>
<div id="org7b6b3d9" 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 12: </span>Primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
</div>
<div id="org0bd0e56" 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 13: </span>Coupling in the primary plant in the task with (dashed) and without (solid) IFF</p>
</div>
<div id="org1358a80" 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 14: </span>Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF</p>
</div>
</div>
</div>
<div id="outline-container-org3afc321" class="outline-3">
<h3 id="org3afc321"><span class="section-number-3">3.5</span> Effect of the Low Authority Control on the Sensibility to Disturbances</h3>
<div class="outline-text-3" id="text-3-5">
<div id="org71868a4" class="figure">
<p><img src="figs/amplified_piezo_iff_disturbances.png" alt="amplified_piezo_iff_disturbances.png" />
</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-orgface252" class="outline-3">
<h3 id="orgface252"><span class="section-number-3">3.6</span> Optimal Stiffnesses</h3>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-25 lun. 11:05</p>
</div>
</body>
</html>