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<title>Amplified Piezoelectric Stack Actuator</title>
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</div><div id="content">
<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="#org8dcb67f">1. Simplified Model</a>
<ul>
<li><a href="#org3b87947">1.1. Parameters</a></li>
<li><a href="#org81d8607">1.2. Identification</a></li>
<li><a href="#orged55dcf">1.3. Root Locus</a></li>
<li><a href="#org02c2e3f">1.4. Analytical Model</a></li>
</ul>
</li>
<li><a href="#org7b08cc3">2. Rotating X-Y platform</a>
<ul>
<li><a href="#org4add324">2.1. Parameters</a></li>
<li><a href="#org8192181">2.2. Identification</a></li>
<li><a href="#org7675381">2.3. Root Locus</a></li>
<li><a href="#org8634d85">2.4. Analysis</a></li>
</ul>
</li>
<li><a href="#orga0a1938">3. Stewart Platform with Amplified Actuators</a>
<ul>
<li><a href="#org1eb7bf5">3.1. Initialization</a></li>
<li><a href="#org31d938a">3.2. Identification</a></li>
<li><a href="#org4dbdefd">3.3. Controller Design</a></li>
<li><a href="#orga2cff39">3.4. Effect of the Low Authority Control on the Primary Plant</a></li>
<li><a href="#orgf448298">3.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
<li><a href="#org781f74e">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="#orgb0a1ae8">1</a>).
The parameters are shown in the table below.
</p>
<div id="orgb0a1ae8" 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-org8dcb67f" class="outline-2">
<h2 id="org8dcb67f"><span class="section-number-2">1</span> Simplified Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org3b87947" class="outline-3">
<h3 id="org3b87947"><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-org81d8607" class="outline-3">
<h3 id="org81d8607"><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="org6546639" 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-orged55dcf" class="outline-3">
<h3 id="orged55dcf"><span class="section-number-3">1.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-1-3">
<div id="org8832ee2" 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-org02c2e3f" class="outline-3">
<h3 id="org02c2e3f"><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="orgc2707e4" 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>
<div id="outline-container-org7b08cc3" class="outline-2">
<h2 id="org7b08cc3"><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-org4add324" class="outline-3">
<h3 id="org4add324"><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-org8192181" class="outline-3">
<h3 id="org8192181"><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="org89aafdf" 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-org7675381" class="outline-3">
<h3 id="org7675381"><span class="section-number-3">2.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-2-3">
<div id="org0e2ba71" 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-org8634d85" class="outline-3">
<h3 id="org8634d85"><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="orgb32e2be" 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-orga0a1938" class="outline-2">
<h2 id="orga0a1938"><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-org1eb7bf5" class="outline-3">
<h3 id="org1eb7bf5"><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-org31d938a" class="outline-3">
<h3 id="org31d938a"><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-org4dbdefd" class="outline-3">
<h3 id="org4dbdefd"><span class="section-number-3">3.3</span> Controller Design</h3>
<div class="outline-text-3" id="text-3-3">
<div id="org4b286db" 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="org85ffb18" 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="orgd336750" 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-orga2cff39" class="outline-3">
<h3 id="orga2cff39"><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="orgf3eee4b" 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="orgdabfde7" 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="orgf756d9c" 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="org1dc0d4a" 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-orgf448298" class="outline-3">
<h3 id="orgf448298"><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="org982cf72" 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-org781f74e" class="outline-3">
<h3 id="org781f74e"><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. 10:45</p>
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3
org/amplified_piezoelectric_stack.org
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@ -200,7 +200,7 @@ exportFig('figs/amplified_piezo_root_locus.pdf', 'width', 'wide', 'height', 'tal
#+RESULTS:
[[file:figs/amplified_piezo_root_locus.png]]
** Analytical Results
** Analytical Model
If we apply the Newton's second law of motion on the top mass, we obtain:
\[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \]
@ -542,7 +542,6 @@ Identification
open('nass_model.slx')
#+end_src
** Initialization
#+begin_src matlab
initializeGround();