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Analysis of IFF with amplified actuators

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Thomas Dehaeze 2020-05-20 16:55:50 +02:00
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-05-20 mer. 16:41 --> <!-- 2020-05-20 mer. 16:56 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Amplified Piezoelectric Stack Actuator</title> <title>Amplified Piezoelectric Stack Actuator</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -36,25 +36,26 @@
<ul> <ul>
<li><a href="#org996fd7c">1. Simplified Model</a> <li><a href="#org996fd7c">1. Simplified Model</a>
<ul> <ul>
<li><a href="#org0dc544d">1.1. Parameters</a></li> <li><a href="#org1e2b26f">1.1. Parameters</a></li>
<li><a href="#org08e3567">1.2. Identification</a></li> <li><a href="#orgac18acf">1.2. Identification</a></li>
<li><a href="#orgbba342e">1.3. Root Locus</a></li> <li><a href="#orgd6c324c">1.3. Root Locus</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgf1a765f">2. Rotating X-Y platform</a> <li><a href="#orgf1a765f">2. Rotating X-Y platform</a>
<ul> <ul>
<li><a href="#org1e2b26f">2.1. Parameters</a></li> <li><a href="#orgd4866c5">2.1. Parameters</a></li>
<li><a href="#orgb0841c5">2.2. Identification</a></li> <li><a href="#orgcfc57a7">2.2. Identification</a></li>
<li><a href="#orgd6c324c">2.3. Root Locus</a></li> <li><a href="#org8d3f9bd">2.3. Root Locus</a></li>
<li><a href="#org069f401">2.4. Analysis</a></li> <li><a href="#org069f401">2.4. Analysis</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org3c74f7f">3. Stewart Platform with Amplified Actuators</a> <li><a href="#org3c74f7f">3. Stewart Platform with Amplified Actuators</a>
<ul> <ul>
<li><a href="#org5a7c6dc">3.1. Initialization</a></li> <li><a href="#org5a7c6dc">3.1. Initialization</a></li>
<li><a href="#orgac18acf">3.2. Identification</a></li> <li><a href="#orgf0cb0e7">3.2. Identification</a></li>
<li><a href="#org14c7063">3.3. Controller Design</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="#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>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -131,8 +132,8 @@ The parameters are shown in the table below.
<h2 id="org996fd7c"><span class="section-number-2">1</span> Simplified Model</h2> <h2 id="org996fd7c"><span class="section-number-2">1</span> Simplified Model</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org0dc544d" class="outline-3"> <div id="outline-container-org1e2b26f" class="outline-3">
<h3 id="org0dc544d"><span class="section-number-3">1.1</span> Parameters</h3> <h3 id="org1e2b26f"><span class="section-number-3">1.1</span> Parameters</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">m = 1; % [kg] <pre class="src src-matlab">m = 1; % [kg]
@ -162,8 +163,8 @@ IFF Controller:
</div> </div>
</div> </div>
<div id="outline-container-org08e3567" class="outline-3"> <div id="outline-container-orgac18acf" class="outline-3">
<h3 id="org08e3567"><span class="section-number-3">1.2</span> Identification</h3> <h3 id="orgac18acf"><span class="section-number-3">1.2</span> Identification</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
Identification in open-loop. Identification in open-loop.
@ -218,8 +219,8 @@ Giff.OutputName = {'Fs', 'x1'};
</div> </div>
</div> </div>
<div id="outline-container-orgbba342e" class="outline-3"> <div id="outline-container-orgd6c324c" class="outline-3">
<h3 id="orgbba342e"><span class="section-number-3">1.3</span> Root Locus</h3> <h3 id="orgd6c324c"><span class="section-number-3">1.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<div id="org85cd6e5" class="figure"> <div id="org85cd6e5" class="figure">
@ -235,8 +236,8 @@ Giff.OutputName = {'Fs', 'x1'};
<h2 id="orgf1a765f"><span class="section-number-2">2</span> Rotating X-Y platform</h2> <h2 id="orgf1a765f"><span class="section-number-2">2</span> Rotating X-Y platform</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
</div> </div>
<div id="outline-container-org1e2b26f" class="outline-3"> <div id="outline-container-orgd4866c5" class="outline-3">
<h3 id="org1e2b26f"><span class="section-number-3">2.1</span> Parameters</h3> <h3 id="orgd4866c5"><span class="section-number-3">2.1</span> Parameters</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">m = 1; % [kg] <pre class="src src-matlab">m = 1; % [kg]
@ -263,8 +264,8 @@ h = 0.2; % [m]
</div> </div>
</div> </div>
<div id="outline-container-orgb0841c5" class="outline-3"> <div id="outline-container-orgcfc57a7" class="outline-3">
<h3 id="orgb0841c5"><span class="section-number-3">2.2</span> Identification</h3> <h3 id="orgcfc57a7"><span class="section-number-3">2.2</span> Identification</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
Rotating speed in rad/s: Rotating speed in rad/s:
@ -313,8 +314,8 @@ end
</div> </div>
</div> </div>
<div id="outline-container-orgd6c324c" class="outline-3"> <div id="outline-container-org8d3f9bd" class="outline-3">
<h3 id="orgd6c324c"><span class="section-number-3">2.3</span> Root Locus</h3> <h3 id="org8d3f9bd"><span class="section-number-3">2.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<div id="orgccd3396" class="figure"> <div id="orgccd3396" class="figure">
@ -412,8 +413,8 @@ We set the stiffness of the payload fixation:
</div> </div>
</div> </div>
<div id="outline-container-orgac18acf" class="outline-3"> <div id="outline-container-orgf0cb0e7" class="outline-3">
<h3 id="orgac18acf"><span class="section-number-3">3.2</span> Identification</h3> <h3 id="orgf0cb0e7"><span class="section-number-3">3.2</span> Identification</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">K = tf(zeros(6)); <pre class="src src-matlab">K = tf(zeros(6));
@ -479,12 +480,57 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
<div id="outline-container-org043ce40" class="outline-3"> <div id="outline-container-org043ce40" class="outline-3">
<h3 id="org043ce40"><span class="section-number-3">3.4</span> Effect of the Low Authority Control on the Primary Plant</h3> <h3 id="org043ce40"><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="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>
</div>
<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>
</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>
</div>
<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>
</div>
</div>
</div>
<div id="outline-container-orgbc2f246" class="outline-3">
<h3 id="orgbc2f246"><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="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>
</div>
<div class="important">
</div>
</div>
</div> </div>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-20 mer. 16:41</p> <p class="date">Created: 2020-05-20 mer. 16:56</p>
</div> </div>
</body> </body>
</html> </html>

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@ -657,8 +657,8 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
*** Identification of the undamped plant :ignore: *** Identification of the undamped plant :ignore:
#+begin_src matlab :exports none #+begin_src matlab :exports none
Kdvf_backup = Kdvf; Kiff_backup = Kiff;
Kdvf = tf(zeros(6)); Kiff = tf(zeros(6));
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
@ -687,7 +687,7 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
#+end_src #+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
Kdvf = Kdvf_backup; Kiff = Kiff_backup;
#+end_src #+end_src
*** Identification of the damped plant :ignore: *** Identification of the damped plant :ignore:
@ -790,6 +790,16 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
linkaxes([ax1,ax2,ax3,ax4],'x'); linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_plant_damped_X.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_iff_plant_damped_X
#+caption: Primary plant in the task space with (dashed) and without (solid) IFF
#+RESULTS:
[[file:figs/amplified_piezo_iff_plant_damped_X.png]]
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(0, 3, 5000); freqs = logspace(0, 3, 5000);
@ -827,6 +837,16 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
linkaxes([ax1,ax2],'x'); linkaxes([ax1,ax2],'x');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_damped_plant_L.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_iff_damped_plant_L
#+caption: Primary plant in the space of the legs with (dashed) and without (solid) IFF
#+RESULTS:
[[file:figs/amplified_piezo_iff_damped_plant_L.png]]
*** Effect of the Damping on the coupling dynamics :ignore: *** Effect of the Damping on the coupling dynamics :ignore:
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(0, 3, 1000); freqs = logspace(0, 3, 1000);
@ -849,6 +869,16 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
ylim([1e-12, inf]); ylim([1e-12, inf]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_damped_coupling_X.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_iff_damped_coupling_X
#+caption: Coupling in the primary plant in the task with (dashed) and without (solid) IFF
#+RESULTS:
[[file:figs/amplified_piezo_iff_damped_coupling_X.png]]
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(0, 3, 1000); freqs = logspace(0, 3, 1000);
@ -869,3 +899,145 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, inf]); ylim([1e-9, inf]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_damped_coupling_L.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_iff_damped_coupling_L
#+caption: Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF
#+RESULTS:
[[file:figs/amplified_piezo_iff_damped_coupling_L.png]]
** Effect of the Low Authority Control on the Sensibility to Disturbances
*** Introduction :ignore:
*** Identification :ignore:
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'nass_model';
%% Micro-Hexapod
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz'); io_i = io_i + 1; % Z Ground motion
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fty_z'); io_i = io_i + 1; % Parasitic force Ty - Z
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_z'); io_i = io_i + 1; % Parasitic force Rz - Z
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fd'); io_i = io_i + 1; % Direct forces
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
#+end_src
#+begin_src matlab :exports none
Kiff_backup = Kiff;
Kiff = tf(zeros(6));
#+end_src
#+begin_src matlab :exports none
Gd = {zeros(length(Ms), 1)};
for i = 1:length(Ms)
initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));
%% Run the linearization
G = linearize(mdl, io);
G.InputName = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
Gd(i) = {G};
end
#+end_src
#+begin_src matlab :exports none
Kiff = Kiff_backup;
#+end_src
#+begin_src matlab :exports none
Gd_iff = {zeros(length(Ms), 1)};
for i = 1:length(Ms)
initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));
%% Run the linearization
G = linearize(mdl, io);
G.InputName = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
Gd_iff(i) = {G};
end
#+end_src
*** Results :ignore:
#+begin_src matlab :exports none
freqs = logspace(0, 3, 5000);
figure;
subplot(2, 2, 1);
title('$D_{w,z}$ to $E_z$');
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), ...
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Dwz'), freqs, 'Hz'))), '--', ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');
subplot(2, 2, 2);
title('$F_{dz}$ to $E_z$');
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]');
subplot(2, 2, 3);
title('$F_{T_y,z}$ to $E_z$');
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Fty_z'), freqs, 'Hz'))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
subplot(2, 2, 4);
title('$F_{R_z,z}$ to $E_z$');
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_disturbances.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_iff_disturbances
#+caption: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied
#+RESULTS:
[[file:figs/amplified_piezo_iff_disturbances.png]]
*** Conclusion :ignore:
#+begin_important
#+end_important