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#+TITLE: Amplified Piezoelectric Stack Actuator
#+SETUPFILE: ./setup/org-setup-file.org
* Introduction :ignore:
The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
The parameters are shown in the table below.
#+name: fig:souleille18_model_piezo
#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
[[file:./figs/souleille18_model_piezo.png]]
#+caption: Parameters used for the model of the APA 100M
| | Value | Meaning |
|-------+-------------------+----------------------------------------------------------------|
| $m$ | $1\,[kg]$ | Payload mass |
| $k_e$ | $4.8\,[N/\mu m]$ | Stiffness used to adjust the pole of the isolator |
| $k_1$ | $0.96\,[N/\mu m]$ | Stiffness of the metallic suspension when the stack is removed |
| $k_a$ | $65\,[N/\mu m]$ | Stiffness of the actuator |
| $c_1$ | $10\,[N/(m/s)]$ | Added viscous damping |
* Simplified Model
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+BEGIN_SRC matlab
simulinkproject('../');
#+END_SRC
#+begin_src matlab
open 'amplified_piezo_model.slx'
#+end_src
** Parameters
#+begin_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]
#+end_src
IFF Controller:
#+begin_src matlab
Kiff = -8000/s;
#+end_src
** Identification
Identification in open-loop.
#+begin_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'};
#+end_src
Identification in closed-loop.
#+begin_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'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
ax1 = subplot(2, 3, 1);
title('$\displaystyle \frac{x_1}{w}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
ax2 = subplot(2, 3, 2);
title('$\displaystyle \frac{x_1}{f}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax3 = subplot(2, 3, 3);
title('$\displaystyle \frac{x_1}{F}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax4 = subplot(2, 3, 4);
title('$\displaystyle \frac{F_s}{w}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
ax5 = subplot(2, 3, 5);
title('$\displaystyle \frac{F_s}{f}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax6 = subplot(2, 3, 6);
title('$\displaystyle \frac{F_s}{F}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_tf_ol_and_cl.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_tf_ol_and_cl
#+caption: Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)
#+RESULTS:
[[file:figs/amplified_piezo_tf_ol_and_cl.png]]
** Root Locus
#+begin_src matlab :exports none :post
figure;
gains = logspace(1, 6, 500);
hold on;
plot(real(pole(G('Fs', 'f'))), imag(pole(G('Fs', 'f'))), 'kx');
plot(real(tzero(G('Fs', 'f'))), imag(tzero(G('Fs', 'f'))), 'ko');
for k = 1:length(gains)
cl_poles = pole(feedback(G('Fs', 'f'), -gains(k)/s));
plot(real(cl_poles), imag(cl_poles), 'k.');
end
hold off;
axis square;
xlim([-2500, 100]); ylim([0, 2600]);
xlabel('Real Part'); ylabel('Imaginary Part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:amplified_piezo_root_locus
#+caption: Root Locus
#+RESULTS:
[[file:figs/amplified_piezo_root_locus.png]]
** Analytical Model
If we apply the Newton's second law of motion on the top mass, we obtain:
\[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \]
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:
\[ x_e = \frac{k_a}{k_e + k_a} w + \frac{1}{k_e + k_a} f + \frac{k_e}{k_e + k_a} x_1 \]
Re-injecting that into the previous equations gives:
\[ x_1 = F \frac{1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + f \frac{\frac{k_e}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \]
\[ F_s = - F \frac{\frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_e k_a}{k_e + k_a} \Big( \frac{ms^2}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) - f \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
#+begin_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'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
ax1 = subplot(2, 3, 1);
title('$\displaystyle \frac{x_1}{w}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('x1', 'w'), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
ax2 = subplot(2, 3, 2);
title('$\displaystyle \frac{x_1}{f}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('x1', 'f'), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax3 = subplot(2, 3, 3);
title('$\displaystyle \frac{x_1}{F}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('x1', 'F'), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax4 = subplot(2, 3, 4);
title('$\displaystyle \frac{F_s}{w}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'w'), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
ax5 = subplot(2, 3, 5);
title('$\displaystyle \frac{F_s}{f}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'f'), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax6 = subplot(2, 3, 6);
title('$\displaystyle \frac{F_s}{F}$')
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'F'), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_simscape_analytical.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:comp_simscape_analytical
#+caption: Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed)
#+RESULTS:
[[file:figs/comp_simscape_analytical.png]]
** Analytical Analysis
For Integral Force Feedback Control, the plant is:
\[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
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:
\[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
It has two complex conjugate zeros at:
\[ z = \pm j \sqrt{\frac{k_1}{m}} \]
And two complex conjugate poles at:
\[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \]
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 $k_e + k_a \approx \text{cst}$).
* Rotating X-Y platform
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+BEGIN_SRC matlab
simulinkproject('../');
#+END_SRC
#+begin_src matlab
open 'amplified_piezo_xy_rotating_stage.slx'
#+end_src
** Parameters
#+begin_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]
#+end_src
#+begin_src matlab
Kiff = tf(0);
#+end_src
** Identification
Rotating speed in rad/s:
#+begin_src matlab
Ws = 2*pi*[0, 1, 10, 100];
#+end_src
#+begin_src matlab
Gs = {zeros(length(Ws), 1)};
#+end_src
Identification in open-loop.
#+begin_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
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
ax1 = subplot(2, 2, 1);
title('$\displaystyle \frac{F_{s,x}}{f_x}$')
hold on;
for i = 1:length(Ws)
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fx'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
ax2 = subplot(2, 2, 2);
title('$\displaystyle \frac{F_{s,y}}{f_x}$')
hold on;
for i = 1:length(Ws)
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fx'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax3 = subplot(2, 2, 3);
title('$\displaystyle \frac{F_{s,x}}{f_y}$')
hold on;
for i = 1:length(Ws)
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fy'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
ax4 = subplot(2, 2, 4);
title('$\displaystyle \frac{F_{s,y}}{f_y}$')
hold on;
for i = 1:length(Ws)
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fy'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplitifed_piezo_xy_rotation_plant_iff.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplitifed_piezo_xy_rotation_plant_iff
#+caption: Transfer function matrix from forces to force sensors for multiple rotation speed
#+RESULTS:
[[file:figs/amplitifed_piezo_xy_rotation_plant_iff.png]]
** Root Locus
#+begin_src matlab :exports none :post
figure;
gains = logspace(1, 6, 500);
hold on;
for i = 1:length(Ws)
set(gca,'ColorOrderIndex',i);
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
set(gca,'ColorOrderIndex',i);
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
for k = 1:length(gains)
set(gca,'ColorOrderIndex',i);
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
plot(real(cl_poles), imag(cl_poles), '.');
end
end
hold off;
axis square;
xlim([-2900, 100]); ylim([0, 3000]);
xlabel('Real Part'); ylabel('Imaginary Part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_xy_rotation_root_locus.pdf', 'width', 'tall', 'height', 'wide');
#+end_src
#+name: fig:amplified_piezo_xy_rotation_root_locus
#+caption: Root locus for 3 rotating speed
#+RESULTS:
[[file:figs/amplified_piezo_xy_rotation_root_locus.png]]
** Analysis
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:
\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
Let's verify that.
#+begin_src matlab
Ws = 2*pi*[140, 160];
#+end_src
#+begin_src matlab :exports none
Gs = {zeros(length(Ws), 1)};
#+end_src
Identification
#+begin_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
#+end_src
#+begin_src matlab :exports none
figure;
gains = logspace(1, 6, 500);
hold on;
for i = 1:length(Ws)
set(gca,'ColorOrderIndex',i);
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
set(gca,'ColorOrderIndex',i);
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
for k = 1:length(gains)
set(gca,'ColorOrderIndex',i);
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
plot(real(cl_poles), imag(cl_poles), '.');
end
end
hold off;
axis square;
xlim([-100, 50]); ylim([0, 150]);
xlabel('Real Part'); ylabel('Imaginary Part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_xy_rotating_unstable_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:amplified_piezo_xy_rotating_unstable_root_locus
#+caption: Root Locus for the two considered rotation speed. For the red curve, the system is unstable.
#+RESULTS:
[[file:figs/amplified_piezo_xy_rotating_unstable_root_locus.png]]
* Stewart Platform with Amplified Actuators
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
simulinkproject('../');
#+end_src
#+begin_src matlab
open('nass_model.slx')
#+end_src
** Initialization
#+begin_src matlab
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'hac-iff');
#+end_src
We set the stiffness of the payload fixation:
#+begin_src matlab
Kp = 1e8; % [N/m]
#+end_src
** Identification
#+begin_src matlab
K = tf(zeros(6));
Kiff = tf(zeros(6));
#+end_src
We identify the system for the following payload masses:
#+begin_src matlab
Ms = [1, 10, 50];
#+end_src
#+begin_src matlab :exports none
Gm_iff = {zeros(length(Ms), 1)};
#+end_src
The nano-hexapod has the following leg's stiffness and damping.
#+begin_src matlab
initializeNanoHexapod('actuator', 'amplified');
#+end_src
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'nass_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
#+end_src
#+begin_src matlab :exports none
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_iff = linearize(mdl, io);
G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
Gm_iff(i) = {G_iff};
end
#+end_src
** Controller Design
#+begin_src matlab :exports none
freqs = logspace(-1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ms)
plot(freqs, abs(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:length(Ms)
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz')))), ...
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-270, 90]);
yticks([-360:90:360]);
legend('location', 'northeast');
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_loop_gain.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_iff_loop_gain
#+caption: Dynamics for the Integral Force Feedback for three payload masses
#+RESULTS:
[[file:figs/amplified_piezo_iff_loop_gain.png]]
#+begin_src matlab :exports none
figure;
gains = logspace(2, 5, 300);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(real(pole(Gm_iff{i})), imag(pole(Gm_iff{i})), 'x', ...
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
set(gca,'ColorOrderIndex',i);
plot(real(tzero(Gm_iff{i})), imag(tzero(Gm_iff{i})), 'o', ...
'HandleVisibility', 'off');
for k = 1:length(gains)
set(gca,'ColorOrderIndex',i);
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
plot(real(cl_poles), imag(cl_poles), '.', ...
'HandleVisibility', 'off');
end
end
hold off;
axis square;
xlim([-400, 10]); ylim([0, 500]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:amplified_piezo_iff_root_locus
#+caption: Root Locus for the IFF control for three payload masses
#+RESULTS:
[[file:figs/amplified_piezo_iff_root_locus.png]]
Damping as function of the gain
#+begin_src matlab :exports none
c1 = [ 0 0.4470 0.7410]; % Blue
c2 = [0.8500 0.3250 0.0980]; % Orange
c3 = [0.9290 0.6940 0.1250]; % Yellow
c4 = [0.4940 0.1840 0.5560]; % Purple
c5 = [0.4660 0.6740 0.1880]; % Green
c6 = [0.3010 0.7450 0.9330]; % Light Blue
c7 = [0.6350 0.0780 0.1840]; % Red
colors = [c1; c2; c3; c4; c5; c6; c7];
figure;
gains = logspace(2, 5, 100);
hold on;
for i = 1:length(Ms)
for k = 1:length(gains)
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
set(gca,'ColorOrderIndex',i);
plot(gains(k), sin(-pi/2 + angle(cl_poles)), '.', 'color', colors(i, :));
end
end
hold off;
xlabel('IFF Gain'); ylabel('Modal Damping');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylim([0, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/amplified_piezo_iff_damping_gain.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:amplified_piezo_iff_damping_gain
#+caption: Damping ratio of the poles as a function of the IFF gain
#+RESULTS:
[[file:figs/amplified_piezo_iff_damping_gain.png]]
Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
#+begin_src matlab
Kiff = -1e4/s*eye(6);
#+end_src
** Effect of the Low Authority Control on the Primary Plant
*** Introduction :ignore:
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'nass_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
#+end_src
#+begin_src matlab :exports none
load('mat/stages.mat', 'nano_hexapod');
#+end_src
*** Identification of the undamped plant :ignore:
#+begin_src matlab :exports none
Kiff_backup = Kiff;
Kiff = tf(zeros(6));
#+end_src
#+begin_src matlab :exports none
G_x = {zeros(length(Ms), 1)};
G_l = {zeros(length(Ms), 1)};
#+end_src
#+begin_src matlab :exports none
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 = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
Gx = -G*inv(nano_hexapod.kinematics.J');
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
G_x(i) = {Gx};
Gl = -nano_hexapod.kinematics.J*G;
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
G_l(i) = {Gl};
end
#+end_src
#+begin_src matlab :exports none
Kiff = Kiff_backup;
#+end_src
*** Identification of the damped plant :ignore:
#+begin_src matlab :exports none
Gm_x = {zeros(length(Ms), 1)};
Gm_l = {zeros(length(Ms), 1)};
#+end_src
#+begin_src matlab :exports none
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 = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
Gx = -G*inv(nano_hexapod.kinematics.J');
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Gm_x(i) = {Gx};
Gl = -nano_hexapod.kinematics.J*G;
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
Gm_l(i) = {Gl};
end
#+end_src
*** Effect of the Damping on the plant diagonal dynamics :ignore:
#+begin_src matlab :exports none
freqs = logspace(0, 3, 5000);
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(1, 1), freqs, 'Hz'))), '--');
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(2, 2), freqs, 'Hz'))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')
ax2 = subplot(2, 2, 2);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(3, 3), freqs, 'Hz'))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('$\mathcal{X}_z/\mathcal{F}_z$')
ax3 = subplot(2, 2, 3);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz')))));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz')))));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(1, 1), freqs, 'Hz')))), '--');
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(2, 2), freqs, 'Hz')))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-270, 90]);
yticks([-360:90:360]);
ax4 = subplot(2, 2, 4);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))), ...
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(3, 3), freqs, 'Hz')))), '--', ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-270, 90]);
yticks([-360:90:360]);
legend('location', 'southwest');
linkaxes([ax1,ax2,ax3,ax4],'x');
#+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
freqs = logspace(0, 3, 5000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(1, 1), freqs, 'Hz'))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))), ...
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_l{i}(1, 1), freqs, 'Hz')))), '--', ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-270, 90]);
yticks([-360:90:360]);
legend('location', 'southwest');
linkaxes([ax1,ax2],'x');
#+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:
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_x{1}(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
plot(freqs, abs(squeeze(freqresp(Gm_x{1}(i, j), freqs, 'Hz'))), '--', 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G_x{1}(1, 1), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gm_x{1}(1, 1), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-12, inf]);
#+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
freqs = logspace(0, 3, 1000);
figure;
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_l{1}(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
plot(freqs, abs(squeeze(freqresp(Gm_l{1}(i, j), freqs, 'Hz'))), '--', 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G_l{1}(1, 1), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gm_l{1}(1, 1), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, inf]);
#+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
** Optimal Stiffnesses
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