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Add analysis of the damped plant with IFF

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Thomas Dehaeze 2020-02-04 18:36:27 +01:00
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@ -66,8 +66,12 @@ The disturbances are:
<<sec:undamped_system>> <<sec:undamped_system>>
** Introduction :ignore: ** Introduction :ignore:
We first look at the undamped system. In this section, we identify the dynamic of the system from forces applied in the nano-hexapod legs to the various sensors included in the nano-hexapod that could be use for Active Damping, namely:
The performance of this undamped system will be compared with the damped system using various techniques. - A relative motion sensor, measuring the relative displacement of each of the leg
- A force sensor measuring the total force transmitted to the top part of the leg in the direction of the leg
- A absolute velocity sensor measuring the absolute velocity of the top part of the leg in the direction of the leg
After that, a tomography experiment is simulation without any active damping techniques.
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -115,6 +119,7 @@ We set the references to zero.
initializeReferences(); initializeReferences();
#+end_src #+end_src
No disturbance is included in the system.
#+begin_src matlab #+begin_src matlab
initializeDisturbances('enable', false); initializeDisturbances('enable', false);
#+end_src #+end_src
@ -143,17 +148,19 @@ First, we identify the dynamics of the system using the =linearize= function.
%% Input/Output definition %% Input/Output definition
clear io; io_i = 1; clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fnl'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Fnl'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Vlm'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Vlm'); io_i = io_i + 1; % Absolute Velocity Outputs
io(io_i) = linio([mdl, '/Compute Error in NASS base'], 2, 'openoutput'); io_i = io_i + 1; % Metrology Outputs
%% Run the linearization %% Run the linearization
G = linearize(mdl, io, 0.5, options); G = linearize(mdl, io, 0.5, options);
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ... G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ... 'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'}; 'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6', ...
'Dxn', 'Dyn', 'Dzn', 'Rxn', 'Ryn', 'Rzn'};
#+end_src #+end_src
We then create transfer functions corresponding to the active damping plants. We then create transfer functions corresponding to the active damping plants.
@ -163,9 +170,16 @@ We then create transfer functions corresponding to the active damping plants.
G_ine = minreal(G({'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'})); G_ine = minreal(G({'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
#+end_src #+end_src
#+begin_src matlab
load('mat/stages.mat', 'nano_hexapod');
G_cart = minreal(G({'Dxn', 'Dyn', 'Dzn', 'Rxn', 'Ryn', 'Rzn'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*inv(nano_hexapod.J');
G_cart.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src
And we save them for further analysis. And we save them for further analysis.
#+begin_src matlab #+begin_src matlab
save('./active_damping/mat/undamped_plants.mat', 'G_iff', 'G_dvf', 'G_ine'); save('./active_damping/mat/undamped_plants.mat', 'G_iff', 'G_dvf', 'G_ine');
save('./active_damping/mat/cart_plants.mat', 'G_cart');
#+end_src #+end_src
*** Obtained Plants for Active Damping *** Obtained Plants for Active Damping
@ -207,7 +221,7 @@ And we save them for further analysis.
#+end_src #+end_src
#+NAME: fig:nass_active_damping_iff_plant #+NAME: fig:nass_active_damping_iff_plant
#+CAPTION: =G_iff=: IFF Plant ([[./figs/nass_active_damping_iff_plant.png][png]], [[./figs/nass_active_damping_iff_plant.pdf][pdf]]) #+CAPTION: =G_iff=: Transfer functions from forces applied in the actuators to the force sensor in each actuator ([[./figs/nass_active_damping_iff_plant.png][png]], [[./figs/nass_active_damping_iff_plant.pdf][pdf]])
[[file:figs/nass_active_damping_iff_plant.png]] [[file:figs/nass_active_damping_iff_plant.png]]
#+begin_src matlab :exports none #+begin_src matlab :exports none
@ -244,8 +258,8 @@ And we save them for further analysis.
#+end_src #+end_src
#+NAME: fig:nass_active_damping_dvf_plant #+NAME: fig:nass_active_damping_dvf_plant
#+CAPTION: =G_dvf=: Plant for Direct Velocity Feedback ([[./figs/nass_active_damping_dvf_plant.png][png]], [[./figs/nass_active_damping_dvf_plant.pdf][pdf]]) #+CAPTION: =G_dvf=: Transfer functions from forces applied in the actuators to the relative motion sensor in each actuator ([[./figs/nass_active_damping_dvf_plant.png][png]], [[./figs/nass_active_damping_dvf_plant.pdf][pdf]])
[[file:figs/nass_active_damping_ine_plant.png]] [[file:figs/nass_active_damping_dvf_plant.png]]
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(0, 3, 1000); freqs = logspace(0, 3, 1000);
@ -281,7 +295,7 @@ And we save them for further analysis.
#+end_src #+end_src
#+NAME: fig:nass_active_damping_inertial_plant #+NAME: fig:nass_active_damping_inertial_plant
#+CAPTION: =G_ine=: Inertial Feedback Plant ([[./figs/nass_active_damping_inertial_plant.png][png]], [[./figs/nass_active_damping_inertial_plant.pdf][pdf]]) #+CAPTION: =G_ine=: Transfer functions from forces applied in the actuators to the geophone located in each leg measuring the absolute velocity of the top part of the leg in the direction of the leg ([[./figs/nass_active_damping_inertial_plant.png][png]], [[./figs/nass_active_damping_inertial_plant.pdf][pdf]])
[[file:figs/nass_active_damping_inertial_plant.png]] [[file:figs/nass_active_damping_inertial_plant.png]]
** Tomography Experiment ** Tomography Experiment
@ -508,7 +522,8 @@ We identify the dynamics for the following sample mass.
ax2 = subplot(2, 1, 2); ax2 = subplot(2, 1, 2);
hold on; hold on;
for i = 1:length(Gm_iff) for i = 1:length(Gm_iff)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), 'DisplayName', sprintf('$M = %.0f$ [kg]', masses(i))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), ...
'DisplayName', sprintf('$M = %.0f$ [kg]', masses(i)));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
@ -1688,7 +1703,16 @@ Also, for the Inertial Sensor, a RHP zero may appear when the spindle is rotatin
#+end_note #+end_note
** Introduction :ignore: ** Introduction :ignore:
Integral Force Feedback is applied on the simscape model. Here, we study the use of *Integral Force Feedback* (IFF) to actively damp the resonances.
The IFF control is applied in a decentralized way: there is on controller for each leg.
The control architecture is represented in figure [[fig:iff_1dof]] where one of the 6 nano-hexapod legs is represented.
#+name: fig:iff_1dof
#+caption: Integral Force Feedback applied to a 1dof system
#+RESULTS:
[[file:figs/iff_1dof.png]]
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -1713,9 +1737,10 @@ Integral Force Feedback is applied on the simscape model.
** Control Design ** Control Design
*** Plant *** Plant
Let's load the previously indentified undamped plant: Let's load the previously identified undamped plant:
#+begin_src matlab #+begin_src matlab
load('./active_damping/mat/undamped_plants.mat', 'G_iff'); load('./active_damping/mat/undamped_plants.mat', 'G_iff');
load('./active_damping/mat/plants_variable.mat', 'masses', 'Gm_iff');
#+end_src #+end_src
Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure [[fig:iff_plant]]). Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure [[fig:iff_plant]]).
@ -1727,8 +1752,8 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to
ax1 = subplot(2, 1, 1); ax1 = subplot(2, 1, 1);
hold on; hold on;
for i=1:6 for i=1:length(masses)
plot(freqs, abs(squeeze(freqresp(G_iff(['Fnlm', num2str(i)], ['Fnl', num2str(i)]), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(-Gm_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@ -1736,14 +1761,16 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to
ax2 = subplot(2, 1, 2); ax2 = subplot(2, 1, 2);
hold on; hold on;
for i=1:6 for i=1:length(masses)
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff(['Fnlm', num2str(i)], ['Fnl', num2str(i)]), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(-Gm_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), ...
'DisplayName', sprintf('$M = %.0f$ [kg]', masses(i)));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]); ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]); yticks([-180, -90, 0, 90, 180]);
legend('location', 'southwest');
linkaxes([ax1,ax2],'x'); linkaxes([ax1,ax2],'x');
#+end_src #+end_src
@ -1760,7 +1787,8 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to
*** Control Design *** Control Design
The controller for each pair of actuator/sensor is: The controller for each pair of actuator/sensor is:
#+begin_src matlab #+begin_src matlab
K_iff = 1000/s; w0 = 2*pi*50;
K_iff = -5000/s * (s/w0)/(1 + s/w0);
#+end_src #+end_src
The corresponding loop gains are shown in figure [[fig:iff_open_loop]]. The corresponding loop gains are shown in figure [[fig:iff_open_loop]].
@ -1772,8 +1800,8 @@ The corresponding loop gains are shown in figure [[fig:iff_open_loop]].
ax1 = subplot(2, 1, 1); ax1 = subplot(2, 1, 1);
hold on; hold on;
for i=1:6 for i=1:length(masses)
plot(freqs, abs(squeeze(freqresp(K_iff*G_iff(['Fnlm', num2str(i)], ['Fnl', num2str(i)]), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(K_iff*Gm_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@ -1781,14 +1809,16 @@ The corresponding loop gains are shown in figure [[fig:iff_open_loop]].
ax2 = subplot(2, 1, 2); ax2 = subplot(2, 1, 2);
hold on; hold on;
for i=1:6 for i=1:length(masses)
plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G_iff(['Fnlm', num2str(i)], ['Fnl', num2str(i)]), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*Gm_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), ...
'DisplayName', sprintf('$M = %.0f$ [kg]', masses(i)));
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]); ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]); yticks([-180, -90, 0, 90, 180]);
legend('location', 'southwest');
linkaxes([ax1,ax2],'x'); linkaxes([ax1,ax2],'x');
#+end_src #+end_src
@ -1814,64 +1844,8 @@ We save the controller for further analysis.
save('./active_damping/mat/K_iff.mat', 'K_iff'); save('./active_damping/mat/K_iff.mat', 'K_iff');
#+end_src #+end_src
*** IFF with High Pass Filter
#+begin_src matlab
w_hpf = 2*pi*10; % Cut-off frequency for the high pass filter [rad/s]
K_iff = 2*pi*200/s * (s/w_hpf)/(s/w_hpf + 1);
#+end_src
The corresponding loop gains are shown in figure [[fig:iff_hpf_open_loop]].
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i=1:6
plot(freqs, abs(squeeze(freqresp(K_iff*G_iff(['Fnlm', num2str(i)], ['Fnl', num2str(i)]), 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=1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G_iff(['Fnlm', num2str(i)], ['Fnl', num2str(i)]), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/iff_hpf_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:iff_hpf_open_loop
#+CAPTION: Loop Gain for the Integral Force Feedback with an High pass filter ([[./figs/iff_hpf_open_loop.png][png]], [[./figs/iff_hpf_open_loop.pdf][pdf]])
[[file:figs/iff_hpf_open_loop.png]]
We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
#+begin_src matlab
K_iff = -K_iff*eye(6);
#+end_src
We save the controller for further analysis.
#+begin_src matlab
save('./active_damping/mat/K_iff_hpf.mat', 'K_iff');
#+end_src
** TODO Identification of the damped plant :noexport: ** TODO Identification of the damped plant :noexport:
*** Initialize the Simulation *** Initialize the Simulation :noexport:
We initialize all the stages with the default parameters. We initialize all the stages with the default parameters.
#+begin_src matlab #+begin_src matlab
initializeGround(); initializeGround();
@ -1884,24 +1858,24 @@ We initialize all the stages with the default parameters.
initializeMirror(); initializeMirror();
#+end_src #+end_src
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg. No disturbances.
#+begin_src matlab
initializeDisturbances('enable', false);
#+end_src
The nano-hexapod is a piezoelectric hexapod.
#+begin_src matlab #+begin_src matlab
initializeNanoHexapod('actuator', 'piezo'); initializeNanoHexapod('actuator', 'piezo');
initializeSample('mass', 50); initializeSample('mass', 50);
#+end_src #+end_src
We set the references to zero. And all the controllers are set to 0.
#+begin_src matlab
initializeReferences();
#+end_src
And all the controllers are set to 0 except for the IFF.
#+begin_src matlab #+begin_src matlab
K = tf(zeros(6)); K = tf(zeros(6));
save('./mat/controllers.mat', 'K', '-append'); save('./mat/controllers.mat', 'K', '-append');
K_ine = tf(zeros(6)); K_ine = tf(zeros(6));
save('./mat/controllers.mat', 'K_ine', '-append'); save('./mat/controllers.mat', 'K_ine', '-append');
K_iff = K_iff; load('./active_damping/mat/K_iff.mat', 'K_iff');
save('./mat/controllers.mat', 'K_iff', '-append'); save('./mat/controllers.mat', 'K_iff', '-append');
K_dvf = tf(zeros(6)); K_dvf = tf(zeros(6));
save('./mat/controllers.mat', 'K_dvf', '-append'); save('./mat/controllers.mat', 'K_dvf', '-append');
@ -1920,25 +1894,23 @@ First, we identify the dynamics of the system using the =linearize= function.
%% Input/Output definition %% Input/Output definition
clear io; io_i = 1; clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fnl'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Fnl'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Compute Error in NASS base'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
%% Run the linearization %% Run the linearization
G = linearize(mdl, io, options); G = linearize(mdl, io, 0.5, options);
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ... G.OutputName = {'Dxn', 'Dyn', 'Dzn', 'Rxn', 'Ryn', 'Rzn'};
'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
#+end_src #+end_src
We then create transfer functions corresponding to the active damping plants.
#+begin_src matlab #+begin_src matlab
G_iff = minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'})); load('mat/stages.mat', 'nano_hexapod');
% G_rmc = minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'})); G_cart_iff = G*inv(nano_hexapod.J');
G_cart_iff.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src #+end_src
And we save them for further analysis. And we save them for further analysis.
#+begin_src matlab #+begin_src matlab
save('./active_damping/mat/plants.mat', 'G_iff', '-append'); save('./active_damping/mat/cart_plants.mat', 'G_cart_iff', '-append');
#+end_src #+end_src
*** TODO Sensitivity to disturbances *** TODO Sensitivity to disturbances
@ -2022,8 +1994,11 @@ As shown on figure [[fig:sensitivity_dist_iff]]:
#+CAPTION: Sensitivity to force disturbances in various stages when IFF is applied ([[./figs/sensitivity_dist_stages_iff.png][png]], [[./figs/sensitivity_dist_stages_iff.pdf][pdf]]) #+CAPTION: Sensitivity to force disturbances in various stages when IFF is applied ([[./figs/sensitivity_dist_stages_iff.png][png]], [[./figs/sensitivity_dist_stages_iff.pdf][pdf]])
[[file:figs/sensitivity_dist_stages_iff.png]] [[file:figs/sensitivity_dist_stages_iff.png]]
*** TODO Damped Plant *** Damped Plant
Now, look at the new damped plant to control. Now, look at the new damped plant to control.
#+begin_src matlab
load('./active_damping/mat/cart_plants.mat', 'G_cart', 'G_cart_iff');
#+end_src
It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure [[fig:plant_iff_damped]]. It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure [[fig:plant_iff_damped]].
@ -2034,59 +2009,57 @@ It damps the plant (resonance of the nano hexapod as well as other resonances) a
ax1 = subplot(2, 2, 1); ax1 = subplot(2, 2, 1);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cart('Dxn', 'Fnx'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cart('Dyn', 'Fny'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cart('Dzn', 'Fnz'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1); set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--'); plot(freqs, abs(squeeze(freqresp(G_cart_iff('Dxn', 'Fnx'), freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--'); plot(freqs, abs(squeeze(freqresp(G_cart_iff('Dyn', 'Fny'), freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--'); plot(freqs, abs(squeeze(freqresp(G_cart_iff('Dzn', 'Fnz'), freqs, 'Hz'))), '--');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
ax2 = subplot(2, 2, 2); ax2 = subplot(2, 2, 2);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cart('Rxn', 'Mnx'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cart('Ryn', 'Mny'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz')))); plot(freqs, abs(squeeze(freqresp(G_cart('Rzn', 'Mnz'), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1); set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--'); plot(freqs, abs(squeeze(freqresp(G_cart_iff('Rxn', 'Mnx'), freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--'); plot(freqs, abs(squeeze(freqresp(G_cart_iff('Ryn', 'Mny'), freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--'); plot(freqs, abs(squeeze(freqresp(G_cart_iff('Rzn', 'Mnz'), freqs, 'Hz'))), '--');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]'); ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]');
ax3 = subplot(2, 2, 3); ax3 = subplot(2, 2, 3);
hold on; hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart('Dxn', 'Fnx'), freqs, 'Hz')))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$');
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart('Dyn', 'Fny'), freqs, 'Hz')))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$');
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart('Dzn', 'Fnz'), freqs, 'Hz')))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$');
set(gca,'ColorOrderIndex',1); set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart_iff('Dxn', 'Fnx'), freqs, 'Hz')))), '--', 'HandleVisibility', 'off');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart_iff('Dyn', 'Fny'), freqs, 'Hz')))), '--', 'HandleVisibility', 'off');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart_iff('Dzn', 'Fnz'), freqs, 'Hz')))), '--', 'HandleVisibility', 'off');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]); yticks([-540:180:540]);
yticks([-180, -90, 0, 90, 180]); legend('location', 'southwest');
legend('location', 'northwest');
ax4 = subplot(2, 2, 4); ax4 = subplot(2, 2, 4);
hold on; hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart('Rxn', 'Mnx'), freqs, 'Hz')))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$');
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart('Ryn', 'Mny'), freqs, 'Hz')))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$');
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart('Rzn', 'Mnz'), freqs, 'Hz')))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$');
set(gca,'ColorOrderIndex',1); set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart_iff('Rxn', 'Mnx'), freqs, 'Hz')))), '--', 'HandleVisibility', 'off');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart_iff('Ryn', 'Mny'), freqs, 'Hz')))), '--', 'HandleVisibility', 'off');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off'); plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_cart_iff('Rzn', 'Mnz'), freqs, 'Hz')))), '--', 'HandleVisibility', 'off');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]); yticks([-540:180:540]);
yticks([-180, -90, 0, 90, 180]); legend('location', 'southwest');
legend('location', 'northwest');
linkaxes([ax1,ax2,ax3,ax4],'x'); linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src #+end_src
@ -2110,10 +2083,12 @@ However, it increases coupling at low frequency (figure [[fig:plant_iff_coupling
for iy = 1:6 for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy); subplot(6, 6, (ix-1)*6 + iy);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(G.G_cart(ix, iy), freqs, 'Hz'))), 'k-'); plot(freqs, abs(squeeze(freqresp(G_cart(ix, iy), freqs, 'Hz'))), 'k-');
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart(ix, iy), freqs, 'Hz'))), 'k--'); plot(freqs, abs(squeeze(freqresp(G_cart_iff(ix, iy), freqs, 'Hz'))), 'k--');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-12, 1e-5]); ylim([1e-13, 1e-4]);
xticks([])
yticks([])
end end
end end
#+end_src #+end_src
@ -3673,8 +3648,8 @@ This Matlab function is accessible [[file:src/prepareTomographyExperiment.m][her
#+begin_src matlab #+begin_src matlab
arguments arguments
args.nass_actuator char {mustBeMember(args.nass_actuator,{'piezo', 'lorentz'})} = 'piezo' args.nass_actuator char {mustBeMember(args.nass_actuator,{'piezo', 'lorentz'})} = 'piezo'
args.sample_mass (1,1) double {mustBeNumeric, mustBePositive} = 50 args.sample_mass (1,1) double {mustBeNumeric, mustBePositive} = 50 % [kg]
args.Ry_period (1,1) double {mustBeNumeric, mustBePositive} = 1 args.Rz_period (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
end end
#+end_src #+end_src
@ -3702,7 +3677,7 @@ The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
We set the references to zero. We set the references to zero.
#+begin_src matlab #+begin_src matlab
initializeReferences('Rz_type', 'rotating', 'Rz_period', args.Ry_period); initializeReferences('Rz_type', 'rotating', 'Rz_period', args.Rz_period);
#+end_src #+end_src
And all the controllers are set to 0. And all the controllers are set to 0.
@ -3762,7 +3737,7 @@ And all the controllers are set to 0.
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
lzoad('mat/stages.mat', 'nano_hexapod'); load('mat/stages.mat', 'nano_hexapod');
G_cart = G*inv(nano_hexapod.J'); G_cart = G*inv(nano_hexapod.J');
G_cart.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'}; G_cart.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src #+end_src

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