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Continue analysis of optimal_stiffness

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Thomas Dehaeze
2020-04-02 21:38:31 +02:00
parent 764b1292eb
commit 9c6d397fe4
12 changed files with 1608 additions and 34 deletions
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@@ -22,7 +22,7 @@
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :tangle ../matlab/optimal_stiffness.m
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:shell :eval no-export
@@ -73,6 +73,7 @@ The overall goal is to design a nano-hexapod that will allow the highest possibl
#+end_src
#+begin_src matlab
load('mat/conf_simulink.mat');
open('nass_model.slx')
#+end_src
@@ -89,20 +90,15 @@ We initialize all the stages with the default parameters.
initializeMirror();
#+end_src
The worst case scenario is a rotation speed of 60rpm with a payload mass of 1Kg.
The worst case scenario is a rotation speed of 60rpm with a payload mass of 10Kg.
#+begin_src matlab
initializeSample('mass', 10);
#+end_src
We don't include gravity nor disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics.
#+begin_src matlab
initializeSimscapeConfiguration('gravity', false);
initializeSimscapeConfiguration('gravity', true);
initializeDisturbances('enable', false);
#+end_src
We set the controller type to Open-Loop, and we do not need to log any signal.
#+begin_src matlab
initializeController('type', 'stabilizing');
initializeLoggingConfiguration('log', 'none');
#+end_src
@@ -121,20 +117,22 @@ We set the controller type to Open-Loop, and we do not need to log any signal.
** Identification when not rotating
We set the range of stiffness that we want to use.
#+begin_src matlab
Ks = logspace(3,9,7)
Ks = logspace(3,9,7); % [N/m]
#+end_src
We don't move any stage and no controller is used.
#+begin_src matlab
initializeReferences();
initializeController();
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
Gk_iff = {zeros(length(Ks))};
Gk_dvf = {zeros(length(Ks))};
Gk_err = {zeros(length(Ks))};
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
for i = 1:length(Ks)
initializeNanoHexapod('k', Ks(i));
@@ -156,6 +154,7 @@ We set the range of stiffness that we want to use.
#+end_src
** Identification when rotating at maximum speed
We now set the reference path such that the Spindle is rotating at 60rpm and such that it is at the zero position at the time of the identification.
#+begin_src matlab
Rz_rpm = 60;
@@ -168,17 +167,20 @@ We set the range of stiffness that we want to use.
t_sim = Rz.time(i_end); % Simulation time before identification [s]
#+end_src
We here use a decentralized controller that is used to stabilize the nano-hexapod until the identification is made.
This controller virtually adds stiffness in each of the nano-hexapod leg.
#+begin_src matlab
k_sta = -1e8;
initializeController('type', 'stabilizing');
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
Gk_wz_iff = {zeros(length(Ks))};
Gk_wz_dvf = {zeros(length(Ks))};
Gk_wz_err = {zeros(length(Ks))};
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
for i = 1:length(Ks)
initializeNanoHexapod('k', Ks(i));
@@ -199,13 +201,17 @@ We set the range of stiffness that we want to use.
end
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
save('mat/optimal_stiffness_Gk_wz.mat', 'Ks', ...
'Gk_iff', 'Gk_dvf', 'Gk_err', ...
'Gk_wz_iff', 'Gk_wz_dvf', 'Gk_wz_err');
#+end_src
** Change of dynamics
** TODO Change of dynamics
- [ ] problem of dynamics at low frequency
Check if gravity is a problem
Think of a before way to identify the dynamics
#+begin_src matlab :exports none
load('mat/optimal_stiffness_Gk_wz.mat');
#+end_src
@@ -397,11 +403,11 @@ Comparison of the coupling from $F_x$ to $D_y$ when rotating at 60rpm to the dir
** 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>>
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
@@ -409,9 +415,337 @@ Comparison of the coupling from $F_x$ to $D_y$ when rotating at 60rpm to the dir
#+end_src
#+begin_src matlab
load('mat/conf_simulink.mat');
open('nass_model.slx')
#+end_src
** Identification of the micro-station compliance
We initialize all the stages with the default parameters.
#+begin_src matlab
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod('type', 'compliance');
#+end_src
We put nothing on top of the micro-hexapod.
#+begin_src matlab
initializeAxisc('type', 'none');
initializeMirror('type', 'none');
initializeNanoHexapod('type', 'none');
initializeSample('type', 'none');
#+end_src
#+begin_src matlab :exports none
initializeReferences();
initializeDisturbances();
initializeController();
initializeSimscapeConfiguration();
initializeLoggingConfiguration();
#+end_src
And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.
#+begin_src matlab :exports noone
%% Name of the Simulink File
mdl = 'nass_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Compliance/Fm'], 1, 'openinput'); io_i = io_i + 1; % Direct Forces/Torques applied on the micro-hexapod top platform
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Compliance/Dm'], 1, 'output'); io_i = io_i + 1; % Absolute displacement of the top platform
%% Run the linearization
Gm = linearize(mdl, io, 0);
Gm.InputName = {'Fmx', 'Fmy', 'Fmz', 'Mmx', 'Mmy', 'Mmz'};
Gm.OutputName = {'Dx', 'Dy', 'Dz', 'Drx', 'Dry', 'Drz'};
#+end_src
#+begin_src matlab :exports none
labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'};
freqs = logspace(1, 3, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gm(1, 1), freqs, 'Hz'))), 'k-', 'DisplayName', labels{1});
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gm(1, i), freqs, 'Hz'))), 'DisplayName', labels{i});
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Compliance');
legend('location', 'northwest');
#+end_src
** Identification of the dynamics with a rigid micro-station
*** Initialization
#+begin_src matlab :exports none
initializeReferences();
initializeDisturbances();
initializeController();
initializeSimscapeConfiguration();
initializeLoggingConfiguration();
initializeSimscapeConfiguration('gravity', false);
#+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;
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En'); io_i = io_i + 1;
#+end_src
#+begin_src matlab
Ks = logspace(3,9,7); % [N/m]
#+end_src
#+begin_src matlab
initializeSample('type', 'rigid', 'mass', 20);
#+end_src
*** Rigid micro-station
#+begin_src matlab
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
initializeAxisc('type', 'rigid');
initializeMirror('type', 'rigid');
#+end_src
#+begin_src matlab :exports none
Gmr_iff = {zeros(length(Ks))};
Gmr_dvf = {zeros(length(Ks))};
Gmr_err = {zeros(length(Ks))};
#+end_src
#+begin_src matlab :exports none
for i = 1:length(Ks)
initializeNanoHexapod('k', Ks(i));
G = linearize(mdl, io);
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
Gmr_iff(i) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
Gmr_dvf(i) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
Jinvt = tf(inv(nano_hexapod.J)');
Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
Gmr_err(i) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt};
end
#+end_src
** Identification of the dynamics with a flexible micro-station
*** Flexible micro-station
#+begin_src matlab
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
#+end_src
#+begin_src matlab :exports none
Gmf_iff = {zeros(length(Ks))};
Gmf_dvf = {zeros(length(Ks))};
Gmf_err = {zeros(length(Ks))};
#+end_src
#+begin_src matlab :exports none
for i = 1:length(Ks)
initializeNanoHexapod('k', Ks(i));
G = linearize(mdl, io);
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
Gmf_iff(i) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
Gmf_dvf(i) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))};
Jinvt = tf(inv(nano_hexapod.J)');
Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
Gmf_err(i) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt};
end
#+end_src
#+begin_src matlab :exports none
save('mat/optimal_stiffness_micro_station_compliance.mat', 'Ks', ...
'Gmr_iff', 'Gmr_dvf', 'Gmr_err', ...
'Gmf_iff', 'Gmf_dvf', 'Gmf_err');
#+end_src
** Obtained Dynamics
#+begin_src matlab :exports none
load('mat/optimal_stiffness_micro_station_compliance.mat');
#+end_src
IFF plant
#+begin_src matlab :exports none
freqs = logspace(-1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmr_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmf_iff{i}('Fnlm1', 'Fnl1'), 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(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmr_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmf_iff{i}('Fnlm1', 'Fnl1'), 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');
xlim([freqs(1), freqs(end)]);
#+end_src
DVF plant
#+begin_src matlab :exports none
freqs = logspace(-1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmr_dvf{i}('Dnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmf_dvf{i}('Dnlm1', 'Fnl1'), 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(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmr_dvf{i}('Dnlm1', 'Fnl1'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmf_dvf{i}('Dnlm1', 'Fnl1'), 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');
xlim([freqs(1), freqs(end)]);
#+end_src
X direction
#+begin_src matlab :exports none
freqs = logspace(-1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), 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(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmr_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmf_err{i}('Ex', 'Fx'), 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');
xlim([freqs(1), freqs(end)]);
#+end_src
Z direction
#+begin_src matlab :exports none
freqs = logspace(-1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmr_err{i}('Ez', 'Fz'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gmf_err{i}('Ez', 'Fz'), 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(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmr_err{i}('Ez', 'Fz'), freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gmf_err{i}('Ez', 'Fz'), 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');
xlim([freqs(1), freqs(end)]);
#+end_src
** Conclusion :ignore:
* Payload "Impedance" Effect
<<sec:payload_impedance>>
@@ -432,6 +766,7 @@ Comparison of the coupling from $F_x$ to $D_y$ when rotating at 60rpm to the dir
#+end_src
#+begin_src matlab
load('mat/conf_simulink.mat');
open('nass_model.slx')
#+end_src
@@ -448,16 +783,17 @@ We initialize all the stages with the default parameters.
initializeMirror();
#+end_src
We don't include gravity nor disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics.
We don't include disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics.
#+begin_src matlab
initializeSimscapeConfiguration('gravity', false);
initializeSimscapeConfiguration('gravity', true);
initializeDisturbances('enable', false);
#+end_src
We set the controller type to Open-Loop, and we do not need to log any signal.
#+begin_src matlab
initializeController('type', 'stabilizing');
initializeController();
initializeLoggingConfiguration('log', 'none');
initializeReferences();
#+end_src
#+begin_src matlab :exports none
@@ -472,27 +808,24 @@ We set the controller type to Open-Loop, and we do not need to log any signal.
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En'); io_i = io_i + 1;
#+end_src
** Change of payload dynamics
** Identification of the dynamics while change the payload dynamics
- Change of mass: from 1kg to 50kg
- Change of resonance frequency: from 50Hz to 500Hz
- The damping ratio of the payload is fixed to $\xi = 0.2$
#+begin_src matlab
initializeReferences();
Ks = logspace(3,9,7) % [N/m]
#+end_src
We identify the dynamics for the following payload masses =Ms= and nano-hexapod leg's stiffnesses =Ks=:
#+begin_src matlab
Ms = [1, 20, 50]; % [Kg]
Ks = logspace(3,9,7); % [N/m]
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
Gm_iff = {zeros(length(Ks), length(Ms))};
Gm_dvf = {zeros(length(Ks), length(Ms))};
Gm_err = {zeros(length(Ks), length(Ms))};
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
for i = 1:length(Ks)
for j = 1:length(Ms)
initializeNanoHexapod('k', Ks(i));
@@ -515,17 +848,18 @@ We set the controller type to Open-Loop, and we do not need to log any signal.
end
#+end_src
We then identify the dynamics for the following payload resonance frequencies =Fs=:
#+begin_src matlab
Fs = [50, 200, 500]; % [Hz]
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
Gf_iff = {zeros(length(Ks), length(Fs))};
Gf_dvf = {zeros(length(Ks), length(Fs))};
Gf_err = {zeros(length(Ks), length(Fs))};
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
for i = 1:length(Ks)
for j = 1:length(Fs)
initializeNanoHexapod('k', Ks(i));
@@ -548,13 +882,12 @@ We set the controller type to Open-Loop, and we do not need to log any signal.
end
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
save('mat/optimal_stiffness_Gm_Gf.mat', 'Ks', 'Ms', 'Fs', ...
'Gm_iff', 'Gm_dvf', 'Gm_err', ...
'Gf_iff', 'Gf_dvf', 'Gf_err');
#+end_src
** Plots
** Change of optimal gain for decentralized control
For each payload, compute the optimal gain for the IFF control.
The optimal value corresponds to critical damping to *all* the 6 modes of the nano-hexapod.
@@ -1061,6 +1394,5 @@ For a stiff nano-hexapod
xlim([freqs(1), freqs(end)]);
#+end_src
** Conclusion :ignore:
* Total Change of dynamics