712 lines
22 KiB
Org Mode
712 lines
22 KiB
Org Mode
#+TITLE: Decentralize control to add virtual mass
|
|
#+SETUPFILE: ./setup/org-setup-file.org
|
|
|
|
* 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
|
|
load('mat/conf_simulink.mat');
|
|
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-dvf');
|
|
#+end_src
|
|
|
|
The nano-hexapod has the following leg's stiffness and damping.
|
|
#+begin_src matlab
|
|
initializeNanoHexapod('k', 1e5, 'c', 2e2);
|
|
#+end_src
|
|
|
|
We set the stiffness of the payload fixation:
|
|
#+begin_src matlab
|
|
Kp = 1e8; % [N/m]
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
K = tf(zeros(6));
|
|
Kdvf = tf(zeros(6));
|
|
#+end_src
|
|
|
|
* Identification
|
|
We identify the system for the following payload masses:
|
|
#+begin_src matlab
|
|
Ms = [1, 10, 50];
|
|
#+end_src
|
|
|
|
Identification of the transfer function from $\tau$ to $d\mathcal{L}$.
|
|
#+begin_src matlab :exports none
|
|
Gm = {zeros(length(Ms), 1)};
|
|
|
|
%% 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;
|
|
|
|
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 = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
|
|
Gm(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
Identification of the Primary plant without virtual add of mass
|
|
#+begin_src matlab :exports none
|
|
G_x = {zeros(length(Ms), 1)};
|
|
G_l = {zeros(length(Ms), 1)};
|
|
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
|
|
%% 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
|
|
|
|
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
|
|
|
|
* Adding Virtual Mass in the Leg's Space
|
|
** Plant
|
|
#+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{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)
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{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/virtual_mass_plant_L.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_plant_L
|
|
#+caption: Transfer function from $\tau_i$ to $d\mathcal{L}_i$ for three payload masses
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_plant_L.png]]
|
|
|
|
** Controller Design
|
|
#+begin_src matlab
|
|
Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
for i = 1:length(Ms)
|
|
isstable(feedback(Gm{i}*Kdvf, eye(6), -1))
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 4, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(1,1), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(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([-180, 180]);
|
|
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/virtual_mass_loop_gain_L.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_loop_gain_L
|
|
#+caption: Loop Gain for the addition of virtual mass in the leg's space
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_loop_gain_L.png]]
|
|
|
|
** Identification of the Primary Plant
|
|
#+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
|
|
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
|
|
GmL_x = {zeros(length(Ms), 1)};
|
|
GmL_l = {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 = {'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'};
|
|
GmL_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
GmL_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
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(GmL_x{i}(1, 1), freqs, 'Hz'))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(GmL_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(GmL_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(GmL_x{i}(1, 1), freqs, 'Hz')))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_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(GmL_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/virtual_mass_L_primary_plant_X.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_L_primary_plant_X
|
|
#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the leg's space
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_L_primary_plant_X.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
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(GmL_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(GmL_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/virtual_mass_L_primary_plant_L.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_L_primary_plant_L
|
|
#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the leg's space
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_L_primary_plant_L.png]]
|
|
|
|
* Adding Virtual Mass in the Task Space
|
|
** Plant
|
|
Let's look at the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$:
|
|
\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \]
|
|
|
|
#+begin_src matlab :exports none
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
|
|
GmX = {zeros(length(Ms), 1)};
|
|
for i = 1:length(Ms)
|
|
GmX(i) = {inv(nano_hexapod.kinematics.J) * Gm{i} * inv(nano_hexapod.kinematics.J')};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(GmX{i}(1, 1), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(GmX{i}(2, 2), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 2, 3);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{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(GmX{i}(2, 2), 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', 'northeast');
|
|
|
|
ax3 = subplot(2, 2, 2);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(GmX{i}(3, 3), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(3, 3), 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,ax3,ax4],'x');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/virtual_mass_plant_X.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_plant_X
|
|
#+caption: Dynamics from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ used for virtual mass addition in the task space
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_plant_X.png]]
|
|
|
|
** Controller Design
|
|
#+begin_src matlab
|
|
KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0]));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
for i = 1:length(Ms)
|
|
isstable(feedback(GmX{i}*KmX, eye(6), -1))
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
LmX = GmX{i}*KmX;
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(LmX(1, 1), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(LmX(2, 2), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax3 = subplot(2, 2, 3);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
LmX = GmX{i}*KmX;
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(1, 1), freqs, 'Hz')))), ...
|
|
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(2, 2), freqs, 'Hz')))), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-360:90:360]);
|
|
legend('location', 'southwest');
|
|
|
|
ax2 = subplot(2, 2, 2);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
LmX = GmX{i}*KmX;
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(LmX(3, 3), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
LmX = GmX{i}*KmX;
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(3, 3), 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([-180, 180]);
|
|
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/virtual_mass_loop_gain_X.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_loop_gain_X
|
|
#+caption: Loop gain for virtual mass addition in the task space
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_loop_gain_X.png]]
|
|
|
|
#+begin_src matlab
|
|
Kdvf = inv(nano_hexapod.kinematics.J')*KmX*inv(nano_hexapod.kinematics.J);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
for i = 1:length(Ms)
|
|
isstable(feedback(Gm{i}*Kdvf, eye(6), -1))
|
|
end
|
|
#+end_src
|
|
|
|
** Identification of the Primary Plant
|
|
#+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
|
|
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
|
|
GmX_x = {zeros(length(Ms), 1)};
|
|
GmX_l = {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 = {'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'};
|
|
GmX_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
GmX_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
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(GmX_x{i}(1, 1), freqs, 'Hz'))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(GmX_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(GmX_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(GmX_x{i}(1, 1), freqs, 'Hz')))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_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(GmX_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/virtual_mass_X_primary_plant_X.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_X_primary_plant_X
|
|
#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the task space
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_X_primary_plant_X.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
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(GmX_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(GmX_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/virtual_mass_X_primary_plant_L.pdf', 'width', 'full', 'height', 'full')
|
|
#+end_src
|
|
|
|
#+name: fig:virtual_mass_X_primary_plant_L
|
|
#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the task space
|
|
#+RESULTS:
|
|
[[file:figs/virtual_mass_X_primary_plant_L.png]]
|