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#+TITLE: Control of the NASS with optimal stiffness
:DRAWER:
#+STARTUP: overview
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+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+ :mkdirp yes
#+PROPERTY: header-args:shell :eval no-export
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* Introduction :ignore:
* Low Authority Control - Decentralized Integral Force Feedback
** Introduction :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)
<<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
We initialize all the stages with the default parameters.
#+begin_src matlab
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
#+end_src
We set the references that corresponds to a tomography experiment.
#+begin_src matlab
initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 1);
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
#+end_src
#+begin_src matlab
initializeController('type', 'hac-iff');
#+end_src
** Identification
#+begin_src matlab
Kx = tf(zeros(6));
Kiff = tf(zeros(6));
#+end_src
#+begin_src matlab
Ms = [1, 10, 50];
Gm_iff = {zeros(length(Ms), 1)};
#+end_src
#+begin_src matlab
initializeNanoHexapod('k', 1e5, 'c', 2e2);
#+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', 200*ones(6,1));
%% 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',[]);
title('Diagonal elements of the Plant');
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
Root Locus
#+begin_src matlab :exports none :post
figure;
gains = logspace(0, 3, 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([-140, 10]); ylim([0, 150]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/opt_stiff_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:opt_stiff_iff_root_locus
#+caption: Root Locus for the
#+RESULTS:
[[file:figs/opt_stiff_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(0, 3, 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
Kiff = -200/s*eye(6);
#+end_src
* Primary Control
** Introduction :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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
** Identification
#+begin_src matlab
Gm_x = {zeros(length(Ms), 1)};
Gm_l = {zeros(length(Ms), 1)};
#+end_src
#+begin_src matlab
load('mat/stages.mat', 'nano_hexapod');
#+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, '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
for i = 1:length(Ms)
initializeSample('mass', Ms(i), 'freq', 200*ones(6,1));
%% 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.J');
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Gm_x(i) = {Gx};
Gl = -nano_hexapod.J*G;
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
Gm_l(i) = {Gl};
end
#+end_src
** Controller in the task space
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'};
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Diagonal elements of the Plant');
ax2 = subplot(2, 2, 3);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))), 'DisplayName', labels{i});
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]);
legend();
ax3 = subplot(2, 2, 2);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Off-Diagonal elements of the Plant');
ax4 = subplot(2, 2, 4);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
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,ax3,ax4],'x');
#+end_src
*** Translation
#+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(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(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(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(Gm_x{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', 'southwest');
linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src
#+begin_src matlab
Kx = tf(zeros(6));
h = 1.5;
Kx(1,1) = 2e6 * ...
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
(s/2/pi/10 + 1)/(s/2/pi/10);
Kx(2,2) = Kx(1,1);
h = 1.5;
Kx(3,3) = 1e7 * ...
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
(s/2/pi/10 + 1)/(s/2/pi/10);
#+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(Gm_x{i}(1, 1)*Kx(1,1), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(2, 2)*Kx(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(Gm_x{i}(3, 3)*Kx(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(Gm_x{i}(1, 1)*Kx(1,1), freqs, 'Hz')))));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(2, 2)*Kx(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(Gm_x{i}(3, 3)*Kx(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', 'southwest');
linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src
*** Rotations
#+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(Gm_x{i}(4, 4), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(5, 5), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
title('$\mathcal{X}_{R_x}/\mathcal{M}_x$, $\mathcal{X}_{R_y}/\mathcal{M}_y$')
ax2 = subplot(2, 2, 2);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(6, 6), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
title('$\mathcal{X}_{R_z}/\mathcal{M}_z$')
ax3 = subplot(2, 2, 3);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(4, 4), freqs, 'Hz')))));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(5, 5), 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(Gm_x{i}(6, 6), 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', 'southwest');
linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src
#+begin_src matlab
h = 1.5;
Kx(4,4) = 1e5 * ...
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
(s/2/pi/10 + 1)/(s/2/pi/10);
Kx(5,5) = Kx(4,4);
h = 1.5;
Kx(6,6) = 2e5 * ...
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
(s/2/pi/10 + 1)/(s/2/pi/10);
#+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(Gm_x{i}(4, 4)*Kx(4,4), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(5, 5)*Kx(5,5), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
title('$\mathcal{X}_{R_x}/\mathcal{M}_x$, $\mathcal{X}_{R_y}/\mathcal{M}_y$')
ax2 = subplot(2, 2, 2);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(6, 6)*Kx(6,6), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
title('$\mathcal{X}_{R_z}/\mathcal{M}_z$')
ax3 = subplot(2, 2, 3);
hold on;
for i = 1:length(Ms)
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(4, 4)*Kx(4,4), freqs, 'Hz')))));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(5, 5)*Kx(5,5), 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(Gm_x{i}(6, 6)*Kx(6,6), 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', 'southwest');
linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src
*** Stability
#+begin_src matlab
for i = 1:length(Ms)
isstable(feedback(Gm_x{i}*Kx, eye(6), -1))
end
#+end_src
** Simulation
** Control in the leg space
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Diagonal elements of the Plant');
ax2 = subplot(2, 2, 3);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$d\\mathcal{L}_%i / \\tau_%i$', i, i));
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]);
legend();
ax3 = subplot(2, 2, 2);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Off-Diagonal elements of the Plant');
ax4 = subplot(2, 2, 4);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
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,ax3,ax4],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ms)
for j = 1:6
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j), freqs, 'Hz'))));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Diagonal elements of the Plant');
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:length(Ms)
for j = 1:6
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j), freqs, 'Hz'))));
end
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
#+begin_src matlab
h = 1.5;
Kl = 5e6 * eye(6) * ...
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
(s/2/pi/10 + 1)/(s/2/pi/10);
#+end_src
#+begin_src matlab
for i = 1:length(Ms)
isstable(feedback(Gm_l{i}(1,1)*Kl(1,1), 1, -1))
end
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:length(Ms)
for j = 1:6
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
end
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)
for j = 1:6
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
end
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
* Simulations
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